Bayesian Methods for Gravitational
Waves and Neural Networks
Philip Graff
Cavendish Astrophysics and Queens’ College
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
Submitted 06 June 2012
In final form 25 June 2012
Declaration
This dissertation is the result of work carried out in the Astrophysics Group
of the Cavendish Laboratory, Cambridge, between October 2008 and June
2012. Except where explicit reference is made to the work of others, the
work contained in this dissertation is my own, and is not the outcome of
work done in collaboration. No part of this dissertation has been submitted
for a degree, diploma or other qualification at this or any other university.
The total length of this dissertation does not exceed sixty thousand words.
Philip Graff
25 June 2012
Cambridge was the place for someone from the Colonies or the
Dominions to go on to, and it was to the Cavendish Laboratory that
one went to do physics.
Sir Aaron Klug
I would like to dedicate this thesis to my loving parents, who have
supported and encouraged me through the years.
Acknowledgements
I acknowledge financial support from the Gates Cambridge Trust, founded
by the Bill and Melinda Gates Foundation, which provided me with the op-
portunity to study in Cambridge. The Cambridge Philosophical Society,
Queens’ College, and the Astrophysics Group of the Cavendish Labora-
tory have also provided financial support for my studies and presentations
at conferences.
I am very grateful to Anthony Lasenby and Mike Hobson for their con-
tinued encouragement and help over the past three and a half years, both
in my projects and in how to perform proper scientific research. I also
thank them for proof-reading this thesis and providing useful feedback. I
would also like to thank Farhan Feroz for all of the help and advice he has
given me. Without him, much of this work would not have been able to be
completed in the time it was. I thank David Titterington for help with lo-
cal computing issues and Stuart Rankin and Andrey Kaliazin for technical
support with the Darwin and COSMOS supercomputing facilities.
I am very happy to have spent the last three and a half years in the Astro-
physics Group, whose members past and present have been great friends.
Whether in the office, in the tea room, at the pub after a hard day’s work,
at a party, or elsewhere, they have been wonderful compatriots in sharing
complaints, jokes, and life as an astrophysicist. The members of Queens’
College MCR, and especially my flatmate for two years Robert Lowe, also
deserve great thanks for making my time in Cambridge as fantastic as it
has been. My other friends in and around Cambridge – fellow Gates Cam-
bridge Scholars, the Newnham College MCR matriculation class of 2009,
the Gaelic Athletics Club, the Cambridge Royals baseball team, and many
more – have also been great sources of fun and support without whom my
Cambridge experience would have been much more boring.
Lastly, I would like to thank my parents, siblings, and friends back in the
USA for their continued support throughout my PhD.
LIGO Scientic Collaboration Acknowledgement
The second chapter of this thesis contains analysis of LIGO data per-
formed by the author as a member of the LIGO Scientic Collaboration.
The author gratefully acknowledges the support of the United States Na-
tional Science Foundation for the construction and operation of the LIGO
Laboratory and the Science and Technology Facilities Council of the United
Kingdom, the Max-Planck-Society, and the State of Niedersachsen/Germany
for support of the construction and operation of the GEO600 detector. The
authors also gratefully acknowledge the support of the research by these
agencies and by the Australian Research Council, the Council of Scientic
and Industrial Research of India, the Istituto Nazionale di Fisica Nucleare
of Italy, the Spanish Ministerio de Educacio´n y Ciencia, the Conselleria
d’Economia, Hisenda i Innovacio of the Govern de les Illes Balears, the
Scottish Funding Council, the Scottish Universities Physics Alliance, The
National Aeronautics and Space Administration, the Carnegie Trust, the
Leverhulme Trust, the David and Lucile Packard Foundation, the Research
Corporation, and the Alfred P. Sloan Foundation.
Abstract
Einstein’s general theory of relativity has withstood 100 years of testing
and will soon be facing one of its toughest challenges. In a few years
we expect to be entering the era of the first direct observations of gravi-
tational waves. These are tiny perturbations of space-time that are gen-
erated by accelerating matter and affect the measured distances between
two points. Observations of these using the laser interferometers, which
are the most sensitive length-measuring devices in the world, will allow
us to test models of interactions in the strong field regime of gravity and
eventually general relativity itself.
I apply the tools of Bayesian inference for the examination of gravitational
wave data from the LIGO and Virgo detectors. This is used for signal
detection and estimation of the source parameters. I quantify the abil-
ity of a network of ground-based detectors to localise a source position
on the sky for electromagnetic follow-up. Bayesian criteria are also ap-
plied to separating real signals from glitches in the detectors. These same
tools and lessons can also be applied to the type of data expected from
planned space-based detectors. Using simulations from the Mock LISA
Data Challenges, I analyse our ability to detect and characterise both burst
and continuous signals. The two seemingly different signal types will be
overlapping and confused with one another for a space-based detector; my
analysis shows that we will be able to separate and identify many signals
present.
Data sets and astrophysical models are continuously increasing in com-
plexity. This will create an additional computational burden for perform-
ing Bayesian inference and other types of data analysis. I investigate the
application of the MOPED algorithm for faster parameter estimation and
data compression. I find that its shortcomings make it a less favourable
candidate for further implementation.
The framework of an artificial neural network is a simple model for the
structure of a brain which can “learn” functional relationships between sets
of inputs and outputs. I describe an algorithm developed for the training of
feed-forward networks on pre-calculated data sets. The trained networks
can then be used for fast prediction of outputs for new sets of inputs. Af-
ter demonstrating capabilities on toy data sets, I apply the ability of the
network to classifying handwritten digits from the MNIST database and
measuring ellipticities of galaxies in the Mapping Dark Matter challenge.
The power of neural networks for learning and rapid prediction is also
useful in Bayesian inference where the likelihood function is computa-
tionally expensive. The new BAMBI algorithm is detailed, in which our
network training algorithm is combined with the nested sampling algo-
rithm MULTINEST to provide rapid Bayesian inference. Using samples
from the normal inference, a network is trained on the likelihood function
and eventually used in its place. This is able to provide significant increase
in the speed of Bayesian inference while returning identical results. The
trained networks can then be used for extremely rapid follow-up analyses
with different priors, obtaining orders of magnitude of speed increase.
Learning how to apply the tools of Bayesian inference for the optimal
recovery of gravitational wave signals will provide the most scientific in-
formation when the first detections are made. Complementary to this, the
improvement of our analysis algorithms to provide the best results in less
time will make analysis of larger and more complicated models and data
sets practical.
Contents
1 Introduction 1
1.1 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 General Relativity Theory . . . . . . . . . . . . . . . . . . . 2
1.1.2 Gravitational Wave Sources . . . . . . . . . . . . . . . . . . 4
1.1.3 Gravitational Wave Detectors . . . . . . . . . . . . . . . . . 8
1.2 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 MCMC and Nested Sampling . . . . . . . . . . . . . . . . . 14
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Analysis of LIGO Data 21
2.1 Simple Inspiral Model . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1 Generating the Signal . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 Calculating the Likelihood . . . . . . . . . . . . . . . . . . . 23
2.1.3 MULTINEST Results . . . . . . . . . . . . . . . . . . . . . . 24
2.2 LIGO Sky Localisation . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 The Injections . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . 31
2.3 The “Big Dog” Event and Coherence Tests . . . . . . . . . . . . . . . 41
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
xiii
CONTENTS
3 The Mock LISA Data Challenges 45
3.1 Cosmic String Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1 Burst waveform models . . . . . . . . . . . . . . . . . . . . 47
3.1.2 MLDC challenge 3.4 search . . . . . . . . . . . . . . . . . . 51
3.1.3 Model selection using Bayesian evidence . . . . . . . . . . . 53
3.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Galactic White Dwarf Binaries . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3 Cosmic Strings in the Presence of a Galactic Foreground . . . . . . . 70
3.3.1 Data and Signal Models . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Comparison of Results . . . . . . . . . . . . . . . . . . . . . 74
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 An Investigation into the MOPED Algorithm 87
4.1 The Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . 88
4.1.1 A Simple Iterative Approach . . . . . . . . . . . . . . . . . . 88
4.2 The MOPED Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2.1 Data Compression with MOPED . . . . . . . . . . . . . . . . 89
4.2.2 Speed Gains with MOPED . . . . . . . . . . . . . . . . . . . 91
4.3 Simple Example With One Parameter . . . . . . . . . . . . . . . . . 92
4.4 Recovery in a 2 Parameter Case . . . . . . . . . . . . . . . . . . . . 94
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Artificial Neural Networks 103
5.1 Network Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1.1 Choosing the Number of Hidden Layer Nodes . . . . . . . . . 105
5.2 Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.2 Data Whitening . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.3 Finding the next step . . . . . . . . . . . . . . . . . . . . . . 109
5.2.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2.5 Autoencoders and Pre-Training . . . . . . . . . . . . . . . . 112
xiv
CONTENTS
5.2.6 The Evidence of a Network . . . . . . . . . . . . . . . . . . 114
5.3 Toy Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3.1 Sinc Function with Noise . . . . . . . . . . . . . . . . . . . . 115
5.3.2 Three-Way Classification . . . . . . . . . . . . . . . . . . . . 116
5.3.3 Autoencoders as Non-Linear PCA . . . . . . . . . . . . . . . 117
5.4 MNIST Handwriting Recognition . . . . . . . . . . . . . . . . . . . 120
5.5 Mapping Dark Matter Challenge . . . . . . . . . . . . . . . . . . . . 125
5.5.1 The Data and Model . . . . . . . . . . . . . . . . . . . . . . 127
5.5.2 Results of Training . . . . . . . . . . . . . . . . . . . . . . . 129
5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 The BAMBI Algorithm 133
6.1 The Structure of BAMBI . . . . . . . . . . . . . . . . . . . . . . . . 134
6.1.1 When to Use the Trained Network . . . . . . . . . . . . . . . 135
6.2 BAMBI Toy Examples . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2.1 Eggbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2.2 Gaussian Shells . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.3 Rosenbrock function . . . . . . . . . . . . . . . . . . . . . . 141
6.3 Cosmological Parameter Estimation with BAMBI . . . . . . . . . . . 142
6.3.1 Updated Timing Comparisons . . . . . . . . . . . . . . . . . 149
6.4 Using Trained Networks for Follow-up in BAMBI . . . . . . . . . . . 151
6.4.1 Exact Error Calculation . . . . . . . . . . . . . . . . . . . . . 152
6.4.2 Fast Error Calculation . . . . . . . . . . . . . . . . . . . . . 157
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7 Conclusions and Future Work 165
7.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.2 Avenues for Future Work . . . . . . . . . . . . . . . . . . . . . . . . 168
References 170
xv
CONTENTS
xvi
Chapter 1
Introduction
I was born not knowing and have had only a
little time to change that here and there.
Richard P. Feynman
1.1 Gravitational Waves
Einstein’s general theory of relativity [1] (GR) predicts the existence of gravitational
waves (GWs), perturbations to the curvature of spacetime that are generated in regions
of strong gravity and will allow scientists to observe the universe through an entirely
new window. The direct detection of gravitational waves will usher in a new era of
astronomy. The GW spectrum in many ways independent of and complementary to
electromagnetic radiation. GWs can be used to probe astrophysical events otherwise
unobservable, such as the merger of two black holes, the inner processes of supernovae,
and the equation of state of neutron star matter. Projects are currently underway across
the world to detect these very weak signals. Indirect evidence for the existence of
gravitational waves exists from observations of binary pulsars spinning in towards each
other as the system loses energy due to the emission of GWs [2].
GR has withstood almost 100 years of testing and verification; however, it is now
facing one of its toughest challenges in gravitational wave searches. Direct observa-
tions may allow for the testing of alternate theories of gravity as a test of the strong
field predictions of general relativity.
1
1. INTRODUCTION
1.1.1 General Relativity Theory
In the linearised regime of general relativity, applicable for weak gravitational fields,
we can assume a flat Minkowski background metric for the spacetime, given by ηab =
diag(−1,1,1,1). The true metric can then be defined as
gab = ηab+hab, ‖ hab ‖ 1. (1.1)
This constrains us to a weak gravitational field and we will refer to hab as the metric
perturbation. We can now derive Einstein’s equations in the linearised regime, where
terms of order h2ab and above will be ignored as they are significantlly small relative to
the first-order perturbation. I will only present the main results of this procedure, full
steps can be found in [3].
We calculate the Einstein tensor with this new metric, discarding small terms of
O(h2ab) and higher, and obtain
Gab =
1
2
(∂c∂bhca+∂
c∂ahbc−hab−∂a∂bh−ηab∂c∂ dhcd +ηabh). (1.2)
We substitute in for the trace-reversed perturbation, h¯ab = hab− 12ηabh (where h = haa)
and choose to satisfy the Lorenz gauge condition that ∂ ah¯ab = 0. Doing so is allowed
because there is freedom to choose the gauge (coordinate system) we want to work in
without affecting the physical observables of the system. Therefore, the Einstein tensor
simplifies to Gab = −12h¯ab, where = ∂c∂ c = ∇2− ∂ 2t . The linearised Einstein
equation is thus
h¯ab =−16piTab. (1.3)
In a vacuum Tab = 0 and this equation describes the propagation of plane waves trav-
eling at the speed of light. These are gravitational waves.
Furthermore, in addition to the Lorenz gauge we can specify the gauge to be purely
spatial and traceless,
htt = hti = 0 and h = 0. (1.4)
This now fully specifies our coordinate system. The combination of these choices
determines that the perturbation is transverse, meaning that ∂ihi j = 0. Therefore, we
call this the transverse-traceless gauge and write it as hT Ti j . Being traceless, h
T T
i j = h¯
T T
i j .
This gauge choice is very useful and leaves only two independent components of the
2
1.1 Gravitational Waves
Figure 1.1: The effect of a passing sinusoidal gravitational wave with period T on a
ring of particles. Image from www.learner.org.
metric perturbation. When orienting our coordinate system to describe waves traveling
along the z axis, these degrees of freedom are given by
hT Txx =−hT Tyy ≡ h+(t− z), (1.5a)
hT Txy = h
T T
yx ≡ h×(t− z). (1.5b)
The two components can now be seen as two independent polarisations, plus and cross,
whose effect on a ring of particles (for a sinusoidal GW source) can be seen in Fig-
ure 1.1. The matter source of a gravitational wave will specify the strengths of its two
polarisations as a function of time.
We are able to detect gravitational waves because they affect the geodesics that
free-falling bodies follow in spacetime. In the transverse-traceless gauge the spacetime
coordinates of a body will remain the same, as they move with the waves, but the proper
separation of two bodies will change as the metric varies with the passing wave. A
straightforward analysis [3] shows that the proper separation of two bodies on the x
axis will oscillate with a fractional length change of
δL
L
' 1
2
hT Txx (t,z = 0), (1.6)
from a wave travelling along the z axis. This formula can be generalized for any
wave propagation direction and any coordinate separation. Because of this effect, the
amplitude of the GW signal is known as the GW strain.
3
1. INTRODUCTION
A complete mathematical background and derivation of general relativity, as well
as its various applications and extensions, can be found in [4].
1.1.2 Gravitational Wave Sources
If we consider a gravitational wave source that is near the origin of our coordinate
system, measure the field at a distance r that is large compared to the spatial extent of
the source, and assume that the constituents of the source are moving slowly compared
to c, then the solution to the linearised Einstein equation can be given by the compact-
source approximation (a full derivation and further details on the following analysis
can be found in [5]),
h¯µν(ct,~x) =−4G
c4r
∫
T µν(ct− r,~y)d3~y. (1.7)
Taking the different components of T µν separately,
∫
T 00d3~y=Mc2 and
∫
T 0id3~y=∫
T i0d3~y = Pic. These components describe the integrated energy and momentum of
the system, respectively, and are therefore constant for a closed system and will not
contribute to the varying gravitational wave strain. Additionally, we are able to trans-
form our coordinates to the source’s centre-of-momentum frame so that
h¯00 =−4GM
c2r
, h¯0i = h¯i0 = 0. (1.8)
Using the transverse property of T µν and Gauss’s law, we can re-write the integral
over the remaining components as∫
T i jd3~y =
1
2c2
d2
dt ′2
∫
T 00yiy jd3~y, (1.9)
where ct ′ = ct− r. Therefore we obtain the quadrupole formula
h¯i j(ct,~x) =−2G
c6r
d2Ii j(ct ′)
dt ′2
, (1.10)
where we define the quadrupole moment tensor of the source as
Ii j(ct) =
∫
T 00(ct,~y)yiy jd3~y. (1.11)
For a given source, this can be used to determine the far-field gravitational radiation.
4
1.1 Gravitational Waves
As an illustration, let us consider two particles of equal mass M moving non-
relativistically in circular orbits of radius a in the z = 0 plane about their common
centre of mass with angular frequency Ω. Since the particles are moving slowly, we
may use the approximation that T 00 ≈ c2ρ , where ρ is the density of the source in its
own reference frame. Therefore,
Ii j(ct) = c2
∫
ρ(ct,~y)yiy jd3~y. (1.12)
It is straightforward to show that for this source
Ii j(ct) = Mc2a2
1+ cos(2Ωt) sin(2Ωt) 0sin(2Ωt) 1− cos(2Ωt) 0
0 0 0
. (1.13)
Substituting this into Equation (1.10) yields
h¯i j(ct,~x) =
8GMa2Ω2
c4r
cos(2Ωt ′) sin(2Ωt ′) 0sin(2Ωt ′) −cos(2Ωt ′) 0
0 0 0
. (1.14)
Considering only the radiative part of hµν gives
h¯µνrad(ct,~x) =
8GMa2Ω2
c4r
0 0 0 0
0 cos(2Ωt ′) sin(2Ωt ′) 0
0 sin(2Ωt ′) −cos(2Ωt ′) 0
0 0 0 0
. (1.15)
We should note at this point two important aspects of our solution. First, the amplitude
scales as 1/r, which means that the GW is a spherical wave. However, at large radii
it can be approximated as a plane wave. Additionally, the 1/r scaling implies that
a factor of X improvement in detector sensitivity will yield the same factor of X in
observable distance and thus X3 in observable volume. Our second comment on the
solution is that the frequency of the GW is twice the orbital frequency of the binary;
this is due to the fact that the radiation is quadrupolar. Monopole and dipole radiation
are zero due to the conservation of mass and momentum, respectively.
We now consider the polarisation of the GW received by an observer. For an ob-
server on the z axis, we begin by noting that Equation (1.15) is already transverse-
traceless so h¯µνT T = h
µν
T T . h+(t) ∝ cos(2Ωt) and h×(t) ∝ sin(2Ωt), corresponding to
right-handed circularly polarised radiation. For an observer on the x axis, however,
5
1. INTRODUCTION
we must first convert Equation (1.15) to the transverse-traceless gauge. We zero the
non-transverse components and then subtract one-half of the trace from the remaining
diagonal components to obtain
(h¯T Trad)
µν(ct,~x) =
4GMa2Ω2
c4r
0 0 0 0
0 0 0 0
0 0 −cos(2Ωt ′) 0
0 0 0 cos(2Ωt ′)
. (1.16)
We observe that the radiation is in the plus polarisation entirely. It is also important to
note that the coefficient in front is a factor of 2 smaller. This combined with the fact
that there is radiation in only one polarisation state implies that the GW energy emitted
in the z direction will be eight times larger than that in the x or y directions, making it
highly anisotropic.
Integrating the energy flux over all angles and averaging over multiple GW periods
yields that the energy lost by the system due to the emission of gravitational waves is
LGW =
128GM2a4Ω6
5c2
. (1.17)
This loss of energy is not considered in the initial analysis where we maintained con-
servation of mass-energy. Waveforms defined in this manner must use a quasi-static
approximation – the evolution of the orbital radius is defined by the internal energy
and this further defines the orbital period through Kepler’s third law. The loss of en-
ergy means that system will evolve inwards (decreasing a) with increasing Ω until
collision/merger. The strong dependence of the emitted energy on frequency results
in this occurring at an increasing rate and creates a burst of radiation at the end. This
is the characteristic “chirp” signal that data analysis pipelines are searching for. For
sources far from merger, the energy lost in gravitational wave emission is low enough
that the frequency evolution can be modeled as an initial frequency f and a frequency
derivative f˙ over a period of observation that is short compared to the time to reach
merger.
In general, this approximation holds for binaries with large separations. As the
binary spirals inwards, further post-Newtonian [6–12] corrections must be made to
the waveform models. These are corrections that expand in powers of v/c, where v
is the transverse velocity of a binary member, and provide more accurate modeling
6
1.1 Gravitational Waves
Figure 1.2: A sample inspiral GW signal that LIGO might observe. Image from
www.ligo.org.
of the phase and amplitude of the GW signal. At higher post-Newtonian orders the
spin of the binary members becomes important and spin-orbit (L ·S1 and L ·S2 terms)
and spin-spin (S1 ·S2 terms) interactions can have significant effects on the waveform
(amplitude and frequency modulation and orbit precession) [13–18]. A sample inspiral
GW signal is shown in Figure 1.2.
Near to the point of merger, the effective-one-body (EOB) approach to waveform
modeling can provide a better template [19–23]. This method reconstitutes the prob-
lem as that of a test particle with some spin spiraling in to a Kerr black hole. Since
accurate numerical simulations of Einstein’s full equations have been performed [24–
29], this waveform can then be attached to a fit of the signal coming from the merger
of the two objects (this has been done for black hole binaries only). A ringdown
waveform derived from black hole perturbation theory and numerical simulations is
then used to complete the signal. This is characterised by constant modes of oscil-
lation exponentially decreasing in amplitude over time. Combining post-Newtonian
and/or EOB theory with numerical relativity simulations and black hole perturbation
theory allows for the construction of complete inspiral-merger-ringdown (IMR) [30–
36] hybrid waveform families that are analytic fits over specific parameter ranges. By
modeling the entire signal, these allow for the extraction of the most information and
signal-to-noise ratio. However, they are limited by our ability to perform numerical
simulations with enough orbits prior to merger and covering a large enough area of the
possible parameter space.
7
1. INTRODUCTION
Similar brief derivations of the quadrupole formula for gravitational wave radiation
from compact binaries can be found in [37, 38]. A more complete derivation and
analysis of the gravitational waves from compact binaries is available in [39] and a
review of the post-Newtonian formalism can be found in [40]. Gravitational waves
are created by a variety of astrophysical sources, such as supernova explosions [41]
and spinning neutron stars [42–47], but compact binaries are the most well-studied. A
review of sources that may be found in high and low frequency ranges can be found
in [48].
Follow-up searches to find GWs associated with observed gamma-ray bursts have
been performed [49] and studies are underway to cross-correlate GW and neutrino
observations [50]. Gravitational wave detections can be used to test general relativity
and measure any deviations from Einstein’s theory [51, 52], such as the existence of
a massive graviton [53] (as GR predicts a massless graviton). Observations can also
be used to test cosmological models by using binary mergers as a standard siren for
luminosity distance and redshift measurements [54–58].
1.1.3 Gravitational Wave Detectors
The Laser Interferometer Gravitational-wave Observatory (LIGO) [59] is a major US
project funded by the National Science Foundation for the detection of GWs. It con-
sists of a pair of observatories located in Livingston, Louisiana and Hanford, Wash-
ington in the United States. The detectors are sensitive in the audio band (∼ 30 Hz
to several kHz) with strain amplitudes as low as 10−21. The main research centres
are located at the California Institute of Technology in Pasadena, California and the
Massachusetts Institute of Technology in Cambridge, Massachusetts.
The basic geometry of each observatory is that of a 90◦ Michelson interferometer.
However, some changes have been made so as to increase their sensitivity. Each arm
consists of a resonant Fabry-Perot cavity, with the two mirrors also acting as the gravi-
tational test masses. The resonance amplifies the effect of a passing gravitational wave
on the phase difference between the two arms. Another mirror is placed between the
laser source and the beamsplitter so as to create a resonant cavity that increases the to-
tal power in the interferometer by not letting light escape back towards the laser—this
is called the power recycling mirror. An input mode cleaner before the PRM removes
8
1.1 Gravitational Waves
unwanted spatial modes from the laser source, leaving only the (l,m) = (0,0) mode, a
single central Gaussian peak. The Livingston observatory has an interferometer with
4km arm lengths, denoted L1. The Hanford observatory has two interferometers in
the same vacuum system; H1 has 4km arms and H2 has 2km arms. In the final Initial
LIGO science run, S5, all detectors operated at design strain sensitivity. The status of
compact binary coalescence searches at that time is given in [60] and descriptions of
the standard analysis pipeline are found in [61, 62].
Enhanced LIGO (eLIGO) [63] was an improvement upon the sensitivity of LIGO
by a factor of approximately two that did not require opening the vacuum chambers.
A plot of the sensitivity measured during eLIGO’s science run (S6) as a function of
GW frequency can be seen in Figure 1.3. At the low frequency end, the sensitivity is
limited by seismic noise. At the high frequency end, the limiting factor is quantum shot
noise from the error in counting the photons that reach the detector; the fractional error
in photon counts on the shorter integration timescales approaches the same fractional
distance to be measured. For eLIGO, a more powerful 35 W laser was installed, the
GW signal was switched to a DC readout that operates slightly off the dark fringe,
and an output mode cleaner was installed to remove unwanted spatial modes that add
shot noise but do not contribute to the GW signal. The S6 science run was performed
over 2009–2010 in this configuration. Despite the many controls, LIGO experiences
transient noise sources that detection pipelines have to rule out as possible signals [64–
67].
The next stage of development is Advanced LIGO (aLIGO) [69–71], which aims
to increase the sensitivity by a further factor of five. aLIGO’s upgrades include adding
a signal recycling mirror that both increases the broadband sensitivity and allows the
sensitivity of the interferometer to be “tuned”. Additionally, the test masses will be
hung from triple pendulums to decrease their sensitivity to seismic noise and reduce
the lower limit on observations to ∼ 10Hz. A 180 W laser will be installed and active
modulators and isolators will be used on the test masses to compensate for thermal
deformation. The test masses are also larger in size at 40 kg, up from 10 kg in Initial
and Enhanced LIGO. The H2 interferometer will be removed and the mirrors sent to
India in support of the building of a detector there by the IndIGO project [72]. A
schematic of the aLIGO setup can be seen in Figure 1.4.
9
1. INTRODUCTION
Figure 1.3: A plot of the strain sensitivities of the LIGO and Virgo interferometers
from the S6 and VSR2/3 science runs [68].
In addition to the LIGO project in the United States, there is a joint Italian-French
observatory, Virgo [73], located in Cascina, Italy. Virgo has a similar setup to LIGO
but with 3km arms and active seismic isolation. Virgo’s instrumental noise is similar
to LIGO’s as seen in Figure 1.3, but with a higher noise floor and better sensitivity
at frequencies below ∼ 50 Hz. Virgo’s second and third science runs were performed
in coincidence with LIGO’s S6. There is also a joint German-British observatory,
GEO600 [74], near Sarstedt, Germany that is part of the LIGO Scientific Collaboration.
GEO600 is primarily used for testing technology that will be used in upcoming LIGO
upgrades, but is currently on “astrowatch” while LIGO and Virgo are offline being up-
graded. Due to its shorter arm lengths and lack of Fabry-Perot cavities in the arms,
GEO’s sensitivity is much lower than LIGO or Virgo. In Japan the TAMA300 [75] ob-
servatory is operating while the KAGRA detector (the Large Cryogenic Gravitational-
wave Telescope) is under construction [76, 77]. Together, LIGO, Virgo, GEO, LCGT,
and IndIGO will form a worldwide network of detectors that will allow for simultane-
ous detection of GWs and improve our ability to determine the parameters of a signal.
10
1.1 Gravitational Waves
Figure 1.4: A diagram of the aLIGO interferometer. ETM = end test mass, ITM =
initial test mass, BS = 50/50 beam splitter, PD = photodetector, PRM = power recycling
mirror, SRM = signal recycling mirror, and MOD = phase modulation. Image from
www.ligo.caltech.edu.
11
1. INTRODUCTION
Studies are also underway for the construction of a third-generation ground-based in-
terferometric detector, the Einstein Telescope [78, 79].
The next generation GW detector under development is ESA’s New Gravitational-
wave Observatory (NGO) [80, 81], a re-scoped version of the Laser Interferometer
Space Antenna project (LISA) [82]. NGO will consist of three satellites (one ‘mother’
and two ‘daughters’) that will be the vertices of an equilateral triangle one million km
on a side. The NGO constellation is planned to orbit the Sun in the same orbit as the
Earth, but behind and at an inclination of 60◦ to the ecliptic. NGO will form an inter-
ferometer that can observe GWs from ∼0.1 mHz up to ∼0.1 Hz. This will allow NGO
to detect the mergers of supermassive black holes [83, 84], quasi-stationary signals
from smaller compact binaries in the galaxy, extreme-mass-ratio inspirals, bursts from
cusps in cosmic strings, and a stochastic GW background [85, 86].
In preparation for the NGO mission, LISA Pathfinder [87] will be launched by
ESA to test new technology. LPF is an advanced geodesics explorer and will be a
proof of principle for NGO. Pathfinder is scheduled for launch in 2014, with NGO
launch currently unsure.
A summary of the use of interferometry for detection of gravitational waves both
on the ground and in space is given by [88].
The near-perfect clock nature of pulsars also allows for the detection of very long
wavelength gravitational waves. As a GW passes between a source pulsar and the
Earth, it periodically distorts the spacetime that the pulsar signal must pass through.
This results in periodic variations in the timing of the pulsar signal at Earth. By mea-
suring many pulsars’ timings to high precision over several years, these variations may
be cross-correlated to indicate the passing of a gravitational wave or the presence of a
GW background [89–91].
The Gravitational Wave International Committee’s roadmap [92] outlines plans for
GW detection in the coming decades.
1.2 Bayesian Inference
Gravitational wave detection poses a major data analysis problem. Most signals in the
collected data streams are expected to be of low SNR. Additionally, there may be many
overlapping signals, not all of the same type, and a non-stationary noise background.
12
1.2 Bayesian Inference
Therefore, we need an efficient and robust method for identifying signals in the data
and determining their physical parameters, both intrinsic and extrinsic. To do this, we
turn to the methods of Bayesian statistics.
1.2.1 Bayes’ Theorem
In our analysis, we wish to estimate the parameters Θ in a model or hypothesis H for
given data D. Bayes’ theorem states that
Pr(Θ|D,H) = Pr(D|Θ,H)Pr(Θ|H)
Pr(D|H) . (1.18)
Pr(Θ|D,H)≡ P(Θ) is the posterior probability distribution, Pr(D|Θ,H)≡L(Θ) is the
likelihood function, Pr(Θ|H) ≡ pi(Θ) is the prior distribution of the parameters, and
Pr(D|H)≡Z is the Bayesian evidence. The evidence is the factor required to normalise
the posterior over Θ, so we can ignore it for parameter estimation and analyse just the
un-normalised posterior. It can be found, however, with the following integral (where
N is the dimension of the parameter space):
Z=
∫
L(Θ)pi(Θ)dN(Θ). (1.19)
To compare two models, H0 and H1, we compare their relative probabilities with
respect to given data. This is known as the odds ratio and is given by
Pr(H1|D)
Pr(H0|D) =
Pr(D|H1)Pr(H1)
Pr(D|H0)Pr(H0) =
Z1
Z0
Pr(H1)
Pr(H0)
. (1.20)
The prior probability ratio between the two models, Pr(H1)/Pr(H0), is taken to be
unity when no further information is available as to which model is more probable.
Thus, it can be seen from Equation (1.20) that the evidence plays a critical role in model
selection as it determines the relative probabilities; Z1/Z0 is known as the evidence ra-
tio. However, calculating the evidence requires the evaluation of a multi-dimensional
integral which is far from trivial, especially in the multimodal and degenerate parame-
ter spaces that we will be exploring.
In cases of both model selection and parameter estimation, we require a sampling
of the parameter space that provides us with information about the shape of the likeli-
hood function. From this information, we can numerically integrate and compare the
13
1. INTRODUCTION
evidence of different models or compare the relative likelihoods (and, as we will see,
local evidences) of different parameter choices. A more detailed review of Bayesian
statistics and its particular applications to cosmology can be found in [93].
1.2.2 MCMC and Nested Sampling
Due to the difficulty in integrating Equation (1.19) for complicated likelihood func-
tions in many dimensions and the computational impracticality of grid approximations,
Markov chain Monte Carlo (MCMC) methods have long been used for computing the
integral and obtaining samples from the posterior distribution. One of the most popu-
lar implementations is the Metropolis-Hastings algorithm [94, 95]. In this algorithm,
an initial point is chosen and steps to new points are chosen according to proposal
distributions. New points are accepted with the following probability:
Pr(xt+1|xt) = min
(
1,
P(xt+1)Q(xt+1,xt)
P(xt)Q(xt ,xt+1)
)
, (1.21)
where xt is the initial point, xt+1 is the new point, and the proposal distribution Q(xt+1,xt)
is the probability of jumping to point xt+1 from point xt . If the point is not accepted
then xt+1 = xt . After an initial “burn-in” series of steps, the resulting distribution of
samples will be equivalent to sampling from P(x). We therefore set P(xt) =L(xt)pi(xt)
to obtain samples from the posterior probability distribution. However, this methodol-
ogy will require many samples and likelihood evaluations, which can be computation-
ally expensive. Additionally, in order to obtain a calculation of the Bayesian evidence,
additional techniques such as simulated annealing (a.k.a. parallel tempering) [96, 97]
are typically used, further increasing the computational cost. Model selection with
MCMC may also be performed using a technique known as “reversible jump” MCMC
(RJMCMC), which allows jumps between different model parameter spaces [98].
Nested sampling [99, 100] is a Monte Carlo method targeted at the efficient cal-
culation of the evidence, but also produces posterior inferences as a by-product. It
calculates the evidence by transforming the multi-dimensional evidence integral into a
one-dimensional integral that is easy to evaluate numerically. This is accomplished by
defining the prior volume X as dX = pi(Θ)dNΘ, so that
X(λ ) =
∫
L(Θ)>λ
pi(Θ)dNΘ, (1.22)
14
1.2 Bayesian Inference
(a) (b)
Figure 1.5: Cartoon illustrating (a) the posterior of a two dimensional problem; and
(b) the transformed L(X) function where the prior volumes Xi are associated with each
likelihood Li. Images from [101].
where the integral extends over the region(s) of parameter space contained within the
iso-likelihood contour L(Θ) = λ . The evidence integral, Equation (1.19), can then be
written as
Z=
∫ 1
0
L(X)dX , (1.23)
where L(X), the inverse of Equation (1.22), is a monotonically decreasing function of
X . Thus, if one can evaluate the likelihoods Li = L(Xi), where Xi is a sequence of
decreasing values,
0 < XM < · · ·< X2 < X1 < X0 = 1, (1.24)
as shown schematically in Figure 1.5, the evidence can be approximated numerically
using standard quadrature methods as a weighted sum
Z=
M
∑
i=1
Liwi, (1.25)
where the weights wi for the simple trapezium rule are given by wi = 12(Xi−1−Xi+1).
An example of a posterior in two dimensions and its associated functionL(X) is shown
in Figure 1.5.
15
1. INTRODUCTION
The summation in Equation (1.25) is performed as follows. The iteration counter
is first set to i = 0 and N “active” (or “live”) samples are drawn from the full prior
pi(Θ), so the initial prior volume is X0 = 1. The samples are then sorted in order of
their likelihood and the smallest (with likelihood L0) is removed from the active set
and replaced by a point drawn from the prior subject to the constraint that the point
has a likelihood L > L0. The corresponding prior volume contained within the iso-
likelihood contour associated with the new live point will be a random variable given
by X1 = t1X0, where t1 follows the distribution Pr(t) = NtN−1 (i.e., the probability
distribution for the largest of N samples drawn uniformly from the interval [0,1]). At
each subsequent iteration i, the removal of the lowest likelihood point Li in the active
set, the drawing of a replacement with L> Li and the reduction of the corresponding
prior volume Xi = tiXi−1 are repeated, until the entire prior volume has been traversed.
The algorithm thus travels through nested shells of likelihood as the prior volume is
reduced. The mean and standard deviation of log(t), which dominates the geometrical
exploration, are:
E[log t] =−1/N, σ [log t] = 1/N. (1.26)
Since each value of log(t) is independent, after i iterations the prior volume will shrink
down such that logXi ≈−(i±
√
i)/N. Thus, one takes Xi = exp(−i/N).
The nested sampling algorithm is terminated when the evidence has been computed
to a pre-specified precision. The evidence that could be contributed by the remaining
live points is estimated as ∆Zi =LmaxXi, where Lmax is the maximum-likelihood value
of the remaining live points, and Xi is the remaining prior volume. The algorithm
terminates when ∆Zi is less than a user-defnied value (we use 0.5 in log-evidence).
Once the evidence Z is found, posterior inferences can be easily generated using
the final live points and the full sequence of discarded points from the nested sam-
pling process, i.e., the points with the lowest likelihood value at each iteration i of the
algorithm. Each such point is simply assigned the probability weight
pi =
Liwi
Z
. (1.27)
These samples can then be used to calculate inferences of posterior parameters such
as means, standard deviations, covariances and so on, or to construct marginalised
posterior distributions.
16
1.3 Thesis Outline
By providing both posterior samples and the value of the evidence, nested sampling
can greatly reduce the number of points at which we need to evaluate the likelihood.
This will provide us with what is necessary for both parameter estimation and model
selection in less time than traditional MCMC methods.
1.2.2.1 The MULTINEST Algorithm
The most challenging task in implementing nested sampling is to draw samples from
the prior within the hard constraint L> Li at each iteration i. The MULTINEST algo-
rithm [101, 102] tackles this problem through an ellipsoidal rejection sampling scheme.
The live point set is enclosed within a set of (possibly overlapping) ellipsoids and a new
point is then drawn uniformly from the region enclosed by these ellipsoids. The ellip-
soidal decomposition of the live point set is chosen to minimize the sum of volumes
of the ellipsoids. The ellipsoidal decomposition is well suited to dealing with poste-
riors that have curving degeneracies, and allows mode identification in multi-modal
posteriors, as shown in Figure 1.6. If there are subsets of the ellipsoid set that do not
overlap with the remaining ellipsoids, these are identified as a distinct mode and sub-
sequently evolved independently. The MULTINEST algorithm has proven very useful
for tackling inference problems in cosmology and particle physics [103–106], typi-
cally showing two orders of magnitude improvement in efficiency over conventional
techniques. More recently, it has been shown to perform well as a search tool for grav-
itational wave data analysis [107]. A different implementation of nested sampling for
LIGO analysis is described in [108].
1.3 Thesis Outline
In Chapter 2 I describe applications of Bayesian inference to LIGO observations of
binary inspirals. I begin with a simple example and proof-of-principle, followed by a
systematic analysis of the ability of a LIGO-Virgo network to localise the position of a
GW source on the sky. These are both performed with MULTINEST for Bayesian infer-
ence. I then consider using coherent versus incoherent model selection to distinguish
a real GW signal from glitches in actual LIGO data.
17
1. INTRODUCTION
(a)
(b)
Figure 1.6: Examples of MULTINEST enclosing live points in sets of ellipsoids. 1000
points have been randomly sampled from (a) a torus and (b) two non-intersecting el-
lipsoids. Images from [102].
18
1.3 Thesis Outline
Chapter 3 describes Bayesian inference for LISA/NGO sources using MLDC sim-
ulations. The first section covers parameter estimation and model selection for burst
sources from cusps on cosmic strings. I then analyse continuous gravitational wave
sources for LISA, where there are many overlapping signals present in the data to the
point of presenting foreground confusion noise at lower frequencies. Finally, I consider
detecting burst sources in the presence of the foreground of continuous GW sources.
At this point I turn to looking at data analysis techniques in a more general sense.
Chapter 4 analyses the benefits and drawbacks of the Multiple Optimised Parame-
ter Estimation and Data compression (MOPED) algorithm. I consider cases where it
works and where it fails and attempt to set a prescription for determining the viability
of a MOPED analysis.
Artificial neural networks are introduced in Chapter 5 as a tool for learning and
performing regression and classification functions. I detail the structure of neural net-
works and the algorithm used for training them on prepared data sets. Several examples
are then provided to illustrate the capabilities of the networks and the training algo-
rithm. These include the learning of analytic functions, classification of complicated
data, and reduction of the dimensionality of data.
In order to further improve tools for general Bayesian inference, neural networks
are integrated into the MULTINEST algorithm to learn the structure of the likelihood
function. They can then be used to make future predictions of the likelihood at new
points. This is especially beneficial for problems where likelihood evaluations are par-
ticularly time-consuming. In addition, the networks may be used for follow-up analy-
ses that can now be performed up to a hundred thousand times faster. The new Blind
Accelerated Multimodal Bayesian Inference (BAMBI) algorithm is demonstrated on a
few toy problems and then applied to the task of cosmological parameter estimation,
where significant increases in speed are demonstrated for initial analyses and excep-
tional speed-ups are achieved in follow-up.
19
1. INTRODUCTION
20
Chapter 2
Analysis of LIGO Data
In theory, there is no difference between theory
and practice. But, in practice, there is.
Jan L. A. van de Snepscheut
In the next few years the Advanced LIGO and Advanced Virgo detectors will be
coming online and collecting science data. During their observation period, one or
more compact binary coalescence events are expected to be detected [71, 109]. These
signals are well-modeled by post-Newtonian theory [6–8, 40], the effective-one-body
approach [19–22], and numerical relativity [24–27], giving us a variety of waveform
models to use to search the data. Additionally, the sensitivity of the LIGO and Virgo
detectors is designed to be optimal for these sources. In this chapter I begin by in-
troducing a simple proof-of-principle for detecting the gravitational waveform from
one of these events using Bayesian inference techniques. I then present work done as
part of the LIGO Scientific Collaboration (LSC) to quantify the ability of Bayesian
inference to determine the sky location of a detected event for use in electromagnetic
follow-up searches. This was done in part with the Parameter Estimation sub-group of
the Compact Binary Coalescence group of the LSC and will form part of a publication
currently in preparation; code from the LSC Algorithm Library (LAL [110]) was used
in generating and analysing this data. I conclude with my contribution to the analysis
of a hardware blind injection into LIGO’s sixth and Virgo’s second science runs [68].
The analysis in this section was presented as a poster at the Ninth Edoardo Amaldi
Conference on gravitational waves [111].
21
2. ANALYSIS OF LIGO DATA
2.1 Simple Inspiral Model
Many stars form as part of a binary system [112]. Over their lifetimes, these stars will
slowly spiral into each other as they lose energy from the emission of gravitational
waves. These predictions have been verified by the discovery of binary pulsar PSR
B1913+16 in 1974 by Hulse and Taylor and subsequent observations [2]. As the two
stars spiral in on each other, their GW signal produces a “chirp”, characterised by
increasing amplitude and frequency until a specific frequency of the last stable orbit,
after which they go through merger and ringdown phases.
2.1.1 Generating the Signal
The sensitivity of the LIGO observatory is best for the inspiral phase of a compact
binary coalescence for objects with sizes on the order of solar masses, so for this
reason and for computational simplicity we consider only the inspiral. We assume
the two black holes are nonspinning and have masses m1 and m2. We therefore define
the total mass, m=m1+m2, symmetric mass ratio, η = (m1m2)/(m1+m2)2, and chirp
mass, M= (m1m2)3/5(m1+m2)−1/5 = mη3/5. Following the approach of Veitch and
Vecchio [113], we take the leading Newtonian quadrupole order of the strain in the
frequency domain. The GW signal can thus be expressed as
h˜( f ;~θ) =
{
AM5/6 f−7/6eıψ( f ;~θ) f ≤ fLSO
0 f > fLSO
, (2.1)
where
ψ( f ;~θ) = 2pi f tc−φc− pi4 +
3
4
(8piM f )−5/3 (2.2)
is the signal phase and fLSO is the frequency of the last stable orbit in the Schwarzschild
spacetime. For a general system this is given by
fLSO = (63/22−1pim)−1, (2.3)
but for simplicity we assume m1 = m2 so we can re-write fLSO as
fLSO = (63/221/5piM)−1. (2.4)
22
2.1 Simple Inspiral Model
In equations (2.1) and (2.2), ~θ refers to the 4-dimensional parameter vector, where our
parameters are
~θ = {A,M, tc,φc}. (2.5)
A is the overall signal amplitude and depends on the source masses, distance, sky
location, inclination angle, and polarisation; tc and φc are the time and phase at coa-
lescence, respectively. In this analysis, geometrical units are being used, so c = G = 1
and therefore M = 4.926×10−6 s and A has the units of s−2. It should be noted that
only the chirp mass, M, is being used in this formula. Therefore, a degeneracy exists
between the two black hole masses—any two masses that produce the same chirp mass
will give the same results with this formula. A more accurate model of the signal that
also uses the symmetric mass ratio, or any other function of the masses, would be able
to break this degeneracy.
The detector noise profile in the frequency domain is modelled as a Gaussian ran-
dom process in the real and imaginary parts, with zero mean and a variance at fre-
quency fk (k = 1, . . . ,K, where K is the number of frequency bins) given by
σ2k =
(
fk
f0
)−4
+1+
(
fk
f0
)2
. (2.6)
This is representative, within a multiplicative constant, of the LIGO interferometers
with f0 = 150 Hz setting the scale for the frequency of maximum sensitivity.
2.1.2 Calculating the Likelihood
We define our likelihood function to be a multi-variate Gaussian, centred on the true
parameter values:
L(~θ) =
K
∏
k=1
[
1
2piσ2k
exp
(
−|h˜k(
~θ)− d˜k|2
2σ2k
)]
. (2.7)
To simplify this, we compute the log-likelihood and subtract out any constants. This
will give us an evidence value that is off by a constant factor, but all local evidences will
have this factor which will be cancelled out in the evidence ratio. The log-likelihood
is therefore
lnL(~θ) =
K
∑
k=1
(
−|h˜k(
~θ)− d˜k|2
2σ2k
)
. (2.8)
23
2. ANALYSIS OF LIGO DATA
Parameter Measured Actual
A (s−1/M) 4.809 ± 0.328 5.00
M (M) 8.000 ± 0.015 8.009
tc (s) 20.0 ± 0.21 ×10−3 20.0
φc (rad) 1.262 ± 0.280 1.00
Table 2.1: Results from the MULTINEST parameter inference on a simple inspiral chirp
signal.
In order to run the nested sampling algorithm MULTINEST [101, 102] on each data
set we need to define the prior volume for it to explore. In order to have the most
generality, we will use a uniform prior range defined over the following intervals:
A ∈ [0,1000] s−1/M, (2.9a)
M ∈ [1,20]M, (2.9b)
tc ∈ [19.5,20.5] s, (2.9c)
φc ∈ [0,2pi) rad. (2.9d)
This is a significantly wider prior range than that used by Veitch and Vecchio.
Their prior ranges were A ∈ [0,10] s−1/M, M ∈ [7.7,8.3]M, tc ∈ [19.9,20.1] s, and
φc ∈ [0,2pi) rad.
2.1.3 MULTINEST Results
MULTINEST was used to analyse the simulated data produced by the model described
in Section 2.1.1 with parameters of ~θ = {5s−1/M,8.009M,20,1}. The results of
the run are given in Table 2.1 and the final marginalized posterior distributions can be
seen in Figure 2.1. These results show that MULTINEST was able to estimate all four
parameter values very closely.
In Figure 2.2 is plotted the log-likelihood function as it varies over the amplitude
and chirp mass for the correct values of tc and φc. We can clearly see a ridge at constant
M and a background drop-off for increasing A andM. In order to ensure sampling from
this ridge, whose peak coincides with the true parameters, an increased number of live
points for MULTINEST need to be used. This had the expected side effect of increasing
24
2.1 Simple Inspiral Model
Figure 2.1: Plots of the one-and two-dimensional marginalized posterior distributions
obtained by MULTINEST. Veritcal red lines and plus signs indicate true input values.
[A] = s−1/M, [M] = M and [φc] =rad.
25
2. ANALYSIS OF LIGO DATA
Figure 2.2: The log-likelihood function for a given range of amplitude and chirp mass
values. The actual values of signal are A = 300s−1/M, M = 3.52M, tc = 20 s, and
φc = 0 rad. tc and φc have been set to their actual values for this calculation.
the computation time. On a single Linux machine, MULTINEST still took less than an
hour to run, even for signals with low SNRs.
If we were to look at the log-likelihood for the prior ranges of [113], the large
background feature in A and M is not significant, as can be seen in Figure 2.3. The
range of chirp masses they explore limits the effect of this background effect and also
causes the ridge to occupy a larger fraction of the sampled space. This makes the
likelihood easier to sample, but limits the detectability to signals from binaries whose
parameters fall within a narrow range.
We can now measure how well MULTINEST can find the true signal embedded in
noise. A natural measure of this is the Bayes factor, the ratio of the posterior odds of
the signal and noise-only models. We assume a prior odds ratio of one so the Bayes
factor is calculated simply by dividing the evidence of the signal model as measured
by MULTINEST and the evidence of the noise-only model. A Bayes factor greater than
one indicates that the signal model is more probable than the noise model; we ask for
a slightly larger Bayes factor to declare a detection. As the Bayes factor increases, the
confidence of our detection grows as well. To measure how detection is affected by
weaker signals, I computed the Bayes factor for several signal-to-noise ratios (SNRs).
26
2.1 Simple Inspiral Model
Figure 2.3: The log-likelihood function for data with an input signal of ~θ =
{5/M,8M,20,0}, where tc = 20 s and φc = 0 rad have been set.
The SNR, specified by ρ , provides a measure of the relative strength of the signal to
the background noise and is calculated using the following formula:
ρ =
√√√√ N∑
k=1
|h˜k(~θ)|2
σ2k
. (2.10)
Varying values of the SNR were obtained by changing the amplitude of the signal.
The chirp mass (total mass and symmetric mass ratio), time of coalescence, and phase
of coalescence were all kept equal so as to minimize any effects other than varying sig-
nal strength. The global evidence of the signal model was given by MULTINEST at the
end of each run, which were performed using the original log-likelihood function and
1000 live points to ensure a thorough sampling of the prior volume. The null evidence
(that of the noise model) was computed by fixing the amplitude to zero. Figure 2.4
shows the log Bayes factor as a function of the measured SNR over a range of SNRs.
We can see that once a certain threshold has been reached by an SNR of 10, we can be
very confident in our detection of a signal. Below an SNR of ∼ 6, however, we cannot
be sure of a signal being present.
27
2. ANALYSIS OF LIGO DATA
Figure 2.4: The log Bayes factor as a function of SNR. Past an SNR of ∼ 10, our
confidence of detection increases exponentially.
This clearly shows that Bayesian inference as performed by MULTINEST is a
strong tool for detecting gravitational wave signals in instrument noise. Being able
to make detections at SNRs below 10 is important as the current instruments do not
expect to have many SNRs above that if a signal is in fact present. For comparison,
see Veitch and Vecchio’s original work on this same problem in [113, 114], which
shows similar results for Bayesian evidences to those obtained here, and their use of
delayed rejection in [115, 116]. MULTINEST has also been used to detect the fully
parameterised waveform from non-spinning supermassive black holes in [107]. The
usefulness of inspiral-only templates for detection of full inspiral-merger-ringdown
waveforms is examined in [117]. The use of MULTINEST for the detection of galaxy
clusters with a fully Bayesian analysis can be found in [118].
2.2 LIGO Sky Localisation
In order to confirm the first detections of gravitational waves and extract the most
science from an observed event, we will want to obtain follow-up observations from
electromagnetic telescopes [119–126]. For certain events, these will be able to confirm
28
2.2 LIGO Sky Localisation
the event occurring as well as provide additional information. For binary black hole
mergers, the lack of an observed signal will confirm theory and the detection of a signal
could indicate the presence of an accretion disk around one or both of the black holes.
To this end, a project was begun in the Bayesian parameter estimation subgroup of the
CBC group in the LSC to determine our ability to localise the position of a source on
the sky and measure how fast we would be able to accomplish this. As this was also a
validation test of the Bayesian codes, we elected to use a simpler model for the inspiral
signals and analytically generate the noise according to a known power spectral density
(PSD).
Studies have already been performed to analyse the ability of GW detector net-
works to localise a source on the sky [127–134], but this is the first systematic Bayesian
study that utilises coherent detection and compares many inference algorithms to con-
firm results and measure speed. The full paper presenting results from the Parameter
Estimation group is in preparation. Other studies, such as [135], have measured statis-
tical uncertainties in measured parameters with the advanced detector network, but use
the Fisher matrix approximation that cannot capture the full nature of posterior degen-
eracies and does not apply at low SNRs. There have also been efforts to characterise
the ability of a detector network to measure other source parameters [136].
2.2.1 The Injections
The injections were performed with the restricted frequency-domain post-Newtonian
waveform model, TaylorF2 [10], at second order in phase (corrections of O(v4)) and
with zeroth order (Newtonian) amplitude. No spin was included in the models in order
to simplify the analysis so there were 9 total parameters: chirp mass, symmetric mass
ratio, luminosity distance, inclination of the orbital plane (angular momentum) to the
line of sight, polarisation, phase at coalescence, time of coalescence at the geocentre,
longitude (related to right ascension), and colatitude (related to declination). The last
two of these are the position on the sky of the source. The parameter vector is thus
given by ~θ = {M,η ,dL, ι ,ψ,φ0, tc,φs,θs}. φs and θs are related to the right ascension
α and declination δ by φs = α −GAST (GAST is the Greenwich Apparent Sidereal
29
2. ANALYSIS OF LIGO DATA
Time) and θs = pi/2−δ . The waveform is given in the source frame by [10]:
h˜+( f ) =
M5/6
dL
f−7/6(1+ cos2(ι))cos(Φ( f )) (2.11a)
h˜×( f ) =−M
5/6
dL
f−7/6 cos(ι)sin(Φ( f )) (2.11b)
Φ( f ) =2pi f tc−φ0− pi4 +
3
128ηv5
× (2.11c)[
1+
20
9
(
743
336
+
11
4
η
)
v2−16piv3+10
(
3058673
1016064
+
5429
1008
η+
617
144
η2
)
v4
]
where v = (piM f )1/3. The waveform is evaluated for frequencies up to the last stable
orbit given by Equation (2.3).
This is measured by each of the LIGO and Virgo detectors with sensitivity patterns
given by:
F+(ψ,φs,θs) =−12(1+ cos
2(θs))cos(2φs)cos(2ψ)− cos(θs)sin(2φs)sin(2ψ),
(2.12)
F×(ψ,φs,θs) =
1
2
(1+ cos2(θs))cos(2φs)sin(2ψ)− cos(θs)sin(2φs)cos(2ψ),
(2.13)
such that the signal measured is given by
h˜ = h˜+F++ h˜×F×. (2.14)
We analyse 200 injections chosen uniformly from the distributions given in Ta-
ble 2.2. Notice that the masses are sampled by m1 and m2 instead of M and η – this is
to allow equal-mass binaries more easily. The additional constraints of m1 > m2 and
m1+m2 ≤ 20M are also applied. tT is the pre-determined and supplied trigger time
of the signal that we use as the basis for a search interval. These same ranges are also
used as the priors for the subsequent data analysis.
The noise PSD was generated in the frequency domain according to the analytic
formula given below with f0 = 150Hz.
S2h( f ) = 9×10−46
((
4.49
f
f0
)−56
+0.16
(
f
f0
)−4.52
+0.52+0.32
(
f
f0
)2)
.
(2.15)
30
2.2 LIGO Sky Localisation
Parameter Units Minimum Maximum
m1 M 1 15
m2 M 1 15
log(dL) Mpc 10 40
cos(ι) – −1 1
ψ rad 0 pi
φ0 rad 0 2pi
tc sec tT −0.1 tT +0.1
φs rad 0 2pi
cos(θs) – −1 1
Table 2.2: Ranges for the sampling distributions of signal parameters. These are used
for both signal generation and prior probabilities.
This same function was used in the analysis for the PSD. Although this does not mir-
ror the non-stationarity and non-Gaussianities in real LIGO noise, it does allow us to
confirm the results of our analysis using statistical measures.
2.2.2 Analysis and Results
The 200 injections were analysed with MULTINEST on LIGO’s Caltech cluster, from
which we were able to obtain converged results on 187 (the remaining 13 were ex-
cluded due to computational issues that kept preventing analysis jobs from finishing).
5000 live points were used to ensure complete sampling of the prior. Run times on a
single processor varied from approximately 2 days to 2 weeks depending on the num-
ber of likelihood evaluations necessary. The loudest and quietest signals had the lowest
sampling efficiency due to sharp peaks or broad degenerate modes in the likelihood
function, respectively, and so took the longest to converge.
A typical posterior distribution is shown in Figure 2.5 for an event with a total
network SNR of 19.6. Strong correlations can be seen between certain parameters,
particularly between distance and inclination and between different pairs of mass pa-
rameters. We can also see that the phase at coalescence is not well constrained and
that there is a multimodal degeneracy in the polarisation of the GW signal. It is impor-
31
2. ANALYSIS OF LIGO DATA
tant to fully explore these modes and degeneracies as the true signal parameters can be
located anywhere within them.
Focusing on the intrinsic parameters of the system, which for this simple model is
limited to M (mc) and η , we see in Figure 2.6 a degeneracy but accurate recovery of
the true parameters within the 68% confidence interval. For the sky location, which is
what we need to measure for electromagnetic follow-up, there is again a degeneracy
but the location has been measured to within a fraction of a steradian and the true
value recovered just outside the 68% confidence interval. The 68% interval contains
less than 50 deg2 and the 95% interval less than 100 deg2; the posterior as mapped on
the entire sky can be seen in Figure 2.7.
For our first test that posteriors were recovered correctly, we look at the fraction
of injections recovered at a given level of each parameter’s integrated one-dimensional
posterior probability. We integrate the posteriors from the minimum to the maximum
of each parameter and record the cumulative distribution function (CDF) value at the
point of the true injected value. We expect the distribution of these values to map
directly to the posterior distribution. If we therefore plot the fraction of injections re-
covered below a given CDF value against CDF values, the distribution should follow
a line corresponding to y = x from 0 to 1. We are essentially comparing the CDF of
the injections with the CDF of the measured posteriors to confirm they are equal. In
Figure 2.8 we show this for all 9 model parameters as well as m1 and m2, which our
prior is defined on. We find that the 1D CDFs all match the expected distributions
very closely, usually to within 2σ error. Some deviations are seen, but these are most
exaggerated for poorly constrained parameters and correlations between parameters
can cause these errors to affect other measurements. To quantitatively measure the dif-
ferences in experimental and theoretical CDFs, we performed a Kolmogorov-Smirnov
(K-S) test on each. This test measures the maximum difference in CDFs to decide the
probability that the data supports the null hypothesis that both CDFs are from the same
distribution. A p-value less than the threshold of 0.05 indicates that we reject the null
hypothesis at 95% confidence. All parameters return p-values above 0.05 as indicated
in the subplot titles. This is a good indication that the recovered posterior distributions
do accurately represent our state of knowledge.
A more thorough check involves measuring the posterior confidence level at which
the injection was recovered in sky position. This interval is is calculated by measuring
32
2.2 LIGO Sky Localisation
Figure 2.5: Marginalised 2D and 1D posterior distributions for event 22 (all parameters
except tc). This event has an SNR of 9.0 in H1, 13.4 in L1, and 11.1 in V1 for a total
network SNR of 19.6. Red vertical lines and crosses indicate the true values. Contours
on 2D plots indicate 68% and 95% confidence intervals.
33
2. ANALYSIS OF LIGO DATA
(a) (b)
Figure 2.6: Marginalised 2D posterior distributions for event 22 in (a) chirp mass and
symmetric mass ratio and (b) sky position. The red cross indicates the true value.
Contours indicate 68% and 95% confidence intervals.
Figure 2.7: The marginalised posterior probability distribution for the sky position of
event 22 as shown across the entire sky. The white × marks the true location.
34
2.2 LIGO Sky Localisation
Figure 2.8: The fraction of injections recovered at or below 1D posterior CDF values.
Error bars are given by
√
p(1− p)/N, where N is the number of injections. K-S test
p-values are given in the titles for each subplot. All parameters pass the test for a
threshold of 0.05.
35
2. ANALYSIS OF LIGO DATA
Figure 2.9: The fraction of injections recovered within high probability density inter-
vals of the posterior as computed over θs and φs. The sag is due to posterior biases and
K-D tree binning.
the accumulated posterior probability in regions with higher probability density than
where the injection is located. We perform a K-D tree [137] binning on the two sky
location variables and sum the posterior density for ‘leaves’ with higher density than
where the injection is found. As in the 1D case, we expect to find X% of injections
within the X% high probability density (HPD) region of the posterior. Figure 2.9
displays our measured recovery compared to the theoretical line. There is a degree
of ‘sag’ below the line, but this is an expected artifact from the process of K-D tree
binning and Poisson error on the number of samples in a certain region.
A final validation of the recovered posterior values involves direct measurement of
the full nine-dimensional HPD interval at which each injection is recovered. We do
this by calculating the log-prior at each posterior sample to add to the log-likelihood
and obtain an non-normalised log-posterior density value. The same value is computed
for the true parameters of the injection. We then count the fraction of posterior sam-
ples that have a higher density than the true parameters and this equals the confidence
interval at which the injection was recovered. We expect the fraction of injections re-
36
2.2 LIGO Sky Localisation
Figure 2.10: The fraction of injections recovered within high probability density in-
tervals of the posterior as computed from posterior probability density values directly.
Error bars are given by
√
p(1− p)/N, where N is the number of injections. The sag is
due to accumulation of biases seen in the 1D cases (Figure 2.8). The K-S test p-value
is 0.179.
covered within a given HPD to follow y= x as in Figure 2.8 if the recovered posteriors
are, in fact, representative of the correct priors and likelihood. This comparison is
shown in Figure 2.10 along with 1σ error bars. The recovery is consistent to within
2σ of the theoretical value for all confidence intervals, but there is a systematic bias
towards lower values. This is likely due to the biases seen in the one-dimensional
CDFs in Figure 2.8 accumulating to push the peak away from the true value beyond
the expected statistical variation. The K-S test results in a p-value of 0.179, so the null
hypothesis that the CDFs are equal is not rejected.
Having passed these tests of the posterior distribution, we now look at the statis-
tics for the areas within certain intervals. Our first consideration is the area contained
within a particular posterior high probability density interval. Using the K-D tree bin-
ning, we calculate the sky area within 68%, 90%, and 95% confidence intervals. In
Figure 2.11 we plot the fraction of injections whose confidence intervals contain a
37
2. ANALYSIS OF LIGO DATA
Figure 2.11: The fraction of injections whose HPD posterior confidence intervals con-
tain the amount of sky area or less. Areas are calculated using K-D tree binning.
certain amout of sky area or less. This allows us to say that if we can search over a
certain amount of sky area, say 100 deg2, then ∼ 80% of all signals will have their
90% confidence intervals contained within that search area. We therefore expect to
place constraints on ∼ 72% of signals. We can also determine that our confidence
regions increase by approximately an order of magnitude in sky area in going from
68% to 90% and again from 90% to 95%. Therefore, using larger confidence intervals
may make the search regions impractically large for electromagnetic follow-up. Pos-
terior distributions tend to be ellipses in right ascension and declination, making them
slightly curved on the sky when extending further from the peak.
The next consideration is the fraction of injections that were recovered within a
given sky area. For this, we use the K-D tree binning to determine the sky area con-
tained within bins with posterior density equal to or greater than that for the bin in
which the true point is located. This comparison is shown in Figure 2.12 and allows us
to determine the amount of sky area we need to be able to search to find counterparts
for a given fraction of signals or conversely, the fraction of signals we can expect to
find counterparts for in a given search area. For example, if we can only search over an
38
2.2 LIGO Sky Localisation
Figure 2.12: The fraction of injections that were recovered within the amount of sky
area or less. Areas are calculated using K-D tree binning.
area of 4 deg2, then we can expect to locate counterparts (where they exist) for 50% of
signals; however, if we wish to identify counterparts for 90% of signals then we will
need to search 300 deg2 of sky. With this information we can optimise our follow-up
search strategies based on either the sky area they can map or the fraction of signals
we wish to recover.
The first two analyses were averaged over all injections. However, we expect to be
able to localise signals with higher SNR to smaller sky areas. Therefore, the final com-
parison considers the sky area as a function of signal-to-noise ratio. Figure 2.13 shows
the sky area contained within the 90% posterior confidence interval as a function of
SNR. It also shows the area within the interval at which each signal was recovered,
but this measurement is subject to statistical variations in the relative positions of the
true location and the posterior peak due to the particular noise realisation. Overall,
we can see in both sets of points that increased SNR does result in smaller sky areas,
with the 90% interval area scaling roughly ∝ 1/SNR2. For SNRs above 25, all 90%
confidence regions contain 100 deg2 or less, but for all signals with SNR below 20
this same interval contains at least 20 deg2. We see then that in order to find electro-
39
2. ANALYSIS OF LIGO DATA
Figure 2.13: The sky area within 90% posterior confidence intervals and the posterior
interval at which an injected signal was recovered as a function of signal-to-noise ratio.
Areas are calculated using K-D tree binning. A line showing A ∝ 1/SNR2 is plotted
for comparison.
magnetic counterparts to weaker signals, we will have to be able to search larger areas
of the sky. Even with very loud signals, 90% confidence intervals still contain over a
square degree, which is a large area for electromagnetic telescopes.
Since our ability to localise sources is fundamentally limited to areas of O(10)
deg2 for 90% confidence intervals, optimised methods for electromagnetic follow-up,
such as those explored in [138], are necessary to increase the chances of observing
a counterpart. Combined detections of both GWs and electromagnetic signals (and
possibly neutrinos) will further confirm source models for GW observations and lead
to improved science. It may be possible to discover more about the internal physics of
supernovae, neutron star mergers that form black holes, and much more.
40
2.3 The “Big Dog” Event and Coherence Tests
2.3 The “Big Dog” Event and Coherence Tests
Throughout 2009 and 2010, LIGO and Virgo were collecting data in their sixth and
second/third science runs, respectively (called S6 and VSR2/3). One of the primary
sources that these detectors could observe and that was searched for in the data is low-
mass compact binary coalescences. These involve binaries with a total mass less than
25M and a minimum component mass of 1M. The CBC group of the LVC has
published a report on the results of this search [68, 139], which made no detections
of gravitational waves. A complementary high-mass search was also performed [140].
A similar analysis was previously performed on the joint observations during S5 and
VSR1 [141–144]. A review of knowledge of event source rates following S5/VSR1 is
presented in [145].
During S6 and VSR2/3, a hardware injection was made into the detectors (by ac-
tuating test mass mirror controls) without the knowledge of the data analysis teams as
part of a “blind injection challenge”. The task was for the detection pipelines to find
the event and then for the analysis groups to correctly identify and characterise the
signal that was injected. The event came to be known as the “big dog” since initial
sky location estimates placed the source in the Canis Major constellation. The analysis
in this section was presented as a poster at the Ninth Edoardo Amaldi Conference on
gravitational waves [111] and used data from the S6 and VSR2 science runs.
One of the major considerations for determining the existence of a real signal is
the possibility of false coincidences. Each of the detectors registered an event trigger
at a similar time, but this could also potentially happen if they each glitched at ap-
proximately the same time. We are able to rule out coincident glitches by performing
a coherent search and comparing the evidence to that of an incoherent search. In the
case of just the LIGO detectors, this can be summarised as:
coherent =⇒ ∫ LHLLdθ = ZHL
incoherent =⇒ ∫ LHdθ1× ∫ LLdθ2 = ZHZL
If each detector did indeed observe the same signal, then the combined likelihood
function will provide a larger evidence ZHL than the product of the evidences from
the separate likelihoods ZHZL. If in fact this was not a real GW signal, then the two
detectors will favour different parameters and the coherent evidence will be severely
penalised. We can consider these as two separate models and compare the Bayesian
41
2. ANALYSIS OF LIGO DATA
evidences, such that if ZHL > ZHZL we will favour the coherent signal model and if
ZHL < ZHZL we favour the incoherent model, which implies no GW signal. A toy
example of this application can be found in [146].
A large number of time-slides were analysed in the calculation of the false alarm
rate for the “big dog” event. These involve taking data from one of the two LIGO de-
tectors and translating it in time such that there can be no coherent GW signal present.
This data is then analysed for potential triggers. Data from Virgo was not used in the
analysis that generated the time-slides and the signal in Virgo was of low SNR so V1
data is not used here, either. In total, 14 triggers were found with SNR or “newSNR”
(SNR modified by χ2 of data minus best-fit template) greater than that of the original
event; all of these contained the injected signal in H1 and a glitch in L1. Details of the
time-slides and newSNR calculation can be found in [68]. These time-slides provide
us with an opportunity to test the Bayesian criterion for comparing coherent and inco-
herent signal models. The 14 time-slide triggers and the “big dog” were each analysed
with the TaylorF2 waveform model at 2, 2.5, and 3.5 post-Newtonian order in phase
(0pN amplitude). Table 2.3 reports the Bayesian evidence ratios calculated for each
trigger using each waveform model.
The significantly negative log-evidence ratios clearly eliminate all time-slide trig-
gers from being coherent signals with probabilities much less than 1%. The most
significant time-slide, TS5, contained a weak signal in L1, so was thus able to disagree
with the signal in H1 the least. L1 on its own returned a very low evidence that would
not likely have passed as a trigger on its own. The cause for low evidence ratios for
the “big dog” event is parameter biases from the fact that we are not using the correct
waveform model. After the true signal was revealed, we found that the injection was
calculated with a time-domain waveform and contained significant spin in the larger
component, something the TaylorF2 waveform was not modeling. Even so, the lowest
coherent probability was still 37.5%, more than enough to prevent us from eliminating
the model as a possibility without further investigation.
This analysis demonstrated the usefulness of Bayesian evidence criteria for model
selection and detection for LIGO. When the noise is difficult to model but uncorre-
lated between separate detectors, comparing the Bayesian evidence from coherent and
incoherent signal models can be used as a further detection threshold to eliminate co-
incident glitches.
42
2.3 The “Big Dog” Event and Coherence Tests
Event log(R) 2pN log(R) 2.5pN log(R) 3.5 pN
BD 1.39 -0.509 1.90
TS1 -17.8 -15.4 -15.1
TS2 -11.5 -10.2 -10.1
TS3 -12.7 -14.3 -13.4
TS4 -11.1 -12.5 -10.8
TS5 -2.96 -3.88 -2.65
TS6 -14.2 -14.2 -14.1
TS7 -14.2 -12.0 -11.7
TS8 -8.38 -8.52 -6.59
TS9 -7.74 -9.21 -9.19
TS10 -14.2 -14.7 -14.0
TS11 -15.2 -13.1 -14.2
TS12 -7.39 -8.06 -8.07
TS13 -10.4 -7.53 -9.00
TS14 -10.6 -11.8 -12.3
Table 2.3: Logs of the Bayesian evidence ratios (R = ZHL/(ZHZL)) comparing coher-
ent and incoherent signal models for the time-slide triggers (TS1-TS14) and “big dog”
event (BD). This analysis clearly rules out all time-slide triggers as signals but is not
definitive with regards to the “big dog”.
43
2. ANALYSIS OF LIGO DATA
2.4 Conclusion
Throughout this chapter I have shown the useufulness of Bayesian inference methods
for the analysis of ground-based gravitational wave detector data. A first toy example
demonstrated the ability of Bayesian techniques to provide accurate parameter estima-
tion as well as a threshold for detection.
An analysis of many simulated signals injected into simulated noise representa-
tive of the LIGO-Virgo network further illustrates this applicability to ground-based
detector data. We are able to obtain accurate posterior inferences about the source
parameters, allowing us to measure intrinsic parameters of the system as well as de-
termine the location of a source on the sky. This latter capability was quantified as it
is important for obtaining electromagnetic follow-up observations to supplement the
information about the source event. We found that the LIGO-Virgo network will be
able to identify 50% of signals to within 4 deg2 and 90% of signals to within 300 deg2.
Louder signals will be more precisely measured on the sky, with the area in a given
posterior confidence interval scaling roughly ∝ 1/SNR2.
Lastly, through analysis of the “big dog” blind injection event in S6 and VSR2, we
were able to test a Bayesian criterion for the veracity of a detected signal. By requiring
that the signal in distant detectors be from the same source, we compared coherent and
incoherent signal models using the Bayesian evidence. Coincident but incoherent sig-
nals were simulated by using time-slides of detector data and the evidence ratio clearly
indicated that these were all incoherent detections, while supporting the hyposthesis
that the injection itself was a real signal.
These two tools of being able to distinguish signals from coincident glitches and
measure the probability distribution for source parameters are vital to future LIGO and
Virgo observations. Once detection claims are being made it is even more important
to be sure that these are in fact real astrophysical signals and not conspiring glitches
in the detectors. Additionally, measuring source parameters will allow for tests of
general relativity and astrophysical source and population models. Determining sky
locations facilitates electromagnetic follow-up observations to obtain even more infor-
mation about a detection and perform multi-messenger astronomy.
44
Chapter 3
The Mock LISA Data Challenges
Measure what can be measured, and make
measureable what cannot be measured.
Galileo Galilei
In addition to ground-based gravitational wave searches, there are long-term plans
for a space-based detector. Originally the joint NASA and ESA project, the Laser
Interferometer Space Antenna (LISA) [82], it is now an ESA-only venture, the New
Gravitational-wave Observatory (NGO) [80, 81]. Preparation for this has involved both
technical development of the satellites as well as exploration of the science case. This
has involved modeling the populations of expected sources in the sensitivity range for
the detector and then preparing and analysing expected observational data [147, 148].
In this chapter I present the analysis of data from a few of the mock data challenges that
were run in order to spur development of data analysis code and demonstrate our ability
to obtain scientific information from future observations. The work in Section 3.1 was
done in collaboration with Jonathan Gair and Farhan Feroz and published in [149];
Section 3.2 was done in collaboration with F. Feroz and Stanislav Babak of the Max
Plank Institute for Gravitational Physics (Albert Einstein Institute, AEI) in Potsdam,
Germany; Section 3.3 was performed in collaboration with F. Feroz, S. Babak, and
Antoine Petiteau also of the AEI.
45
3. THE MOCK LISA DATA CHALLENGES
3.1 Cosmic String Cusps
In preparation for the future launch of a space-based gravitational wave detector, such
as LISA or NGO, development of data analysis algorithms has been a field of active
research. To this end, LISA algorithm development has been encouraged by the Mock
LISA Data Challenges (MLDCs) [150]. Round number 3, completed in April 2009,
included data sets containing galactic white dwarf binaries (see Section 3.2), super-
massive black hole binary mergers, extreme-mass-ratio inspirals, cosmic string cusps,
or a stochastic background added to instrumental noise. In [149], we applied MULTI-
NEST to the problem of detecting and characterising cosmic string cusps (MLDC 3.4).
A summary of all submissions for MLDC 3 as well as plans for MLDC 4 can be found
in [151]. A separate analysis of MLDC 3.4 is described in [152], where the authors
focus on the problems of the degeneracies inherent in this signal type’s posterior dis-
tribution.
Cosmic string cusps in the observable frequency range of LISA are a burst source
with durations of hundreds of seconds or less, which is short on the timescale of a LISA
mission lasting up to two years. However, they are just one type of potential source for
burst signals. Since they can be physically modelled we can use a matched filtering
search with expected waveforms. There may be bursts of gravitational waves in the
LISA data stream from other sources and the question arises as to whether we would
be able to detect them and if we can distinguish cosmic string bursts from bursts due
to these other sources. The Bayesian evidence that MULTINEST computes is one tool
that can be used to address model selection. Evidence has been used for gravitational
wave model selection to test the theory of relativity in ground based observations [114],
and, in a LISA context, to distinguish between an empty data set and one containing
a signal [153]. Calculation of an evidence ratio requires a second model for the burst
as an alternative to compare against. We chose to use a sine-Gaussian waveform as a
generic alternative burst model, as this is one of the burst models commonly used in
LIGO data analysis (see for instance [154]). We find that the evidence is a powerful
tool for characterising bursts — for the majority of detectable bursts, the evidence
ratio strongly favours the true model over the alternative. While the sine-Gaussian
model does not necessarily well describe all possible un-modelled bursts, these results
46
3.1 Cosmic String Cusps
suggest that the evidence can be used to correctly identify any cosmic string bursts that
are present in the LISA data stream.
3.1.1 Burst waveform models
3.1.1.1 Cosmic Strings
Gravitational waves can be generated by cosmic strings through the formation of cusps,
where portions of the string are traveling at nearly the speed of light [155, 156]. Such
radiation is highly beamed and when viewed along the emission axis is linearly po-
larised and takes a simple power-law form, h(t) ∝ |t − tc|1/3 [155]. When viewed
slightly off-axis, the waveform is still approximately linearly polarised but the cusp
spectrum is rounded off and decays rapidly for frequencies above fmax ∼ 2/(α3L),
where α is the viewing angle and L is the dimension of the feature generating the
cusp [155, 157]. The particular model for the frequency domain waveform adopted in
the MLDC is given by [150]
|h( f )|=
{
A f−4/3 f < fmax
A f−4/3exp
(
1− ffmax
)
f > fmax
. (3.1)
In addition, the MLDC waveforms include a fourth-order Butterworth filter to mitigate
dynamic-range issues associated with inverse Fourier transforms. We adopt the same
ansatz as the MLDC, namely that the Fourier domain waveform amplitude is given by
|h+| =
A f−4/3
(
1+
(
flow
f
)2)−4
f < fmax
A f−4/3
(
1+
(
flow
f
)2)−4
exp
(
1− ffmax
)
f > fmax
|h×| = 0 (3.2)
and the phase by exp(2piı f tc), where tc is the burst time.
3.1.1.2 Sine Gaussians
A sine-Gaussian waveform is centred on a particular frequency, and has exponentially
suppressed power at nearby frequencies. We choose to consider a linearly polarised
47
3. THE MOCK LISA DATA CHALLENGES
sine-Gaussian, for which the waveform magnitudes in the frequency domain are given
by
|h+|= A2
√
Q2
2pi f 2c
exp
(
−Q
2
2
(
f − fc
fc
)2)
, |h×|= 0 (3.3)
where A is the dimensionless amplitude, fc is the frequency of the sinusoidal oscil-
lation, and Q is the width of the Gaussian envelope. The phase of the wave is again
exp(2piı f tc), where tc is the central burst time. In the time domain, the sine-Gaussian
is a small burst “packet” of particular frequency, fc, with the number of cycles in the
burst determined by Q.
3.1.1.3 Detector model
To include the LISA response we made use of the static LISA model as described
in [158]. This approximation is valid for burst sources, as LISA does not move signif-
icantly over the typically short duration of the bursts, which is of the order of 1000s
or less. The static LISA response model was also adopted for cosmic string bursts
in [157].
Three optimal detection time-delay interferometry (TDI) channels, A, E and T , can
be constructed from the LISA data stream [159]. The noise is uncorrelated within and
across these channels. For the search of the MLDC data, the A, E and T channels were
constructed as linear combinations of the three Michelson channels, X , Y and Z, that
were provided in the data release.
A = 1√
2
(Z−X), (3.4a)
E = 1√
6
(X−2Y +Z), (3.4b)
T = 1√
3
(X +Y +Z). (3.4c)
48
3.1 Cosmic String Cusps
The power spectral densities of the noise in the three channels are given by
SA( f ) = SE( f ) = 16sin2(2pi f tL)
(
2
(
1+ cos(2pi f tL)+ cos2(2pi f tL)
)
Spm( f )
+(1+ cos(2pi f tL)/2)Ssn f 2
)
(3.5)
ST ( f ) = 16sin2(2pi f tL)
(
2
(
1−2cos(2pi f tL)+ cos2(2pi f tL)
)
Spm( f )
+(1− cos(2pi f tL))Ssn f 2
)
(3.6)
Spm( f ) =
(
1+
(
10−4Hz
f
)2) Sacc
f 2
(3.7)
where tL = 16.678s is the light travel time along one arm of the LISA constella-
tion, Sacc = 2.5× 10−48Hz−1 is the proof mass acceleration noise and Ssn = 1.8×
10−37Hz−1 is the shot noise.
The LISA data stream will also contain a confusion noise foreground from unre-
solved white dwarf binaries in our galaxy. These were not included in the noise model
for the MLDC round 3.4, and so we have ignored the confusion noise contribution in
the current work. In practice, we found that the T channel was too noisy to be used in
the MLDC search and did not contribute any significant SNR, so we used the A and E
channels only for data analysis.
3.1.1.4 Likelihood evaluation
The space of gravitational waveform signals possesses a natural scalar product [160,
161],
〈h |s〉= 2
∫ ∞
0
d f
Sn( f )
[
h˜( f )s˜∗( f )+ h˜∗( f )s˜( f )
]
, (3.8)
where
h˜( f ) =
∫ ∞
−∞
dt h(t)e2piı f t (3.9)
is the Fourier transform of the time domain waveform h(t). The quantity Sn( f ) is
the one-sided noise spectral density of the detector. For a family of sources with
waveforms h(t;~λ ) that depend on parameters ~λ , the output of the detector, s(t) =
h(t; ~λ0) + n(t), consists of the true signal h(t; ~λ0) and a particular realisation of the
49
3. THE MOCK LISA DATA CHALLENGES
Parameter Minimum Maximum
log(A) −23.3 −21
fmax (Hz) 0.001 0.5
tc (s) T0 T0+∆T
sin(θ) −1 1
φ (rad) 0 2pi
ψ (rad) 0 2pi
Table 3.1: Prior probability distributions for the cosmic string cusp model.
noise, n(t). Assuming that the noise is stationary and Gaussian, the logarithm of the
likelihood that the parameter values are given by~λ is
logL
(
~λ
)
=C− 1
2
〈
s−h
(
~λ
)∣∣∣s−h(~λ)〉 , (3.10)
and it is this log-likelihood that is evaluated at each point in the search. The constant,
C, depends on the dimensionality of the search space, but its value is not important
as we are only interested in the relative likelihoods of different points. As mentioned
earlier, the LISA data stream has several independent data channels. The total multi-
channel likelihood is obtained by summing the scalar product in Equation (3.8) over
the A and E channels that are used.
3.1.1.5 Priors
The parameter space over which to search for signals must also be specified. For the
searches with the cosmic string model, we used uniform priors for each parameter that
covered the signal space from which the MLDC sources were drawn. These correspond
to the ranges in Table 3.1. Here, θ is the sky colatitude, φ is the sky azimuthal angle,
and ψ is the polarization. The data stream was divided into segments for the search
and T0 is the start time of the particular data segment being searched and ∆T is that
segment’s length.
For the sine-Gaussian model, we maintained the approach of using uniform prior
ranges while making sure that all signals would fall within our specified bounds. The
priors used are given in Table 3.2. T0, ∆T , θ , φ , and ψ are the same physical quantities
50
3.1 Cosmic String Cusps
Parameter Minimum Maximum
log(A) −22.3 −18
fc (Hz) 0.001 0.5
Q 1 10
tc (s) T0 T0+∆T
sin(θ) −1 1
φ (rad) 0 2pi
ψ (rad) 0 2pi
Table 3.2: Prior probability distributions for the sine-Gaussian model.
as in the cosmic string model; the prior on log(A) was modified in order to cover the
same range of SNRs for the signals.
3.1.2 MLDC challenge 3.4 search
Mock LISA Data Challenge 3.4 [150] consisted of a data set of 221 samples at a ca-
dence of 1s, containing GW burst signals from cosmic string cusps [155, 156]. The
signals occurred as a Poissonian random process throughout the data set, with an ex-
pectation value of five events. To analyse the data, we divided the data stream into
segments of 32768s in length, with 2048s of overlap between neighbouring segments.
The overlap was chosen to ensure that each of the bursts, which were of maximum
duration of ∼ 1000s, would be entirely contained in at least one segment. To avoid
artefacts in the Fourier domain, we used Welch windowing before performing the Fast
Fourier Transform (FFT) of each segment. We also analysed data segments of the same
length, offset from the first set by 16384s, to check no signals had been missed and to
verify that the detected signals were not edge artefacts. We also repeated the search
with segments of 16384s and 8192s in length to verify our results. MULTINEST was
run with 4000 live points on each segment, using 8 3GHz Intel Woodcrest processors
in parallel; the algorithm took approximately 5 minutes to run on a single segment
when no signal was present, and twice this when one was present. The search of the
offset and shorter segments returned all of the same detections. We ran the search on
the MLDC training data set to test the algorithm, and then on the challenge data set.
51
3. THE MOCK LISA DATA CHALLENGES
To assess the quality of our solutions we will not only use the recovered parameters
for a source, but also the signal-to-noise ratio (SNR) of the recovered solution. The
degeneracies in the parameter space mean that a waveform can match the true wave-
form very well with very different parameter values. The recovered SNR is a measure
of how much of a residual would be left in the data stream if the recovered parameters
were used to subtract a given source from the data set. If the recovered SNR is close to
the true SNR we can say that we have correctly identified the waveform of the source
even if the parameters are quite different.
3.1.2.1 Challenge data
Three signals were found in the challenge data set at tc ∼ 6× 105s, tc ∼ 1.07× 106s
and tc ∼ 1.6× 106s and this was the correct number of signals. We label these as
source candidates 3, 2 and 1 respectively for consistency with the MLDC key file. For
source candidates 1 and 3, MULTINEST returned a flat distribution in fmax, thereby
leading us to conclude that the actual maximum was above the Nyquist frequency of
0.5 Hz. We identified several modes in the posterior in each case, and used the local
Bayesian evidence to characterise these. We decided to submit the two modes of high-
est evidence; these generally corresponded to two antipodal sky solutions. However,
for source candidate 3 there was a third mode of almost equal local evidence, and so
we submitted that mode as well. The middle source, 2, was found to have a break
frequency of 0.0011 Hz, very close to the minimum value in the prior range of 0.001
Hz. At this low frequency, LISA is not able to resolve the sky position for a burst
source. As expected, we found very broad posteriors for all parameters other than the
maximum frequency. As the posterior was not well separated into modes, we chose
to submit only one set of parameters in this case, which we took to be the maximum
a-posteriori parameters.
In Table 3.3 we list the true parameters for the three sources in the challenge data
set (available at [162]), and the parameters recovered by MULTINEST. For simplicity
we only include the parameters for the mode of highest local evidence, not all of the
modes submitted. The agreement between the true and recovered parameters is broadly
consistent with our expectations, and the true parameters lie within a few standard
deviations of the means in most cases. For source 1, tc is quite far from the true value
52
3.1 Cosmic String Cusps
by this measure. However, this is due to the correlated degeneracies between tc, θ
and φ . The true time of coalesence was consistent with the full recovered posterior
distribution. Additionally, for source 1 the true value of fmax is considerably below
Nyquist, while our analysis recovered a flat distribution indicative of a value above
Nyquist. We carried out further checks, but there is no indication that the data favours
a particular value of fmax in this case.
In Figure 3.1 we show the true waveforms, in the A channel, for the three sources
present in the data set, and overlay them with the waveforms corresponding to each of
the modes that we included in our submission. We see clearly that, in all three cases,
the recovered waveforms reproduce the true waveforms well, with small differences
that arise from noise fluctuations in the detector. We expect the recovered SNR to be
a better measure of the quality of the solution than the values of the recovered pa-
rameters. In Table 3.4 we list the true SNRs in the A and E channels, and the SNRs
recovered by each of the solutions included in the submission. We have clearly recov-
ered all of the SNR of the true signal, and the residual after subtraction would be at the
level of instrumental noise. We thus regard the search as a success.
In Figure 3.2 we show 1D and 2D marginalised posterior probability distributions
for the second of the sources present in the challenge data set. We see that in this case
fmax can be measured very well, since it is far below Nyquist, but degeneracies in sky
position and time of coalesence are clearly evident. The sky position degeneracies are
particularly severe since the source is at very low frequency.
3.1.3 Model selection using Bayesian evidence
The MLDC search demonstrated that MULTINEST is able to correctly identify and
characterize bursts from cosmic string cusps in LISA data. However, cosmic string
cusps produce a very particular waveform signature. LISA might also detect bursts
of gravitational radiation from other sources, such as supernovae of hypermassive ob-
jects, or even unmodelled sources. A search of the LISA data using a cosmic string
cusp model could conceivably detect these other types of burst. In order to make sci-
entific statements it is important to be able to say what the most likely origin for an
observed burst could be. The Bayesian evidence provides a tool for carrying out model
53
3. THE MOCK LISA DATA CHALLENGES
Source A (×10−21) ψ (rad) θ (rad) φ (rad) fmax (Hz) tc (×106s)
1
True 0.866 3.32 0.556 3.71 0.0296 1.6022
Recovered 1.65 2.82 0.349 6.01 0.5∗ 1.6030
Mean 1.13 2.70 −0.141 4.38 0.27 1.6030
Std. Dev. ±0.22 ±0.14 ±0.30 ±2.6 ±0.14 ±0.000071
2
True 2.79 5.12 −0.444 3.17 0.00108 1.0727
Recovered 2.97 0.271 −0.893 5.12 0.00108 1.0732
Mean 2.75 0.91 −0.00804 5.31 0.00108 1.0733
Std. Dev. ±0.63 ±0.28 ±0.52 ±0.46 ±0.000048 ±0.00019
3
True 0.854 4.66 −0.800 0.217 6.15 0.60000
Recovered 1.14 0.927 −0.562 5.45 0.5∗ 0.59991
Mean 1.16 0.962 −0.488 5.48 0.24 0.59992
Std. Dev. ±0.18 ±0.16 ±0.16 ±0.18 ±0.14 ±0.000076
Table 3.3: Parameters of the three signals present in the challenge data set, and the
parameters recovered by MULTINEST for the mode of highest local evidence identified
by the search. The parameters are amplitude, A, polarisation, ψ , sky colatitude, θ ,
and longitude, φ , break frequency, fmax, and time of coalesence at the Solar System
barycentre, tc. In the row labelled “Recovered” we quote the maximum likelihood
parameters of the mode with highest evidence identified by MULTINEST. In the rows
labelled “Mean” and “Std. Dev.” we quote the mean and standard deviation in that
parameter, as computed from the recovered posterior for the mode of highest evidence.
Where the value of fmax is marked by a ∗, the distribution in this parameter was very
flat, indicating that fmax was probably above Nyquist. In these cases, we set fmax to
the Nyquist frequency of 0.5Hz for the maximum likelihood parameters, but still quote
the actual recovered mean and standard deviation.
54
3.1 Cosmic String Cusps
1.6024 1.6024 1.6025 1.6025 1.6026 1.6026 1.6027 1.6027 1.6028
x 106
ï3
ï2.5
ï2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
x 10ï20
Time (s)
A
Ch
an
ne
l A
m
pli
tu
de
True
Recovered 1
Recovered 2
(a)
1.072 1.0725 1.073 1.0735 1.074 1.0745
x 106
ï0.5
0
0.5
1
1.5
2
2.5
x 10ï21
Time (s)
A
Ch
an
ne
l A
m
pli
tu
de
True
Recovered
(b)
5.995 5.9955 5.996 5.9965 5.997 5.9975 5.998 5.9985 5.999
x 105
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
2.5
x 10ï20
Time (s)
A
Ch
an
ne
l A
m
pli
tu
de
True
Recovered 1
Recovered 2
Recovered 3
(c)
Figure 3.1: Plots comparing the true waveforms with the waveforms for the various
modes recovered by MULTINEST and included in our submission for the analysis of
the challenge data set. (a) Source 1, (b) Source 2, and (c) Source 3.
55
3. THE MOCK LISA DATA CHALLENGES
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Figure 3.2: Two-dimensional marginalised posteriors as recovered by MULTINEST in
the search for the second of the cosmic string bursts in the MLDC challenge data set.
The parameters, from top-to-bottom and left-to-right, are colatitude, longitude, burst
time, burst amplitude, burst break frequency and waveform polarization. At the top of
each column we also show the one-dimensional posterior for the column parameter.
56
3.1 Cosmic String Cusps
Recovered SNR
Source tc(s) Channel True SNR Mode 1 Mode 2 Mode 3
1 1.6×106 A
E
41.0
14.5
41.2
14.5
41.0
14.6
N/A
2 1.1×106 A
E
30.7
13.9
30.7
13.9
N/A N/A
3 6×105 A
E
18.8
36.9
18.9
36.7
18.5
37.1
18.4
36.8
Table 3.4: Signal-to-noise ratios for the three sources in the challenge data set. We
show the true SNRs in the A and E channels, plus the SNRs for all of the modes we
submitted for analysis.
selection of this type, although it relies on having two alternative hypotheses to com-
pare. As an alternative to a cosmic string model, we consider a sine-Gaussian burst,
which is a generic burst waveform commonly used in burst analysis for ground-based
gravitational wave detectors [154]. We can use the evidence ratio to compare these two
models when analysing a data stream containing a cosmic string burst or containing a
sine-Gaussian burst. There are two stages to this analysis — 1) determining the signal-
to-noise ratio at which we begin to be able to detect a signal of each type in a LISA
data set; 2) determining the SNR at which the Bayesian evidence begins to favour the
true model over the alternative.
3.1.3.1 Detection threshold
To determine the SNR needed for detection of a burst source with the MULTINEST al-
gorithm, we generated a sequence of data sets containing a particular source and Gaus-
sian instrumental noise generated using the theoretical noise spectral densities given in
Equation (3.7), but varying the burst amplitude between the data sets to give a range
of SNRs between 5 and 15. Note that here and in the following we quote total SNRs
for the A and E channels combined, ρ2Total = ρ
2
A + ρ
2
E . Each data set was searched
with MULTINEST and the global log-evidence of the data set, as computed by MULTI-
NEST, was recorded. This was repeated for 8 different cosmic string sources and 9 dif-
ferent sine-Gaussian sources. The parameters for the cosmic string sources were taken
57
3. THE MOCK LISA DATA CHALLENGES
from the parameters of the five training and three blind sources injected in the MLDC
data sets, as these covered the parameter space of possible signals nicely. The im-
portant parameters of the sine-Gaussian model are the frequency and the burst width,
so we considered three different values of each, and constructed waveforms for the
nine possible combinations, while choosing a random value for the sky position and
waveform polarisation in each case.
In Figure 3.3 we show the global log-evidence of the data as a function of the SNR
of the injected signal for the cosmic string burst sources. Figure 3.4 shows the corre-
sponding results for the sine-Gaussian bursts. The detection threshold for cosmic string
bursts is at an SNR of 5–8 depending on the source parameters. Most of the sources
have a significant log-evidence (greater than 3) at an SNR of 7, but the SNR required
for detection is higher for the first training source and the second blind source. These
are the two sources with the lowest values of the break frequency fmax, which makes
the waveforms much smoother and simpler in the time domain. This may explain why
it is more difficult to distinguish them from noise. The MLDC data sets had a prior
for the source SNR, in a single Michelson channel, of ρ ∈ [10,100], so these results
suggest we would be able to detect any source drawn from the MLDC prior.
The sine-Gaussian bursts require a slightly higher SNR for detection, of 6–9, but
this increase in threshold is only of the order of 1 in SNR. The sine-Gaussian signals
are in general much simpler in form than the cosmic strings and therefore it is perhaps
unsurprising that it requires a higher SNR to distinguish them from noise. In this case,
it was the sources with highest frequency, fc = 0.49Hz, that were most difficult to
detect. However, since the Nyquist frequency of the data sets was 0.5Hz, the difficulty
of detection may have arisen because the signal was partially out of band.
3.1.3.2 Model selection
To explore model selection, we used MULTINEST to search the same data sets de-
scribed above, but now using the alternative model, i.e., we searched the cosmic string
data sets using the sine-Gaussian likelihood and vice versa. For sufficiently high SNR,
in both cases the alternative model was able to successfully detect a signal in the data
set. Typically, the best-fit sine-Gaussian signal to a cosmic string source has low Q and
a frequency that matches the two lobes of the cosmic string burst. The parameter Q sets
58
3.1 Cosmic String Cusps
Figure 3.3: The Bayesian log-evidence for the data set as a function of the SNR for
eight different cosmic string cusp signals. These signals had the same parameters,
other than amplitude, as the various sources used in the training and blind data sets
of the MLDC. The labels in the key are consistent with labels in Table 3.3 and the
solutions at [162].
Figure 3.4: As Fig. 3.3, but now showing results for nine different sine-Gaussian sig-
nals, with frequency, f , and width, Q, as shown.
59
3. THE MOCK LISA DATA CHALLENGES
2.0596 2.0597 2.0598 2.0599 2.06 2.0601 2.0602 2.0603
x 106
−20
−15
−10
−5
0
5
x 10−21
Time (s)
A
ch
an
ne
l a
m
pli
tu
de
Comparison of Recovered Signals for a Cosmic String
True Cosmic String
Recovered Sine Gaussian
Recovered Cosmic String
Figure 3.5: Typical example of confusion when searching a cosmic string data set with
the wrong model. The plot shows a comparison of the injected cosmic string signal to
the best-fit signals found by MULTINEST using the cosmic string model as templates
and using the sine-Gaussian model as templates.
the number of cycles in the sine-Gaussian wave packet, and so a sine-Gaussain with
Q ∼ 2 most closely resembles a cosmic string event, which typically has two cycles.
This is illustrated for a typical case in Figure 3.5. Similarly, the best-fit cosmic string
source to a sine-Gaussian signal matches the central two peaks of the sine-Gaussian
waveform as well as possible. A typical case is shown in Figure 3.6.
From the results of these searches it is possible to construct the evidence ratio
of the two models for each of the data sets. In Figure 3.7 we show the ratio of the
Bayesian evidence for the cosmic string model to that of the sine-Gaussian model
when searching the cosmic string data sets. We see that for some signals the evidence
ratio starts to significantly favour the true model, i.e., the cosmic string, at an injected
SNR of ∼ 5, which is the point at which we first start to be able to detect the cosmic
string burst at all. However, for the two low frequency sources, training source 1
and blind source 2, the evidence ratio only starts to favour the true model at SNR∼
11 or higher. This partially reflects the higher SNR required to detect the source,
but also includes the confusion caused by the signal very closely resembling a sine-
Gaussian. We conclude that when a cosmic string burst is loud enough to be detected,
then the evidence favours the interpretation of the event as a cosmic string burst, with
60
3.1 Cosmic String Cusps
0.998 0.999 1 1.001 1.002 1.003 1.004 1.005 1.006
x 105
−5
−4
−3
−2
−1
0
1
2
3
4
x 10−20
Time (s)
A
ch
an
ne
l a
m
pli
tu
de
Comparison of Recovered Signals for a Sine Gaussian
True Sine Gaussian
Recovered Sine Gaussian
Recovered Cosmic String
Figure 3.6: As Figure 3.5, but we now show a typical confusion example for searches
of the sine-Gaussian data sets. We compare the injected sine-Gaussian signal to the
best-fit signals recovered by MULTINEST using both the cosmic string and the sine
Gaussian models.
lower frequency sources requiring additional SNR. Since MLDC sources have an SNR
greater than 10 in each Michelson channel, they should be clearly distinguishable from
sine-Gaussian bursts.
In Figure 3.8 we show the ratio of the evidence of the sine-Gaussian model to that of
the cosmic string model when searching the data sets containing sine-Gaussian signals.
These are distinguishable from cosmic strings at SNRs of ∼ 6–9. This slightly higher
SNR in order to correctly choose the sine-Gaussian model reflects the fact that we need
a somewhat higher SNR to detect the sine-Gaussians in the first place. The only case
for which the evidence ratio does not begin to favour the sine-Gaussian model at the
point where the source becomes detectable is the case with f = 0.002 Hz and Q = 2.
This is a sine-Gaussian signal with only two smooth oscillations, and so it does look
rather like a low frequency cosmic string event. Even in that case, the evidence begins
clearly to favour the correct model for SNRs of ∼ 12 and higher.
61
3. THE MOCK LISA DATA CHALLENGES
Figure 3.7: The ratio of the Bayesian evidence for the cosmic string model to that of
the sine-Gaussian model when searching a data set containing a cosmic string burst
source. We show the Bayesian evidence ratio as a function of signal SNR for a variety
of different cosmic string sources.
Figure 3.8: As Fig. 3.7, but now showing the ratio of the Bayesian evidence for the
sine-Gaussian model to that of the cosmic string model when searching a data set
containing a sine-Gaussian burst source.
62
3.1 Cosmic String Cusps
3.1.4 Discussion
We have considered the use of the multi-modal nested sampling algorithm MULTI-
NEST for detection and characterisation of cosmic string burst sources in LISA data.
As a search tool, the algorithm was able successfully to find the three cosmic string
bursts that were present in the MLDC challenge data set. These sources were cor-
rectly identified in the sense that the full signal-to-noise ratio of the injected source
was recovered, and a posterior distribution for the parameters obtained. The maximum
likelihood and maximum a-posteriori parameters were not particularly close to the true
parameters of the injected signals, but this was a consequence of the intrinsic degen-
eracies in the cosmic string model parameter space and in all cases the true parameters
were consistent with the recovered posterior distributions.
In controlled studies, we found that the SNR threshold required for detection of the
cosmic string bursts was ∼ 5–8, depending on the burst parameters. Bursts with a low
break-frequency require a higher SNR to detect than those with high break frequen-
cies. We also explored the detection of sine-Gaussian bursts and in that case the SNR
required for detection was slightly higher, being typically 6–9, with sources having
frequency close to Nyquist being more difficult to detect.
MULTINEST is designed to evaluate the evidence of the data under a certain hy-
pothesis, and this can be used to compare possible models for the burst sources. LISA
may detect bursts from several different sources, and it is important for scientific in-
terpretation that the nature of the burst be correctly identified. We used the Bayesian
evidence as a tool to choose between two different models for a LISA burst source —
the cosmic string model and the sine-Gaussian model, which was chosen to represent
a generic burst. The Bayesian evidence works very well as a discriminator between
these two models. The evidence ratio begins to clearly favour the correct model over
the alternative at the same SNR that the sources become loud enough to detect in the
first place.
The usefulness of MULTINEST as a search tool in this problem is a further illus-
tration of the potential utility of this algorithm for LISA data analysis, as previously
demonstrated in a search for non-spinning SMBH binaries [107]. Other algorithms
based on Markov Chain Monte Carlo techniques have also been applied to the search
for cosmic strings [157]. Both approaches performed equally well as search tools in
63
3. THE MOCK LISA DATA CHALLENGES
the last round of the MLDC. MULTINEST was not designed primarily as a search al-
gorithm, but as a tool for evidence evaluation, and this work has demonstrated the
utility of the Bayesian evidence as a tool for model selection in a LISA context. Other
problems where the evidence ratio approach could be applied include choosing be-
tween relativity and alternative theories of gravity as explanations for the gravitational
waves observed by LISA, or choosing between different models for a gravitational
wave background present in the LISA data set. The Bayesian evidence was previously
used in a LIGO context as a tool to choose between alternative theories of gravity [114]
and in a LISA context to distinguish a data set containing a source from one containing
purely instrumental noise [153].
In the context of interpretation of LISA and NGO burst events, what we have con-
sidered here is only part of the picture. We have shown that we are able to correctly
choose between two particular models for a burst, and this can easily be extended
to include other burst models. However, LISA/NGO might also detect bursts from un-
modelled sources. In that case, algorithms such as MULTINEST which rely on matched
filtering would find the best fit parameters within the model space, but a higher intrin-
sic SNR of the source would be required for detection. In such a situation, we would
like to be able to say that the source was probably not from a model of particular type,
e.g., not a cosmic string burst. There are several clues which would provide an indi-
cation that this was the case. The sine-Gaussian model is sufficiently generic that we
would expect it, in general, to provide a better match to unmodelled bursts than the
cosmic string model, which has a very specific form. Therefore, we could say that if
the evidence ratio favoured the cosmic string model over the sine-Gaussian model it
was highly likely that the burst was in fact a cosmic string and not something else.
Similarly, if we found that several of the alternative models had almost equal evidence,
but the SNR was quite high, it would be indicative that the burst was not described by
any of the models. We have seen that at relatively moderate SNRs, when the signal is
described by one of the models, the evidence clearly favours the true model over an
alternative. If we found that two models gave almost equally good descriptions of the
source, it would suggest that the burst was not fully described by either of them. A
third clue would come from the shape of the posterior for the source parameters. The
cosmic string waveform space contains many degeneracies, but these can be charac-
terised theoretically for a given choice of source parameters. If the signal was not from
64
3.2 Galactic White Dwarf Binaries
a cosmic string, we might find that the structure of the posterior was modified. Fi-
nally, some techniques have been developed for the Bayesian reconstruction of generic
bursts [163] which could also be applied in a LISA/NGO context. The usefulness of
these various approaches can be explored further by analysing data sets into which nu-
merical supernovae burst waveforms have been injected. While the necessary mass of
the progenitor is probably unphysically high for a supernova to produce a burst in the
LISA/NGO frequency band, such waveforms provide examples of unmodelled burst
signals on which to test analysis techniques. The final LISA/NGO analysis will em-
ploy a family of burst models to characterize any detected events. The work described
here demonstrates that the Bayesian evidence will be a useful tool for choosing be-
tween such models, and MULTINEST is a useful tool for computing those evidences.
3.2 Galactic White Dwarf Binaries
The Milky Way Galaxy has a population of order tens of millions of objects that can
be classified as compact binaries. These include separated binaries of two objects that
interact only gravitationally and interacting binaries which have mass transfer from
one object to the other. Separated binaries may consist of two white dwarfs, two neu-
tron stars, or one of each. Types of interacting binaries include AM CVn stars (white
dwarf accreting mass from a companion) and ultra-compact X-ray binaries (neutron
star accreting mass from a companion). Mock LISA Data Challenge 3.1 [150] uses
a population model which is described by [164]. It consists of ∼ 26 million detached
binaries and∼ 34 million interacting binaries, each modeled as two point masses in cir-
cular orbit with increasing or decreasing orbital frequency depending on the dominant
process (gravitational radiation or mass transfer, respectively).
3.2.1 Data Model
The plus and cross polarizations of the binary signal are given by [150]
h+(t) =A(1+ cos2 ι)cos
[
2pi( f t+ f˙ t2/2)+φ0
]
, (3.11)
h×(t) =−2A(cos ι)sin
[
2pi( f t+ f˙ t2/2)+φ0
]
, (3.12)
A =
4(GM)5/3
c4DL
(pi f )2/3, (3.13)
65
3. THE MOCK LISA DATA CHALLENGES
where M is the chirp mass as defined before, f˙ is the constant frequency derivative,
φ0 is the phase at t = 0, and ι is the inclination of the binary to our line of sight. A
fast-slow decomposition of the LISA orbital motion can also be used to simulate the
detector phase measurements in the frequency domain, as described in [165].
This work was performed as an extension to that done in [166]. In that paper,
the authors use an F-statistic to analytically maximise over most amplitude and phase
factors and then perform a template bank search followed by parameter refinement
with the Nelder-Mead optimisation algorithm [167]. The search is done in the four-
dimensional space defined by the initial frequency, f , frequency drift, f˙ , and sky lati-
tude and longitude, β and λ .
The challenge data set was broken into 0.1 mHz frequency bands to simplify the
analysis; each band was separated with Butterworth bandpass filters as described in
Section V of [166]. In the initial analysis for each frequency band, the loudest signal
is subtracted and then the search is performed again until no signal can be identified.
This process is able to handle the large number of signals present in the data, espe-
cially below ∼ 7mHz, but can be very time-consuming at higher frequencies where
larger template banks are needed to ensure a minimum correlation with any signals
present. We therefore aimed to use the ability of MULTINEST to identify multiple
signals present in each band and evaluate local Bayesian evidences and a posterior
maximum for each.
Each bandpassed frequency range was analysed for signals with frequencies in
the centre of the band. The prior on f was U[ fc− 0.06, fc + 0.06]mHz. Since the
bandpasses were performed at 0.1 mHz intervals, this allowed for 0.01 mHz of overlap
on either side while not searching too close to the edges of the filter. The frequency
drift, f˙ for detached binaries evolving via gravitaitonal radiation is given by
f˙ =
24
5pi
(
GM
2c3
)5/3
(2pi f )11/3. (3.14)
Therefore, in order to set prior limits on this parameter, we assumed a uniform prior
over [(−0.25) f˙max, f˙max]. f˙max is calculated with a maximum chirp mass of 1.5M at
the highest frequency in the prior. A minimum f˙ of −1/4 times the maximum is used
because as [165, 168] describe, contact binaries with mass transfer are driven to longer
periods with similar frequency evolution to Equation (3.14), but with opposite sign and
66
3.2 Galactic White Dwarf Binaries
slightly lower magnitude. We have the benefit of knowing the true signals present in
the data and these prior bounds were confirmed to include them all. The sky position
prior is taken to be uniform over the sphere: uniform in λ ∈ [0,2pi] and uniform in
cosβ ∈ [−1,1].
The noise in each TDI channel can be taken as approximately constant over the
frequency band and is calculated at the central frequency using the formulas provided
in [166].
3.2.2 Results
We searched 71 frequency ranges, spanning 4.99–12.11mHz. The density of signals
is much greater at lower frequencies so we expect more modes of the posterior to be
identified in those bands. Modes were clustered by starting frequency only. Once the
analyses were complete, the posteriors were evaluated to determine how many signals
were found and with what parameter values. This was done by collecting all posterior
modes with local log-evidences above 100. If any two were found to have frequencies
within 0.16µHz and frequency derivative within 2.6×10−14Hz/s, the one with larger
evidence was considered. These constraints on saying two signals are potentially the
same are determined by 10/Tobs and 100/T 2obs, where Tobs is the total two year obser-
vation time. This served to eliminate potential false detections and secondary max-
ima. Maximum likelihood parameter values were saved for each detected signal; very
narrow modes meant that these did not differ significantly from maximum posterior
values. In total, 754 signals were identified in the 4.99–12.11mHz range this way.
Using the criteria from [166] to determine the true/false positive rate for these de-
tections we find that the majority of signals found are true. The frequency resolution is
given by the width of bins in Fourier space, which is given by ∆ f = 1/Tobs ' 16nHz.
The resolution in the frequency drift is the difference at which over the total obser-
vation time the frequency will drift by the resolution in frequency; this is given by
∆ f˙ = 1/T 2obs ' 2.5× 10−16Hz/s. We therefore require our reported solutions to be
correct within the resolution possible. However, in this frequency range we find only
about one quarter of the 2924 total signals present. At higher frequency bands we find
all or nearly all of the signals but the recovery decreases as more signals add to the
confusion. Figure 3.9 shows the number of true signals in the MLDC3.1 key (bright
67
3. THE MOCK LISA DATA CHALLENGES
Figure 3.9: Number of true, recovered, and correctly detected signals in each frequency
band analysed.
signals that can be measured above the background) as well as the number of reported
and correct detections in each frequency band. Complementary to this, Figure 3.10
shows the purity and completeness of the reported detections from the analysis of each
frequency band. The purity is the percent of reported detections that are correct and
completeness is the percentage of all true signals that are correctly detected. The purity
is roughly consistent across all frequency bands, staying above 80% most of the time
and only once dipping just below 70%. Completeness is very poor at low frequencies,
with values as low as 10%. However, at frequencies above 10mHz most signals present
are recovered.
Values of f and f˙ for true signals present in the data versus those recovered are
shown in Figure 3.11. The first plot shows all signals within the 4.99–12.11mHz band,
while the second and third plots show the upper and lower 0.5mHz ranges, respectively.
We can compare in these last two the relative density of signals and how that has an
effect on the completeness of the detections and the rate of false positives (indicated
by a red ×). In the range of less dense signals we not only recover a larger fraction of
68
3.2 Galactic White Dwarf Binaries
Figure 3.10: The purity and completeness of detections for all frequency bands anal-
ysed. The purity is the percent of reported detections that are correct and completeness
is the percentage of all true signals that are correctly detected.
69
3. THE MOCK LISA DATA CHALLENGES
signals present, but also with smaller errors and, in this case, no false positives.
For all correctly recovered signals we can consider the recovery errors. Figure 3.12
plots histograms of the difference in the reported values of the measured parameters
versus their true values. f and f˙ are reported in units of the resolution size. We find that
f and f˙ are found to high precision, with most values being correct to within a fraction
of the resolution. This is possible as the signals will not be entirely in one frequency
bin over the entire observation period and the relative magnitudes can provide this extra
resolution. In addition, the sky location is found to good precision with most errors
much less than a radian.
3.2.3 Discussion
This analysis has shown that MULTINEST, and nested sampling in general, is a useful
tool for detecting multiple signals present in a single data set. Even in the presence
of a much larger number of signals causing confusion, the purity of recovered signals
is not adversely affected. This will be very useful for analysis of eLISA/NGO data
that will have many white dwarf binary signals present. In order to improve the purity
and completeness of these detections, it will likely be necessary to implement a multi-
stage analysis where the loudest detected signals are subtracted and the residual data
re-analysed for more signals. This type of pipeline has been tested for white dwarf
binary sources for LISA [169]. At lower frequencies it may also be necessary to use
narrower frequency bands for analysis to lower the number of signals present. The
status of detection at this time is very promising as this analysis may be performed
very rapidly and in parallel to detect multiple signals at a time.
3.3 Cosmic Strings in the Presence of a Galactic Fore-
ground
When a space-based GW detector is eventually launched, it will observe many signals
from various sources all at the same time. As considered in the previous section, 3.2,
there may be a foreground of galactic white dwarf binary signals. The impact of this
will depend on the final sensitivity of the eLISA/NGO detector. Here we consider the
data from the fourth round of the Mock LISA Data Challenge in an attempt to recover
70
3.3 Cosmic Strings in the Presence of a Galactic Foreground
(a) (b)
(c)
Figure 3.11: Plots comparing the values of f and f˙ for true signals present in MLDC
3.1 challenge data in the 4.99–12.11mHz range. True positive detections are indicated
by a blue +, false positives by a red ×. (a) Displays the entire frequency range, (b) the
upper 0.5mHz, and (c) the lower 0.5mHz. Dashed green lines indicate the maximum
and minimum for the f˙ prior.
71
3. THE MOCK LISA DATA CHALLENGES
Figure 3.12: The error in recovered parameters for correctly detected signals. f and f˙
are reported in terms of the resolution size.
72
3.3 Cosmic Strings in the Presence of a Galactic Foreground
cosmic string bursts in the presence of no galaxy, a reduced galactic component, and
the full galaxy foreground.
The data generation procedure is described in [151]. With the training data key
file for MLDC 4, we generated data sets with just the cosmic strings, instrumental
noise, and one of three options for the level of inclusion for galactic white dwarf (WD)
binaries. At the lowest level, no WDs are included. In the middle level, a confusion
foreground of millions of WD binaries that cannot be separated from each other is
added. These have an approximate power spectral density given by Equation (3.15),
which was calculated in [170] (Equation 14) for the specific model of the galactic
stellar population used in MLDC 4.
Sconf( f ) =
10−44.62 f−2.3 10−4.0 < f ≤ 10−3.0
10−50.92 f−4.4 10−3.0 < f ≤ 10−2.7
10−62.80 f−8.8 10−2.7 < f ≤ 10−2.4
10−89.68 f−20.0 10−2.4 < f ≤ 10−2.0
0 else
m2Hz−1 (3.15)
Finally, the full WD galaxy population, including the confusion foreground and thou-
sands of louder, separable binaries is added.
3.3.1 Data and Signal Models
The full MLDC 4 data spanned approximately 2 years and was sampled at a 1.875s
cadence, yielding 225 total samples. Cosmic string signals were injected at times ran-
domly sampled from a uniform distribution over the entire span, with a number that
was sampled from a Poisson distribution with a mean of 20 signals (22 were actually
injected). In order to efficiently search the data for these sparse signals, we segmented
the data into stretches that were 61440s long with 15360s of overlap (each start was
46080s after previous), to avoid missing signals potentially at the edges. Additionally,
we applied a Tukey window to each segment to avoid edge effects in the Fourier trans-
form; the Tukey window was used in place of the Welch window that was implemented
in Section 3.1.2 as it ensures that no data point was windowed by more than 50% with
the given overlap (the minimum overlap from the Welch window was ∼ 25%). There
were 1366 segments in total for each data set.
The instrumental noise model was the same as in MLDC3, so Equations (3.5), (3.6),
and (3.7) were used. Additionally, for the data sets that included either the reduced
73
3. THE MOCK LISA DATA CHALLENGES
or full galaxy population we added the galactic foreground confusion noise given by
Equation (3.15). Values from Equation (3.15) were further multiplied by a factor to
bring them into the same units as the instrumental noise and our data values.
S′conf( f ) = 64pi
2 f 2t2L sin
2(2pi f tL)×Sconf( f ) (3.16)
3.3.2 Comparison of Results
3.3.2.1 No Galaxy
In this data set, only instrumental noise and cosmic strings were present. MULTINEST
was run on each segment of data and modes found with a local log-evidence of more
than 7 were recorded as potential detections. This limit is chosen due to the analy-
sis in Section 3.1.3.1. Additionally, the global log-evidence from each segment was
recorded. Of the 1366 segments, 1339 contained no GW signals. The 22 signals were
present in 27 of the segments due to overlap. When compiling the list of detected sig-
nals, any two within 998s of each other were considered to be the same signal; this
time was chosen as it is approximately equal to the light travel time across the diam-
eter of the Earth’s orbit, along which LISA would travel. If a signal was found twice,
the mode with higher log-evidence was chosen to give the evidence and parameters
of detection. This always selected the segment in which the signal was further from
the edge and thus suffered less from windowing. Each signal was recovered to high
accuracy with log-evidences of 70 or greater. In Figure 3.13 we plot the receiver-
operator curve (ROC) for using the log-evidence as a threshold for detection and the
true and false positive rates as a function of this threshold. The ROC is perfect, which
indicates that there is a clear detection threshold for these signals in the presence of
only instrumental noise and that all signals pass. Figure 3.14 shows a histogram of the
global log-evidences from noise-only segments and a plot of the same log-evidences
for each segment versus time. The frequency distribution of log-evidences follows an
inverse-χ2 distribution and there is no time dependency, as expected. All segments
that contained only noise had log-evidences of 4 or less, with 98.5% having ln(Z)< 0
and thus supporting the noise-only model.
74
3.3 Cosmic Strings in the Presence of a Galactic Foreground
Figure 3.13: (Top) ROC for cosmic strings in instrumental noise, showing true positive
rate vs. false positive rate as the threshold varies. (Bottom) The true and false positive
detection rates as a function of the log-evidence. ln(Z) is adjusted by adding a constant
to all values so that the minimum is greater than 1.
75
3. THE MOCK LISA DATA CHALLENGES
Figure 3.14: (Top) Histogram of global log-evidences for the segments that do not
contain a cosmic string GW signal. These fit an inverse-χ2 distribution and 98.5%
have ln(Z) < 0. (Bottom) The log-evidence values of noise-only segments over time
(segment number). No time dependence is present.
76
3.3 Cosmic Strings in the Presence of a Galactic Foreground
Figure 3.15: (Top) Histogram of global log-evidences for the segments that do not
contain a cosmic string GW signal with the reduced galaxy population. (Bottom) The
log-evidence values of noise-only segments over time (segment number). There is a
clear time-dependence as the detector’s sensitivity sweeps over the galaxy.
3.3.2.2 Reduced Galaxy
The second data set analysed contained the reduced population of galactic binaries in
addition to instrumental noise. MULTINEST was run and we recorded detections and
evidences as before. Figure 3.15 shows the histogram and time-dependent plot of log-
evidences from segments that were known not to contain signals. The histogram does
not follow the inverse-χ2 distribution expected if Equation (3.15) correctly accounts
for the presence of the galactic foreground confusion noise. We can also see in the
time-dependent plot that the distribution of ln(Z) is not time-independent.
This time-dependence of the magnitude of the foreground confusion noise is due
to the varying sensitivity of the LISA detector to signals coming from the galaxy over
the course of the simulated mission. To account for this, the foreground confusion
noise was multipled by a time-varying factor that applied the necessary modulation and
77
3. THE MOCK LISA DATA CHALLENGES
Figure 3.16: Power spectral density of a data segment containing only instrumental
and reduced galaxy foreground confusion noise. On top are plotted the instrumental
and instrumental plus reduced galaxy noise models.
increase in PSD when the detector is most sensitive to the galaxy. The new confusion
noise is given, up to an unimportant normalisation factor, by
S′′conf =
(
1+ sin2
(
2pi
tm
1 year
))
×S′conf, (3.17)
where tm is the time of the mid-point of a data segment. Figure 3.16 shows the power
spectral density from a data segment containing no cosmic string signal, only instru-
mental and reduced galactic foreground confusion noise. The instrumental and in-
strumental plus reduced galaxy confusion noise models have been plotted on top for
comparison.
The analysis was re-run with this new model for the galactic foreground and the
resulting histogram and time-dependent plot of ln(Z) from segments without a cos-
78
3.3 Cosmic Strings in the Presence of a Galactic Foreground
Figure 3.17: (Top) Histogram of global log-evidences for the segments that do not
contain a cosmic string GW signal with the reduced galaxy population and modulated
foreground noise model. These fit the correct distribution much more closely and
99.6% have ln(Z)< 0. (Bottom) The log-evidence values of noise-only segments over
time (segment number). The time-dependence is much less significant.
mic string signal now have the features we expect, as seen in Figure 3.17. The time-
dependence is nearly gone, but what remains does not significantly affect further re-
sults. The ROC along with true and false positive rates as a function of detection
threshold are shown in Figure 3.18. Again the ROC is perfect and the detection thresh-
old set is consistent with the results from Section 3.1.3.1.
Figures 3.19 and 3.20 show the true and recovered waveforms in the A and E
channels for a selection of four of the 22 signals (true in solid black, recovered in
dashed red). Differences from the true signal are a result of the addition of instrumental
and galactic foreground noise as well as windowing for signals that were nearer to the
edge of a segment. Low-frequency signals, such as signals 2 and 6 in Figure 3.19, were
often recovered with higher precision due to their simpler structure and the fact that
the entire signal is within the frequency sensitivity of the detector. Signals with higher
79
3. THE MOCK LISA DATA CHALLENGES
Figure 3.18: (Top) ROC for cosmic strings in instrumental and galactic foreground
confusion noise, showing true positive rate vs. false positive rate as the threshold
varies. (Bottom) The true and false positive detection rates as a function of the log-
evidence. ln(Z) is adjusted by adding a constant to all values so that the minimum is
greater than 1.
80
3.3 Cosmic Strings in the Presence of a Galactic Foreground
Figure 3.19: A comparison of the true and recovered waveforms in the A and E chan-
nels for two of the 22 signals. The true signal is in solid black and the recovered is in
dashed red. These signals have a low value of fmax.
fmax show more discrepancies, as can be seen for signals 3 and 14 in Figure 3.20.
3.3.2.3 Full Galaxy
The final data set analysed consisted of instrumental noise, the full galaxy population
of white dwarf binaries, and the cosmic strings we were looking for. The modulated
foreground confusion noise model given in Equation (3.17) was used in this analysis,
which was performed in the same way as for the previous two data sets. In this sce-
nario, we do not expect the noise model to account for all contaminating sources and
thus we will be presented with many false detections.
Due to the presence of loud signals across all frequencies in each data segment,
81
3. THE MOCK LISA DATA CHALLENGES
Figure 3.20: A comparison of the true and recovered waveforms in the A and E chan-
nels for two of the 22 signals. The true signal is in solid black and the recovered is in
dashed red. These signals have a high value of fmax.
82
3.3 Cosmic Strings in the Presence of a Galactic Foreground
Figure 3.21: (Top) Histogram of global log-evidences for the segments that do not
contain a cosmic string GW signal with the full galaxy population and modulated fore-
ground noise model. (Bottom) The log-evidence values of noise-only segments over
time (segment number).
evidences for all segments were significantly increased. In Figure 3.21 we show the
histrogram and time scatter plot of log-evidences of segments that do not contain a
cosmic string GW signal. There is clearly a much larger base level that will obscure
finding true signals. The ROC along with true and false positive rates for detecting
segments with signals present are shown in Figure 3.22. It is clear that some level of
contamination will be present if we wish to set a threshold that does not exclude true
cosmic string signals.
3.3.2.4 Summary
The results presented in this section clearly show that perfect identification and excel-
lent recovery of cosmic string burst signals is possible in the presence of instrumental
noise and reduced galactic foreground. The population of galactic WD binaries with
83
3. THE MOCK LISA DATA CHALLENGES
Figure 3.22: (Top) ROC for cosmic strings in instrumental noise with the full galactic
population, showing true positive rate vs. false positive rate as the threshold varies.
(Bottom) The true and false positive detection rates as a function of the log-evidence.
ln(Z) is adjusted by adding a constant to all values so that the minimum is greater than
1.
84
3.4 Discussion
the loudest signals removed forms a confusion foreground that, when appropriately
modeled, can be accounted for and will not affect detection of cosmic string bursts.
However, when the full WD population is left in the data, there is significant contami-
nation and keeping this to a manageable amount will result in potentially missing some
cosmic string signals. Spurious signals may be identified by comparing to either a sine-
Gaussian or constant frequency model, using the Bayesian evidence to determine if one
of these is preferred over the cosmic string burst model.
This demonstrates the need to identify and subtract out of the data the loudest WD
binaries detected by LISA, or any similar detector, in order to be able to detect other
signals underneath. Doing so should not be affected by cosmic strings still present in
the data since the latter are transient while the former are present over the entire two
year data span. A main caveat is that this analysis was performed without the presence
of SMBH or EMRI signals, which are transient but significantly longer (typically of
duration three months or longer) and potentially much louder than any cosmic string
bursts. SMBH signals can have SNRs of thousands when coalescing during the ob-
servation period; EMRIs, however, build up an SNR of similar magnitude to cosmic
strings (typically 10 to 100) but over a duration of months. Detection and removal of
these sources in the presence of all others will need to be performed as well. We have
shown, however, that cosmic string bursts can be detected even in the presence of the
full galactic WD population and that the Bayesian evidence can be used as a detection
threshold. Like any detection statistic, some false positives may be included and will
require further analysis to be filtered out.
3.4 Discussion
The eLISA/NGO detector that is eventually launched will be able to observe multiple
signals that evolve on very different time scales. From bursts that last only a few
minutes to supermassive black hole binaries that coalesce over months to galactic white
dwarf binaries that are essentially monochromatic over a two year observation, these
signals will need to be detected and characterised from a single data stream. Data
analysis methods investigated here that utilise nested sampling will be essential in
obtaining the most information from the data as possible.
85
3. THE MOCK LISA DATA CHALLENGES
Segmenting data to look for transients will enable us to separate these short-lived
signals from the more constant sources that require time to build up a significant SNR.
Modeling of other sources present in the data in the noise estimation will further im-
prove our ability to recover the signals we are looking for. In searching for cosmic
string burst signals with a reduced WD binary foreground, the quality of parameter
estimation was not adversely affected as the additional sources were well-modeled by
the noise PSD and the signals of interest were of sufficiently high SNR. Analysing fre-
quency bands for near-monochromatic signals such as white dwarf binaries can yield
many detections that will not be influenced by noise at other frequencies. Subtracting
the loudest signals and re-analysing should return a high rate of detection. For data
with other signals present, analysing the two years of observation separately and re-
quiring coherent detection should reduce the effect of transient sources passing through
the frequency bands of interest.
The source environment for eLISA/NGO will require many diverse techniques for
separating out the different signal types which can also vary over orders of magnitude
in SNR. Some of these methods have been explored here and present promising results
for future data analysis pipeline development.
86
Chapter 4
An Investigation into the MOPED
Algorithm
Science is what we understand well enough to
explain to a computer.
Donald Knuth
The analysis of gravitational wave data involves measuring a signal that is given
by thousands of data points but is only defined by 17 or fewer individual parame-
ters. This drastic increase in the number of values to be measured slows down like-
lihood calculations and requires large amounts of computer memory. Additionally,
estimation and inversion of noise covariance matrices needs increased computational
resources. Simplifying assumptions are used to reduce these requirements, but remov-
ing the need analytically without losing information would be preferable. With this
goal in mind, I investigated implementing the Multiple Optimised Parameter Estima-
tion and Data compression (MOPED) algorithm for use in gravitational wave burst
searches for LISA.
I begin this chapter by introducing the general likelihood function and a simple
method of decreasing its complexity in Section 4.1. I then introduce MOPED in Sec-
tion 4.2 and define the MOPED likelihood function along with comments on the poten-
tial speed benefits of MOPED. In Section 4.3 I introduce my astrophysical scenario of
gravitational wave burst signals where we found that MOPED did not accurately por-
tray the true likelihood function. In Section 4.4 I expand upon this scenario to another
87
4. AN INVESTIGATION INTO THE MOPED ALGORITHM
where MOPED is found to work and to two other scenarios where it does not. I present
a discussion of the criteria under which I believe MOPED will accurately represent the
likelihood in Section 4.5, as well as a discussion of an implementation of a solution
provided by [171]. Work presented in this chapter has been published in [172].
4.1 The Likelihood Function
We begin by defining our data as a vector, x. Our model describes x by a signal plus
random noise,
x = u(θ T )+n(θ T ), (4.1)
where the signal is given by a vector u(θ ) that is a function of the set of parameters θ =
{θi} defining our model, and the true parameters are given by θ T . The noise is assumed
to be Gaussian with zero mean and noise covariance matrix N jk =
〈
n jnk
〉
, where the
angle brackets indicate an ensemble average over noise realisations. In general this
matrix may also be a function of the parameters θ , as in the case of galaxy spectra
explored in [173] where noise in a frequency bin depends on the amplitude of the
signal in the bin; however, in the examples explored in this chapter we will not be
considering this fully general case in a GW context. The full likelihood for N data
points in x is given by
L(θ ) =
1
(2pi)N/2
√|N(θ )| exp
{
−1
2
[x−u(θ )]TN−1(θ )[x−u(θ )]
}
. (4.2)
At each point, then, this requires the calculation of the determinant and inverse of
an N ×N matrix. Both scale as N3, so even for smaller datasets this can become
cumbersome.
4.1.1 A Simple Iterative Approach
One can perform a simple approximation to avoid performing the O(N3) matrix oper-
ations repeatedly by following the steps below.
1. Choose a fiducial point from the prior, θ F .
2. Calculate N−1(θ F) and |N(θ F)|.
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4.2 The MOPED Algorithm
3. Locate the peak of the posterior probability distribution of θ assuming constant
N−1(θ F) and |N(θ F)| instead of evaluating these at each point θ .
4. Take the posterior maximum as the new fiducial point and repeat steps 2 and 3.
Continue doing so until convergence is reached; the final posterior distribution
is the solution.
This method will provide a quick approximation of the true likelihood function and
only requires one matrix inversion and determinant calculation per iteration.
4.2 The MOPED Algorithm
Multiple Optimised Parameter Estimation and Data compression (MOPED; [173]) is a
patented algorithm for the compression of data and the speeding up of the evaluation of
likelihood functions in astronomical data analysis and beyond. It becomes particularly
useful when the noise covariance matrix is dependent upon the parameters of the model
and so must be calculated and inverted at each likelihood evaluation. However, such
benefits come with limitations. Since MOPED only guarantees maintaining the Fisher
matrix of the likelihood at the chosen fiducial point, multimodal and some degenerate
distributions will present a problem. In this chapter I report on some of the limita-
tions of the application of the MOPED algorithm. In the cases where MOPED does
accurately represent the likelihood function, however, its compression of the data and
consequent much faster likelihood evaluation does provide orders of magnitude im-
provement in runtime. In [173], the authors demonstrate the method by analysing the
spectra of galaxies and in [174] they illustrate the benefits of MOPED for estimation
of the CMB power spectrum. The problem of “badly” behaved likelihoods was found
by [171] for the problem of light transit analysis; nonetheless, the authors present a
solution that still allows MOPED to provide a large speed increase.
4.2.1 Data Compression with MOPED
Full details of the MOPED method are given in [173], here we will only present a
limited introduction.
89
4. AN INVESTIGATION INTO THE MOPED ALGORITHM
MOPED allows one to eliminate the need for the full matrix inversion by compress-
ing the N data points in x into M data values, one for each parameter of the model. Ad-
ditionally, MOPED creates the compressed data values such that they are independent
and have unit variance, further simplifying the subsequent likelihood calculation to an
O(M) operation. Typically, M N so this gives us a significant increase in speed. A
single compression is done on the data, x, and then again for each point in parameter
space where we wish to compute the likelihood. The compression is done by generat-
ing a set of weighting vectors, bi(θ F) (i = 1 . . .M), from which we can generate a set
of MOPED components from the theoretical model and data,
yi(θ F)≡ bi(θ F) ·x = bTi (θ F)x. (4.3)
Note that the weighting vectors must be computed at some assumed fiducial set of
parameter values, θ F . The only choice that will truly maintain the likelihood peak is
when the fiducial parameters are the true parameters, but obviously we will not know
these in advance for real analysis situations. Thus, we can choose our fiducial model
to be anywhere and iterate the procedure, taking our likelihood peak in one iteration
as the fiducial model for the next iteration. This process will converge very quickly,
and may not even be necessary in some instances. For our later examples, since we
do know the true parameters we will use these as the fiducial (θ F = θ T ) in order to
remove this as a source of confusion (all equations, however, are written for the more
general case). Note that the true parameters, θ T , will not necessarily coincide with
the peak θˆO of the original likelihood or the peak θˆM of the MOPED likelihood (see
below).
The weighting vectors must be generated in some order so that each subsequent
vector (after the first) can be made orthogonal to all previous ones using Gram-Schmidt
orthonormalisation. We begin by writing the derivative of the model with respect to
the ith parameter as ∂u∂θi
∣∣∣
θ F
= u,i(θ F). This gives us a solution for the first weighting
vector, properly normalised, of
b1(θ F) =
N−1(θ F)u,1(θ F)√
uT,1(θ F)N−1(θ F)u,1(θ F)
. (4.4)
The first compressed value is y1(θ F) = bT1 (θ F)x and will weight up the data combina-
tion most sensitive to the first parameter. The subsequent weighting vectors are made
90
4.2 The MOPED Algorithm
orthogonal by subtracting out parts that are parallel to previous vectors and are then
normalised. The resulting formula for the remaining weighting vectors is
bm =
N−1u,m−∑m−1q=1 (uT,mbq)bq√
uT,mN−1u,m−∑m−1q=1 (uT,mbq)2
, (4.5)
where m= 2 . . .M and all values are evaluated at θ F . Weighting vectors generated with
Equations (4.4) and (4.5) form an orthonormal set with respect to the noise covariance
matrix so that
bTi (θ F)N(θ F)b j(θ F) = δi j. (4.6)
This means that the noise covariance matrix of the compressed values yi is the identity,
which significantly simplifies the likelihood calculation. The new likelihood function
is given by
LMOPED(θ ) =
1
(2pi)M/2
exp
{
−1
2
M
∑
i=1
(yi(θ F)−〈yi〉(θ ;θ F))2
}
, (4.7)
where yi(θ F) = bTi (θ F)x represents the ith compressed data value and 〈yi〉(θ ;θ F) =
bTi (θ F)u(θ ) represents the ith compressed signal value. This is a much easier likeli-
hood to calculate and is time-limited by the generation of a new signal template instead
of the inversion of the noise covariance matrix. The peak value of the MOPED likeli-
hood function is not guaranteed to be unique as there may be other points in the original
parameter space that map to the same point in the compressed parameter space; this is
a characteristic that we will investigate.
The MOPED likelihood will closely approximate the true likelihood function. Most
importantly, the peak value will remain the same when the true parameters (or very
similar) are used for the fiducial. Additionally, in the limiting case where the noise
covariance matrix does not depend on the parameters, the Fisher matrix describing
the curvature of the likelihood surface at the peak is identical to the original. How-
ever, properties of the likelihood away from the peak value are not guaranteed to be so
well-behaved.
4.2.2 Speed Gains with MOPED
MOPED implicity assumes that the covariance matrix,N, is independent of the param-
eters θ . With this assumption it is appropriate to compare MOPED to the approximate
91
4. AN INVESTIGATION INTO THE MOPED ALGORITHM
method described in Section 4.1.1. Therefore, a full likelihood calculation with N data
points would require an O(N2) operation at each point in parameter space (or O(N)
if N is diagonal). In MOPED, however, the compression of the theoretical data is an
O(NM) linear operation followed by an O(M) misfit calculation, leading to an overall
complexity of O(NM)+O(M) ' O(NM) for each likelihood calculation. Thus, one
loses on speed if N is diagonal but gains by a factor of N/M otherwise. For the data
sets we will analyze, N/M > 100. We begin, though, by assuming a diagonal N for
simplicity, recognizing that this will cause a speed reduction but that it is a necessary
step before addressing a more complex noise model.
One can iterate the parameter estimation procedure if necessary, taking the maxi-
mum likelihood or posterior point found as the new fiducial and re-analyzing (perhaps
with tighter prior constraints) as in the approximate method suggested in the previous
section. This procedure is recommended for MOPED in [173] but is not always found
to be necessary.
MOPED has the additional benefit that the weighting vectors, bi, need only be
computed once provided the fiducial model parameters are kept constant over the anal-
ysis of different data sets. Computed compressed parameters, 〈yi〉, can also be stored
for reference and require significantly less memory than storing the entire theoretical
data set.
4.3 Simple Example With One Parameter
In order to demonstrate some of the limitations of the applicability of the MOPED
algorithm, we will consider a few test cases. These originate in the context of gravita-
tional wave data analysis for LISA since it is in this scenario that we first discovered
such cases of failure. The full problem is seven-dimensional parameter estimation, but
we have fixed most of these variables to their known true values in the simulated data
set in order to create a lower-dimensional problem that is simpler to analyse.
We consider the case of a sine-Gaussian burst signal present in the LISA detector.
The short duration of the burst with respect to the motion of LISA allows us to use
the static approximation to the response. As in Section 3.1.1.2, the frequency-space
92
4.3 Simple Example With One Parameter
waveform is described by [149]
h˜( f ) = AQf exp
{
−12Q2( f− fcfc )
2
}
exp(2piıt0 f ). (4.8)
Here A is the dimensionless amplitude factor; Q is the width of the Gaussian envelope
of the burst measured in cycles; fc is the central frequency of the oscillation being mod-
ulated by the Gaussian envelope; and t0 is the central time of arrival of the burst. This
waveform is further multiplied by a projection factor dependent on the sky position of
the burst source, θ and φ , and the burst polarisation, ψ , with respect to the detector.
The one-sided noise power spectral density of the LISA detector is given by [149]
Sh( f ) = 16sin2(2pi f tL)×(
2
(
1+ cos(2pi f tL)+ cos2(2pi f tL)
)
Spm( f )
+(1+ cos(2pi f tL)/2)Ssn f 2
)
, (4.9)
Spm( f ) =
(
1+
(
10−4Hz
f
)2) Sacc
f 2
, (4.10)
where tL = 16.678s is the light travel time along one arm of the LISA constella-
tion, Sacc = 2.5× 10−48Hz−1 is the proof mass acceleration noise and Ssn = 1.8×
10−37Hz−1 is the shot noise. This is independent of the signal parameters and all
frequencies are independent of each other, so the noise covariance matrix is constant
and diagonal. This less computationally expensive example allows us to show some
interesting properties.
We begin by taking the one-dimensional case where the only unknown parameter
of the model is the central frequency of the oscillation, fc. We set Q = 5 and t0 =
105s; we then analyze a 2048s segment of LISA data, beginning at t = 9.9× 104s,
sampled at a 1s cadence. For this example, the data were generated with random noise
(following the LISA noise power spectrum) at an SNR of ∼ 34 with fc,T = 0.1 Hz
(thus fc,F = 0.1 Hz for MOPED). The prior range on the central frequency is uniform
from 10−3 Hz to 0.5 Hz. 10,000 samples uniformly spaced in fc were taken and their
likelihoods calculated using both the original and MOPED likelihood functions. The
log-likelihoods are shown in Figure 4.1. Note that the absolute magnitudes are not
important but the relative values within each plot are meaningful. Both the original
and MOPED likelihoods have a peak close to the input value fc,T .
93
4. AN INVESTIGATION INTO THE MOPED ALGORITHM
Figure 4.1: The original and MOPED log-likelihoods as a function of fc for the chosen
template.
We see, however, that in going from the original to MOPED log-likelihood, the
latter also has a second peak of equal height at an incorrect fc. To see where this
peak comes from, we look at the values of the compressed parameter 〈y1〉( fc; fc,F) as
it varies with respect to fc as shown in Figure 4.2. The true compressed value peak
occurs at fc ' 0.1 Hz, where y1( fc,F) = 〈y1〉( fc; fc,F). However, we see that there
is another frequency that yields this exact same value of 〈y1〉( fc; fc,F); it is at this
frequency that the second, incorrect peak occurs. By creating a mapping from fc to
〈y1〉( fc; fc,F) that is not one-to-one, MOPED has created the possibility for a second
solution that is indistinguishable in likelihood from the correct one. This is a very
serious problem for parameter estimation.
4.4 Recovery in a 2 Parameter Case
Interestingly, we find that even when MOPED fails in a one-parameter case, adding
a second parameter may actually rectify the problem, although not necessarily. If we
now allow the width of the burst, Q, to be a variable parameter, there are now two
94
4.4 Recovery in a 2 Parameter Case
Figure 4.2: The value of the MOPED compressed parameter as a function of the orig-
inal frequency parameter.
orthogonal MOPED weighting vectors that need to be calculated. This gives us two
compressed parameters for each pair of fc and Q. Each of these may have its own
unphysical degeneracies, but in order to give an unphysical mode of equal likelihood
to the true peak, these degeneracies will need to coincide. In Figure 4.3, we plot the
loci in ( fc,Q) space where 〈yi〉(θ ;θ F) = 〈yi〉(θˆM;θ F) as θ ranges over fc and Q
values. We can clearly see the degeneracies present in either variable, but since these
only overlap at the one location, near to where the true peak is, there is no unphysical
second mode in the MOPED likelihood. Hence, when we plot the original and MOPED
log-likelihoods in Figure 4.4, although the behaviour away from the peak has changed,
the peak itself remains in the same location and there is no second mode.
Adding more parameters, however, does not always improve the situation as the
signal varies in different ways for each. We now consider the case where Q is once
again fixed to its true value and we instead make the polarisation of the burst, ψ , a vari-
able parameter. There are degeneracies in both of these parameters and in Figure 4.5
we plot the loci in ( fc,ψ)-space where the compressed values are each equal to the
value at the maximum MOPED likelihood point. These two will necessarily intersect
at the maximum likelihood solution, near the true value ( fc = 0.1 Hz and ψ = 1.3 rad),
95
4. AN INVESTIGATION INTO THE MOPED ALGORITHM
Figure 4.3: Loci of 〈y1〉(θ ;θ F) = 〈y1〉(θˆM;θ F) and 〈y2〉(θ ;θ F) = 〈y2〉(θˆM;θ F) in
the parameter space θ = { fc,Q}. The one intersection is the true maximum likelihood
peak.
but a second intersection is also apparent. This second intersection will have the same
likelihood as the maximum and be another mode of the solution. However, as we can
see in the left plot of Figure 4.6, this is not a mode of the original likelihood function.
MOPED has, in this case, created a second mode of equal likelihood to the true peak.
For an even more extreme scenario, we now fix to the true ψ and allow the time of
arrival of the burst t0 to vary (we also define ∆t0 = t0− t0,T ). In this scenario, the loci in
( fc,∆t0)-space where 〈yi〉(θ ;θ F) = 〈yi〉(θˆM;θ F) are much more complicated. Thus,
we have many more intersections of the two loci than just at the likelihood peak near
the true values and MOPED creates many alternative modes of likelihood equal to that
of the original peak. This is very problematic for parameter estimation. In Figure 4.7
we plot these loci so the multiple intersections are apparent. Figure 4.8 shows the
original and MOPED log-likelihoods, where we can see the single peak becoming a
set of peaks.
96
4.4 Recovery in a 2 Parameter Case
Figure 4.4: Contours of the original and MOPED log-likelihoods (left and right, re-
spectively). The MOPED likelihood has been multiplied by a constant factor so that
its peak value is equal to the peak of the original likelihood. Contours are at 1, 2, 5, 10,
20, 30, 40, 50, 75, and 100 log-units below the peak going from the inside to outside.
97
4. AN INVESTIGATION INTO THE MOPED ALGORITHM
Figure 4.5: Loci of 〈y1〉(θ ;θ F) = 〈y1〉(θˆ ;θ F) and 〈y2〉(θ ;θ F) = 〈y2〉(θˆ ;θ F) in the
parameter space θ = { fc,ψ}.
4.5 Discussion
What we can determine from the previous two sections is a general rule for when
MOPED will generate additional peaks in the likelihood function equal in magni-
tude to the true one. For an M-dimensional model, if we consider the (M − 1)-
dimensional hyper-surfaces where 〈yi〉(θ ;θ F) = 〈yi〉(θˆM;θ F), then any point where
these M hyper-surfaces intersect will yield a set of θ -parameter values with likelihood
equal to that at the peak near the true values. We expect that there will be at least one
intersection at the likelihood peak corresponding to approximately the true solution.
However, we have shown that other peaks can exist as well. The set of intersections of
contour surfaces will determine where these additional peaks are located. This degen-
eracy will interact with the model’s intrinsic degeneracy, as any degenerate parameters
will yield the same compressed values for different original parameter values.
Unfortunately, there is no simple way to find these contours other than by mapping
out the 〈yi〉(θ ;θ F) values, which is equivalent in procedure to mapping the MOPED
likelihood surface. The benefit comes when this procedure is significantly faster than
mapping the original likelihood surface. The mapping of 〈yi〉(θ ;θ F) can even be per-
98
4.5 Discussion
Figure 4.6: Contours of the original and MOPED log-likelihoods (left and right, re-
spectively). The MOPED likelihood has been multiplied by a constant factor so that
its peak value is equal to the peak of the original likelihood. Contours are at 1, 2, 5, 10,
20, 30, 40, 50, 75, and 100 log-units below the peak going from the inside to outside.
99
4. AN INVESTIGATION INTO THE MOPED ALGORITHM
Figure 4.7: Loci of 〈y1〉(θ ;θ F) = 〈y1〉(θˆ ;θ F) and 〈y2〉(θ ;θ F) = 〈y2〉(θˆ ;θ F) in the
parameter space θ = { fc,∆t0}. We can see many intersections here other than the true
peak.
formed before data is obtained or used, if the fiducial model is chosen in advance; this
allows us to analyse properties of the MOPED compression before applying it to data
analysis. If the MOPED likelihood is mapped and there is only one contour intersec-
tion, then we can rely on the MOPED algorithm and will have saved a considerable
amount of time, since MOPED has been demonstrated to provide speed-ups of a factor
of up to 107 in [174]. However, if there are multiple intersections then it is necessary to
map the original likelihood to know if they are due to degeneracy in the model or were
created erroneously by MOPED. In this latter case, the time spent finding the MOPED
likelihood surface can be much less than that which will be needed to map the original
likelihood, so relatively little time will have been wasted. If any model degeneracies
are known in advance, then we can expect to see them in the MOPED likelihood and
will not need to find the original likelihood on their account.
One possible way of determining the validity of degenerate peaks in the MOPED
likelihood function is to compare the original likelihoods of the peak parameter values
with each other. By using the maximum MOPED likelihood point found in each mode
and evaluating the original likelihood at this point, we can determine which one is
100
4.5 Discussion
Figure 4.8: Contours of the original and MOPED log-likelihoods (left and right, re-
spectively). The MOPED likelihood has been multiplied by a constant factor so that
its peak value is equal to the peak of the original likelihood. Contours are at 1, 2, 5, 10,
20, 30, 40, 50, 75, and 100 log-units below the peak going from the inside to outside.
101
4. AN INVESTIGATION INTO THE MOPED ALGORITHM
correct. The correct peak and any degeneracy in the original likelihood function will
yield similar values to one another, but a false peak in the MOPED likelihood will have
a much lower value in the original likelihood and can be ruled out. This means that a
Bayesian evidence calculation cannot be obtained from using the MOPED likelihood;
however, the algorithm was not designed to be able to provide this.
The solution for this problem presented in [171] is to use multiple fiducial mod-
els to create multiple sets of weighting vectors. The log-likelihood is then averaged
across these choices. Each different fiducial will create a set of likelihood peaks that
include the true peaks and any extraneous ones. However, the only peaks that will be
consistent between fiducials are the correct ones. Therefore, the averaging maintains
the true peak(s) but decreases the likelihood at incorrect values. This was tested with
20 random choices for θ F for the two-parameter models presented and was found to
leave only the true peak at the maximum likelihood value. Other, incorrect, peaks are
still present, but at log-likelihood values five or more units below the true peak. When
applied to the full seven parameter model, however, the SNR threshold for signal re-
covery is increased significantly, from ' 10 to ' 30.
The MOPED algorithm for reducing the computational expense of likelihood func-
tions can, in some examples, be extremely useful and provide orders of magnitude of
improvement. However, as we have shown, this is not always the case and MOPED
can produce erroneous peaks in the likelihood that impede parameter estimation. It is
important to identify whether or not MOPED has accurately portrayed the likelihood
function before using the results it provides. Some solutions to this problem have been
presented and tested but further implementation was not pursued.
102
Chapter 5
Artificial Neural Networks
The computer was born to solve problems that
did not exist before.
Bill Gates
Many calculations in gravitational wave physics, and astrophysics in general, are
very computationally expensive. If a calculation needs to be performed a large number
of times then it can slow down simulations and analysis significantly. One example
in gravitational wave physics is the calculation of waveforms for binary black hole
mergers where the full spin dynamics of both black holes are considered. A single
waveform can take on the order of seconds to calculate, or even longer in the case of
numerical relativity simulations. For this and other problems, we set out to develop a
generic algorithm for the training of artificial neural networks in order to facilitate and
expedite these repeated difficult computations.
Artificial neural networks (NNs) are a method of computation loosely based on
the structure of a brain. They consist of a group of interconnected nodes, which pro-
cess information that they receive and then pass this product along to other nodes via
weighted connections. We will consider only feed-forward NNs, for which the struc-
ture is directed; an input layer of nodes passes information to an output layer via zero,
one, or many “hidden” layers in between. A network is able “learn” a relationship
between inputs and outputs given a set of training data and can then make predictions
of the outputs for new input data. Further introduction can be found in [175].
103
5. ARTIFICIAL NEURAL NETWORKS
In section 5.1 I describe the general structure of NNs and how they may be trained
on a specific problem. Section 5.2 will then detail the procedure used to train NNs
to solve a particular task. This is then applied to some toy examples in Section 5.3.
Applying NNs to learning how to classify images of handwritten digits is described
in Section 5.4 and to the problem of measuring the shape of sheared, blurred, and
pixelated images of galaxies in Section 5.5. Work in this chapter was performed in
collaboration with Farhan Feroz.
5.1 Network Structure
A multilayer perceptron artificial neural network (NN) is the simplest type of network
and consists of ordered layers of perceptron nodes that pass scalar values from one
layer to the next. The perceptron is the simplest kind of node, and maps an input
vector x ∈ℜn to a scalar output f (x;w,θ) via
f (x;w,θ) = θ +
n
∑
i=1
wixi, (5.1)
where w = {wi} and θ are the parameters of the perceptron, called the “weights” and
“bias”, respectively. For a 3-layer NN, which consists of an input layer, a hidden layer,
and an output layer as shown in Figure 5.1, the outputs of the nodes in the hidden and
output layers are given by the following equations:
hidden layer: h j = g(1)( f
(1)
j ); f
(1)
j = θ
(1)
j +∑
l
w(1)jl xl, (5.2)
output layer: yi = g(2)( f
(2)
i ); f
(2)
i = θ
(2)
i +∑
j
w(2)i j h j, (5.3)
where l runs over input nodes, j runs over hidden nodes, and i runs over output
nodes. The functions g(1) and g(2) are called activation functions and must be bounded,
smooth, and monotonic for our purposes. We use g(1)(x) = 1
/
(1+ e−x) = sig(x) (sig-
moid) and g(2)(x) = x; the non-linearity of g(1) is essential to allowing the network to
model non-linear functions.
The weights and biases are the values we wish to determine in our training (de-
scribed in Section 5.2). As they vary, a huge range of non-linear mappings from inputs
to outputs is possible. In fact, a universal approximation theorem [176] states that a
104
5.1 Network Structure
Figure 5.1: A 3-layer neural network with 3 inputs, 4 hidden nodes, and 2 outputs.
Image courtesy of Wikimedia Commons.
NN with three or more layers can approximate any continuous function as long as the
activation function is locally bounded, piecewise continuous, and not a polynomial
(hence our use of sigmoid g(1), although other functions would work just as well, such
as a tanh). By increasing the number of hidden nodes, we can achieve more accuracy
at the risk of overfitting to our training data. To expand the NN to include more hidden
layers, we mirror Equation (5.2) for each connection from one hidden layer to the next,
each time using the same g(1) activation function. The final hidden layer will connect
to the output layer with Equation (5.3).
5.1.1 Choosing the Number of Hidden Layer Nodes
An important choice when training a NN is the number of hidden layer nodes to use.
The optimal number and organisation into one or more layers is a complex relationship
between the number of training data points, the number of inputs and outputs, and the
complexity of the function to be trained. Choosing too few nodes will mean that the
NN is unable to learn the relationship to the highest possible accuracy; choosing too
many will increase the risk of overfitting to the training data and will also slow down
the training process. As a general rule, we find that a NN should not need more hidden
nodes than the number of training data values used (each with a vector of inputs and
outputs).
105
5. ARTIFICIAL NEURAL NETWORKS
We choose to use 5–10 nodes in the hidden layer in our toy examples (see Sec-
tion 5.3). These choices allow the network to model the complexity of the function
without unnecessary work. In practice, it will be better to slightly over-estimate the
number of hidden nodes required. There are checks built in to prevent over-fitting (de-
scribed in Section 5.2) and the additional training time will not be a large penalty if an
optimal network can be obtained in an early attempt. The optimal NN structure can be
determined by comparing the Bayesian evidence of different trained NNs as calculated
in Section 5.2.6.
5.2 Network Training
In training a NN, we wish to find the optimal set of network weights and biases that
maximise the accuracy of predicted outputs. However, we must be careful to avoid
overfitting to our training data at the expense of making predictions for input values
the network has not been trained on. The NN can be trained from a random initial state
or can be pre-trained, a procedure which will be discussed in Section 5.2.5. But first
we will discuss the general process of NN training. The set of training data inputs and
outputs, D= {x(k), t(k)}, is provided by the user. Approximately 75% should be used
for actual NN training and the remainder provided as a validation set of data that will
be used to determine convergence to avoid overfitting. This ratio of 3:1 gives plenty of
information for training but still leaves a representative subset of the data for checks to
be made.
5.2.1 Overview
The weights and biases we will collectively call the network parameter vector a. We
can now consider the probability that a given set of network parameters is able to
reproduce the known training data outputs – representing how well our NN model of
the original function reproduces the true values. For problems of regression (fitting
the model to a function), this gives us a log-likelihood function for a, depending on a
standard χ2 error function, given by
log(L(a;σ )) =−K log(2pi)
2
−
N
∑
i=1
log(σi)− 12
K
∑
k=1
N
∑
i=1
[
t(k)i − yi(x(k);a)
σi
]2
, (5.4)
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5.2 Network Training
where N is the number of outputs, K is the number of data points and y(x(k);a) is
the NN’s predicted output vector for the input vector x(k) and network parameters a.
The values of σ = {σi} are hyper-parameters of the model that describe the standard
deviation (error size) of each of the outputs.
For a classification network that aims to learn the probabilities that a set of inputs
belongs to a set of output classes, the outputs are transformed according to the softmax
procedure in order that they are all non-negative and sum to one.
y′i(x
(k);a) =
exp
(
yi(x(k);a)
)
∑Nj=1 exp
(
y j(x(k);a)
) (5.5)
The classification likelihood is then given by the cross entropy function [175],
log(L(a;σ )) =−
K
∑
k=1
N
∑
i=1
t(k)i log(y
′
i(x
(k);a)). (5.6)
In this scenario, the true and predicted output values are probabilities (in [0,1]). In the
true outputs, all are zero except for the correct output which has a value of one. For
classification networks, the σ hyper-parameters do not factor into the log-likelihood.
Our algorithm considers the parameters a to be probabilistic with a log-prior dis-
tribution given by
log(S(a;α)) =−α
2 ∑i
a2i . (5.7)
α is a hyper-parameter of the model, called the “regularisation constant”, that gives
the relative influence of the prior and the likelihood.
The posterior probability of a set of NN parameters is thus
Pr(a;α,σ ) ∝ L(a;σ )×S(a;α). (5.8)
The network training begins by setting random values for the weights, sampled
from a normal distribution with zero mean. The initial value of σ is set by the user
and can be set on either the true log-likelihood values themselves or on their whitened
values (as defined in Section 5.2.2). The only difference between these two settings is
the magnitude of the error used. The algorithm then calculates a large initial estimate
for α ,
α =
|∇ log(L)|√
Mr
, (5.9)
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5. ARTIFICIAL NEURAL NETWORKS
where M is the total number of weights and biases (NN parameters) and r is a rate set
by the user (0 < r ≤ 1, default r = 0.1) that defines the size of the “confidence region”
for the gradient. This formula for α sets larger regularisation (a.k.a. “damping”) when
the magnitude of the gradient of the likelihood is larger. This relates the amount of
“smoothing” required to the steepness of the function being smoothed. The rate factor
in the denominator allows us to increase the damping for smaller confidence regions
on the value of the gradient. This results in smaller, more conservative steps that are
more likely to result in an increase in the function value but results in more steps being
required to reach the optimal weights.
The training then uses conjugate gradients to calculate a step, ∆a, that should be
taken (see Section 5.2.3). Following a step, adjustments to α and σ may be made be-
fore another step is calculated. The methods for calculating the initial α value and then
determining subsequent adjustments of α and/or σ are as developed for the MEMSYS
software package, described in [177].
5.2.2 Data Whitening
Most of the time, it is prudent to “whiten” the data before training a network. Whiten-
ing normalises the input and/or output values according to a specified format, which
makes it easier to train from the initial weights which are small and centred on zero.
The network weights in the first and last layers can then be “unwhitened” after training
so that the network will be able to process the original inputs and outputs.
Standard whitening will transform each input to a standard normal distribution by
subtracting the mean and dividing by the standard deviation over all elements in the
training data.
x(k)′l =
x(k)l − xl
σl
(5.10a)
xl =
1
K
K
∑
k=1
x(k)l (5.10b)
σ2l =
1
K−1
K
∑
k=1
(x(k)l − xl)2 (5.10c)
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5.2 Network Training
Data can also be whitened by bringing all values into [0,1].
x(k)′l =
x(k)l −mink(x(k)l )
maxk(x
(k)
l )−mink(x(k)l )
(5.11)
These two functions are normally performed separately on each input, but can be cal-
culated across all inputs if the inputs are related. The mean, standard deviation, min-
imum, or maximum would then be computed over all inputs for all data values. The
same whitening functions are also used for whitening the outputs. Since both functions
consist of subtracting an offset and multiplying by a scale factor, they can easily be per-
formed and reversed. To unwhiten network weights the inverse transform is applied,
with the offset and scale determined by the source input node or target output node.
Outputs for a classification network are not whitened since they are already simple
probabilities.
5.2.3 Finding the next step
In order to find the most efficient path to an optimal set of parameters, we perform
conjugate gradients using second-order derivative information. Newton’s method gives
the second-order approximation of a function,
f (a+∆a)≈ f (a)+(∇ f (a))T∆a+ 1
2
(∆a)TB∆a, (5.12)
where B is the Hessian matrix of second derivatives of f at a. In this approximation,
the maximum of f will occur when
∇ f (a+∆a)≈ ∇ f (a)+B∆a = 0. (5.13)
Solving this for ∆a gives us
∆a =−B−1∇ f (a). (5.14)
Iterating this stepping procedure will eventually bring us to a local maximum value
of f . For our purposes, the function f is the log-posterior distribution of the NN
parameters and hence Equation (5.12) is a Gaussian approximation to the posterior.
The Hessian of the log-posterior is the regularised (“damped”) Hessian of the log-
likelihood function, where the prior – whose magnitude is set by α – provides the
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5. ARTIFICIAL NEURAL NETWORKS
regularisation. If we define the Hessian matrix of the log-likelihood as H, then B =
H+αI (I being the identity matrix). Regularisation increases the probability that the
local maximum found is also the global maximum.
Using the second-order information provided by the Hessian allows for more effi-
cient steps to be made, since curvature information can extend step sizes in directions
where the gradient varies less and shorten where it is varying more. Additionally, using
the Hessian of the log-posterior instead of the log-likelihood adds the regularisation of
the prior. As mentioned before, this helps prevent the algorithm from getting stuck in a
local maximum by smoothing out the function being explored. It also aids in reducing
the region of confidence for the gradient information which will make it less likely that
a step results in a worse set of parameters.
Given the form of the log-likelihood, Equation (5.4), is a sum of squares (plus
a constant), we can also save computational expense by utilising the Gauss-Newton
approximation of its Hessian, given by
Hi j =−
K
∑
k=1
(
∂rk
∂ai
· ∂rk
∂a j
+ rk · ∂
2rk
∂ai∂aj
)
≈−
K
∑
k=1
(
∂rk
∂ai
· ∂rk
∂a j
)
, (5.15)
where
rk =
t(k)−y(x(k);a)
σ
. (5.16)
The drawback of using second-order information is that calculation of the Hessian
is computationally expensive and requires large storage, especially so in many dimen-
sions as we will encounter for more complex networks. In general, the Hessian is
not guaranteed to be positive semi-definite and so may not be invertible; however, the
Gauss-Netwon approximation does have this guarantee and does not require calculat-
ing second derivatives. Inversion of the very large matrix will still be computationally
expensive.
As noted in [178] however, we only need products of the Hessian with a vector to
compute the solution, never actually the full Hessian itself. To calculate these approx-
imate Hessian-vector products, we use a fast approximate method given in [179, 180].
This method takes advantage of the approximation made in Equation (5.13). If we
110
5.2 Network Training
choose ∆a = rv where v is any vector and r is a small number we can solve for the
Hessian-vector product,
Bv≈ ∇ f (a+ rv)−∇ f (a)
r
. (5.17)
This is a simple approximation for the product and can be computed in about the same
time as required to find the gradient, which must be calculated anyway. In order to
make this an exact method and less susceptible to numerical errors, we take the limit
as r→ 0. Doing so gives us
Bv =
∂
∂ r
∇ f (a+ rv)
∣∣∣∣
r=0
. (5.18)
We can therefore define the Rv{·} operator,
Rv{ f (a)} ≡ ∂∂ r f (a+ rv)
∣∣∣∣
r=0
, (5.19)
such that Bv = Rv{∇ f (a)}. This operator is then applied to all of the same equations
and procedures used to calculate the gradient. These equivalent functions have the
same cost as a gradient calculation, which is an O(M) operation for M weights. This
is a vast improvement over calculating and storing the entire Hessian matrix.
By combining all of these methods, second-order information is practical to use
and significantly improves the rate of convergence of NN training.
5.2.4 Convergence
Following each step, the posterior, likelihood, correlation, and error squared values are
calculated for the training data and the validation data that was not used in training
(calculating the steps). Convergence to a best-fit set of parameters is determined by
maximising the posterior, likelihood, correlation, or negative of the error squared of the
validation data, as chosen by the user. This prevents overfitting as it provides a check
that the network is still valid on points not in the training set. We use the error squared
as the default function to maximise as it does not include the model hyper-parameters
in its calculation and is less prone to problems with zeros than the correlation.
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5. ARTIFICIAL NEURAL NETWORKS
5.2.5 Autoencoders and Pre-Training
Autoencoders are a specific type of neural network wherein the inputs are mapped to
themselves through one or more hidden layers. These typically have a symmetric setup
of several hidden layers with a central layer containing fewer nodes than there are in-
puts. The NN can be split in half, with one part mapping the inputs to this central layer
and the second part mapping the central layer weights to the outputs (same as original
inputs). These two parts are called the “encoder” and “decoder” respectively and map
either to or from a reduced basis set of “feature vectors” embodied in the central layer.
This is akin to a non-linear principle component analysis (PCA). The non-linearity
should allow for more complex relationships and more information to be contained in
the same number of components/feature vectors. The individual feature vectors can
be found by decoding (1,0,0, ...,0), (0,1,0, ...,0), and so on. A basic diagram of an
autoencoder is shown in Figure 5.2. The three inputs (x1,x2,x3) are mapped to them-
selves via three symmetric hidden layers, with 2 nodes in the central layer. The weights
of the central layer (z1,z2) are feature vector weights for the reduced non-linear basis
set.
Autoencoders are notoriously difficult to train. A broad local maximum exists
wherein all outputs are the average value of the inputs. Geoffrey Hinton, Simon Osin-
dero, and Yee-Whye Teh developed a method for “pre-training” networks to obtain a
set of weights near the true global maximum [181]. This method was created with
symmetric autoencoders in mind. The map from the inputs to the first hidden layer
and then back is treated symmetrically and the weights are adjusted through a num-
ber of “epochs”, gradually reducing the reproduction error. This is repeated for the
first to second hidden layer and so on until the central layer is reached. The network
weights can then be “unfolded” by using the transpose for the symmetric connections
in the decoding half to provide a decent starting point for the full training to begin.
This is shown in Figure 5.2, where the W1 and W2 weights matrices are defined by
pre-training. More details can be found in [181] and [182] has useful diagrams and
explanations.
This pre-training can also be used for non-autoencoder networks. All layers of
weights, except for the final one that connects the last hidden layer to the outputs,
are pre-trained as if they were the first half of a symmetric autoencoder. However,
112
5.2 Network Training
Figure 5.2: Schematic diagram of an autoencoder. The 3 input values are being en-
coded to 2 feature vectors. Pre-training defines the W1 and W2 weight matrices to
provide a starting point for fine-tuning in training.
113
5. ARTIFICIAL NEURAL NETWORKS
the network weights are not unfolded; instead the final layer of weights is initialised
randomly as would have been done without pre-training. In this way, the network
“learns the inputs” before mapping to a set of outputs.
When an autoencoder is pre-trained, the activation function to the central hidden
layer is made linear and the activation function from the final hidden layer to the out-
puts is made sigmoidal. Regular networks that are pre-trained continue to use the
original activation functions. Examples of the use of pre-training for autoencoders and
other networks will be provided in the following sections of this chapter.
5.2.6 The Evidence of a Network
As we are determining the NN’s optimal weights from a Bayesian formalism with like-
lihood and prior functions, we can also define the Bayesian evidence of the network
model. This may be used to compare different types of networks and determine the op-
timal NN structure. The best NN model will have a larger evidence than other models
and will, in general, be the smallest network that provides the best predictions possible
for the given data. We approximate the evidence as deriving from a single Gaussian
peak about the optimal weights found as in [183]. In practice, this approximation is
close enough to the true value to provide a good guide. From [183] the evidence for an
optimised network is given by
log(Znetwork) = log(S(aMP;α))+ log(L(aMP);σ )− 12 log(|HMP|)
+
M
2
log(α)− 1
2
N
∑
i=1
log(σi)− N2 log(2pi), (5.20)
where ‘MP’ indicates the value at the posterior maximum and M and N are defined as
before (number of NN weights and outputs, respectively). There can be significant un-
certainty in this measurement due to the statistical method of calculating log(|H|) that
avoids calculating H itself. However, later examples will display trends that indicate
when an optimal network structure has been obtained.
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5.3 Toy Examples
Figure 5.3: Comparisons of the true and predicted values for the sinc function on the
training and validation data sets.
5.3 Toy Examples
5.3.1 Sinc Function with Noise
The first toy example we examined was a simple regression problem. In this, we gen-
erate 200 samples of x ∈ U[−5pi,5pi] for which we evaluate a modified sinc function,
y(x) =
sin(x)
x
+0.04x, (5.21)
and then add Gaussian noise with zero mean and a standard deviation of 0.05. The
noise is added to make the learning of the function more difficult and prevent any
exact solution being possible. The samples are split, using a random selection of 150
for training and 50 for validation. We use Equation (5.10) to whiten the inputs and
outputs and train a network with 7 hidden layer nodes, obtaining correlations of greater
than 99.3%. A comparison of the true and predicted outputs is shown in Figure 5.3.
The optimal number of hidden layer nodes can be determined by comparing the
calculated evidence from networks with different numbers of hidden nodes. These
results are shown in Table 5.1. The evidence clearly increases until we reach 6 hidden
115
5. ARTIFICIAL NEURAL NETWORKS
Nhid Avg. Err. Sqr. Correlation % ln(Znet)±25
3 7.08×10−3 98.23 −279
4 6.44×10−3 98.38 −256
5 3.20×10−3 99.26 −161
6 2.93×10−3 99.31 −142
7 2.76×10−3 99.35 −143
8 2.86×10−3 99.34 −144
9 2.87×10−3 99.34 −143
10 2.73×10−3 99.36 −137
11 2.75×10−3 99.36 −134
12 2.92×10−3 99.32 −154
13 2.77×10−3 99.35 −134
14 2.72×10−3 99.37 −170
15 2.82×10−3 99.35 −183
Table 5.1: Results for training regression networks on a modified sinc function with
noise.
nodes and then levels off to within its own measurement error. We can say then that any
increase in accuracy for larger networks is offset by the increased complexity of the
network. This additional complexity does not contribute enough additional accuracy
beyond 13 nodes, when the evidence begins to drop significantly again.
5.3.2 Three-Way Classification
Radford Neal created a three-way classification data set [184] for testing his own al-
gorithm for NN training. In this data set, four variables x1, x2, x3, and x4 are sampled
from U[0,1] 1000 times each. If the two-dimensional Euclidean distance between
(x1,x2) and (0.4,0.5) is less than 0.35, the point is placed in class 0; otherwise, if
0.8x1 + 1.8x2 < 0.6, the class was set to 1; and if neither of these conditions is true,
the class was set to 2. Note that the values of x3 and x4 play no part in the classifica-
tion. Gaussian noise with standard deviation 0.1 was then added to the input values.
Figure 5.4 shows the data set with classifications. Approximately 75% of data was
used for training and the remaining 25% for validation. A network was trained using 8
116
5.3 Toy Examples
Figure 5.4: The classifications for all data points (training and validation).
True Class Number
Predicted Class (%)
0 1 2
0 282 84.0 4.96 11.0
1 93 14.0 82.8 3.2
2 386 7.0 1.3 91.7
Table 5.2: Classifications for the toy training data set.
hidden layer nodes with the inputs whitened using Equation (5.10). In total, 87.8% of
training data points and 85.4% of validation points were correctly classified. The sum-
mary of classifications is given in Tables 5.2 and 5.3. These results compare well with
Neal’s own original results [184] and are similar to classifications based on applying
the original criteria to the new points that have noise added.
5.3.3 Autoencoders as Non-Linear PCA
As a first test, we trained an autoencoder network on the three-way classification in-
puts. For this we used hidden layers of 10 and then 4 units, so the final network
117
5. ARTIFICIAL NEURAL NETWORKS
True Class Number
Predicted Class (%)
0 1 2
0 99 75.7 6.1 18.2
1 19 21.1 78.9 0.0
2 121 5.0 0.8 94.2
Table 5.3: Classifications for the toy validation data set.
was 4+10+4+10+4. This network is not reducing the dimensionality, but does per-
form a non-linear transformation. Without pre-training, even this relatively simple
autoencoder with 188 weights continuously achieves an average error squared value of
0.0950269, which corresponds to each output being equal to the average value of that
particular input across all data points. However, with pre-training we can obtain an av-
erage error squared of 0.00352961, which corresponds to a correlation of 99.7%. After
just one step of training the error squared is already below the best without pre-training.
Reducing the central layer to 3 nodes does not affect the results without pre-training;
with pre-training we are now only able to obtain an error squared of 0.00621163, which
is a correlation of 92.6%. Clearly some significant information is lost, which agrees
with the fact that there are four independent inputs.
To provide a quick comparison with traditional principle component analysis (PCA),
we use the example of data points sampled from a multivariate Gaussian distribution.
The eigenvalues and eigenvectors of this data represent the components that a PCA
analysis would measure and use for data compression. In the first example, a 3D non-
singular covariance matrix is used to generate samples from a multivariate Gaussian.
The eigenvalues and eigenvectors of the covariance matrix calculated from these sam-
ples are then computed, to compare with the analytic values. As expected, these match
very closely. Autoencoders with an increasing number of hidden layer nodes (in a
single layer) are then trained. In the simplest case of a single hidden layer node, we
expect that the one feature vector that will be represented will be the eigenvector with
the largest eigenvalue, as this captures as much information as possible. For two hid-
den layer nodes, the two feature vectors will now be a linear combination of the two
eigenvectors with the two largest eigenvalues, so that the same plane in the 3D space is
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5.3 Toy Examples
Nhid Avg. Err. Sqr. Correlation % ln(Znet)
1 0.00191 94.06 5984
2 5.73×10−4 98.08 7510
3 5.93×10−5 99.82 10243
Table 5.4: Results for training autoencoder networks on a 3D multivariate Gaussian.
spanned. Finally, with three hidden layer nodes the three feature vectors should span
the 3D space and be able to generate a nearly 100% correlation.
Pretraining was used as, even in these very small networks, it is easy to fall into the
large local maximum of each output always containing the average value. Table 5.4
shows the results obtained when training autoencoders with increasing hidden layer
nodes. As more nodes are added, there is a significant decrease in the average error
squared as well as increasing correlation, up to 99.82%. The evidence value for the net-
work also increases, giving another indication of a better fit. Furthermore, the feature
vector from the network with a single hidden layer node was indeed an eignenvector
with the highest eigenvalue and the feature vectors from the two hidden layer nodes
network spanned approximately the same plane as the two eigenvectors with highest
eigenvalues.
Additionally, we tested our ability to determine the optimal number of hidden layer
nodes for an autoencoder when additional, redundant, information is provided. To ac-
complish this, a 3D multivariate Gaussian was rotated into 5D space. Therefore, there
were only three independent combinations of the five provided values. The known
optimal number of hidden nodes is thus also three and the autoencoders reflect that
clearly in the evidences shown in Tabke 5.5. Once three hidden nodes are used, adding
more neither decreases the error squared or increases the correlation; however, the ev-
idence does decrease as Occam’s razor applies a penalty for using more parameters
without improving the fit to the data.
In both of these examples, it is found that with one hidden layer node the feature
vector that is represented is equivalent to the primary eigenvector of the covariance
matrix of samples, exactly consistent with a PCA analysis. Having shown that PCA-
like results can be reproduced on a simple example, we can now apply autoencoders
119
5. ARTIFICIAL NEURAL NETWORKS
Nhid Avg. Err. Sqr. Correlation % ln(Znet)
1 0.00613 79.6 −6293
2 0.00127 96.0 12430
3 4.87×10−5 99.86 17164
4 4.87×10−5 99.86 16823
5 4.87×10−5 99.86 16859
Table 5.5: Results for training autoencoder networks on a 3D multivariate Gaussian in
5D space.
for finding non-linear features in more complicated data sets by using larger network
structures.
5.4 MNIST Handwriting Recognition
The MNIST database of handwritten digits is a subset of a larger set available from
NIST (National Institute for Standards and Technology). It consists of 60,000 training
and 10,000 validation samples of handwritten digits. The images have been size-
normalised and centred in 28×28 pixel greyscale images. They are publicly available
at [185], which also provides more information on the generation of the data set and re-
sults from previous analyses by other researchers. This data set has become a standard
for testing of machine learning algorithms. The learning task is to correctly identify
the digit written in each image, with the chosen digit being given by the output class
(0 to 9) with the highest probability. Some sample digits are shown in Figure 5.5.
We have trained several different networks on this data set, using pre-training for
any hidden layers. All networks whitened the inputs using Equation (5.11) across
all inputs. Additionally, all networks had 784 inputs (each pixel of the image) and
10 outputs (one for each possible digit). Deeper and larger networks were able to
obtain the best results. Results are summarised in Table 5.6, where the error rates are
those calculated on the validation set of images. These can be compared with results
referenced at [185], which obtain error rates as low as 0.35% [186] but more typically
between 1% and 5% [187].
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5.4 MNIST Handwriting Recognition
Figure 5.5: Sample handwritten digits from the MNIST database.
Hidden Layer Nodes Error Rate (%)
0 8.08
100 4.15
250 3.00
1000 2.38
300+30 2.83
500+50 2.62
1000+300+30 2.31
500+500+2000 1.76
Table 5.6: Error rates for different networks trained to predict the MNIST data set.
Training with larger networks was stopped before full convergence due to the large
computational cost for marginal improvements.
121
5. ARTIFICIAL NEURAL NETWORKS
True Count
Predicted Digit (%)
0 1 2 3 4 5 6 7 8 9
0 980 99.1 0.0 0.2 0.1 0.0 0.3 0.2 0.1 0.0 0.0
1 1135 0.0 99.0 0.1 0.3 0.0 0.3 0.1 0.1 0.2 0.0
2 1032 0.2 0.1 97.6 0.9 0.1 0.1 0.2 0.4 0.5 0.0
3 1010 0.0 0.0 0.0 98.4 0.0 0.4 0.0 0.3 0.7 0.2
4 982 0.0 0.0 0.2 0.0 98.3 0.0 0.5 0.0 0.0 1.0
5 892 0.1 0.0 0.0 1.1 0.0 98.0 0.3 0.0 0.3 0.1
6 958 0.3 0.2 0.0 0.2 0.2 0.3 98.3 0.0 0.4 0.0
7 1028 0.0 0.1 0.5 0.5 0.1 0.0 0.0 98.0 0.2 0.6
8 974 0.2 0.0 0.1 0.8 0.1 0.4 0.0 0.1 97.7 0.5
9 1009 0.2 0.1 0.0 0.5 0.5 0.2 0.1 0.4 0.2 97.8
Table 5.7: Classifications for the MNIST blind data set. (Rows may not add up to
exactly 100 due to rounding.)
In Table 5.7 we provide the classification rates made the best-performing (784 +
500 + 500 + 2000 + 10) network on the blind data set. From this we can see how the
network is distributing its correct and incorrect predictions. To further illustrate, in
Figure 5.6 we show a sample selection of the digits predicted incorrectly. Some are
indeed hard even for a human to distinguish but some are unclear as to why they were
mis-identified.
A pair of auto-encoder networks were also trained on the data set, one with hid-
den layers of 1000+ 300+ 30+ 300+ 1000 (called AE-30) and another with hidden
layers of 1000+500+50+500+1000 (called AE-50). Both had 784 inputs/outputs
to match the image size. The AE-30 network was able to obtain an average total error
squared of only 4.64 on reproducing the digits and AE-50 obtained an average total
error squared of 3.29. The networks encoded each image as 30 or 50 feature vector
weights, respectively, and then decoded these weights to produce an image of the digit
comparable to the original. The error squared values are comparable to those obtained
by Hinton and Salakhutdinov in [182]. The feature vectors from the AE-30 network
are shown in Figure 5.7 – each one being obtained by setting a single decoder input
weight to one and all others to zero. Non-linear weighted combinations of these fea-
122
5.4 MNIST Handwriting Recognition
Figure 5.6: A sample of digits predicted incorrectly by the 784+500+500+2000+10
network. The true and predicted digit values are given above each image (true →
predicted).
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5. ARTIFICIAL NEURAL NETWORKS
Figure 5.7: Feature vectors of the MNIST handwriting samples for the AE-30 network.
The total error squared from these features is 4.64.
tures are used to reproduce each of the handwriting samples to within a small error.
We are able to use the feature vector weights of the encoded writing samples for
classification. All of the training data images were passed through the two encoders to
obtain the sets of 30 or 50 feature vector weights. Networks with 30 or 50 inputs and
the ten classification outputs were then trained on this compressed data set. Results of
the NN training are given in Table 5.8.
An autoencoder that calculated only two feature vectors using a deep network
(1000+ 500+ 250+ 2 symmetric hidden layers) was also trained. Although this net-
work was significantly less able to reproduce the original digits, having an average
total error squared of 31.0, we can plot the values of the two feature vector weights for
124
5.5 Mapping Dark Matter Challenge
Encoder Hidden Layers Error Rate (%)
AE-30
0 9.57
10 6.39
30 3.03
100+50+10 2.55
AE-50
0 8.68
10 6.61
50 2.65
100+50+10 2.71
Table 5.8: Error rates for different networks trained to predict the MNIST data set
based on encoded feature vector weights.
each of the classes to illustrate the classification differences. We do so in Figure 5.8 for
the 10,000 blind data points. There is some clear overlap between digits with similar
shapes, but some digits do occupy distinct regions of the parameter space (particularly
1 in the top right, some 0s in the bottom right, and many 2s in the middle right).
Through this example, we have shown the NN training algorithm to be able to
handle the training of both deep classification and autoencoder networks. Using pre-
training and second-order gradient descent we trained several networks to classify im-
ages of handwritten digits to a high degree of accuracy. Although this may be an easy
task for a human brain, it is quite difficult for a computer to learn. Additionally, we
were able to train autoencoder networks with millions of weights in order to reduce
the dimensionality of the information we were attempting to classify from 784 pix-
els to 30 or 50 non-linear feature vector weights. These reduced basis sets retained
enough information about the original images to reproduce them to within small er-
rors, thus enabling us to perform classification using these weights. This classification
was nearly as accurate as our best-performing network trained on the full images.
5.5 Mapping Dark Matter Challenge
The Mapping Dark Matter Challenge was presented on www.kaggle.com as a sim-
plified version of the GREAT10 Challenge [188]. In this problem, each data value
125
5. ARTIFICIAL NEURAL NETWORKS
Figure 5.8: Scatterplot distribution of feature vector weights for blind data set digits
as given from the encoding half of the 784+1000+500+250+2 symmetric autoencoder.
This is comparable to Figure 3B in [182].
126
5.5 Mapping Dark Matter Challenge
consists of an image of a galaxy and an image of a corresponding star. Each image is
48× 48 pixels and greyscale. The galaxy has some ellipticity that has been obscured
by a point-spread function and statistical noise. The star is a point source that has
also been transformed by the same point-spread function and statistical noise. The
trained network must take the galaxy and star images as inputs and predict the galaxy’s
ellipticity. The training data set contained 40,000 image pairs and a challenge (vali-
dation) data set contained 60,000 image pairs. During the challenge, the solutions for
the validation data set were kept secret and participating teams submitted their predic-
tions. Further details on the challenge and descriptions of the top results can be found
in [189].
5.5.1 The Data and Model
Each galaxy image is an ellipse with a simple brightness profile. This is then convolved
with a point-spread function which blurs the ellipse in a similar way to telescope obser-
vations. There is also Poisson noise degrading the image. Accompanying each galaxy
image is a star image, which is a point source convolved with the same point-spread
function and similar noise. A sample galaxy and star image pair is shown in Figure 5.9.
The ellipticity of a galaxy, which is essentially an ellipse, is given by two param-
eters, e1 and e2. These measure the relative lengths of the major and minor axes and
the angle of the ellipse and may vary in [−1,1]. Equation (5.22) gives these definitions
with a, b, and θ shown in Figure 5.10.
e1 =
a−b
a+b
cos(2θ) (5.22a)
e2 =
a−b
a+b
sin(2θ) (5.22b)
Further details about the data set can be found at the challenge’s webpage [190].
The website also gives the unweighted quadrupole moments (UWQM) formula for
calculating the ellipticity. However, as the competition organisers note, this formula
will not provide very good measurements as it does not account for the point-spread
function. Our aim was to use NNs to provide a better way of measuring the ellipticity.
127
5. ARTIFICIAL NEURAL NETWORKS
(a) (b)
Figure 5.9: Example pair of (a) galaxy and (b) star images for the Mapping Dark
Matter challenge. Each image is 48×48 greyscale pixels.
Figure 5.10: Definition of ellipse measurements for Equation (5.22). Image
from [190].
128
5.5 Mapping Dark Matter Challenge
Data Set Hidden Layers RMSE
Full Images
0 0.0224146
2 0.0186944
5 0.0184237
10 0.0182661
Cropped Star
0 0.0175578
2 0.0176236
5 0.0175945
10 0.0174997
50 0.0172977
50+10 0.0171719
Galaxy Only
0 0.0234740
2 0.0234669
5 0.0236508
10 0.0226440
Table 5.9: Root mean square error rates on ellipticity predictions for different networks
trained on the MDM data sets and evaluated on the 60,000 challenge image pairs.
5.5.2 Results of Training
The quality of a network’s predictions are measured by the root mean squared error of
its predictions of the ellipticities for the 60,000 challenge galaxies. Better predictions
will result in lower values of the RMSE. The magnitude of the problem and the size
of the dataset limited the ability to train large networks due to immense computational
cost. Therefore, for this demonstration we only trained relatively smaller networks, but
used three different data sets: (A) the full galaxy and star images, (B) the full galaxy
image and a centrally cropped star image, and (C) just the full galaxy image. 75%
of the provided training data was used for the training subset and the remaining 25%
was used for validation. We originally trained on just 25% of the training subset and
without using pre-training. Networks were then fine-tuned using the entire training
subset. RMSE values for trained networks evaluated on the challenge set are given in
Table 5.9.
These scores show naive first results that already perform very well; the standard
129
5. ARTIFICIAL NEURAL NETWORKS
software package SourceExtractor scores 0.0862191 on this test data and UWQM
scores 0.1855600. The results could potentially be further improved by fitting pro-
files to the images and using these fits for training, which would reduce the number
of inputs by about two orders of magnitude. Additionally, an autoencoder could be
trained to allow us to decompose the images into feature vectors and use the corre-
sponding weights for training, thereby reducing the dimensionality and the impact of
noise in the images.
By reducing the number of inputs without affecting the information content – in
this problem by cropping the star images to the central 24×24 pixels – we were able
to improve the precision of predictions and lower our RMSE. Due to its better per-
formance on the initial networks trained, the “CroppedStar” data set was also used
to train two slightly larger networks with 50 and 50+ 10 hidden layer nodes. These
continued to show improving predictions, indicating that even more complex networks
could further improve ellipticity measurements. The best results obtained, with an
RMSE of 0.0171719, compare well with the competition winners who had an RMSE
of 0.0150907 [189, 190] and used a mixture of methods that included NNs. This scored
would have placed us in 32nd place out of 66 teams that entered.
The “GalaxyOnly” data set shows us that removing the star images does not al-
low the NN to account for the point-spread function and therefore gives a significant
increase in the RMSE as might be expected.
The RMSE on the challenge data was slightly increased relative to that obtained on
initial training data because it was generated with a non-zero mean ellipticity (actual
mean of 0.01). Since the network is only able to predict what it has been trained on, the
statistical differences between the two data sets led to a slightly increased prediction
error on average.
5.6 Discussion
In this chapter I have introduced the framework of artificial neural networks used for
machine learning. By using an efficient and robust algorithm, we are able to train large
and deep networks. We use second-order information to allow convergence in fewer
steps and prevent overfitting with prior information and checks on separate validation
data. The algorithm is demonstrated on toy examples of regression, classification, and
130
5.6 Discussion
autoencoder networks. The framework was then applied to handwriting classification
in determining digits from the MNIST database. Shallow and deep classification net-
works as well as autoencoders were used to accomplish this machine learning task.
Lastly, NNs were applied to measuring the ellipticity of noisy and convolved galaxy
images in the Mapping Dark Matter Challenge. NNs proved to perform well given the
raw, un-treated data.
The learning power of NNs combined with the training algorithm described is a
powerful tool for use in further astrophysics projects, such as the generation of grav-
itational wave signals, as well as computational tasks of other types that involve re-
peated and complex functions. This includes evaluations of likelihoods for Bayesian
inference, an application that is investigated in the next chapter.
131
5. ARTIFICIAL NEURAL NETWORKS
132
Chapter 6
The BAMBI Algorithm
G-d does not care about our mathematical
difficulties. He integrates empirically.
Albert Einstein
Bayesian methods of inference are widely used in astronomy and cosmology and
are gaining popularity in other fields, such as particle physics. Some uses have al-
ready been described in Chapters 2 and 3. At each point in parameter space, Bayesian
methods require the evaluation of a likelihood function describing the probability of
obtaining the data for a given set of model parameters. For some cosmological and
particle physics problems each such function evaluation takes up to tens of seconds.
MCMC applications may require millions of these evaluations, making them pro-
hibitively costly. MULTINEST is able to reduce the number of likelihood function
calls by an order of magnitude or more, but further gains can be achieved if we are
able to speed up the evaluation of the likelihood itself.
An artificial neural network is ideally suited for this task. A universal approx-
imation theorem [176] assures us that we can accurately and precisely approximate
the likelihood with a three-layer, feed-forward NN. The training of NNs is one of the
most widely studied problems in machine learning, so techniques for learning the like-
lihood function are well established. In the Blind Accelerated Multimodal Bayesian
Inference (BAMBI) algorithm [191], we incorporate NNs with MULTINEST. Samples
from MULTINEST are used to train a NN or set of NNs on the likelihood function
133
6. THE BAMBI ALGORITHM
which can subsequently be used to predict new likelihood values in a tiny fraction of
the time originally required. We implement the algorithm described in Chapter 5 for
our NN training.
In this chapter, Section 6.1 will describe the structure of the new algorithm, BAMBI.
Section 6.2 then demonstrates this algorithm performing on a few toy examples. The
algorithm is applied to the computationally costly problem of cosmological parameter
estimation in Section 6.3. For this same problem, we then demonstrate rapid follow-
up analyses in Section 6.4. Work in this chapter was performed in collaboration with
Farhan Feroz and large portions are published in [191].
6.1 The Structure of BAMBI
The Blind Accelerated Multimodal Bayesian Inference (BAMBI) algorithm combines
nested sampling and neural networks. After a specified number of new samples from
MULTINEST have been obtained (specified by the updInt run parameter), BAMBI
uses these to train a regression network on the log-likelihood function. Approximately
80% of the samples are used for training and the remaining 20% are used for the val-
idation set. These values are slightly different from the previous chapter to give more
information for training on the likelihood, but at the sacrifice of a smaller validation
set. After convergence to the optimal NN weights, we test that the network is able to
predict likelihood values to within a specified tolerance level. If not, sampling con-
tinues using the original log-likelihood until enough new samples have been made for
training to be resumed. Once a network is trained that is sufficiently accurate, its pre-
dictions are used in place of the original log-likelihood function for future samples in
MULTINEST. Consistency checks are made to ensure the NN is not making predic-
tions outside the range of data on which it was trained. Using the network reduces the
log-likelihood evaluation time from seconds to microseconds, allowing MULTINEST
to complete analysis much more rapidly. As a bonus, the user also obtains a network
or set of networks that are trained to easily and quickly provide more log-likelihood
evaluations near the peak if needed, or in subsequent analyses.
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6.1 The Structure of BAMBI
6.1.1 When to Use the Trained Network
The optimal network possible with a given set of training data may not be able to
predict log-likelihood values accurately enough, so an additional criterion is placed on
when to use the trained network. This requirement is that the root-mean-square error
of log-likelihoods evaluations by the network is less than a user-specified tolerance,
tol. When the trained network does not pass this test, then BAMBI will continue
using the original log-likelihood function to obtain updInt/2 new samples to generate
a new training data set of the last updInt accepted samples. Network training will then
resume, beginning with the weights that it had found as optimal for the previous data
set. Since samples are generated from nested contours and each new data set contains
half of the previous one, the saved network will already be able to produce reasonable
predictions on this new data; resuming therefore enables us to save time as fewer steps
will be required to reach the new optimum weights.
Once a NN is in use in place of the original log-likelihood function its evalua-
tions are taken to be the actual log-likelihoods, but checks are made to ensure that
the network is maintaining its accuracy. If the network makes a prediction outside of
[mintraining(logL)−tol,maxtraining(logL)+tol], then that value is discarded and the
original log-likelihood function is used for that point. Additionally, the central 95th per-
centile of the output log-likelihood values from the training data used is calculated and
if the network is making > 95% of its predictions outside of this range then it will be
re-trained. To re-train the network, BAMBI first substitutes the original log-likelihood
function back in and gathers the required number of new samples from MULTINEST.
Training then commences, resuming from the previously saved network. These crite-
ria ensure that the network is not trusted too much when making predictions beyond
the limits of the data it was trained on, as we cannot be sure that such predictions are
accurate.
The flow of sampling and training within BAMBI is demonstrated by the flowchart
given in Figure 6.1. Red boxes indicate sampling with the original likelihood function
and we want to minimise the time spent in these. The brown box is time spent training
the NN and so does not advance the Bayesian inference; time here should be minimized
as well. Lastly, the green box is sampling done with a trained NN and this is where we
135
6. THE BAMBI ALGORITHM
Figure 6.1: A flowchart depicting the transitions between sampling and NN training
within BAMBI. N is given by updInt from MULTINEST.
want to maximise usage, so that a larger percentage of the log-likelihood evaluations
are done rapidly.
6.2 BAMBI Toy Examples
In order to demonstrate the ability of BAMBI to learn and accurately explore multi-
modal and degenerate likelihood surfaces, we first tested the algorithm on a few toy
examples. The eggbox likelihood has many separate peaks of equal likelihood, mean-
ing that the network must be able to make predictions across many different areas of
the prior. The Gaussian shells likelihood presents the problem of making predictions
in a very narrow and curving region. Lastly, the Rosenbrock function gives a long,
curving degeneracy that can be extended to higher dimensions. They all require high
accuracy and precision in order to reproduce the posterior truthfully and each presents
unique challenges to the NN in learning the log-likelihood. It is important to note
that running BAMBI on these problems required more time than the straightforward
analysis; this was as expected since the actual likelihood functions are simple analytic
expressions that do not require much computational expense.
136
6.2 BAMBI Toy Examples
Figure 6.2: The eggbox log-likelihood surface, given by Equation (6.1).
6.2.1 Eggbox
This is a standard example of a very multimodal likelihood distribution in two dimen-
sions. It has many peaks of equal value, so the network must be able to take samples
from separated regions of the prior and make accurate predictions in all peaks. The
eggbox log-likelihood [102] is given by
log(L(x,y)) =
(
2+ cos( x2)cos(
y
2)
)5
, (6.1)
where we take a uniform prior U[0,10pi] for both x and y. The structure of the surface
can be seen in Figure 6.2.
We ran the eggbox example in both MULTINEST and BAMBI, both using 4000
live points. For BAMBI, we used 4000 samples for training a network with 50 hidden
nodes. These values were chosen after some initial testing to give the NN sufficient
complexity and data to learn the function. In Table 6.1 we report the evidences re-
covered by both methods as well as the true value obtained analytically from Equa-
tion (6.1). Both methods return evidences that agree with the analytically determined
value to within the given error bounds. Figure 6.3 compares the posterior probability
distributions returned by the two algorithms via the distribution of lowest-likelihood
points removed at successive iterations by MULTINEST. We can see that they are
identical distributions; therefore, we can say that the use of the NN did not reduce
the quality of the results either for parameter estimation or model selection. During
137
6. THE BAMBI ALGORITHM
Method log(Z)
Analytical 235.88
MULTINEST 235.859±0.039
BAMBI 235.901±0.039
Table 6.1: The log-evidence values of the eggbox likelihood as found analytically and
with MULTINEST and BAMBI.
(a) (b)
Figure 6.3: Points of lowest likelihood of the eggbox log-likelihood from successive
iterations as given by (a) MULTINEST and (b) BAMBI.
the BAMBI analysis 51.3% of the log-likelihood function evaluations (∼ 70,000 to-
tal) were done using the NN; if this were a more computationally expensive function,
significant speed gains would have been realised.
6.2.2 Gaussian Shells
The Gaussian shells likelihood function has low values over most of the prior, except
for thin circular shells that have Gaussian cross-sections. We use two separate Gaus-
sian shells of equal magnitude so that this is also a mutlimodal inference problem.
Therefore, our Gaussian shells likelihood is
L(x) = circ(x;c1,r1,w1)+ circ(x;c2,r2,w2), (6.2)
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6.2 BAMBI Toy Examples
Figure 6.4: The Gaussian shell likelihood surface, given by Equations (6.2) and (6.3).
where each shell is defined by
circ(x;c,r,w) =
1√
2piw2
exp
[
−(|x− c|− r)
2
2w2
]
. (6.3)
We used values of r1 = r2 = 2, w1 = w2 = 0.1, c1 = (−3.5,0), and c2 = (3.5,0). This
is shown in Figure 6.4.
As with the eggbox problem, we analysed the Gaussian shells likelihood with both
MULTINEST and BAMBI using uniform priors U[−6,6] on both dimensions of x.
MULTINEST sampled with 1000 live points and BAMBI used 2000 samples for train-
ing a network with 100 hidden nodes (again, tests were performed to find suitable
values for NN learning). In Table 6.2 we report the evidences recovered by both meth-
ods as well as the true value obtained analytically from Equations (6.2) and (6.3). The
evidences are both consistent with the true value. Figure 6.5 compares the posterior
probability distributions returned by the two algorithms (in the same manner as with
the eggbox example). Again, we see that the distribution of returned values is nearly
identical when using the NN, which BAMBI used for 18.2% of its log-likelihood func-
tion evaluations (∼ 10,000 total). This is a significant fraction, especially since they
are all at the end of the analysis when exploring the peaks of the distribution.
139
6. THE BAMBI ALGORITHM
Method log(Z)
Analytical −1.75
MULTINEST −1.768±0.052
BAMBI −1.757±0.052
Table 6.2: The log-evidence values of the Gaussian shell likelihood as found analyti-
cally and with MULTINEST and BAMBI.
(a) (b)
Figure 6.5: Points of lowest likelihood of the Gaussian shell likelihood from successive
iterations as given by (a) MULTINEST and (b) BAMBI.
140
6.2 BAMBI Toy Examples
Figure 6.6: The Rosenbrock log-likelihood surface given by Equation (6.4) with N = 2.
6.2.3 Rosenbrock function
The Rosenbrock function is a standard example used for testing optimization as it
presents a long, curved degeneracy through all dimensions. For our NN training, it
presents the difficulty of learning the likelihood function over a large, curving region
of the prior. We use the Rosenbrock function to define the negative log-likelihood, so
the log-likelihood function is given in N dimensions by
log(L(x)) =−
N−1
∑
i=1
[
(1− xi)2+100(xi+1− x2i )2
]
. (6.4)
Figure 6.6 shows how this appears for N = 2.
We set uniform priors of U[−5,5] in all dimensions and performed analysis with
both MULTINEST and BAMBI with N = 2 and N = 10. For N = 2, MULTINEST sam-
pled with 2000 live points and BAMBI used 2000 samples for training a NN with 50
hidden-layer nodes. With N = 10, we used 2000 live points, 6000 samples for network
training, and 50 hidden nodes. Tests ensured that these settings allowed good sam-
pling of the prior and ample information and NN structure to allow training to a high
enough accuracy. Table 6.3 gives the calculated evidences returned by both algorithms
as well as the analytically calculated values from Equation (6.4) (there does not exist
an analytical solution for the 10D case so this is not included). Figure 6.7 compares
the posterior probability distributions returned by the two algorithms for N = 2. For
141
6. THE BAMBI ALGORITHM
Method log(Z)
Analytical 2D −5.804
MULTINEST 2D −5.799±0.049
BAMBI 2D −5.757±0.049
MULTINEST 10D −41.54±0.13
BAMBI 10D −41.53±0.13
Table 6.3: The log-evidence values of the Rosenbrock likelihood as found analytically
and with MULTINEST and BAMBI.
N = 10, we show in Figure 6.8 comparisons of the marginalised two-dimensional pos-
terior distributions for 12 variable pairs. We see that MULTINEST and BAMBI return
nearly identical posterior distributions as well as consistent estimates of the evidence.
For N = 2 and N = 10, BAMBI was able to use a NN for 64.7% and 30.5% of its
log-likelihood evaluations respectively (∼ 40,000 total for N = 2 and ∼ 1.3×106 for
N = 10). Even when factoring in time required to train the NN this would have resulted
in large decreases in running time for a more computationally expensive likelihood
function. In the N = 2 case the network was trained twice, with re-training required
because the first training did not achieve sufficient accuracy. In the N = 10 case the
network was trained 12 times since the first 10 times did not achieve enough accu-
racy and the first network used needed re-training once the sampling moved further up
in log-likelihood contours and the distribution of predictions no longer resembled the
data on which it was trained.
6.3 Cosmological Parameter Estimation with BAMBI
While likelihood functions resembling our previous toy examples do exist in real phys-
ical models, we would also like to demonstrate the usefulness of BAMBI on simpler
likelihood surfaces where the time of evaluation is the critical limiting factor. One
such example in astrophysics is that of cosmological parameter estimation and model
selection.
We implement BAMBI within the standard COSMOMC code [192], which by de-
fault uses MCMC sampling. This allows us to compare the performance of BAMBI
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6.3 Cosmological Parameter Estimation with BAMBI
(a) (b)
Figure 6.7: Points of lowest likelihood of the Rosenbrock likelihood for N = 2 from
successive iterations as given by (a) MULTINEST and (b) BAMBI.
Figure 6.8: Marginalised 2D posteriors for the Rosenbrock function with N = 10. The
12 most correlated pairs are shown. MULTINEST is in solid black, BAMBI in dashed
blue. Inner and outer contours represent 68% and 95% confidence levels, respectively.
143
6. THE BAMBI ALGORITHM
with other methods, such as MULTINEST, COSMONET, INTERPMC, PICO, Particle
Swarm Optimization, and others [102, 193–198]. In this work, we will only report
performances of BAMBI and MULTINEST, but these can be compared with reported
performance from the other methods.
Bayesian parameter estimation in cosmology requires evaluation of theoretical tem-
perature and polarisation CMB power spectra (Cl values) using cosmological evolution
codes such as CAMB [199]. These evaluations can take on the order of tens of seconds
depending on the cosmological model. The Cl spectra are then compared to WMAP
and other observations for the likelihood function. Considering that tens of thousands
of these evaluations will be required, this is a computationally expensive step and a
limiting factor in the speed of any Bayesian analysis. With BAMBI, we use samples to
train a NN on the combined likelihood function which will allow us to avoid evaluating
the full power spectra. This has the benefit of not requiring a pre-computed sample of
points as in COSMONET and PICO, which is particularly important when including
new parameters or new physics in the model. In these cases we will not know in ad-
vance where the peak of the likelihood will be and it is around this location that the
most samples would be needed for accurate results.
The set of cosmological parameters that we use as variables and their prior ranges
are given in Table 6.4; the parameters have their usual meanings in cosmology [200].
The prior probability distributions are uniform over the ranges given. A non-flat cos-
mological model incorporates all of these parameters, while we set ΩK = 0 for a flat
model. We use w = −1 in both cases. The flat model thus represents the standard
ΛCDM cosmology. We use two different data sets for analysis: (1) CMB observa-
tions alone and (2) CMB observations plus Hubble Space Telescope constraints on
H0, large-scale structure constraints from the luminous red galaxy subset of the SDSS
and the 2dF survey, and supernovae Ia data. The CMB dataset consists of WMAP
seven-year data [200] and higher resolution observations from the ACBAR, CBI, and
BOOMERanG experiments. The COSMOMC website [192] provides full references
for the most recent sources of these data.
Analyses with MULTINEST and BAMBI were run on all four combinations of
models and data sets. MULTINEST sampled with 1000 live points and an efficiency
of 0.5, both on its own and within BAMBI; BAMBI used 2000 samples for training
144
6.3 Cosmological Parameter Estimation with BAMBI
Parameter Min Max Description
Ωbh2 0.018 0.032 Physical baryon density
ΩDMh2 0.04 0.16 Physical cold dark matter density
θ 0.98 1.1
100× Ratio of the sound horizon to angular
diameter distance at last scattering
τ 0.01 0.5 Reionisation optical depth
ΩK −0.1 0.1 Spatial curvature
ns 0.8 1.2 Spectral index of density perturbations
log[1010As] 2.7 4
Amplitude of the primordial spectrum of
curvature perturbations
ASZ 0 2 Amplitude of the Sunyaev-Zel’dovich spectrum
Table 6.4: The cosmological parameters and their minimum and maximum values.
Uniform priors were used on all variables. ΩK was set to 0 for the flat model.
a NN on the likelihood, with 50 hidden-layer nodes for both the flat model and non-
flat model. These values were chosen based on the toy examples and analysis runs
confirmed their viability. Table 6.5 reports the recovered evidences from the two al-
gorithms for both models and both data sets. It can be seen that the two algorithms
report equivalent values to within statistical error for all four combinations. In Fig-
ures 6.9 and 6.10 we show the recovered one- and two-dimensional marginalised pos-
terior probability distributions for the non-flat model using the CMB-only data set.
Figures 6.11 and 6.12 show the same for the non-flat model using the complete data
set. We see very close agreement between MULTINEST (in solid black) and BAMBI
(in dashed blue) across all parameters. The only exception is ASZ since it is uncon-
strained by these models and data and is thus subject to large amounts of variation in
sampling. The posterior probability distributions for the flat model with either data set
are extremely similar to those of the non-flat flodel with setting ΩK = 0, as expected,
so we do not show them here.
A by-product of running BAMBI is that we now have a network that is trained to
predict likelihood values near the peak of the distribution. To see how accurate this net-
work is, in Figure 6.13 we plot the error in the prediction (∆ log(L) = log(Lpredicted)−
log(Ltrue)) versus the true log-likelihood value for the different sets of training and val-
145
6. THE BAMBI ALGORITHM
Algorithm Model Data Set log(Z)
MULTINEST ΛCDM CMB only −3754.58±0.12
BAMBI ΛCDM CMB only −3754.57±0.12
MULTINEST ΛCDM all −4124.40±0.12
BAMBI ΛCDM all −4124.11±0.12
MULTINEST ΛCDM+ΩK CMB only −3755.26±0.12
BAMBI ΛCDM+ΩK CMB only −3755.57±0.12
MULTINEST ΛCDM+ΩK all −4126.54±0.13
BAMBI ΛCDM+ΩK all −4126.35±0.13
Table 6.5: Evidences calculated by MULTINEST and BAMBI for the flat (ΛCDM) and
non-flat (ΛCDM+ΩK) models using the CMB-only and complete data sets. The two
algorithms are in close agreement in all cases.
Figure 6.9: Marginalised 1D posteriors for the non-flat model (ΛCDM+ΩK) using
only CMB data. MULTINEST is in solid black, BAMBI in dashed blue.
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6.3 Cosmological Parameter Estimation with BAMBI
Figure 6.10: Marginalised 2D posteriors for the non-flat model (ΛCDM+ΩK) using
only CMB data. The 12 most correlated pairs are shown. MULTINEST is in solid black,
BAMBI in dashed blue. Inner and outer contours represent 68% and 95% confidence
levels, respectively.
Figure 6.11: Marginalised 1D posteriors for the non-flat model (ΛCDM+ΩK) using
the complete data set. MULTINEST is in solid black, BAMBI in dashed blue.
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6. THE BAMBI ALGORITHM
Figure 6.12: Marginalised 2D posteriors for the non-flat model (ΛCDM+ΩK) using
the complete data set. The 12 most correlated pairs are shown. MULTINEST is in
solid black, BAMBI in dashed blue. Inner and outer contours represent 68% and 95%
confidence levels, respectively.
idation points that were used. What we show are results for networks that were trained
to sufficient accuracy in order to be used for making likelihood predictions; this results
in two networks for each case shown. We can see that although the flat model used the
same number of hidden-layer nodes, the simpler physical model allowed for smaller
error in the likelihood predictions. Both final networks (one for each model) are signif-
icantly more accurate than the specified tolerance of a maximum average error of 0.5.
In fact, for the flat model, all but one of the 2000 points have an error of less than 0.06
log-units. The two networks trained in each case overlap in the range of log-likelihood
values on which they trained. The first network, although trained to lower accuracy, is
valid over a much larger range of log-likelihoods. The accuracy of each network in-
creases with increasing true log-likelihood and the second network, trained on higher
log-likelihood values, is significantly more accurate than the first.
The final comparison, and perhaps the most important, is the running time required.
The analyses were run using MPI parallelisation on 48 processors. We recorded the
time required for the complete analysis, not including any data initialisation prior to
initial sampling. We then divide this time by the number of likelihood evaluations
performed to obtain an average time per likelihood (twall clock, sec×NCPUs/Nlog(L) evals).
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6.3 Cosmological Parameter Estimation with BAMBI
Figure 6.13: The error in the predicted likelihood (∆ log(L) = log(Lpredicted) −
log(Ltrue)) for the BAMBI networks trained on the flat (top row) and non-flat (bot-
tom row) models using the complete data set. The left column represents predictions
from the first NN trained to sufficient accuracy; the right column are results from the
second, and final, NN trained in each case. The flat and non-flat models both used 50
hidden-layer nodes.
Therefore, time required to train the NN is still counted as a penalty factor. If a NN
takes more time to train, this will hurt the average time, but obtaining a usable NN
sooner and with fewer training calls will give a better time since more likelihoods
will be evaluated by the NN. The resulting average times per likelihood and speed
increases are given in Table 6.6. Although the speed increases appear modest, one must
remember that these include time taken to train the NNs, during which no likelihoods
were evaluated. This can be seen in that although 30–40% of likelihoods are evaluated
with a NN, as reported in Table 6.7, we do not obtain the full equivalent speed increase.
We are still able to obtain a significant decrease in running time while adding in the
bonus of having a NN trained on the likelihood function.
6.3.1 Updated Timing Comparisons
The NN training algorithm code was updated to improve memory efficiency and to
use single-precision variables by default. Following these updates, we decided to re-
run the analyses on the cosmological parameter estimation to confirm that results were
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6. THE BAMBI ALGORITHM
Model Data set MULTINEST BAMBI Speed
tL (s) tL (s) factor
ΛCDM CMB only 2.394 1.902 1.26
ΛCDM all 3.323 2.472 1.34
ΛCDM+ΩK CMB only 12.744 9.006 1.42
ΛCDM+ΩK all 12.629 10.651 1.19
Table 6.6: Time per likelihood evaluation, factor of speed increase from MULTINEST
to BAMBI (tMN/tBAMBI).
Model Data set %log(L) Equivalent Actual
with NN speed factor speed factor
ΛCDM CMB only 40.5 1.68 1.26
ΛCDM all 40.2 1.67 1.34
ΛCDM+ΩK CMB only 34.2 1.52 1.42
ΛCDM+ΩK all 30.0 1.43 1.19
Table 6.7: Percentage of likelihood evaluations performed with a NN, equivalent speed
factor, and actual factor of speed increase.
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6.4 Using Trained Networks for Follow-up in BAMBI
Model Data set MULTINEST BAMBI Speed
tL (s) tL (s) factor
ΛCDM CMB only 2.70 2.02 1.34
ΛCDM all 3.69 2.46 1.50
ΛCDM+ΩK CMB only 13.56 9.33 1.45
ΛCDM+ΩK all 13.85 10.36 1.34
Table 6.8: Time per likelihood evaluation, factor of speed increase from MULTINEST
to BAMBI (tMN/tBAMBI).
not adversely affected by use of single-precision. As before, a single hidden layer
with 50 nodes was used for all four pairings of model and data and a tolerance of 0.5
was set for accepting a network as sufficiently accurate. MULTINEST used 1000 live
points and 2000 points were used for network training. After completing, all calculated
evidences were confirmed to agree between MULTINEST and BAMBI and previous
results. Table 6.8 is like Table 6.6 but for the new runs. The reduction in time spent
training a network is reflected in the speed factors which have in all cases increased
from the initial comparisons.
6.4 Using Trained Networks for Follow-up in BAMBI
A major benefit of BAMBI is that following an initial run the user is provided with a
trained NN, or set of NNs, that model the log-likelihood function. These can be used
in a subsequent analysis with different priors to obtain much faster results. This is a
comparable analysis to that of COSMONET [193, 194], except that the NNs here are
a product of an initial Bayesian analysis where the peak of the distribution was not
previously known. No prior knowledge of the structure of the likelihood surface was
used to generate the networks that are now able to be re-used. In this section we will
examine two methods for calculating the error in a network’s prediction and provide
results for performing a follow-up analysis on the cosmological parameter estimation
problems.
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6. THE BAMBI ALGORITHM
6.4.1 Exact Error Calculation
When multiple NNs are trained and used in the initial BAMBI analysis, we must deter-
mine which network’s prediction to use in the follow-up. The approximate uncertainty
error in a NN’s prediction of the value y(x;a) (x denoting input parameters, a the NN
weights and biases as before) that models the log-likelihood function is given by [201]
as
σ2 = σ2pred+σ
2
ν , (6.5)
where
σ2pred = g
TB−1g (6.6)
and σ2ν is the variance of the noise on the output from the network training (the network
hyper-parameter σ 2 on the single output defined in Equation (5.4)). In Equation (6.6),
B is the Hessian of the log-posterior as before, and g is the gradient of the NN’s pre-
diction with respect to the weights about their maximum posterior values,
g =
∂y(x;a)
∂a
∣∣∣∣
x,aMP
. (6.7)
This uncertainty of the prediction is calculated for each training and validation data
point used in the initial training of the NN for each saved NN. The threshold for ac-
cepting a predicted log-likelihood from a network is then set to be 1.2 times the max-
imum uncertainty value found for points in its training and validation data sets. We
can therefore ignore the value of σ2ν in Equation (6.5) as it is a constant value for all
inputs and we only ever compare predictions of a network to previous predictions of
that same network.
A log-likelihood is calculated by first making a prediction with the final NN to be
trained and saved and then calculating the error for this prediction. If the error, σpred, is
greater than that NN’s threshold, then we consider the previous trained NN. We again
calculate the predicted log-likelihood value and error to compare with its threshold.
This continues to the first NN saved until a NN makes a sufficiently confident predic-
tion. If no NNs can make a confident enough prediction, then we set log(L) = −∞.
This is justified because the NNs are trained to predict values in the highest likelihood
regions of parameter space and if a set of parameters lies outside their collective region
of validity, then it must not be within the region of highest likelihoods. This effectively
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6.4 Using Trained Networks for Follow-up in BAMBI
limits the prior and so will bias evidences upwards but not significantly affect parame-
ter estimation.
To demonstrate the speed-up potential of using the NNs, we first ran an analysis of
the cosmological parameter estimation using both models and both data sets. This time,
however, we set the tolerance of the NNs to 1.0 instead of 0.5, so that they would be
valid over a larger range of log-likelihoods and pass the accuracy criterion sooner. Each
analysis produced two trained NNs. We then repeated each of the four analyses, but
set the prior ranges to be uniform over the region defined by xmax(log(L))±4σ , where
σ is the vector of standard deviations of the marginalised one-dimensional posterior
probabilities.
In Figure 6.14 we show predictions from the two trained NNs on the two sets of
validation data points in the case of the non-flat model using the complete data set. In
the left-hand column, we can see that the first NN trained is able to make reasonable
predictions on its own validation data as well as on the second set of points, in red
crosses, from the second NN’s validation data. The final NN is able to make more pre-
cise predictions on its own data set than the initial NN, but is unable to make accurate
predictions on the first NN’s data points. The right-hand column shows the error bar
sizes for each of the points shown. For both NNs, the errors decrease with increasing
log-likelihood. The final NN has significantly lower uncertainty on predictions for its
own validation data, which enables us to set the threshold for when we can trust its
prediction. The cases for the other three sets of cosmological models and data sets are
very similar to this one. This demonstrates the need to use the uncertainty error mea-
surement in determining which NN prediction to use, if any. Always using the final
NN would produce poor predictions away from the peak and the initial NN does not
have sufficient precision near the peak to properly measure the best-fit cosmological
parameters. But by choosing which NN’s prediction to accept, as we have shown, we
can quickly and accurately reproduce the log-likelihood surface for sampling. Fur-
thermore, if one were interested only in performing a re-analysis about the peak, then
one could use just the final NN, thereby omitting the calculational overhead associated
with choosing the appropriate network.
For this same case, we plot the one- and two-dimensional marginalised posterior
probabilities in Figures 6.15 and 6.16, respectively. Although the priors do not cover
exactly the same ranges, we expect very similar posterior distributions since the priors
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6. THE BAMBI ALGORITHM
Figure 6.14: Predictions with uncertainty error bars for NNs saved by BAMBI when
analysing the non-flat model using the complete data set. The left-hand side shows
predictions with errors for the two NNs on their own and the other’s validation data
sets. Each network’s own points are in blue plusses, the other NN’s points are in red
crosses. As many error bars are too small to be seen, the right-hand side uses the same
colour and label scheme to show the magnitudes of the error bars from each NN on the
predictions.
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6.4 Using Trained Networks for Follow-up in BAMBI
Figure 6.15: Marginalised 1D posteriors for the non-flat model (ΛCDM+ΩK) using
the complete data set. BAMBI’s initial run is in solid black, the follow-up analysis in
dashed blue.
are sufficiently wide as to encompass nearly all of the posterior probability. We see
this very close agreement in all cases.
Calculating the uncertainty error requires calculating approximate inverse Hessian-
vector products which slow down the process. We sacrifice a large factor of speed
increase in order to maintain the robustness of our predictions. Using the same method
as before, we computed the time per likelihood calculation for the initial BAMBI run
as well as the follow up; these are compared in Table 6.9. We can see that in addition
to the initial speed-up obtained with BAMBI, this follow-up analysis obtains an even
larger speed-up in time per likelihood calculation. This speed-up is especially large
for the non-flat model, where CAMB takes longer to compute the CMB spectra. The
speed factor also increases when using the complete data set, as the original likelihood
calculation takes longer than for the CMB-only data set; NN predictions take equal
time regardless of the data set.
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6. THE BAMBI ALGORITHM
Figure 6.16: Marginalised 2D posteriors for the non-flat model (ΛCDM+ΩK) using
the complete data set. The 12 most correlated pairs are shown. BAMBI’s initial run is
in solid black, the follow-up analysis in dashed blue. Inner and outer contours represent
68% and 95% confidence levels, respectively.
Model Data set Initial Follow-up Speed
tL (s) tL (s) factor
ΛCDM CMB only 1.635 0.393 4.16
ΛCDM all 2.356 0.449 5.25
ΛCDM+ΩK CMB only 9.520 0.341 27.9
ΛCDM+ΩK all 8.640 0.170 50.8
Table 6.9: Time per likelihood evaluation, factor of speed increase from BAMBI’s
initial run to a follow-up analysis.
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6.4 Using Trained Networks for Follow-up in BAMBI
6.4.2 Fast Error Calculation
One possible way to avoid the computational cost of computing error bars on the pre-
dictions is that suggested by [201] (see Section 9). One can take the NN training data
and add Gaussian noise and train a new NN, using the old weights as a starting point.
Performing many realisations of this will quickly provide multiple NNs whose aver-
age prediction will be a good fit to the original data and whose standard deviation from
this mean will measure the error in the prediction. This will reduce the time needed
to compute an error bar since multiple NN predictions are faster than a single inverse
Hessian-vector product. The steps are given as:
1. Start with the converged network with weights w∗ trained on true data set D∗ =
{x(m), t(m)}. Estimate the Gaussian noise level of the residuals using σ2 =
∑m(t(m)− y(x(m),w∗))2/N.
2. Define a new data set D1 by adding Gaussian noise of zero mean and variance
σ2 to the outputs in D∗.
3. Train a NN on D1 using w∗ as a starting point. Training should converge rapidly
as the new data set is only slightly different from the original. Call the new
network w1.
4. Repeat Steps 2 and 3 multiple times to find an ensemble of networks w j.
5. Use each of the networks w j to make a prediction for a given set of inputs. The
error on the predicted value can be estimated as the standard deviation of this set
of values.
In addition to these steps, we include the option for the user to add a random Gaus-
sian offset to the saved weights read in before training is performed on the new data set
(Step 3). This offset will aid the training in moving the optimisation from a potential
local maximum in the posterior distribution of the network weights, but its size must
be chosen for each problem. We are thus adding noise to both the training data and the
saved network weights before training a new network whose posterior maximum will
be near to, but not exactly the same as, the original network’s.
The training of the additional networks is performed by a separate program and the
resulting error estimates can be compared with those from the exact, slow calculation
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6. THE BAMBI ALGORITHM
Figure 6.17: Comparison of NN prediction error calculations for the 2D Gaussian
shells likelihood from the exact, slow method (Equation (6.6)) and the fast method
with different weight offsets applied. Fast method calculations employ the original
network and networks from 29 realisations of Gaussian noise added to the training
data outputs for a total of 30 networks used in the error estimate.
(Equation (6.6)) described in the previous section (6.4). Time taken for this additional
training of networks depends only on the size of the networks and the training data
sets, not on the complexity of the likelihood function being modeled.
The method for estimating error was initially tested on two of the problems already
analysed with BAMBI – the 2D Gaussian shells and the non-flat cosmology using the
complete data set. Comparisons of the error estimates from different choices of the
random offset applied to saved weights and from the exact, slow method are shown
in Figures 6.17 and 6.18. The estimates from the fast method just described use the
original network plus 29 different realisations of Gaussian noise added to the outputs
to give a total of 30 networks, which results in total training time of less than one hour
when run on 24 parallel CPUs.
From Figures 6.17 and 6.18 we can see that the value of the offset applied has a
significant effect on the size of the resulting error estimates. In general, larger offsets
result in larger error estimates. However, the general trend of increasing error for lower
log-likelihoods (further from the peak) is followed in all cases. We can determine
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6.4 Using Trained Networks for Follow-up in BAMBI
Figure 6.18: Comparison of NN prediction error calculations for the non-flat cosmol-
ogy with complete data set likelihood from the exact, slow method (Equation (6.6))
and the fast method with different weight offsets applied (Gaussian random variable
with mean 0 and standard deviation as noted). Fast method calculations employ the
original network and networks from 29 realisations of Gaussian noise added to the
training data outputs for a total of 30 networks used in the error estimate.
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6. THE BAMBI ALGORITHM
from Figure 6.17 that an offset with standard deviation s ∼ 0.01 is best for the 2D
Gaussian shells and from Figure 6.18 that an offset with s∼ 0.0 is best for the non-flat
cosmology with the complete data set. Therefore, we find that the value of the best
offset is roughly inversely proportional to the prior weight, s ∝ 1/α , from the best fit
network w∗. In the 2D Gaussian shells example α ' 5 and in the non-flat cosmology
example α ' 54000. Therefore, the non-flat cosmology will require a significantly
smaller offset to accurately reproduce the prediction error values. Since the absolute
error values are not used for determining the quality of a fit, but rather the relative
magnitudes, the fast calculations do not need to be too precise. We suggest using an
offset with a standard deviation of s. 1/α as this will yield usable error estimates.
An additional option has been included at this stage to not discard points whose
NN predictions fail error-checking; instead the original likelihood function will be
evaluated at these points. Discarding the points (assigning them log(L) =−∞) essen-
tially removes them from the prior and will only affect points further from the peak
where the networks are not all well trained. As such it will bias the evidence upwards
but should not generally affect parameter estimation as the discarded points will be
of significantly lower likelihood so as to be in the far tails of the distribution. These
points will also be heavily weighted towards early in the analysis before the likelihood
contours move well within the NNs’ region of confident predictions.
To observe the potential speed increases with this method of error estimation, we
re-analysed the cosmological models and data sets and then performed both the slow
and fast methods of error calculation in separate follow-up analyses on restricted prior
ranges. Our initial analyses were performed with the updated version of the NN train-
ing algorithm mentioned in Section 6.3.1. The follow-up analysis limited the prior to
xmax(log(L))±4σ , where σ is the vector of standard deviations of the marginalised one-
dimensional posterior probabilities as in the previous section. When training networks
for the fast error approximation, an offset with standard deviation s ≈ 1/α was used
for an ensemble of 20 total networks per saved network.
Table 6.10 shows the average time taken per log-likelihood function evaluation
when this follow-up was performed with MULTINEST and with BAMBI using both
the slow error calculation and the fast method. This timing includes any MULTINEST
operations as well as loading the saved networks in the beginning and preparing to use
them (it does not include COSMOMC data initialisation). In all cases where points
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6.4 Using Trained Networks for Follow-up in BAMBI
Method Model Data
tL (ms) tL (ms) factor factor
(discard) (discard)
MULTINEST
ΛCDM
CMB 2775 — — —
all 3813 — — —
ΛCDM+ΩK
CMB 12830 — — —
all 10980 — — —
BAMBI slow
ΛCDM
CMB 558.7 378.1 4.96 7.33
all 639.5 393.4 5.96 9.68
ΛCDM+ΩK
CMB 1823.8 97.67 7.03 131.4
all 1181.5 219.8 10.86 49.95
BAMBI fast
ΛCDM
CMB 3.883 0.2077 714.7 13360
all 28.01 0.2146 136.1 17770
ΛCDM+ΩK
CMB 1105 0.08449 11.61 151900
all 402.9 0.2032 27.25 54040
Table 6.10: Time per likelihood evaluation in a follow-up analysis and speed factors
with respect to MULTINEST. Average times without discarding bad predictions are
limited by the number of calls to the original likelihood function.
were not discarded, the BAMBI evidences matched the MULTINEST values to within
statistical error. When discarding points, only the non-flat model with CMB-only data
calculated an evidence that did not agree; this was due to this being the only case with
only a single trained NN. In the other three cases, two trained networks were available
to use. These three cases therefore required fewer points to be calculated with the
original likelihood or discarded (< 2% for nonflat model and < 0.2% for flat model)
but took additional time to load and find the best network prediction. The analysis
using the non-flat model and CMB-only data, that had only a single network, runs more
rapidly when points are discarded, but requires more likelihood samples and picks up
a small error in the evidence calculation of approximately one log-unit (compared to
an evidence uncertainty of approximately a tenth of a log-unit). When not discarding,
this model and data combination required∼ 10% of points to have their log-likelihood
computed with the original function.
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6. THE BAMBI ALGORITHM
Overall, the massive gains in speed from using the fast method along with discard-
ing points (that the networks cannot predict precisely enough) of O(104) to O(105)
make it worthwhile to use this as an initial follow-up procedure. Should there be too
many discarded points, the analysis may be performed again without discarding to
compute a more reliable evidence value but still obtaining a speed increase of O(10) to
O(100). We can thus obtain accurate posterior distributions and evidence calculations
orders of magnitude faster than originally possible. Follow-up without discarding is
limited by the number of calls to the original likelihood function while follow-up with
discarding is limited by the number of stored networks being used.
6.5 Summary
We have introduced and demonstrated a new algorithm for rapid Bayesian data anal-
ysis. The Blind Accelerated Multimodal Bayesian Inference algorithm combines the
sampling efficiency of MULTINEST with the predictive power of artificial neural net-
works to reduce significantly the running time for computationally expensive prob-
lems.
The first applications we demonstrated are toy examples that demonstrate the abil-
ity of the NN to learn complicated likelihood surfaces and produce accurate evidences
and posterior probability distributions. The eggbox, Gaussian shells, and Rosenbrock
functions each present difficulties for Monte Carlo sampling as well as for the train-
ing of a NN. With the use of enough hidden-layer nodes and training points, we have
demonstrated that a NN can learn to accurately predict log-likelihood function values.
We then apply BAMBI to the problem of cosmological parameter estimation and
model selection. We performed this using flat and non-flat cosmological models and
incorporating only CMB data and using a more extensive data set. In all cases, the
NN is able to learn the likelihood function to sufficient accuracy after training on early
nested samples and then predict values thereafter. By calculating a significant fraction
of the likelihood values with the NN instead of the full function, we are able to reduce
the running time by a factor of up to 1.50 and potentially more. This is in comparison to
use of MULTINEST only, which already provides significant speed-ups in comparison
to traditional MCMC methods [102].
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6.5 Summary
Through all of these examples we have shown the capability of BAMBI to increase
the speed at which Bayesian inference can be done. This is a fully general method and
one need only change the settings for MULTINEST and the network training in order
to apply it to different likelihood functions. For computationally expensive likelihood
functions, the network training takes less time than is required to sample enough train-
ing points and sampling a point using the network is extremely rapid as it is a simple
analytic function. Therefore, the main computational expense of BAMBI is calculat-
ing training points as the sampling evolves until the network is able to reproduce the
likelihood accurately enough.
With the trained NNs, we can perform additional analyses using the same likeli-
hood function but different priors and save large amounts of time in sampling points
with the original likelihood and in training a NN. Follow-up analyses using already
trained NNs provide much larger speed increases, with factors of 4 to 11 obtained
for cosmological parameter estimation relative to BAMBI speeds when not discarding
poorly estimated points, and 7 to 130 when discarding using the slow exact method.
Having trained an ensemble of networks to provide error estimates, the speed-ups pos-
sible increase drastically to potentially O(102) times faster when not discarding and
O(104) to O(105) times faster when discarding. The limiting factor in these runs is
the fraction of points early in the analysis that the NNs are unable to make sufficiently
accurate predictions for and thus require the use of the original likelihood function or
re-sampling. The calculation of the error of predictions is a flat cost based on the size
of the NN and data set regardless of the original likelihood function, so the more com-
putationally expensive the original likelihood function is the more benefit is gained
from using NNs.
The NNs trained by BAMBI for cosmology cover a larger range of log-likelihoods
than the one trained for COSMONET. This allows us to use a wider range of priors
for subsequent analysis and not be limited to the four-sigma region around the max-
imum likelihood point. By setting the tolerance for BAMBI’s NNs to a larger value,
fewer NNs with larger likelihood ranges can be trained, albeit with larger errors on
the predictions. Allowing for larger priors requires us to test the validity of our NNs’
approximations, which ends up slowing the overall analysis when no networks can
provide a precise enough prediction.
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6. THE BAMBI ALGORITHM
Since BAMBI uses a NN to calculate the likelihood at later times in the analysis
where we typically also suffer from lower sampling efficiency (harder to find a new
point with higher likelihood than most recent point removed), we are more easily able
to implement Hamiltonian Monte Carlo [202, 203] for finding a proposed sample.
This method uses gradient information to make better proposals for the next point that
should be sampled. Calculating the gradient is usually a difficult task, but with the NN
approximation they are very fast and simple. This improvement may be investigated
in future work.
As larger data sets and more complicated models are used in cosmology, particle
physics, and other fields, the computational cost of Bayesian inference will increase.
The BAMBI algorithm can, without any pre-processing, significantly reduce the re-
quired running time for these inference problems. In addition to providing accurate
posterior probability distributions and evidence calculations, the user also obtains NNs
trained to produce likelihood values near the peak(s) of the distribution that can be
used in extremely rapid follow-up analysis.
164
Chapter 7
Conclusions and Future Work
Somehere, something incredible is waiting to
be known.
Carl Sagan
7.1 Summary of Results
Einstein’s general theory of relativity predicts the existence of gravitational waves,
which are perturbations to the curvature of spacetime caused by the acceleration of
matter and propagate at the speed of light. The periodic strain signal they carry affects
measured distances between points in space, enabling their passing to be detected by
laser interferometers. These signals originate in the regimes of strongest gravity, al-
lowing for one of the first tests of GR where the full non-linear equations must be
used.
Binary star systems emit gravitational radiation and thus lose energy and fall in
toward one another in a characteristic inspiral, generating a “chirp” signal observable
by ground-based detectors such as LIGO and Virgo. By approximating the evolution
of these systems we can obtain waveform functions that model the signal we expect to
observe. Using the tools of Bayesian inference, we are then able to measure probability
distributions for source parameters given these waveform models and detector data.
In Chapter 2 we consider these binary systems and test the posterior probability
distributions given by the MULTINEST algorithm. We then use these posteriors to
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7. CONCLUSIONS AND FUTURE WORK
quantitatively analyse the ability of a network of ground-based detectors to localise
the position of a source on the sky. We do this for a sample of injected waveforms
into simulated noise from which we can infer the area of sky that will need to be
observed in electromagnetic follow-ups in order to find a counterpart signal. We then
use Bayesian model selection criteria for ruling out incoherent signals in LIGO data.
This is performed in the context of the “big dog” blind hardware injection, where
the coherent signal model was shown to be favoured over the incoherent; this same
statistic ruled out manufactured incoherent signals, displaying its ability to differentiate
between real gravitational waves and detector glitches.
Further advances in gravitational wave detection will be realised by the launch of
a space-based laser interferometer (formerly LISA and now NGO). This will allow for
the detection of GWs at much lower frequencies and with much higher SNRs. Simu-
lated data was generated in the MLDCs in order to encourage the development of data
analysis code for LISA and demonstrate the community’s ability to extract scientific
information from the kind of data that was expected. In Chapter 3 I analyse some of
this data, initially looking for burst signals in long stretches of instrument noise. We
are able to use Bayesian criteria to detect and characterise signals and demonstrate the
ability to perform model selection to determine the correct injected signal waveform.
A similar analysis is then performed on LISA data containing a large number of con-
tinuous sources. Despite the confusion, we are able to accurately find many signals at
once, with near-constant purity across a large range of frequencies and high complete-
ness at higher frequencies where there are fewer signals present. Lastly we combine
these two in order to attempt to detect burst signals in the presence of many continuous
sources. Again Bayesian criteria prove to be powerful in separating true from false
detections.
As more detailed signal models and larger data sets are analysed, performing Bayesian
inference will become a more cumbersome task. To assist in simplifying our data anal-
ysis procedures, I investigated the application of the MOPED (Multiple Optimised
Parameter Estimation and Data compression) algorithm. I found, as detailed in Chap-
ter 4, that in some cases the algorithm will work, but as the analysis involves more
parameters, there is the potential for additional modes in the likelihood to be falsely
created, each with equal likelihood to that of the peak. A solution is tested, but the
total overhead is found to not warrant further implementation.
166
7.1 Summary of Results
An ever-increasingly popular tool for handling large computations is artificial neu-
ral networks. These mimic the structure of a brain by passing information through a
set of interconnected nodes with the ability to “learn” a function by adjusting connec-
tion weights. In Chapter 5 I consider the application of feed-forward networks, where
nodes are arranged in ordered layers. These can be trained on a pre-calculated set of
data points from the function to be learned, which can either involve regression or clas-
sification. We implement a Bayesian optimisation algorithm that uses second-order in-
formation on the log-posterior function in order to find the optimal set of weights in as
few steps as possible. We also prevent over-fitting to the precise training data set pro-
vided by using regularisation and a validation data set. The capabilities of the network
are first demonstrated on a few toy problems. The network training algorithm is then
applied to solve two larger and more complex examples. In the first, we identify hand-
written digits from the MNIST database, which involves finding patterns in images to
make ideal classifications. In the second, we measure the ellipticities of sheared and
convolved galaxies, a difficult task addressed by the Mapping Dark Matter challenge.
As its name suggests, if this procedure can be done accurately and rapidly it may be
applied to large surveys of galaxies and can help in mapping dark matter distributions
throughout the universe and testing cosmological models. Our neural network training
algorithm is able to learn both of these data sets very well; prediction error on a further
sample of handwritten digits produces an error rate of less than 2% and the error in
galaxy ellipticity measurements in reduced by a factor of ∼ 5 compared to previous
standard methods.
In Chapter 6 I look to applying neural networks directly to Bayesian inference. The
Blind Accelerated Multimodal Bayesian Inference (BAMBI) algorithm is described,
wherein samples of the log-likelihood function from MULTINEST are used to train a
neural network. This can then be used in place of the original log-likelihood once it
is able to make sufficiently accurate predictions. I first test BAMBI on a few toy like-
lihoods to ensure that the neural network is able to learn the function from a reason-
able number of samples before MULTINEST’s sampling converges. We then address
the task of performing cosmological parameter estimation. This is a problem with a
well-defined likelihood function and common data sets through the CosmoMC soft-
ware package. Likelihood function calls normally take on the order of seconds each
167
7. CONCLUSIONS AND FUTURE WORK
so analysis may take many CPU days. With BAMBI, however, we are able to demon-
strate speed increases of 50%, indicating that ∼ 33% of likelihood samples were per-
formed using a trained neural network. Even more impressive gains in speed come
from follow-up analyses with different priors. We can re-use the trained NNs, thereby
avoiding the original likelihood function entirely. Doing so with a rapid method of
calculating the error in NN predictions yields posterior inferences O(104) to O(105)
times faster than using only MULTINEST. BAMBI’s ability to decrease the computa-
tional expense of initial Bayesian inference and perform incredibly rapid follow-up in
a fully general way means it can be applied to all types of parameter estimation and
model selection problems.
7.2 Avenues for Future Work
Analysis of data from ground-based GW detectors in the advanced era will need to
utilise more advanced waveform families. Inspirals that include the full effects of spin
in the phase and amplitude corrections will provide the best measurements of compact
binary signals we hope to detect. Additionally, the inclusion of merger and ringdown
stages to create a hybrid waveform should provide the most information about a GW
source. Creating these waveform models and analysing their different parameter es-
timation biases is important before and during the advanced detector era where de-
tections are expected. Model selection will prove to be important when determining
which waveform type best fits an observed signal and for filtering out detector glitches.
As the LISA mission has been reconfigured to NGO, we must reconsider the sci-
entific potential of the new detector. Data analysis pipelines will need to work even
harder as there will be lower SNR for NGO-observed signals than for LISA.
Neural networks and the BAMBI algorithm can play a large role in these projects.
NNs can be applied to computing very difficult gravitational waveforms in order to
more rapidly model systems. BAMBI will greatly increase the speed at which Bayesian
analysis may be performed on GW signals, providing sky location and other parameter
estimates with significantly less lag time from the initial trigger. We are always look-
ing to improve our NN training and BAMBI and will continue to develop these two
algorithms to add more functionality and improve the quality and speed of results.
168
7.2 Avenues for Future Work
There is an optimistic future for the search for gravitational waves and the scien-
tific potential they hold. With the right tools we can efficiently and rapidly analyse
detector data to find and characterise any signals present. These tools are not limited
to just gravitational wave astronomy, but may be applied to solve computational or data
analysis problems in a wide number of fields.
169
7. CONCLUSIONS AND FUTURE WORK
170
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