May 11, 2012Preprint typeset using LATEX style emulateapj v. 12/16/11

INTERPOLATION TECHNIQUES FOR MCMC PARAMETER ESTIMATION ON COMPACT BINARYCOALESCENCE GRAVITATIONAL-WAVE SIGNALS

Daniel J. StevensWeinberg College of Arts and Sciences, Northwestern University, Evanston IL

May 11, 2012

ABSTRACT

With gravitational-wave detection on the horizon, astronomers look for ways of extracting usefulinformation from a detected gravitational wave. Like its electromagnetic cousin, a gravitational wavecarries important information about the characteristics of its source, and these characteristics can berecovered through numerical analysis. Using one promising technique known as a Metropolis-HastingsMarkov Chain Monte Carlo (MCMC) simulation, astronomers can produce a probability distributionover an entire parameter space describing gravitational wave signals; given the gravitational wavedata, the MCMC produces a sequence of parameter samples whose distribution converges to theprobability density on parameter space implied by the data. Although an MCMC simulation willproduce the equilibrium probability distribution in an infinite amount of time, a simulation that runsfor a finite amount of time may not. This work focuses on using a kD-tree sorting structure toimprove MCMC sampling. We show that a simple sampling method effectively recovers an accurateprobability distribution in two dimensions but performs worse than a non-interpolated run in ninedimensions. We explain how dimensionality issues and correlations between the nine parameters –which are not taken into account by the simple sampling method – can cause the simple samplingmethod to yield inaccurate distributions, and we compare these results to those from an interpolatedMCMC simulation with a more sophisticated sampling method which takes correlation into account.Improving the convergence of an MCMC simulation through interpolation would allow for faster, morefrequent analysis of gravitational wave signals as well as higher confidence in recovered probabilitydistributions.

1. INTRODUCTION

1.1. Gravitational Waves and Compact BinaryInspirals

The detection and analysis of gravitational waves isa top priority for the astrophysical community. A pre-diction of Einstein’s theory of general relativity, grav-itational waves are ripples in space-time generated bythe acceleration of massive objects. Of particular inter-est are gravitational waves created by the coalescenceof a pair of compact objects, particularly neutron star-neutron star (NS-NS), neutron star-black hole (NS-BH)and black hole-black hole (BH-BH) pairs. In such a bi-nary system, the compact objects lose orbital energy inthe form of gravitational waves, and the objects spiralinwards (Peters & Mathews 1963). The amplitude andfrequency of the signal increase as the objects move closerto each other, culminating in a strong chirp in the sig-nal as the objects coalesce (Creighton 2003). Figure 1shows a simulated inspiral waveform with the character-istic chirp at the end.

A gravitational wave from a compact binary inspiralcan provide significant information about the wave’s pro-genitor. Using Bayesian inference techniques, we canrecover the mass ratio of the compact objects; the or-bital phase, polarization, inclination, declination andright-ascension of the system; the mass of the inspiral-ing object when the system emits the gravitational-wavechirp (chirp mass) the distance to the system; the timeat which the system coalesced; and the projection ofeach object’s spin along each of the three spatial dimen-

dan.stevens@u.northwestern.edu

Figure 1. Simulated gravitational wave from compact binary in-spiral. The frequency and amplitude increase over time as theobjects grow closer to each other; the objects eventually coalesce,resulting in the ”chirp” seen at t = 1 sec (Stuver 2012).

sions. Precise recovery of these parameters can help usplace constrains on models of binary population synthesis(Abadie et al. 2010). Moreover, knowing the sky loca-tion of a binary merger from the detected gravitationalwave can allow for detection of the merger’s electromag-netic radiation; This information can be used to calculatecosmological distances (Bloom et al. 2009); in the spe-cific case of a NS-NS binary merger, the waveforms aresimple enough to yield precise distance measurements(Creighton 2003).

The Laser Interferometer Gravitational-wave Observa-tory (LIGO) will be sensitive to gravitational waves fromthese and other sources. LIGO uses three laser interfer-ometers – two in Hanford, Washington and a third inLivingston, Louisiana – to detect the waves. As a gravi-tational wave propagates through the detector, one 4kmarm of an interferometer will expand while the other con-tracts, and the difference between the two arms revealsthe form of the gravitational wave. Figure 2 shows the

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Figure 2. Detector signal sensitivity as a function of frequencyfor the first five completed LIGO test runs. The solid black lineis the goal curve – the target sensitivity for LIGO before upgrades(LIGO Scientific Collaboration, 2012).

strain sensitivity in the LIGO detectors as a function offrequency in the past five runs.

Ongoing upgrades to LIGO, called Advanced LIGO,will improve the sensitivity of the gravitational-wavedetectors. Advanced LIGO is predicted to have afactor of ten increase in sensitivity to most signals,yielding a factor of 1,000 increase in the volume ofdetectable sources (Creighton 2003). In partnershipwith the Virgo gravitational-wave detector in Italy,Advanced LIGO should detect an estimated 40 NS-NSinspiral events per year (Abadie et al. 2010) as well as13 NS-BH and 500 BH-BH events per year (Creighton2003). Such a large number of predicted detectableevents makes the development and implementation ofefficient parameter estimation methods a critical priority.

1.2. Markov Chain Monte CarloParameter-Estimation Simulations

We are interested in estimating the parameters ofa gravitational-wave emitting compact binary inspiral.The set of all mathematically-possible combinations ofparameters form a parameter space. This space can haveseveral dimensions, depending on the physical nature ofthe wave’s source (Mandel 2010). For our analysis, thispaper focuses on non-spinning binary inspirals, resultingin a nine-dimensional space of parameters as discussedin the previous section. To estimate the parameters ofa compact binary inspiral, we have developed a MarkovChain Monte Carlo (MCMC) algorithm to find the poste-rior probability-density function (PDF) of a source’s pos-sible parameters. The MCMC algorithm uses Bayesianinference to estimate, given data d from a gravitational-wave detector, the PDF over the entire parameter space.For our data, we inject a simulated waveform into sim-ulated detector noise. We Fourier transform the data toexamine it as a function of frequency rather than time,as discussed in van der Sluys et al. (2008). From Bayes’theorem, the posterior for a proposed set of parameters~λ given the waveform data d is given by

p(~λ|d) =p(~λ)L(d|~λ)

p(d). (1)

p(~λ) is the prior probability distribution of the param-eters – our initial guess for the relative probabilities of

each set of parameters. p(d) is the evidence, and L(d|~λ) isthe likelihood, or the conditional probability of the detec-tor data given the set of parameters. We can constrainour prior based on the most extreme sources to whichAdvanced LIGO will be sensitive as well as physical con-straints. For example, we can assign zero probability tonegative values of the chirp mass, and we can also as-sume a uniform distribution on the sky location param-eters; the latter simplification follows from the assump-tion that the distribution of gravitational-wave sourcesis isotropic (Mandel 2010). The evidence can be ignoredin our calculations; p(d) is constant, being a property ofthe model, and the term will drop out as we consider theratio of posteriors of two parameter sets (Mandel 2010).Thus, we can re-express Equation (1) as

p(~λ|d) ∝ p(~λ)L(d|~λ). (2)

The likelihood for a given detector follows from the as-sumption that the detector noise is stationary and Gaus-sian, with RMS magnitude given by S(f) (see Figure 1.2)at frequency f (van der Sluys et al. 2008). Given a data

set d(f) and waveform model m(~λ, f), is

L(d|~λ) ∝ exp

(−2

∫ ∞0

| d(f)− m(~λ, f) |2

Sn(f)df

). (3)

We need not worry about the constant of proportionalityin (3), since we will be considering ratios of likelihoods.Since we assume that the noise in a given detector isindependent from the noise in other detectors, Equation(4) becomes

p(~λ|d) ∝ p(~λ)

N∏i=1

Li(d|~λ), (4)

where N is the number of detectors (van der Sluys et al.2008). For this paper, we consider data from the twoHanford detectors and the Livingston detector, so N = 3.

The MCMC algorithm samples the posterior distribu-tion as follows: the MCMC starts with a set of parame-

ters, ~λi, in the parameter space. The algorithm proposes

a new set of new parameters ~λi+1 via a jump proposal– some prescribed method for exploring the parameterspace which we are free to choose, so long as it can pro-pose points anywhere in the parameter space (Metropo-lis et al. 1953). The MCMC calculates the probabilities

P (~λi → ~λi+1) and P (~λi+1 → ~λi) of jumping from one setof parameters to the other, also known as the jump prob-

abilities; P (~λi → ~λi+1) is the forward jump probability,

and P (~λi+1 → ~λi) is the backward jump probability. Tosample the posterior PDF, the jump probabilities mustsatisfy detailed balance (Press et al. 2007):

p(~λi)P (~λi → ~λi+1) = p(~λi+1)P (~λi+1 → ~λi). (5)

Once the jump probabilities and posteriors have beencalculated, the algorithm then calculates the probabilityPacc of accepting the proposed parameters, called theacceptance probability:

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Pacc =p(~λi+1)P (~λi+1 → ~λi)

p(~λi)P (~λi → ~λi+1)(6)

The proposed parameters are accepted if the accep-tance probability is larger than some random number kbetween zero and one, with the random number beinggenerated anew every iteration. If the proposed param-eters are accepted, then the points are recorded and be-come the current parameters for the next iteration; oth-erwise, the original parameters are repeated in the chain,

i.e. ~λi+1 = ~λi. In this way, the current state dependsonly on the previous state, which gives the algorithmits Markovian nature. These accepted parameters givean estimate of the equilibrium posterior PDF over theparameter space; our MCMC code records the currentparameters every 100 iterations. The recovered posteriorPDF will eventually converge to the true equilibrium dis-tribution if the jump proposals satisfy detailed balance(Metropolis et al. 1953).

1.3. MCMC Convergence Issues

This guaranteed convergence theorem, however, doesnot guarantee convergence in finite time. The MCMC al-gorithm, while being able to sample the entire parameterspace, will preferentially sample high-likelihood regions– regions of the parameter space in which the posteriorPDF has local maxima. The parameter space for gravita-tional wave signals is often multi-modal, containing sev-eral separated local likelihood maxima. For an MCMCsimulation that runs for a finite duration, it is possiblefor the simulation to spend a disproportionate amount oftime sampling one maximum while under-sampling an-other maximum; in this case, the recovered posterior dis-tribution will not be a reasonable approximation to thetrue posterior distribution, as the MCMC has not con-verged.

We use parallel tempering to try to sample the entireparameter space while still exploring regions of high like-lihood in detail. For parallel tempering, several MCMCchains run simultaneously, each with a different temper-ature T . For each chain, the acceptance probability be-comes

Pacc =

(p(~λi+1)P (~λi+1 → ~λi)

p(~λi)P (~λi → ~λi+1)

)1/T

. (7)

In this way, higher-temperature chains still preferjumps to high-likelihood regions, but these chains aremore likely to jump around the entire parameter spaceand explore different modes in the probability distribu-tion (van der Sluys et al. 2008). By swapping param-eters between chains, we allow the simulation to bothexplore wide regions of the parameter space as well asthe high-likelihood regions. We swap between chains oftemperatures Tm and Tn, Tm < Tn, whenever(

Ln

Lm

) 1Tm− 1

Tn

> k, (8)

where k is again a random number between zero and one(van der Sluys et al. 2008). The lowest temperature isset to Tmin = 1, and this chains acceptance probability

becomes Equation (6). The remaining temperature val-ues depend on the signal-to-noise ratio (SNR); a higherSNR leads to a higher maximum temperature (van derSluys et al. 2008). The number of chains is a compro-mise; fewer chains improves MCMC simulation compu-tation speed, whereas more chains yield more efficienttemperature swaps (van der Sluys et al. 2008).

Parallel tempering, however, is computationally ex-pensive. Each temperature chain in a parallel temper-ing setup is an individual MCMC; our code runs eightchains, which considerably increases the hardware andtime needed to produce results. A less-expensive method,which is the focus of this paper, is to interpolate be-tween the MCMC simulation data. This method pro-poses jumps based on accepted points from a previousor ongoing MCMC run. Two variations of this approachare discussed below.

2. INTERPOLATIVE MCMC JUMP PROPOSALS

2.1. kD-Tree Jump Proposal

2.1.1. Overview

The first attempt at interpolating the data from anMCMC simulation uses a jump proposal based on a kD-tree. A kD-tree is a sorting algorithm that partitionsan entire k-dimensional space into cells (or leaves). Inthe first step, the algorithm places a set of k-dimensionaldata points into a single cell that contains the entire pa-rameter space. The algorithm then further partitions thespace according to the following recursion (Farr & Man-del 2011):

1. If the given cell contains a single point,stop.

2. If the given cell contains more than onepoint, partition the current cell into two sub-cells by choosing a dimension x and dividingthe points in half along this dimension. Inthis way, every point on one side of this di-vision has a smaller x-coordinate than everypoint on the other side of the division.

3. Repeat the previous two steps for eachsub-cell and store the resulting kD-trees assub-trees of the current cell.

See Figure 3 for a two-dimensional illustration of thiskD-tree sorting method.

To use kD-tree interpolation as the basis for a jumpproposal, we store accepted parameters from a previousMCMC, or from the past history of an ongoing MCMC,and sort them into a kD-tree. To select parameters, thealgorithm uniformly chooses a set of parameters fromthis sorted data set. The proposal then finds the largestcell containing this point and at most N total points.The choice of N is a compromise: smaller N allows usto explore the local structure of the posterior with bet-ter precision, whereas larger N enables us to smooth outstatistical fluctuations. For our simulations, we set N= 64 (Weinberg 2009). From within this cell, the pro-

posal selects a set of proposed parameters ~λi+1 by gen-erating a point from anywhere within this cells bounds.The MCMC performs this search to establish the ini-

tial parameters ~λ0, and then the jump proposal uses this

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Figure 3. A kD-tree generated around a set of two-dimensionalpoints, distributed normally about the origin (Farr & Mandel2011).

method to select proposed parameters at every iterationof the simulation. For this proposal, the forward jumpprobability is

P (~λi → ~λi+1) =Ncell

NV, (9)

where Ncell is the total number of points in the tree (i.e.the total number of points from the original run) and V

is the volume of the cell containing ~λi+1.The backward jump probability is calculated by finding

the cell whose bounds would contain ~λi. This probabilityis

P (~λi+1 → ~λi) =N ′cellNV ′

, (10)

where N ′cell is the number of previously-accepted points

which lie within the bounds of the cell containing ~λi andV ′ is the volume of that cell. The acceptance probabilityis still given by Equation (6), but accepted parametersare not added to the tree; they are simply recorded andbecome the current parameters for the next iteration.

This interpolated proposal can still sample the en-tire parameter space, but should better explore high-likelihood regions, as most of the original points shouldbe in high-likelihood region. Moreover, this proposalstill satisfies detailed balance and retains the Markovianproperty.

For this jump proposal, we inject a simple waveformthat simulates a non-spinning compact binary inspiral.The non-interpolating MCMC produces a posterior thatconverges quickly to the true posterior, making it idealfor testing the convergence of the first kD-tree jump pro-posal.

2.1.2. Two-Dimensional Test Simulation and Results

We first tested the kD-tree jump proposal in two-dimensional space, covering the chirp mass (mc) andthe mass ratio (eta), but by limiting the MCMC to es-timating only two parameters, we improve the compu-tational speed of the MCMC. We ran one MCMC simu-lation without the kD-tree jump proposal and took the

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Figure 4. Mass ratio posterior PDFs for the original (blue) andinterpolated (yellow) MCMC simulations.

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Figure 5. Chirp mass posterior PDFs for the original (blue) andinterpolated (yellow) MCMC simulations.

data from the lowest-temperature chain; we then sortedthese parameters it into a two-dimensional kD-tree forthe interpolating simulation, which ran for a comparablenumber of iterations. Figure 4 shows the mass ratio pos-terior PDFs for the 2D original and interpolated simula-tions, while Figure 5 shows the corresponding chirp massposterior PDFs. For both the mass ratio and the chirpmass, the posterior recovered with the interpolated jumpproposal appears to have the same shape as the poste-rior recovered with the original jump proposal. However,for both parameters, the posterior appears to be shiftedto the right. By inspection, the interpolated posteriorsmatch the original posteriors closely enough to warranta more rigorous run.

2.1.3. Full Non-spinning Simulation and Results

After the two-dimensional simulation, we tested thekD-tree jump proposal in a full nine-dimensional sim-ulation. Using the same injected waveform as be-fore, we took the accepted parameters from thelowest-temperature chain of a non-interpolating, nine-dimensional run, sorted them into a kD-tree, and ranthe MCMC using only the kD-tree jump proposal. Fig-ure 6 shows compares the recovered mass ratio and chirpmass posteriors for the original and interpolated runs.In higher dimensions, the kD-tree jump proposal yieldsconsiderably imprecise posteriors. From Figure 6, thekD-tree proposal recovers maxima for values of the chirpmass and mass ratio which are significantly differentfrom the values which yield local maxima in the non-interpolated runs.

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Figure 6. Posterior PDFs for the chirp mass (top) and massratio (bottom). The kD-tree proposal (yellow) fails to sample thelocal maxima accurately in a nine-dimensional simulation, despitemoderate success in the previous two-dimensional simulation.

One possible cause for the inaccurate sampling is thesheer number of points required to fill in a reasonablesubdivision of the parameter space. To subdivide each di-mension of a d-dimensional parameter space about once,the approximate number of points N needed is given bythe relation

d ∼ log2(N). (11)

From Equation (11), the kD-tree should have at least4 = 22 points to subdivide the two-dimensional param-eter space once along each dimension; the kD-tree willsplit the points along one dimension such that two pointslie on either side of the split, and then the tree will sub-divide these cells by splitting along the second dimensionsuch that exactly one point lies in each bottom-level cell.Similarly, a single partitioning of the nine-dimensionalparameter space along each dimension requires that thekD-tree contain about 29 = 512 points. Similarly, theapproximate number of points needed to subdivide theparameter space along each dimension n times is

nd ∼ log2(N). (12)

A single division along each dimension does not consti-tute a ”reasonable” subdivision of the parameter spacesunder investigation, however. Given that the posteriorPDFs are roughly a factor of 10 smaller in most dimen-sions, being bounded by physical constraints, the totalvolume of the PDF is ∼ 10−d of the total volume ofthe d-dimensional parameter space. In two-dimensionalspace, the PDF volume is 1

100 the total volume. Assum-ing that each division from a kD-tree cuts the currentcell in half, splitting along each of the two dimensions

yields cells that have(12

)2= 1

4 the volume of the top-

Figure 7. A non-interpolated point density cross-section in theinclination-distance (iota-dist) plane. The parameters are notaligned orthogonally with respect to the distance and inclinationaxes; hence, no matter how small we make the cells, the kD-treejump proposal will likely propose parameters away from the neigh-borhood of points.

level cell. Thus, to achieve a reasonable subdivision ofthe space, we would need to split along each dimensionthree times to yield cells whose volumes are roughly 1

64of the total volume. From Equation (12), this procedurewould require approximately N = 23·2 = 64 points.

Similarly, the total volume of the PDF in nine-dimensional space is 10−9 of the total volume. Creat-ing cells of comparable size would require approximatelythree splits along each dimension, yielding a relative cell

volume of(12

)3·9. This subdivision requires, from Equa-

tion (12), approximately 23·9 ≈ 134, 000, 000 points. Thecomputational cost of running an MCMC long enoughto acquire so many data points makes this kD-tree jumpproposal inefficient, if not impractical, for simulations inhigh-dimensional parameter spaces.

Additionally, neither the kD-tree nor the associatedjump proposal take into account orientation of the pointsin a given cell. Figure 7 shows the density of accepted pa-rameters in the inclination-distance plane from the non-interpolated simulation. This cross-section shows thatthese parameters do not align with the distance and iotacoordinate axes. However, the kD-tree cells do align withthe coordinate axes by construction. This can lead to thejump proposal proposing parameters which lie outside ofthe neighborhood of the sorted parameters, thereby inac-curately sampling the posterior along these dimensions.

2.2. Principal Component Cell Jump Proposal

2.2.1. Overview

A more sophisticated jump proposal built upon thekD-tree sorting structure is the Principal-component Cell(PCC) jump proposal. As in the aforementioned kD-treejump proposal, the PCC proposal will draw uniformlyfrom the existing points. Once a point is selected, theproposal finds the largest cell containing this point andcontaining no more than N total points; as with the pre-vious proposal, N = 64 for our simulations. The proposalthen shifts the cell, translating its center to the meanof all the points in the cell, and aligns the cell bound-aries along the principal axes of the cloud of points inthe cell. The principal axes are determined by calcu-

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-2 -1 1 2

-3

-2

-1

1

2

3

Figure 8. A two-dimensional illustration of the principal-component cell. The larger box aligned with the axes is the normalkD-tree cell containing the given points, while the tighter box isthe principal-component cell.

lating the eigenvectors of the covariance matrix of thecell’s points. The new point is chosen uniformly fromwithin the bounds of the principal-component cell, sub-ject to the restriction that it lie in the original cell as well.Hence, this jump proposal draws a tighter box around thepoints in the cell compared to the box from the previ-ous proposal (see Figure 8), which was aligned with thecoordinate axes.

The forward jump probability for this jump proposalis given by

P (~λi → ~λi+1) =Ncell

N(Vcell ∩ VPCC), (13)

where Ncell is the total number of points in the tree,

Vcell is the volume of the original cell containing ~λi+1

and VPCC is the volume of the principal-component cell

containing ~λi+1. From Equation (13), if the proposedparameters lie outside of the bounds of the original cell,then the forward jump probability vanishes.

The backward jump probability is computed in a sim-ilar fashion by finding the principal-component cell thatwould contain the current set of parameters;.

P (~λi+1 → ~λi) =N ′cell

N(V ′cell ∩ V ′PCC), (14)

where N ′cell is the number of points in the original cell

containing ~λi, V′cell is the volume of the original cell

containing ~λi and V ′PCC is the volume of the principal-

component cell containing ~λi. Note, however, that theprincipal-component cell need not contain the currentparameters; since the principal-component cells do notspan the entire parameter space, the current parametersmay lie in a region of the original cell that does not inter-sect the principal-component cell. In this case, the jumpproposal rejects the proposed step.

If the proposed parameters are accepted, they areadded to the tree, and the proposal updates the meanpoint and covariance matrix for each cell to which thenew point is added; the method by which these valuesare updated is detailed in the next section. In addition,each cell is flagged to have its eigenvectors re-computedif the cell is used in a future call of the PCC jump pro-posal. The eigenvectors are updated immediately oncethe cell is chosen.

This jump proposal, like the simpler kD-tree proposaldiscussed in Section 2.1, should better explore high-likelihood regions. Currently, the intersection of the PCCcell and the original cell volumes is not calculated incomputing the forward and backward jump probabili-ties. Instead, only the PCC cell volume is used for thecalculation, although the algorithm does verify that theproposed parameters lie inside the original cell. If theproposed parameters lie outside of the original cell, thenthe calculated proposal probability is incorrect and theproposed jump is rejected. Since the regions of VPCC

which lie outside of Vcell are likely a small fraction of thetotal volume, we do not expect this to be a significantconcern.

2.2.2. Markovian and Asymptotically-markovian JumpProposals

In the PCC jump proposal, accepted parameters areadded to the kD-tree, altering the structure of the treeas well as the means and covariance matrices for the cellsto which the new parameter set belongs. At first glance,this seems to violate the requirement that a jump pro-posal be Markovian – the requirement that the currentstate of the system depends only on the most-recent pre-vious state. However, a jump proposal need only beasymptotically Markovian to be a valid MCMC jumpproposal. If each change to a jump proposals state mod-ifies the proposal by successively smaller amounts, then,after sufficiently many iterations, the system will behavein a nearly-Markovian manner (ter Braak & Frugt 2008).

To see how this jump proposal is asymptoticallyMarkovian, examine the mean point and covariance ma-trix of a given cell. As we add a new point to a cell,the jump proposal updates the cells mean point ~µ andcovariance matrix S according to the recursive relations

~µi+1 = ~µi +(~λi+1 − ~µi)

i+ 1(15)

and

Smni+1 =

i

i+ 1Smni +

1

i+ 1(~λmi+1−~µm

i+1)(~λni+1−~µni ), (16)

where the superscripts indicate the corresponding entryin the matrix or vector. For the mean vector, as i→∞,the second term on the right-hand side of Equation (15)becomes arbitrarily small; hence, for sufficiently largei, ~µi+1 ≈ ~µi. For the covariance matrix, as i → ∞,the second term on the right-hand side of Equation (16)approaches zero, while the first term approaches Smn

i .Hence, changes to the PCC jump proposal state becomesuccessively smaller, so the PCC jump proposal is asymp-totically Markovian.

2.2.3. Non-spinning Simulation and Results

We tested the PCC jump proposal on a new nine-dimensional, non-spinning simulation. For comparison,we ran the simulation with the same waveform withoutthe PCC jump proposal. Comparison plots for the PCC-proposal simulation and the non-interpolated simulationare given in Figure 9. Not only does the PCC proposalperform much better in sampling the chirp mass andmass ratio posteriors’ maxima than the simpler kD-treejump proposal, but it also recovers peaks in the other

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parameters’ posteriors with considerable accuracy.

Figure 10 shows the cumulative parameter acceptanceratio (the number of parameters accepted over the totalnumber of proposed parameters) versus the number ofproposed parameters, in hundreds. The MCMC rejectsa significant amount of parameters proposed by the PCCjump proposal. We can tolerate such a low acceptance

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Figure 9. Plots of non-spinning inspiral simulations using theinterpolating PCC jump proposal (yellow) and non-interpolatingjump proposal (purple). These plots compare the recovered poste-riors for chirp mass (mchirp), mass ratio (eta), declination (dec),right ascension (ra), inclination (iota), orbital phase (phi orb), po-larization (psi) and distance (dist).

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Figure 10. The cumulative parameter acceptance ratio for thePCC jump proposal as a function of the number of steps (in hun-dreds).

ratio; the goal of the PCC jump proposal is to explorehigh-likelihood regions in detail, so we do not want tojump around the parameter space too frequently. Over-all, the principal-component cells seem to solve the high-dimensionality problem, at least in the nine-dimensionalparameter space of non-spinning inspirals.

3. CONCLUSIONS AND FUTURE WORK

From the trial MCMC runs, we note that the origi-nal kD-tree jump proposal is insufficient for interpolat-ing high-likelihood regions of a high-dimensional param-eter space. The kD-tree proposals inability to recoveran accurate posterior can be explained by the excep-tionally large number of points that need to be sortedinto the tree to fill in a reasonable subdivision of nine-dimensional parameter space. Moreover, the cells usedfor this proposal are not tight enough, leading to a lowerparameter acceptance rate as ”bad” parameters are pro-posed too frequently. The PCC jump proposal, on theother hand, more accurately recovers the posterior andexplores posterior maxima in nine-dimensional parame-ter space. This contrast is likely due to the PCC pro-posal drawing points from cells that are tighter and bet-ter aligned with the points in the cell as opposed tothe standard kD-tree cells. Additionally, the PCC jumpproposal’s ability to hop between modes in the nine-dimensional posterior is vastly improved over that of theoriginal kD-tree jump proposal.

The PCC jump proposal appears to work sufficientlywell for non-spinning inspirals. Pending satisfactory re-sults from additional testing and more robust statisti-cal analysis, the PCC proposal should be an efficientcomplement to, or replacement for, the computationally-expensive parallel tempering method currently employedby our group. Given the results in nine dimensions, thisproposal should be tested on inspirals in which one orboth of the compact objects spin. Estimating the spinparameters along with the other nine parameters yieldsa 15-dimensional parameter space. Posteriors in thisspace can be even more multi-modal, and the potentialfor high-dimensionality issues increases significantly.

Additionally, the kD-tree data structure itself is a pow-erful sorting method, and it could form the basis of other

data-analysis techniques. The kD-tree structure could beused for clustering analysis, nearest-neighbor analysis,and other geometric comparisons of MCMC simulationdata.

4. ACKNOWLEDGMENTS

First and foremost, I would like to thank ProfessorVicky Kalogera. Vicky welcomed me into the LIGOgroup at Northwestern while I was only a sophomore,giving me my first undergraduate research project as wellas funding to pursue it during the summer; that projectbecame the foundation for this senior thesis. Her men-torship as a teacher, a major adviser and a research ad-viser have shaped, in no small way, this thesis and mypost-undergraduate plans.

I would also like to extend my sincerest gratitude toWill Farr, who has played a significant part in the in-ception and development of this project. Will’s guidanceand encouragement have been invaluable throughout myundergraduate research career. This project was my firstreal exposure to numerical research in astrophysics, andWill’s patience and willingness to help me learn as I wentcannot be understated.

Additionally, I wish to thank the other members ofthe Northwestern LIGO group for welcoming me into thecollaboration and for their assistance with this project.Specifically, I want to thank Ilya Mandel for his helpin the beginning stages of the project and his contin-ued investment and insight throughout. I would also liketo thank Vivien Raymond for his assistance with dataanalysis following the nine-dimensional MCMC with thekD-tree jump proposal.

Finally, I must thank my family, friends and fellowundergraduate physics and astronomy researchers theirencouragement and support, both direct and indirect.

This project was funded by Vicky Kalogera during thesummer of 2010 and by a Northwestern University Sum-mer Undergraduate Research Grant during the summerof 2011.

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