Streaming motion in Leo I Separating signal from background
A statistical application in astronomy: Streaming motion in Leo I
Bodhisattva Sen
DPMMS, University of Cambridge, UK Columbia University, New York, USA [email protected]
ETH Zurich, Switzerland
31 May, 2012 Streaming motion in Leo I Separating signal from background Outline
1 Streaming motion in Leo I Modeling Threshold models
2 Separating signal from background Method Theory Streaming motion in Leo I Modeling Separating signal from background Threshold models
What is streaming motion? Local Group dwarf spheroidal (dSph) galaxies: small, dim
Is Leo I in equilibrium or tidally disrupted by Milky Way?
Such a disruption can give rise to streaming motion: the leading and trailing stars move away from the center of the main body of the perturbed system Streaming motion in Leo I Modeling Separating signal from background Threshold models
Data on 328 stars (Y , Σ): Line of sight velocity; std. dev. of meas. error (R, Θ): Projected position, orthogonal to line of sight Contaminated with foreground stars (in the line of sight)
Magnitude of streaming motion Likely to increase beyond a threshold radius Likely to be aligned with the major axis of the system Streaming motion in Leo I Modeling Separating signal from background Threshold models
Statistical questions Is streaming motion evident in Leo I?
If so, how can it be described and estimated?
To what extent can it be described by a threshold model?
Findings [Sen et al. (AoAS, 2009)]: The magnitude of streaming motion appears to be modest, (nearly) significant at 5% level
Appears consistent with a threshold model
Difficult to identify the threshold radius precisely Streaming motion in Leo I Modeling Separating signal from background Threshold models Outline
1 Streaming motion in Leo I Modeling Threshold models
2 Separating signal from background Method Theory Streaming motion in Leo I Modeling Separating signal from background Threshold models
2 Yi = ν(Ri , Θi ) + i + δi ; i ∼ N(0, σ ) independent of (R, Θ) 2 Measurement error: δi |Σi ∼ N(0, Σi )
Cosine Model: ν(r, θ) = ν + λ(r) cos θ; λ ↑
n 2 X {Yi − v − u(Ri ) cos Θi } (ν, ˆ λˆ ) = arg min v∈ ,u↑ σ2 + Σ2 R i=1 i 1 n 2 X ˆ 2 2 σˆ = [{Yi − νˆ − λ(Ri ) cos Θi } − Σi ] n i=1
λ measures the effect of streaming; λ ↑ cos θ determines the deviation from the major axis Streaming motion in Leo I Modeling Separating signal from background Threshold models
λˆ Simulated effect of streaming Streaming motion in Leo I Modeling Separating signal from background Threshold models
Is streaming evident? Test λ = 0 using log-likelihood ratio statistic No streaming: ν(r, θ) ≡ ν, a constant P-value ≈ 0.055 (using permutation test)
How much is streaming at radius r? Need point-wise confidence intervals for λ d Result: n1/3{λˆ(r) − λ(r)} → ηC. Need to estimate η – tricky!
Need ways to by-pass the estimation of η.
Likelihood ratio (LR) based test or bootstrap methods
LR is pivotal and can be used to construct CI for λ(r) (Banerjee and Wellner, AoS, 2001) Streaming motion in Leo I Modeling Separating signal from background Threshold models
Likelihood ratio based method Banerjee and Wellner, AoS, 2001
Test H0 : λ(r) = ξ0
n 2 n 2 X {Yi − v − u(Ri ) cos Θi } X {Yi − v − u(Ri ) cos Θi } ∆SSE(r, ξ0) = min − min v∈ ,u(r)=ξ ,u↑ σ2 + Σ2 v∈ ,u↑ σ2 + Σ2 R 0 i=1 i R i=1 i
d Under H0, ∆SSE(r, ξ0) → D
D does not contain any nuisance parameters
Invert this sequence of hypotheses tests (by varying ξ0) to get a CI for λ(r) Streaming motion in Leo I Modeling Separating signal from background Threshold models
Bootstrap methods Want to bootstrap n1/3{λˆ(r) − λ(r)}
Efron’s bootstrap fails
We claim that the bootstrap estimate does not have any weak limit (Sen et al., AoS, 2010)
Smoothed bootstrap works
LR Bootstrap 0.90 0.95 0.90 0.95 ˆ r0 L U CP L U CP L U L U λ 400 0 3.54 .901 0 3.86 .952 0 3.57 0 3.57 1.92 500 0.10 4.50 .882 0 5.02 .936 0 3.58 0 3.90 1.98 600 0.26 6.66 .827 0 7.30 .897 0 3.30 0 3.64 1.98 700 0.36 8.88 .913 0.05 9.56 .961 0 4.26 0 4.69 1.99 750 1.85 8.88 .906 1.37 9.56 .952 0.44 7.86 0 8.37 5.37 Streaming motion in Leo I Modeling Separating signal from background Threshold models Outline
1 Streaming motion in Leo I Modeling Threshold models
2 Separating signal from background Method Theory Streaming motion in Leo I Modeling Separating signal from background Threshold models Threshold models
Goal: To determine a “threshold” in the domain of the function where some “activity” takes place
Could either be a rapid change in the function value or a discontinuity
Find the threshold radius in the Leo I
( 0 0 ≤ r ≤ d , Model: ν(r, θ) = ν + λ(r) cos θ where λ(r) = 0 > 0 d0 < r ≤ 1.
Two approaches: change point versus split point Streaming motion in Leo I Modeling Separating signal from background Threshold models
Change point models ν(r, θ) = ν + βψ(r − ρ) cos θ 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
x x
ψ(x) = 1{x ≥ 0} ψ(x) = max(0, x) Streaming motion in Leo I Modeling Separating signal from background Threshold models
Change point models ν(r, θ) = ν + βψ(r − ρ) cos θ ρ is the change-point 2 Pn {Yi −v−βψ(Ri −r) cos Θi } Minimize: SSE(v, β, r) = i=1 2 2 σˆ +Σi
SSE(vˆ, β,ˆ r) − SSE(vˆ, β,ˆ ˆr) Streaming motion in Leo I Modeling Separating signal from background Threshold models
Split point model Do not assume specific form for λ(r); λ ↑
Find the closest stump to λ (under mis-specification)
R r 2 R τ 2 κ(b, r) = 0 λ (s)ds + r [λ(s) − b] ds
Population parameters: (β, γ) = arg min κ(b, r)
γ is the split point
Estimate λ by λˆ to estimate γ Streaming motion in Leo I Modeling Separating signal from background Threshold models
Rescaled version of the criterion function
Asymptotics for the split point: n1/3-rate of convergence? Streaming motion in Leo I Method Separating signal from background Theory Outline
1 Streaming motion in Leo I Modeling Threshold models
2 Separating signal from background Method Theory Streaming motion in Leo I Method Separating signal from background Theory Problem
Most astronomical data sets are polluted to some extent by foreground/background objects (“contaminants/noise”) that can be difficult to distinguish from objects of interest (“member/signal”)
Contaminants may have the same apparent magnitudes, colors, and even velocities as signal stars
How do you separate out the “signal” stars?
We develop an algorithm for evaluating membership (estimating parameters & probability of an object belonging to the signal population) Streaming motion in Leo I Method Separating signal from background Theory
Example Data on stars in nearby dwarf spheroidal (dSph) galaxies
Data: (X1i , X2i , V3i , σi , ΣMgi , ...) Velocity samples suffer from contamination by foreground Milky Way stars Streaming motion in Leo I Method Separating signal from background Theory Approach
Our method is based on the Expectation-Maximization (EM) algorithm
We assign parametric distributions to the observables; derived from the underlying physics in most cases
The EM algorithm provides estimates of the unknown parameters (mean velocity, velocity dispersion, etc.)
Also, probability of each star belonging to the signal population; see Walker et al. (2008) Streaming motion in Leo I Method Separating signal from background Theory Streaming motion in Leo I Method Separating signal from background Theory A toy example
Suppose N ∼ Poisson(b + s) is the number of stars observed
s = rate for observing a signal star
b = the foreground rate
Given N = n, we have W1,..., Wn ∼ fb,s where data n bfb(w)+sfs(w) {Wi = (X1i , X2i , V3i , σi )}i=1, and fb,s(w) = b+s
We assume that fb and fs are parameterized (modeled by the underlying physics) probability densities Streaming motion in Leo I Method Separating signal from background Theory
For galaxy stars The stellar density (number of stars per unit area) falls exponentially with radius, R The distribution of velocity given position is assumed to be 2 2 normal with mean µ and variance σ + σi
For foreground stars The density is uniform over the field of view
The distribution of velocities V3i is independent of position (X1i , X2i )
We adopt V3i from the Besancon´ Milky Way model (Robin et al. 2003), which specifies velocity distributions of Milky Way stars along a given line of sight Streaming motion in Leo I Method Separating signal from background Theory Outline
1 Streaming motion in Leo I Modeling Threshold models
2 Separating signal from background Method Theory Streaming motion in Leo I Method Separating signal from background Theory
Model Suppose N ∼ Poisson(b + s) is the number of stars observed s = rate for observing a signal star b = the foreground rate
Given N = n, we have W1,..., Wn ∼ fb,s where data n bfb(w)+sfs(w) {Wi = (X1i , X2i , V3i , σi , ΣMgi ,...)}i=1, and fb,s(w) = b+s
We assume that fb and fs are parameterized (modeled by the underlying physics) probability densities; β = (s, b, µ, σ2,...)
We would ideally like to maximize the likelihood
N Y bfb(Wi ) + sfs(Wi ) L(β) = (Difficult!) b + s i=1 Streaming motion in Leo I Method Separating signal from background Theory
The Likelihood
Let Yi be the indicator of a foreground star, i.e., Yi = 1 if the i’th star is a foreground star, and Yi = 0 otherwise
b Note that Yi ’s are i.i.d. Bernoulli( b+s )
Let Z = (W, Y, N) be the complete data [where W = (W1, W2,..., Wn) and Y = (Y1, Y2,..., Yn)]
The likelihood for the complete data can be written as
N N Yi 1−Yi (b + s) Y b fb(Wi ) s fs(Wi ) LC (β) = e−(b+s) N! b + s b + s i=1
Log-likelihood: lC (β) Streaming motion in Leo I Method Separating signal from background Theory Algorithm
Start with some initial estimates of the parameter β [in our simple example β = (s, b, µ, σ2,...)]
E-step: Evaluates the expectation of the log-likelihood given the observed data under the current estimates of the unknown parameters C Evaluate Q(β, βn) = E [l (β)|W, N] b βbn
M-step:
Maximizes the expectationQ (β, βbn) with respect to β
Iterate until the estimates stabilize (which is guaranteed!) Streaming motion in Leo I Method Separating signal from background Theory
The output of the algorithm
We obtain estimates of the unknown parameters βbn
Can also be used to give estimated probabilities that the i’th star is a signal
sˆˆf (W ) p (i) = Pˆ{Y = 0|Data} = s i mem i ˆ ˆˆ sˆfs(Wi ) + bfb(Wi ) These probabilities can later be used as weights
2 ˆ Pn {Yi −v−u(Ri ) cos Θi } Example: (ˆν, λ) = arg minv∈R,u↑ i=1 2 2 pmem(i) σ +Σi Streaming motion in Leo I Method Separating signal from background Theory
Summary Developed flexible methods to detect streaming motion in dSph
Methods rely on monotone function estimation
Change-point models versus split-point models
Foreground contamination addressed using a version of the EM algorithm for finite mixture models
Can get estimated probability that the i’th star is a signal, which can later be used as weights
Various variants of the EM algorithm can be used to fit more complex models
Thank you! Questions? Streaming motion in Leo I Method Separating signal from background Theory
References Banerjee, M. & Wellner, J. (2001). AoS, 29, 1699- 1731. Cule, M., et al. (2010). JRSS-B, 72, 545-600. Sen, B., et al. (2010). AoS, 38, 1953-1977. Sen, B., et al. (2009). AoAS, 3, 96-116. Walker, M., et al. (2008). Astronomical Journal, 137, 3109. Robin, A. C., et al. (2003). Astron. Astrophys., 409, 523. Streaming motion in Leo I Method Separating signal from background Theory Scenario I
Introduce a non-parametric component Velocity dispersion was assumed constant; now can model it as a function of projected radiusR Needs a tuning parameter to find σ(r) Streaming motion in Leo I Method Separating signal from background Theory Scenario II
Do not assume exponential density profile Assume that as you move from the center of the galaxy, the chance of observing a signal star decreases Streaming motion in Leo I Method Separating signal from background Theory A further extension [Cule et al. (JRSS-B, 2010)]
Pk Pk Mixture model f (x) = j=1 πj fj (x); j=1 πj = 1 p Model f1, f2,..., fk as log-concave densities on R Qn Maximize i=1{π1f1(Xi ) + π2f2(X1) + ... + πk fk (Xi )} over πi ’s and fi ’s log-concave No tuning parameter required – completely non-parametric