-
Constructing Probabilistic Lightcurve Templates for
BayesianObservation Scheduling Methods
Ana-Maria Staicu
Department of Statistics, North Carolina State
[email protected]
WGII: David Jones, Sujit Ghosh, AMS, Ashish Mahabal, Jogesh
Babu, JamesLong
May 9, 2017
1 / 28
-
I Periodic variable stars are very interesting phenomena in
Universe andhave attracted high interest
I Learning their type can inform knowledge about the
Universe
I The stars are observed only at few times; thus it is important
to schedulea future observation time that allows to correctly
recover the star’s class
2 / 28
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Eclipsing binaries
https://www.eso.org
3 / 28
https://www.eso.org
-
Example eclipsing binary training data
4 / 28
-
WGII subgroup 3 overview: aims and scope
Overall aim: schedule observations tomaximize correct lightcurve
classification
Method:
I Develop probabilistic templates asclass specific priors
I Choose observations that separateposterior fits under
different classes
Initial breakdown into manageableinvestigations:
I Using templates to find periods
I Given period, using templates to(further) separate classes,
andschedule new observations
CRTS data, Drake et al. 2014
5 / 28
-
Model framework
Notation: Yl(t) is the lightcurve for object l and t ∈ [0, 1].
Consider a finitebasis function {B1(t), . . . ,BJ(t)} in [0,
1].
I Bayesian hierarchical model for class-specific lightcurves
Yl(til) =J∑
j=1
αljBj(til) + σl�il
αl = (αl1, . . . , αlJ)T ∼ N(β,Σ)
β ∼ N(0, IJ); Σ ∼ Inv −Wishart(J, 0.25IJ)�il ∼ N(0, 1); σl ∼ Inv
− Gamma(3, 1)
I For new Y (·) use posterior distribution π(α|Y (·) ∈ class c)
to classify itI For class c, predict Ŷ c(·) := Ŷ c(·)|{Y (·) ∈
class c} based onπ(α|Y (·) ∈ class c) and the basis function
{Bj(t)}j
I Let c1, c2 be the most probable classes for Y (·). Criterion
for optimalfuture scheduling time that separates classes c1, c2
is
Tc1,c2 := argmaxt∈[0,1]|Ŷc1 (t)− Ŷ c2 (t)|
Optimal time := mode of the distribution of Tc1,c2 .
6 / 28
-
Model framework
Notation: Yl(t) is the lightcurve for object l and t ∈ [0, 1].
Consider a finitebasis function {B1(t), . . . ,BJ(t)} in [0,
1].
I Bayesian hierarchical model for class-specific lightcurves
Yl(til) =J∑
j=1
αljBj(til) + σl�il
αl = (αl1, . . . , αlJ)T ∼ N(β,Σ)
β ∼ N(0, IJ); Σ ∼ Inv −Wishart(J, 0.25IJ)�il ∼ N(0, 1); σl ∼ Inv
− Gamma(3, 1)
I For new Y (·) use posterior distribution π(α|Y (·) ∈ class c)
to classify it
I For class c, predict Ŷ c(·) := Ŷ c(·)|{Y (·) ∈ class c}
based onπ(α|Y (·) ∈ class c) and the basis function {Bj(t)}j
I Let c1, c2 be the most probable classes for Y (·). Criterion
for optimalfuture scheduling time that separates classes c1, c2
is
Tc1,c2 := argmaxt∈[0,1]|Ŷc1 (t)− Ŷ c2 (t)|
Optimal time := mode of the distribution of Tc1,c2 .
6 / 28
-
Model framework
Notation: Yl(t) is the lightcurve for object l and t ∈ [0, 1].
Consider a finitebasis function {B1(t), . . . ,BJ(t)} in [0,
1].
I Bayesian hierarchical model for class-specific lightcurves
Yl(til) =J∑
j=1
αljBj(til) + σl�il
αl = (αl1, . . . , αlJ)T ∼ N(β,Σ)
β ∼ N(0, IJ); Σ ∼ Inv −Wishart(J, 0.25IJ)�il ∼ N(0, 1); σl ∼ Inv
− Gamma(3, 1)
I For new Y (·) use posterior distribution π(α|Y (·) ∈ class c)
to classify itI For class c, predict Ŷ c(·) := Ŷ c(·)|{Y (·) ∈
class c} based onπ(α|Y (·) ∈ class c) and the basis function
{Bj(t)}j
I Let c1, c2 be the most probable classes for Y (·). Criterion
for optimalfuture scheduling time that separates classes c1, c2
is
Tc1,c2 := argmaxt∈[0,1]|Ŷc1 (t)− Ŷ c2 (t)|
Optimal time := mode of the distribution of Tc1,c2 .
6 / 28
-
Model framework
Notation: Yl(t) is the lightcurve for object l and t ∈ [0, 1].
Consider a finitebasis function {B1(t), . . . ,BJ(t)} in [0,
1].
I Bayesian hierarchical model for class-specific lightcurves
Yl(til) =J∑
j=1
αljBj(til) + σl�il
αl = (αl1, . . . , αlJ)T ∼ N(β,Σ)
β ∼ N(0, IJ); Σ ∼ Inv −Wishart(J, 0.25IJ)�il ∼ N(0, 1); σl ∼ Inv
− Gamma(3, 1)
I For new Y (·) use posterior distribution π(α|Y (·) ∈ class c)
to classify itI For class c, predict Ŷ c(·) := Ŷ c(·)|{Y (·) ∈
class c} based onπ(α|Y (·) ∈ class c) and the basis function
{Bj(t)}j
I Let c1, c2 be the most probable classes for Y (·). Criterion
for optimalfuture scheduling time that separates classes c1, c2
is
Tc1,c2 := argmaxt∈[0,1]|Ŷc1 (t)− Ŷ c2 (t)|
Optimal time := mode of the distribution of Tc1,c2 .
6 / 28
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User specified basis fns (cubic B-spline, J = 13)
0.0 0.5 1.0 1.5 2.0
0.5
0.0
−0.
5
100 draws of prior mean
Phase
Nor
mal
ized
mag
nitu
de
0.0 0.5 1.0 1.5 2.0
0.5
0.0
−0.
5100 draws from prior for coefficients
Phase
Nor
mal
ized
mag
nitu
de
0.0 0.5 1.0 1.5 2.0
0.5
0.0
−0.
5
Phase
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Phase
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EA class (B-splines)
0.0 0.5 1.0 1.5 2.0
0.5
0.0
−0.
5
100 draws of prior mean
Phase
Nor
mal
ized
mag
nitu
de
0.0 0.5 1.0 1.5 2.0
0.5
0.0
−0.
5
100 draws from prior for coefficients
Phase
Nor
mal
ized
mag
nitu
de
0.0 0.5 1.0 1.5 2.0
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−0.
5
Phase
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EW class (B-splines)
I Left plot: range of depths of smaller eclipse is a concern
since thetemplate assumes all lightcurves are essentially of the
same type
7 / 28
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Suppose we have templates / class priors e.g.
0.0 0.2 0.4 0.6 0.8 1.0
1.0
0.5
0.0
−0.
5−
1.0
Template medians
Phase
Nor
mal
ized
mag
nitu
de
EWEARRabRRcRRdRS_CVnLPV
8 / 28
-
New lightcurve with 50 observations (down-sampled)
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0.0 0.2 0.4 0.6 0.8 1.0
16.6
16.5
16.4
16.3
16.2
16.1
16.0
Original RRab class lightcurve
Phase
Mag
nitu
de
9 / 28
-
Posterior draws under each class (true class is RRab)
I Red: mean of prior for true class
I Grey: draws from prior of true class
I Blue: draws from posterior of class indicated in panel
title
10 / 28
-
Calculate initial posterior class probabilities (test data)
EW lightcurves:
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EW EA RRab RRc RRd RS_CVn LPV
0.0
0.2
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1.0
Class
Pos
terio
r pr
ob
RRc lightcurves:
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EW EA RRab RRc RRd RS_CVn LPV0.
00.
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81.
0
Class
Pos
terio
r pr
ob
11 / 28
-
Confusion matrix classification (test data)
I We can find a confusion matrix based on the assignments with
the highestposterior probabilities
ClassifiedTrue class EW EA RRab RRc RRd RS CVn LPV
EW 81 8 0 0 0 11 0EA 4 88 0 0 0 8 0
RRab 1 0 82 0 3 13 1RRc 0 0 3 34 40 18 5RRd 0 0 3 6 44 31 16
RS CVn 1 3 0 0 5 72 19LPV 0 1 0 1 23 50 25
12 / 28
-
Maximum separation distribution (RRc)
I Left plot: posterior draws of fit for single lightcurveI Green
lines: times of maximum separation between posterior draws of
fits
under RRc (red) and EA (blue)I Mode of this distribution of
maximum separation times – to schedule a new
observation
I Right plot: RRc class-wide distribution of optimal observation
time fordistinguishing from EA class based on many lightcurves
I Indicates possible general strategy for distinguishing these
two classes
13 / 28
-
Scheduling criterion
The green lines on the previous slides are the posterior draws k
= 1, . . . , 100 ofthe maximum separation of means
T (k)c1,c2 = argmaxt∈[0,1]
∣∣∣Ŷ c1,(k)(t)− Ŷ c2,(k)(t)∣∣∣Scheduling methods: identify the
two most probable classes. Consider thefollowing competitive
approaches to select t̂c1,c2
1. the mode of the distribution Tc1,c2
2. random value from the variable Tc1,c2
3. using the results from a class-class comparison from a
previous study, ifavailable (e.g. using the results displayed in
the previous fig, right plot)
4. random value from Uniform(0, 1) - näıve approach
14 / 28
-
New observations (RRab)
I Chosen to separate two most probable classes
I New observations simulated using approximation to full
lightcurve
0.0 0.5 1.0 1.5 2.0
1.0
0.5
0.0
−0.
5−
1.0
Phase
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mal
ized
mag
nitu
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RRd (0.33)RRab (0.48)
● Post. modePost drawClass−wide optRandom
Circle: mode of green lines(from previous slide)
Triangle: random selection ofgreen line
Square: mode of distributionon right of previous slide
Star: random uniform(0,1)
15 / 28
-
Limitations of current approach
I Optimal selection performance depends on an accurate
construction ofprobabilitstic templates especially for
distinguishing classes that are similar
I User specified basis - requires large number of functions J to
capture thecomplexity of the lightcurves. How to select J ?
I Large J involves high computational burden
I Alternative: data-driven bases functions to construct the
probabilistictemplates
16 / 28
-
Probabilistic templates using data-driven basis (FPCA)
I Use ideas from Functional Principal Component Analysis
(FPCA)
I Model for class-specified lightcurves (also known as
Karhunen-Loève)
Yl(til) = µ(til) +K∑
k=1
ξlkφk(til) + �il
{φk(t)}k − orthogonal basisξlk − basis coefficients uncorrelated
over kξlk ∼ (0, λk);λ1 ≥ λ2 ≥ . . . ≥ 0�il ∼ (0, σ2)
µ(t) - smooth function - is the class-specific mean curve
{φk(t), k ≥ 1} eigenbasis of the smooth covarianceΣ(t, t′) :=
cov{Yl(t),Yl(t′)}
Σ(t, t′) =∑k≥1
λkφk(t)φk(t′)
∫φ2k(t)dt = 1 and
∫φk(t)φk′(t)dt = 0 for k 6= k ′
17 / 28
-
FPCA basis
Consider {Yl(til) : i}l the set of lightcurves in the same
classI Estimate the mean function by univariate smoothing,
µ̂(t)
I Estimate the covariance function by bivariate smoothing, Σ̂(t,
t′)
Eigenanalysis of Σ̂(t, t′) yields {λ̂k , φ̂k(t)}k{φ̂k(t)}k - is
orthogonal basis and λ̂1 ≥ λ̂2 ≥ . . . ≥ 0
I Ŷl(t) = µ̂(t) +∑K
k=1 φ̂k(t)ξ̂lk
where ξ̂lk =∫{Yl(t)− µ̂(t)}φ̂k(t)dt (numerical approx)
18 / 28
-
Illustration using EA class (1000 curves, K=7 for PVE=90%)
0.0 0.2 0.4 0.6 0.8 1.0
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Phase
0.0 0.2 0.4 0.6 0.8 1.0
−3
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0 1 2 3 4
−0.
20−
0.15
−0.
10−
0.05
0.00
0.05
Basis coeff, k= 3
depth ratio
19 / 28
-
Recall example eclipsing binary training data
I SAMSI-ICTS meeting discussion: Matthew Graham pointed out
thatthe eclipse depth ratio does not only take a few discrete
values and thishas caused difficulties for templates before
I Prompted investigation into incorporating eclipse depth ratio
into templateconstruction
20 / 28
-
Illustration using EA class: split the sample according to
binning the depth
0.0 0.2 0.4 0.6 0.8 1.0
0.4
0.2
0.0
−0.
2−
0.4
Phase
Ligh
tcur
ves
0.0 0.2 0.4 0.6 0.8 1.00.
40.
20.
0−
0.2
−0.
4
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Ligh
tcur
ves
0.0 0.2 0.4 0.6 0.8 1.0
0.4
0.2
0.0
−0.
2−
0.4
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Ligh
tcur
ves
0.0 0.2 0.4 0.6 0.8 1.0
0.4
0.2
0.0
−0.
2−
0.4
Phase
Ligh
tcur
ves
0.0 0.2 0.4 0.6 0.8 1.0
−3
−2
−1
01
2
Bin 1 : 1−st eigenfn ( 39.45 %)
Phase
0.0 0.2 0.4 0.6 0.8 1.0
−3
−2
−1
01
23
Bin 2 : 1−st eigenfn ( 44.48 %)
Phase
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Bin 3 : 1−st eigenfn ( 49.62 %)
Phase
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Bin 4 : 1−st eigenfn ( 55.96 %)
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15
Basis coeff, k= 1
depth ratio
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3
Basis coeff, k= 1
depth ratio
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15
Basis coeff, k= 1
depth ratio
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Basis coeff, k= 1
depth ratio
21 / 28
-
Probabilistic templates using FPCA and other covariate info
I Common FPCA assumes that the curves have the same distribution
!
I Account for additional ligthcurve info this when constructing
theprobabilistic template. For eclipsing binaries, construct FPCA
basis usingfor depth ratio
Recal Karhunen-Loève representation of Yl(t) (using FPCA basis
):
Yl(t) = µ(t) +∑k
ξlkφk(t)
Extend the framework to incorporate additional covariate (call
it zl):
I Approach I: Incorporate covariate in the mean only, µ(t,
z)
I Approach II: Incorporate covariate in mean µ(t, z) + eigenfns
{φk (t, z) : k}
Similar ideas to Jiang and Wang (2010)
22 / 28
-
Approach I: Yl(t; zl) = µ(t, zl) +∑
k φk(t)ξk(zl) + Noisetl
Illustration using EA lightcurves
0.0 0.2 0.4 0.6 0.8 1.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Estimated mean fn (I) EA
Phase
log
dept
h ra
tio
−0.4
−0.2
0.0
0.2
0.4
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
1−st eigenfunction ( 47.17 %)
Phase
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
2−nd eigenfunction ( 13.61 %)
Phase
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
3−rd eigenfunction ( 9.16 %)
Phase
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0 1 2 3
−0.
10.
00.
10.
2
Basis coeff, k= 1
depth ratio
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�
