Sampler of Interdisciplinary Measurement Error and Complex Data Problems

Transcription

1 Sampler of Interdisciplinary Measurement Error and Complex Data Problems James Long April 22, / 28

2 Time Domain Astronomy and Variable Stars An Observation on Heteroskedasticity and Misspecified Models Period Estimation and Classification for M33 Miras 2 / 28

3 Outline Time Domain Astronomy and Variable Stars An Observation on Heteroskedasticity and Misspecified Models Period Estimation and Classification for M33 Miras 3 / 28

4 Periodic Variable Stars Periodic variables: Stars that repeat brightness variation over a fixed period Time (Days) Star observed n = 367 times. Data for star is D = {t i, m i, σ i } n i=1. Observe star brightness mi at time t i with uncertainty σ i. 4 / 28

5 Folded Light Curve of Periodic Variable Folded light curve: Brightness versus time modulo period Phase (Days) 5 / 28

6 The OGLE III Survey [5] Collected 100,000s periodic variables in Large Magellanic Cloud Periodic variables belong to different classes Class is related to astrophysical reason for variation Two Examples Mira Variable Time (Days) Phase (Days) RR Lyrae AB Variable Time (Days) Phase (Days) 6 / 28

7 Size of Variable Star Data Sets is Growing Hipparcos ( ): 2712 OGLE (1992 present): 100,000s DES (ongoing): 10 million LSST (starting 2020): 1 billion Data sets of varying quality: Time (Days) 7 / 28

8 Outline Time Domain Astronomy and Variable Stars An Observation on Heteroskedasticity and Misspecified Models Period Estimation and Classification for M33 Miras 8 / 28

9 Folded Light Curve using Two Sinusoidal Models Time (Days) Phase (Days) 9 / 28

10 Period Estimation for Variable Stars Common Model: where ɛ i N(0, σ 2 i m i = β 0 + K a k sin(t i ωk + φ k ) + ɛ i k=1 ) Zechmeister [6], Schwarzenberg [3] Maximum likelihood estimator: ω = argmin ω min φ,a,β 0 n i=1 ( m i β 0 K k=1 a k sin(ωt i k + φ k ) σ i ) 2 10 / 28

11 Question Misspecified models are common and can be useful. Heteroskedasticity in responses is common. Typically we weight observations by inverse of variance. Question: Is this weighting helpful when the model is misspecified? 11 / 28

12 Correct Model: Weighted Fit Magnitude True Light Curve Estimate Time 12 / 28

13 Correct Model: Unweighted Fit Magnitude True Light Curve Estimate Time 13 / 28

14 Summary The fitted curve (orange line) is close to observations with small error (small σ i ). This is good when the actual light curve variation is sinusoidal. Question: What happens for misspecified models (light curves that are not actually sinusoids)? 14 / 28

15 Misspecified Model Weighted Fit Magnitude True Light Curve Estimate Time 15 / 28

16 Misspecified Model Unweighted Fit Magnitude True Light Curve Estimate Time 16 / 28

17 Application to Variable Star Period Estimation g band light curves of 238 bright sources in Stripe 82 SDSS-III Downsampled all light curves to 10,20,30, and 40 observations Simulates difficult period recovery settings encountered by PanStarrs, DES Compare period estimation using weighted, unweighted estimators 17 / 28

18 Results Fraction of periods estimated correctly for different models (K = 1, 2, 3) and using weights (Σ 1 ) and unweighted (I ). K = 1 K = 2 K = 3 n Σ 1 I Σ 1 I Σ 1 I Conclusion: Ignoring heteroskedasticity can improve model fits. 18 / 28

19 Notes linear model case: x i f X, σ i f σ, σ i = y i = f (x i ) + ɛ i where ɛ i N(0, σ 2 i ) β argmin β x i E[(f (x) x T β) 2 ] = E[xx T ] 1 E[xf (x)]. β = (X T Σ 1 X ) 1 X T Σ 1 Y not efficient. Adaptively choose optimal weights: (σ 2 i + ) 1 close connections with Y. Ma [1, 2] many more details: Parameter Estimation for Misspecified Regression Models with Heteroskedastic Errors 19 / 28

20 Outline Time Domain Astronomy and Variable Stars An Observation on Heteroskedasticity and Misspecified Models Period Estimation and Classification for M33 Miras 20 / 28

21 Collaboration Astronomy Lucas Macri Wenlong Yuan Statistics Shiyuan He Jianhua Huang James Long 21 / 28

22 Period Luminosity Relation for Miras in the LMC W IV Cepheid Fundamental Mode Cepheid 1st Overtone RR Lyrae A Miras O rich Miras C rich period Time (Days) 22 / 28

23 PL Relation for Miras in M33 Estimating PL Relation requires: Estimating periods and luminosities accurately. Classifying stars. Challenging case: Time (Days) 23 / 28

24 Sinusoid Fit to LMC Mira Time (Days) Phase (Days) 24 / 28

25 Fit to M33 Mira Time (Days) Phase (Days) Improve sinusoidal model to accurately estimate periods with M / 28

26 Gaussian Process Fit to OGLE Mira Time (Days) 26 / 28

27 Bayes Factors for Separating Different Classes 27 / 28

28 Bibliography I [1] Yanyuan Ma, Jeng-Min Chiou, and Naisyin Wang. Efficient semiparametric estimator for heteroscedastic partially linear models. Biometrika, 93(1):75 84, [2] Yanyuan Ma and Liping Zhu. Doubly robust and efficient estimators for heteroscedastic partially linear single-index models allowing high dimensional covariates. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(2): , [3] A Schwarzenberg-Czerny. Fast and statistically optimal period search in uneven sampled observations. The Astrophysical Journal Letters, 460(2):L107, [4] Branimir Sesar, Željko Ivezić, Skyler H Grammer, Dylan P Morgan, Andrew C Becker, Mario Jurić, Nathan De Lee, James Annis, Timothy C Beers, Xiaohui Fan, et al. Light curve templates and galactic distribution of rr lyrae stars from sloan digital sky survey stripe 82. The Astrophysical Journal, 708(1):717, [5] A Udalski, MK Szymanski, I Soszynski, and R Poleski. The optical gravitational lensing experiment. final reductions of the ogle-iii data. Acta Astronomica, 58:69 87, [6] M Zechmeister and M Kürster. The generalised lomb-scargle periodogram-a new formalism for the floating-mean and keplerian periodograms. Astronomy & Astrophysics, 496(2): , / 28