A Bayesian method for the analysis of deterministic and stochastic time series

Transcription

1 A Bayesian method for the analysis of deterministic and stochastic time series Coryn Bailer-Jones Max Planck Institute for Astronomy, Heidelberg DPG, Berlin, March 2015

2 Time series modelling heteroscedastic, asymmetric noise on time and signal non-uniform time sampling measured signal, y measured time, s Measured data D j =(s j,y j ) and uncertainties j =( sj, y j ) Model M with parameters Likelihood of single data point: integrate over unknown true time (t) and signal (z) Z P (D j j,,m)= P (D j t j,z j, j) t j,z j {z } Measurement model P (t j,z j,m) {z } Time series model dt j dz j

3 Model comparison Likelihood of all data points is P (D,,M)= Y j P (D j j,,m) Evidence is the likelihood marginalized over the parameter prior P (D,M)= Z P (D,,M) {z } likelihood P ( M) {z } prior More robust alternative is the leave-one-out cross validation likelihood P (D j D j,,m)= Z P (D j j,,m) {z } likelihood d P ( D j, j,m) d {z } posterior L CV = j=j Y j=1 P (D j D j,,m) Calculate integrals by MCMC sampling of posterior

4 Time series model Deterministic mean plus stochastic variation of constant variance P (z j t j,,m)= 1 p 2! e (z j (t j )) 2 /2! 2 Gaussian (t j )= a 2 cos[2 ( t + )] + b sinusoidal true signal z true time t red solid: deterministic component red dashed: standard deviation of stochastic component black: true data

5 Time series model Ornstein-Uhlenbeck process A Stationary, Markov, Gaussian process dz(t) = 1 z(t)dt + c1/2 N (t;0,dt) c relaxation time di usion constant P (z j t j,,m)= 1 p 2 Vz e (z j µ z ) 2 /2V z with µ z = z 0 V z = c 2 (1 2 ) 9 = ; where = e (t t 0)/ for t>t 0

6 Examples of OU process realizations relaxation time, signal time different randomisations

7 Luminosity variations in ultra cool dwarf stars 2m0345 2m m1145a 2m1145b m1146 2m1334 signal / mag calar sdss sori31 sori sori time / hrs

8 Luminosity variations in ultra cool dwarf stars Models compared: constant (variability just due to measurement noise) constant with Gaussian stochastic component sinusoid with Gaussian stochastic component OU process

9 Luminosity variations in ultra cool dwarf stars 2m0345 2m m1145a 2m1145b OU process m1146 2m1334 Sinusoid (8.3h, 13.3h) signal / mag calar sdss Sinusoid + stochastic sori31 sori sori time / hrs

10 Periodicity in biodiversity over past 550 Myr? Rohde & Muller 2005 no. genera (- cubic fit) time BP / Myr periodic model with additional fitted Gaussian noise black = data red = model fit no. genera (- cubic fit) stochastic process (OU process) CV likelihood is much higher for this model time BP / Myr

11 Summary a Bayesian method for modelling times series arbitrary time sampling and error models deterministic and stochastic times series use of cross-validation likelihood, a robust alternative to the evidence applications light curves of some very cool stars (and quasars) evolve stochastically no evidence for periodic variation of biodiversity over past 550 Myr more information and software: tinyurl.com/ctsmod

12 Ultra cool dwarf model comparison results Table 4. Log (base 10) LOO-CV likelihood of each model relative to that for the no-model for each light curve (log L LOO CV log L NM ). Light curve OUprocess Off+Stoch Sin Sin+Stoch Off+Sin+Stoch No-model p-value 2m e-4 2m e-4 2m1145a <1e-9 2m1145b e-3 2m e-3 2m e-9 sdss e-5 calar e-4 sori e-5 sori e-3 sori e-9 Notes. The penultimate column gives the value of the log likelihood for the no-model, log L NM.Thelastcolumnisthep-valueforthehypothesis test from BJM.

13 Parameter posterior PDFs: frequency, ν / hr amplitude, a / mag phase φ black = posterior red = prior

14 Parameter posterior PDFs: 2m1145a τ / hr b / mag c / mag 2 hr black = posterior red = prior µ[z 1 ] / mag V 1 2 [z 1 ] / mag

15 Parameter posterior PDFs: 2m offset, b / mag frequency, ν / hr amplitude, a / mag black = posterior red = prior phase φ standard deviation, ω / mag