Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/astro193/VLK_slides_apr20.pdf Дата изменения: Mon Apr 20 20:19:47 2015 Дата индексирования: Sun Apr 10 11:58:03 2016 Кодировка:

Astro 193 : 2015 Apr 20
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Follow-up
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Bayesian Fourier Components HW 11 on Periodograms is online HW 12: on exposure time

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Gaussian Processes, CAR models Bayesian Blocks


Bayesian Blocks
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Studies in Astronomical Time Series Analysis. V. Bayesian Blocks, A New Method to Analyze Structure in Photon Counting Data, Scargle, J.D., 1998, ApJ 504, 405 Studies in Astronomical Time Series Analysis. VI. Bayesian Block Representations, Scargle, J.D., Norris, J.P., Jackson, B., & Chiang, J., 2013, ApJ 764, 167 Structure in the Three-Dimensional Galaxy Distribution. I. Methods and Example Results, Way, M.J., Gazis, P.R., & Scargle, J.D., 2011, ApJ 727, 48 Structure in the 3D Galaxy Distribution. II. Voids and Watersheds of Local Maxima and Minima, Way, M.J., Gazis, P.R., & Scargle, J.D., 2015, ApJ 799, 95 Bayesian Blocks: Detecting and Characterizing local variability in Time Series, Scargle, J.,D., Norris, J.P., Jackson, B., & Chiang, J., 2012, Astrophysics Source Code Library, ascl:1209.001, http://ascl.net/1209.001

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Bayesian Model Selection again
Models Mk, k={1..K} p(
(k)

|D,Mk) = p(D|(k) ,Mk) p((k)|Mk) / p(D|Mk)

p(Mk|D) = p(D|Mk) p(Mk) / p(D) p(D|Mk) = d Odds ratio: Oij = p(Mi|D)/p(Mj|D) = [p(D|Mi)p(Mi)] / [p(D|Mj)p(Mj)]
(k)

p(D|(k),Mk) p((k)|Mk)


Bayesian Blocks : Basics
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Principled Kernel Density Estimation Model a light curve as piecewise constant segments Break a segment into two blocks and ask whether the maximum likelihood is higher for the broken blocks vs the unbroken block In practice: find maximum fitness after adding a "data cell", keep track of maxima, and find optimum changepoints after looking at whole data set
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Use log(likelihood) as fitness functions. e.g., in a block k of duration T(k) that contains N(k) photons, logL(k) = N(k) log - T(k), and the maximum likelihood solution is = N(k)/T(k), so logL(k) = N(k)[logN(k)-logT(k)] + N(k) Fitness is block additive: i.e., fitness of a partition is the sum of fitnesses of blocks that make up the partition Find the last changepoint, peel it off, and repeat, to extract all the blocks

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Stopping rules: use prior on number of changepoints, or impose penalty on adding a new block
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prior p(Ncp) Ncp, with log 73.53p0N

-0.478

-4


Scargle et al. 2011

Bayesian Blocks : Dynamic Programming


Scargle et al. 2011

Bayesian Blocks : Partitions and subpartitions


Scargle et al. 2011

Bayesian Blocks : Finding changepoints


Homework 13: Bayesian Blocks Due Wed Apr 29

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For the time-tagged data of RT Cru from Homework 10, construct a Bayesian Blocks representation.