Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/astro193/VLK_slides_apr13.pdf Дата изменения: Mon Apr 13 19:40:08 2015 Дата индексирования: Sun Apr 10 11:57:51 2016 Кодировка: Поисковые слова: m 43

Astro 193 : 2015 Apr 13
·

Follow-up
· ·

Projects, etc. Orthonormal Wavelet Basis

· · · ·

Fourier Transforms: Aliasing Lomb-Scargle Periodograms BayesFT Gaussian Processes, CAR Models


Class Logistics
·

What we expect from the Projects
·

use data; apply techniques we have discussed in class (OK to go beyond); infer or compute an astronomically/astrophysically interesting quantity 5-slide, 10-minute presentation on May 4 and 6 from 2pm-4pm, 15 min each
· ·
th th

·

sequence TBD, will be announced Friday May 1

st

introduce the problem; describe the data; describe the analysis; show results; draw conclusions.

·

CHASC AstroStatistics Collaboration
· · ·

Mailing list : http://hea-www.harvard.edu/AstroStat/mailinglists.html Present projects to statisticians Suggest items to add to Stat jargon dictionary : http://hea-www.harvard.edu/ AstroStat/statjargon.html

·

Review class on Apr 15. Bring your questions.
·

ALSO: Give a one minute description of your project to the class. No slides.


Orthonormal Wavelet Basis
· · ·

Orthogonal, normalized, and complete e.g., Discrete Haar wavelet mn(x) = 2
-m/2

([x-n2 ]/2 )

-m

-m

(x) = 1 for 0 x < 1/2 = ­1 for 1/2 x < 1 = 0 otherwise
· · ·

= =

nn' mm'

Difference between function f(xi) and its wavelet decomposition jk cjk jk(x) can be made arbitrarily small


Fourier Transforms: Aliasing and Lomb-Scargle


Bayesian Periodogram
· · · ·

Bayesian Spectrum Analysis and Parameter Estimation, Larry Bretthorst Bayesian Logical Data Analysis for the Physical Sciences, Phil Gregory (Ch 13) Data D = {di, i=1..N}; Model f(t) = B1 cos(t) + B2 sin(t) + N(0,І) Schuster Periodogram C() = | R() =
j=1..N j=1..N

dj exp[itj] |І R()І+I()І dj sin(tj)

dj cos(tj), I() =
j=1..N

j=1..N

·

Likelihood p(D|B1,B2,,) = =
­N

(1/) exp[-(dj-f(tj))І/2І]

exp[-(NQ/2І)]

Q = d І - (2/N)[B1R() + B2I()] + (B1І+B2І)/2



Bayesian Periodogram
· ·

Bayesian Spectrum Analysis and Parameter Estimation, Larry Bretthorst Bayesian Logical Data Analysis for the Physical Sciences, Phil Gregory (Ch 13) Data D = {di, i=1..N}; Model f(t) = B1 cos(t) + B2 sin(t) + N(0,І) Schuster Periodogram C() = | R() =
j=1..N j=1..N

· ·

dj exp[itj] |І R()І+I()І dj sin(tj)
­N

dj cos(tj), I() =
j=1..N

j=1..N

·

Likelihood p(D|B1,B2,,) =

(1/) exp[-(dj-f(tj))І/2І] =

exp[-(NQ/2І)]

Q = d І - (2/N)[B1R() + B2I()] + (B1І+B2І)/2
·

integrating over flat priors on B1 and B2, p(|D)
-N+2

exp[-(N/2 )(d І-C())І] exp[C()/І]
­(N-2)/2

2

·

integrating over Jeffrey's prior on (1/), p(|D) [1 - 2C()/Nd І] Uncertainty, 4 / [ N (S/N)]

·


Gaussian Processes and CAR Models