Lecture 5 (Periodogram analysis for regular and irregular data)
Lecture 5 (Periodogram analysis for regular and irregular data)
Lecture 5 (Periodogram analysis for regular and irregular data)
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Statistical tools in astrophysicsCourse contents: Elementary probability theory <strong>and</strong> statistics Probability distributions <strong>and</strong> Bayes theorem Hypothesis testing <strong>and</strong> significance Least-squares estimation: OLS, GLS, χ 2 minimization, non-linear LS Estimation <strong>and</strong> confidence intervals Principle of Maximum Likelihood Pseudor<strong>and</strong>om numbers Monte Carlo <strong>and</strong> resampling methods <strong>Periodogram</strong> <strong>analysis</strong> <strong>for</strong> <strong>regular</strong> <strong>and</strong> ir<strong>regular</strong> <strong>data</strong>this lecture Graphical methodsASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.1måndag den 6 december 2010
Spectral densityLet h(t) be a continuously varying, stationary signal with ⟨h(t)⟩ = 0(⟨⟩ is the time average). The variance of the signal isThe power spectral density (PSD) specifies how the variance is dividedamong different frequencies:P( f )area =fASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.3måndag den 6 december 2010
<strong>Periodogram</strong>In practice you can only get an estimate of the power spectral density.The periodogram could be such an estimate <strong>for</strong> a time series of length T.Most important limitations of the periodogram compared to the continuous PSD: Finite T implies finite frequency resolution f = 1/T (<strong>and</strong> minimum frequency). Finite number of <strong>data</strong> points N implies that at most N (independent) frequencycomponents can be estimated.From a statistical viewpoint the periodogram is a statistic (it’s computed from the<strong>data</strong>), <strong>and</strong> like other sample statistics is has uncertainties.To decide if a signal is periodic or not can be treated as a hypothesis test.We will consider separately: <strong>Periodogram</strong>s <strong>for</strong> evenly sampled <strong>data</strong> ⇒ discrete Fourier trans<strong>for</strong>m <strong>Periodogram</strong>s <strong>for</strong> unevenly sampled <strong>data</strong> ⇒ Lomb-Scargle periodogramASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.4måndag den 6 december 2010
Reminder on Fourier Trans<strong>for</strong>msASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.5måndag den 6 december 2010
Time series <strong>and</strong> the Discrete Fourier Trans<strong>for</strong>mASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.6måndag den 6 december 2010
The Discrete Fourier Trans<strong>for</strong>m <strong>and</strong> its inverseASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.7måndag den 6 december 2010
The periodogramASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.8måndag den 6 december 2010
Example periodograms, noise-free <strong>data</strong>ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 9måndag den 6 december 2010
Example periodograms, noise-free <strong>data</strong>ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 10måndag den 6 december 2010
Example periodograms, noise-free <strong>data</strong>Aliasing: f > f c is reflected in f cASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 11måndag den 6 december 2010
Example periodograms, noise-free <strong>data</strong>Longer time span ⇒ higher frequency resolutionASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 12måndag den 6 december 2010
Example periodograms, noise-free <strong>data</strong>Non-integer number of periods ⇒ frequency leakageASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 13måndag den 6 december 2010
Example periodograms, noise-free <strong>data</strong>Longer time span ⇒ higher resolution ⇒ less frequency leakageASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 14måndag den 6 december 2010
Example periodograms, noisy <strong>data</strong>White noise ⇒ "constant" power (values follow exponential distribution)ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 15måndag den 6 december 2010
Example periodograms, noisy <strong>data</strong>Periodic signal + white noise ⇒ "constant" power + peakASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 16måndag den 6 december 2010
Example periodograms, noisy <strong>data</strong>More <strong>data</strong> points ⇒ less noise power per frequency pointASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 17måndag den 6 december 2010
Unevenly sampled <strong>data</strong> (1/5)ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.18måndag den 6 december 2010
Unevenly sampled <strong>data</strong> (2/5)ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.19måndag den 6 december 2010
Unevenly sampled <strong>data</strong> (3/5)ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.20måndag den 6 december 2010
Unevenly sampled <strong>data</strong> (4/5)ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.21måndag den 6 december 2010
Unevenly sampled <strong>data</strong> (5/5)ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.22måndag den 6 december 2010
Example periodograms, noisy <strong>data</strong>, unevenly sampledn = 320;mjd = r<strong>and</strong>([n 1])*320;R<strong>and</strong>om sampling ⇒ good frequency coverage (no aliasing!)ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 23måndag den 6 december 2010
Checking found period (p) by folding the <strong>data</strong>Phase (φ) of sample j isIs scatter about the mean curve consistent with errors?ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 24måndag den 6 december 2010
Example periodograms, noisy <strong>data</strong>, unevenly sampledn = 320;mjd = r<strong>and</strong>([n 1])*320;<strong>for</strong> j = 1:nif (mod(mjd(j),1)>0.5)mjd(j) = mjd(j) - 0.5;endendNon-r<strong>and</strong>om sampling ⇒ look out <strong>for</strong> frequency window effects!ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 25måndag den 6 december 2010
AAPS uses the 3.9 mAnglo-AustralianTelescope (AAT)at Siding SpringObservatory(New South Wales)<strong>and</strong> the UCLESspectrograph.240 nearby solar-typestars (V < 8) aremonitored.Some 25 planets found.ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p. 26måndag den 6 december 2010
Example of AAPS <strong>data</strong> (HD 27442)ASTM21 Statistical tools in astrophysics<strong>Lecture</strong> 5, p.27måndag den 6 december 2010
Change language