Lecture 15

1. Lecture 15 • Time series (aka Light Curves) • Time Tagged Event (TTE) data • Frequentist vs Bayesian variability analysis. • The discrete Fourier transform (DFT) • Periodograms • 2D Fourier transforms • The Hankel transform

2. Time series (aka light curves) • Binned or unbinned. (Nearly everything is binned at some level, so quasi unbinned is perhaps more appropriate.) • Some varieties encountered: • Periodic (more usually, quasi-periodic) signals • Mira or Cepheid variables • Pulsars • Bursts or flares • Fundamental question: is the source variable or not. • This is just another signal-detection problem. • We can use the Null Hypothesis approach.

3. (Quasi) unbinned data: Time Tagged Events. • Often encountered in high-energy astronomy, in which individual photons are counted. • Basic information is the arrival time of the photon. • Also recorded may be its energy & direction. • In HE astronomy one often speaks, somewhat cautiously, of events rather than photons • because not all detections may be caused by a photon. • Basic quantity: the count rate (events s-1). • The arrival times are a Poisson process (ie independent, uncorrelated).

4. Variability detection example • Let us use a Cash likelihood ratio statistic in an attempt to test the null hypothesis against a simple model, for TTEs in an interval T. • The null hypothesis says that the expectation value for the count rate is constant – ie that there is no variability. This model can be written • For the non-null model, let us choose a 2-block model:

5. In pictures:

6. 2-block comparison continued • There are 3 model parameters: • the average rate A; • the ‘change point’ Tc; • the fraction f of the total counts in block 1. • Note that the NH model, m1, can be expressed by a particular combination of parameters of m2. (Ie when f=Tc/T.) As mentioned when the Cash statistic was first described, this is necessary for the theory to be valid.

7. Generic likelihood for TTE data? • A cunning trick: we introduce a ‘virtual bin’ of arbitrary size Δt. • The hope is that this Δt will cancel from the final expression. • An integer number of events will fall within the bin. • Hence the probability P(n,t) for an given number n in a bin centred on t will be Poisson: where

8. In the limit Δt ->0: • Λ→m(t) Δt • P(n,t) → 0 for n>1. • In other words, if we make the bin small enough, the only possibilities are that it contains either 0 or 1 event. • P(0,t) = exp[-m(t) Δt] • P(1,t) = m(t) Δt exp[-m(t) Δt]

9. The total likelihood: • If there are M virtual bins (ieM=T/ Δt) then the total likelihood is the product of the probabilities in each bin: • It’s fairly easy to show that this gives where N is the total number of events (ie, photons) detected in the time T. • This expression still depends on the arbitrary parameter Δt. But with a likelihood ratio, we expect this to cancel out.

10. The Cash likelihood ratio: • Note the Δts cancel as desired; so do the exponential terms. • With the 2-block model this becomes: where N1 is the number of events in block 1 and τ=Tc/T.

11. Pros and cons of this ‘frequentist’ approach. • With this we can do the following: • Find the best-fit values of A, τ and f. • Use the Cash theory to calculate a value for the probability of the Null Hypothesis. • What we can’t do with it is • Obtain full, joint probability distributions for A, τ and f; • Make use of prior knowledge of same; • Obtain the relative probabilities for the 1- vs 2-block models as a whole. • For this we need a Bayesian approach.

12. A Bayesian approach • Bayesian statistics is not just some different way to approach probabilistic problems, it is the formally correct way to approach them. • The difficulties seem to be • conceptual (the human brain is very poor at formally correct statistical reasoning!) • practical evaluation of the integrals. • Gregory and Loredo ApJ 398, 146 (1992) • Read sections 1 and 2 carefully, about 5 times, and you will begin to understand Bayesian parameter estimation. • My approach follows theirs closely.

13. Approach to Bayes’ theorem. • Remember joint and conditional probability densities in slide 18 of lecture 3. • This works also for probabilities for discrete propositions A and B: • Suitable propositions for our situation: • AD: “Such-and-such a pattern of events was detected.” • BMi: “The data are explained by model i.”

14. Approach to Bayes’ theorem. • If we include an explicit dependence on environmental information I, this gives: • Rearrangement gives Bayes’ theorem: • In words:

15. Global likelihood for all models • This, the denominator of Bayes’ theorem, can be thought of as a normalizing constant. Ie: = sum over (prior x likelihood) for all models in the class. • Ensures that a sum over all posteriors (the LH side of Bayes’ theorem) equals 1.

16. Odds ratio • It can be difficult to calculate the denominator, the global likelihood for the entire class of models, from first principles. • Better to form a ratio of the posterior probabilities for 2 models. P(D|I) will then cancel out. • This is called a Bayesian Odds Ratio:

17. How to calculate the likelihood for model Mi. • If we have no prior reason to favour one model over another, we can set the ratio of priors to 1. • That leaves the problem of calculating the global likelihood P(D|M,I) for each model. • This itself plays the role of denominator, or normalizing constant, in a ‘deeper’ version of Bayes’ theorem, one which deals with the parameter values:

18. Bayes’ theorem applied to the probability density of the parameters. • Note: • I have suppressed the Is for compactness. • I’ve used small p for continuous functions of the parameters (probability densities), but large P for single numbers (probabilities). • We want the LH side, when integrated over Θ, to equal 1. Hence…

19. Global likelihood for model M • In words:

20. A Bayesian procedure: Deduce the likelihood function for each parameter. GL for the competing model Decide on priors for each parameter. Decide on global priors. Integrate the product of (prior x likelihood) to get the global likelihood (GL) for the model. Apply Bayes’ Theorem: Form an Odds Ratio to compare the two models. Posterior joint probability density for the parameters.

21. Applied to the present problem: • Likelihood function: use the ones we already worked out for the 1- and 2-block models: • Prior distributions: • Choose p(A|M)=1/Amax between 0 and Amax. • Amax is arbitrary, but it will cancel. • Choose p(f|M)=1 and p(τ|M)=1 between 0 and 1.

22. Expressions for global likelihood: • Priors for the models: choose them equal. • Odds ratio therefore is

23. Points to note: • The odds ratio doesn’t depend on Δt, T or Amax • Even a very simple model + simple priors can generate a complicated integral. • This particular one can be evaluated – it is just a long, fiddly business. • From Bayes’ theorem, the posterior probability density evaluates to • This is proportional to the Cash formula, but now we know it is correct.

24. Example

25. The DFT • Related to Fourier expansion. • Cyclic: Fk+mxN=Fk. • Use with caution! It will not approximate a FT without special treatment: • ‘Zero padding’ • Vignetting. • High N. • A fast algorithm exists – the Fast Fourier Transform (FFT). • For this, N should be a product of low primes.

26. Zero-padding example Fourier transform

27. Zero-padding example Discrete Fourier transform (DFT)

28. Zero-padding example DFT with zero padding

29. Problems with uneven sampling Back to the full FT

30. Problems with uneven sampling FT of unevenly-sampled data

31. Problems with uneven sampling • Lomb-Scargle periodogram. • Much literature on: • finding peaks • calculating uncertainties. • See eg Press et al, chapter 13.8

32. 2D Fourier pairs fringes  point (delta function).

33. 2D Fourier pairs higher spatial frequency  further from the origin.

34. 2D Fourier pairs multiplication  convolution.

35. 2D Fourier pairs gaussian  gaussian.

36. ‘Natural’ vs ‘DFT’ origins

37. The Hankel transform • If a function has radial symmetry, eg then so does its 2D Fourier transform: • In this case, a short cut to the 2D FT is the Hankel transform: • J0 is the Bessel function of order zero.

38. Hankel transform example