1 / 24

Lecture 8

Lecture 8. Source detection. Different sorts of model. All models. Background + signal. Background + many similar signals. b + s vs b + Σ s i. s may often be assumed to be: slowly varying with r ; with compact support. Source detection.

jasia
Download Presentation

Lecture 8

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 8 • Source detection

  2. Different sorts of model All models Background + signal Background + many similar signals

  3. b + svsb + Σsi • s may often be assumed to be: • slowly varying with r; • with compact support

  4. Source detection • The basic idea is related to Null Hypothesis testing… • But if the sources can be assumed to be localized, we can cut the data up and test each source-sized bit at a time. • sliding window. • Some missed jargon: • the probability at the intercept • is called the P-value (you can • google it) Survival function

  5. Testing the NH: • Not all tests are equally good at finding signals! • Eg Cash statistic is better than χ2 (in circumstances where the Cash test is appropriate – eg bkg is a subset of the signal model). • Cash stat makes use of knowledge about the signal shape – in general any stat which does similar (eg a matched filter) will also perform well. • There is an infinite variety of ‘statistics’ to choose from.

  6. Source detection. • If the SF probability in each patch (the P-value) is smaller than a previously chosen cutoff, we can call this a positive detection. • BUT! Note that there is no certainty. • Sometimes the null model will by chance give a large χ2 => ‘false positives.’ For given data, background and cutoff, there will be a fixed number of false positives expected in the source list. • => ‘reliability’. More on this later. • Sometimes a real source will give a small null-hypothesis χ2 => ‘false negatives’, real sources which are missed. • => ‘completeness’. More on this later.

  7. Problems with the NH approach: • We don’t have exact knowledge of the background. • Have to estimate it either from • separate data – in which case we need the separate data! (Don’t always have the luxury.) • or from the same data… but this may be dominated by the source... • Or our background model may be wrong. • Same issues as other model fitting. In particular: • χ2 has to be used with care when the noise is Poisson.

  8. But where are the sources? • Applying some sort of NH test in a sliding window will return a new random signal – now correlated.. • Finding the sources consists rather of looking for peaks in this random signal. • The simplest example is when the noise is uncorrelated and the source peaks have width=0.

  9. Looking for sources 1 channel at a time: • In each channel, we test the NH with N=1. • Since there are no fitted parameters, υ=1 also. • If the source occupies a single channel, this procedure is optimal. • If, however, the source is spread over several channels (as is usual), this procedure is not efficient. • We want a statistic which uses the maximum amount of information about the source shape.

  10. A generic source-detection algorithm • We shall assume that: • The data is ‘binned’ (eg CCD data). • We have a good independent estimate of the background. • The sources are sparsely distributed – such that we can deal with them one at a time. • The shape of the source profile is known. • The source position is unknown. • The source amplitude is unknown (but >0).

  11. Generic source-detection algorithm: The algorithm has 3 steps: 1: Calculate a sliding-window map. 2: Find the peaks in this map. Choose a Pcutoff 3: For each peak, calculate the probability that it could arise by chance from the background (the null hypothesis P-value). P < Pcutoff? No Yes Sources Rejects

  12. 1: The sliding window. y U y U y U

  13. 1: The sliding window. • For each position of the sliding window, a single number U is calculated from the values falling within the window. • The output is a map of the U values. • The intent is to: • Raise the signal-to-noise • Improve sensitivity • Amplify the sources at the expense of the noise. • Sliding-window processing only has value when the source has a width > 1 pixel. • Edges need special treatment. Same thing.

  14. 1: Window functions • A weighted sum (= a convolution). • Simplest with all weights = 1: “sliding box”. • Optimum weights – a “matched filter”: • For uniform Gaussian noise, wopt = s. • Trickier to optimize for Poisson noise. • Per-window null-hypothesis χ2. • With either an independent value of bkg (in which case degrees of freedom = number of pixels Nw in the window), or… • …one fitted from the data (deg free = Nw-1). • Likelihood (same bkg provisions as χ2).

  15. 1: Window functions Parent function Data

  16. 1: Window functions Parent function Chi squared, size=100 Matched filter, size=10 Log-likelihood, size=100

  17. 2: Peak finding Gaussian noise, convolved with a gaussian filter. …don’t get the gaussians mixed up!

  18. 2: Peak finding • How best to do it? • There’s no single neat prescription. • Naive prescription: • Pixel i is a peak pixel if yi > any other y within a patch of pixels from i-j to i+j. • This probably looks familiar to you. • But what value to choose for j? • Things to avoid are: • j too small – results in more than 1 peak per source; • j too large – misses a close adjacent source.

  19. 2: Peak finding Box too small: Box too large:

  20. 3: Decision time – is it a source or not? • To calculate a P-value we need the probability distribution of peaks in the post-window map of U values (given the null hypothesis). • This is not the same as the probability distribution of the original data values… • …nor is it even the same as the probability distribution of U values. • In fact, little work seems to have been done on ppeaks. (Though there is quite a lot on the distribution of extrema – not quite the same thing.)

  21. 3: The decision ‘Map’ vs ‘peak’ distributions for Gaussian noise. Black: all pixels Red: peaks

  22. 3: Cash to the rescue • A practical recipe for applying Cash to source detection goes as follows: • Choose a window area surrounding each peak. • Within this window, calculate Lnull with model mi = bi (the background map values). • Calculate Lbest by fitting a model • Degrees of freedom ν = 1 (the amplitude) + d (the dimensions of the spatial fit). • The Cash statistic 2(Lbest-Lnull) behaves like χ2 with 1+d deg. free. mi = bi + θ1s(ri – θr)

  23. 3: Cash to the rescue • The only difficult point (which is a problem for every method) is to calculate the fraction of pixels which are peaks. • Monte Carlo • Possibly a Fourier technique? • Also, don’t want to use the fit for final parameter values. A Mighell fit is better. From my 2009 Cash paper.

  24. What is the best detection method? From my 2009 Cash paper.

More Related