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• • • • Lecture 6 A little bit more about source detection. Parameter fitting – to measure the source. Mr Bayes gets to put his point of view at last. How to assess our source list when we have it. NASSP Masters 5003F - Computational Astronomy - 2009 Significance • We have talked about the probability of the Null Hypothesis as a way of deciding which detections to label as sources. • Significance is another way to talk about this. • “X-sigma detection” means amp = X times the standard deviation of the noise. • Often used loosely and questionably. • Related to the concept of confidence intervals (see later). • For Gaussian noise, σ • 5-sigma => NH P-value ~ 3*10-7 NASSP Masters 5003F - Computational Astronomy - 2009 What is the best detection method? From my 2009 Cash paper. NASSP Masters 5003F - Computational Astronomy - 2009 Now back to fitting. • We want good values of the parameters. – Amplitude terms and units: • Counts, count rate (instrumental units) • Brightness • Magnitudes in a given filter band (eg B magnitudes) • Flux S – 10-26 W m-2 Hz-1 = 1 Jansky (radio) – erg cm-2 s-1 (x-ray) – Position terms and units: • RA and dec • arcsec offset from a reference • detector coordinates. NASSP Masters 5003F - Computational Astronomy - 2009 Confidence intervals - uncertainties. • Frequentist interpretation: – Given a confidence interval enclosing probability P: – Parent values of θ will fall within the interval a fraction P of the time. • Can ‘cut the cake’ an infinite number of ways. • Symmetric, width 1σ => P = 0.68 for Gaussian noise. NASSP Masters 5003F - Computational Astronomy - 2009 Confidence intervals. • Uncertainties with multi-dimensional data: – fit a paraboloid to the maximum; – find 68% confidence contour (an ellipse); – σ12, σ22, σ122 etc from the ellipse equation. – Alternatively, this covariance matrix is the inverse of the curvature matrix for χ2. NASSP Masters 5003F - Computational Astronomy - 2009 General problems with fitting: • When some of the θs are ‘near degenerate’. – Solution: avoid this. • When the model is wrong – or, several different models fit equally well. – Solution: F-test (sometimes). Supposedly restricted to the case in which 2 models differ by an additive component. NASSP Masters 5003F - Computational Astronomy - 2009 Degenerate θs NASSP Masters 5003F - Computational Astronomy - 2009 Competing models • Both models give moderately ok chi2, but clearly neither really describes what is happening. Ie we don’t understand the physics here. NASSP Masters 5003F - Computational Astronomy - 2009 Bayesian vs Frequentist Frequentist: Data y Unknown true parent f Assume parent ^f Monte Carlo or analysis Calculated p(y) p(y|f): Prob. dens’y that parent f gives y. Bayesian: p(f|y), prob. dens’y of parent f given data y. Datayy Markov Chain Monte Carlo Bayes’ theorem Prior knowledge of p(f) NASSP Masters 5003F - Computational Astronomy - 2009 Bayesian statistics – a bare outline. • Bayes’ theorem: – p(Θ|y,I) is called the ‘posterior distribution’ of the model parameters, – p(y|Θ) is the probability distribution of the data given the model, – p(Θ|I) contains ‘prior’ knowledge about Θ, – and p(y|I) is a normalizing constant. • Hopefully some examples next week… NASSP Masters 5003F - Computational Astronomy - 2009 Markov Chain Monte Carlo (MCMC) • (To some extent this may just be grandiose terminology…shh…) • A Monte Carlo you know about – it is just a machine for generating random numbers having a particular distribution. • The Markov Chain bit means that you have a loop as follows: Rules Random number Starting value • If you set the rules up correctly, no matter what the starting value, the random numbers converge to the desired distribution. NASSP Masters 5003F - Computational Astronomy - 2009 How to analyse the source catalog. • Things we want to know: – Reliability: • The number of false positives. – Completeness: • The fraction of real sources we are finding. People say things such as “the survey is essentially complete at a flux greater than so-and-so.” – Sensitivity: • Broadly speaking this is the flux at which we are only detecting 50% of the sources. • These are often not very exactly defined terms. NASSP Masters 5003F - Computational Astronomy - 2009 A source-detection Monte Carlo: • Things to note from this plot: 1. Fainter sources become more numerous… 2. …until a cutoff value of S. 3. Measured ^S is scattered about true S. 4. The ^S distribution is biased at low S. NASSP Masters 5003F - Computational Astronomy - 2009 Log N – log S. Input fluxes • In a ‘Euclidean universe’, • Therefore • But Olbers’ paradox says there must be a cutoff. – This is observed in several actual surveys. => large-scale structure. NASSP Masters 5003F - Computational Astronomy - 2009 Eddington bias Red: ‘measurable’ fluxes • Happens because measured flux ^S is random – it is scattered about the true value. • The result is a ‘blurring’ of the ‘true’ logN-logS. • Because usually n(S) has a negative slope, this blurring inflates the number of sources. NASSP Masters 5003F - Computational Astronomy - 2009 How does our catalog shape up? Blue: ‘true’ detections • The really interesting things in the logNlogS curve always seem to be happening just at our sensitivity limit. 2 things to do: – Persuade ESA/NASA etc to spend $$$$ on a bigger and better telescope; • What do you mean, “don’t be ridiculous?” – Ok then, let’s lower the Pcutoff. • But… NASSP Masters 5003F - Computational Astronomy - 2009 …Don’t forget the falsies. Cyan: false detections. • Next episode: Confusion, dynamic range. NASSP Masters 5003F - Computational Astronomy - 2009 From R L White et al (1997). NASSP Masters 5003F - Computational Astronomy - 2009 From M Tajer et al (2004). NASSP Masters 5003F - Computational Astronomy - 2009