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Lecture 6

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. Significance.

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Lecture 6

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  1. 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.

  2. 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, • 5-sigma => NH P-value ~ 3*10-7 σ

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

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. Degenerate θs

  9. Competing models • Both models give moderately ok chi2, but clearly neither really describes what is happening. Ie we don’t understand the physics here.

  10. Bayesian vs Frequentist Frequentist: Unknown true parent f Data y Assume parent ^f Monte Carlo Calculated p(y) or analysis p(y|f): Prob. dens’y that parent f gives y. Bayesian: Data y y p(f|y), prob. dens’y of parent f given data y. Markov Chain Bayes’ theorem Monte Carlo Prior knowledge of p(f)

  11. 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…

  12. 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: • If you set the rules up correctly, no matter what the starting value, the random numbers converge to the desired distribution. Rules Random number Starting value

  13. 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.

  14. A source-detection Monte Carlo: • Things to note from this plot: • Fainter sources become more numerous… • …until a cutoff value of S. • Measured ^S is scattered about true S. • The ^S distribution is biased at low S.

  15. Log N – log S. • In a ‘Euclidean universe’, • Therefore • But Olbers’ paradox says there must be a cutoff. • This is observed in several actual surveys. => large-scale structure. Input fluxes

  16. Eddington bias • 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. Red: ‘measurable’ fluxes

  17. How does our catalog shape up? • The really interesting things in the logN-logS 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… Blue: ‘true’ detections

  18. …Don’t forget the falsies. • Next episode: Confusion, dynamic range. Cyan: false detections.

  19. From R L White et al (1997).

  20. From M Tajer et al (2004).

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