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Astro 193 : 2015 Apr 22
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Follow-up
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David van Dyk visit on Mon Apr 27 The Q: Feedback is open Homework 11: significance CAR : likelihood Homework 13 alteration

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Upper Limits Deconvolution


Upper Limits
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Critical point: cannot talk about an upper limit without invoking the process of detection Limits are placed on model intensity, not on measured counts Type I -level significance decides whether something can be detected
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smaller the , fewer false positives, and harder to detect weak signals

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Type II -level power sets how conservative your limit is
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there is a probability of 1- that a source at the limit will remain undetected


Upper Limits: Homework 12
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Recall: 100 counts in a background region, 19 counts in a source region of area 1/10 that of the background. (Identical exposure times, but any difference can be subsumed into the area.) Typical astronomer assumes Gaussian regime and uses Gehrel's approximation ( = 1+(N+0.75)) to the error.
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C = S + B = 19 & B = 100/10 = 10, so S = C-B = 9 SІ = CІ+BІ = (1+(19+0.75))І + (1+(100+0.75))І/100) = 30.9 S/N = 9/5.55 1.62

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If we use Poisson variance (І=N), SІ = 19+1 S/N2 If we ignore uncertainty in background, S/N2.05 If we set a threshold based only on the background, 3 + B = 19.49, a source is barely, but not detected. If there is a downward 4 fluctuation in the background and we ignore the uncertainty on it, S=19-6=13, S/N3

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Upper Limits: Likelihood of the Background
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nB=100, r=10 ln p(nB|B,r) = nB ln(rB) - rB - ln(nB+1)


Upper Limits: Detection Threshold for MLE background


Upper Limits: Detection Threshold for alt background


Upper Limits: Detection Threshold w. background uncertainty


Upper Limits: Detection Probability


Upper Limits: Controlling for Power


Upper Limits


Upper Limits: Adaptive threshold

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If you choose something like S/N as the detection criterion, its value will depend on nS, the counts observed in the source region.


Deconvolution
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Chapter 10.2 of Robinson


Deconvolution: Richardson-Lucy

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Bayesian-Based Iterative Method of Image Restoration, Richardson, W.H., 1972, J. Optical Soc. of America, v62, p55 An iterative technique for the rectification of observed distributions, Lucy, L.B., 1974, AJ 79, 745 Generalized Cross-Validation as a Stopping Rule for the Richardson-Lucy Algorithm, Perry, K.M., and Reeves, S.J., 1994, in The Restoration of HST Images and Spectra II, STScI, Eds. R.J. Hanisch and R.L. White

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Deconvolution: Richardson-Lucy
Suppose you have an observed image O, which is the true image A, convolved with telescope blurring P, and with noise . O = A*P + p(AI|OJ) = p(OJ|AI)p(AI) / p(OJ), with p(OJ)=K p(OJ|AK) p(AK) similarly, p(AI) = J p(AIOL) = L p(AI|OJ)p(OJ) p(AI) = J p(OJ|AI) p(AI) p(OJ) / [K p(OJ|AK) p(AK)] = p(AI) J p(OJ|AI) p(OJ) / [K p(OJ|AK) p(AK)] p(AJ) = aJ/KaK ; p(OJ) = oJ/KoK ; p(OJ|AK) = P Start with aI(0)=1 aI
(r+1) J,K

; KaK=KoK

= aI(r) J oJ PJ,I / K P

J,K

aK

(r)


Deconvolution: Maximum Entropy
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Entropy, S = ­ I pI log p

I I

Throw N events into M bins, nI = Nf

number of ways to get given configuration, g = N!/J nJ! S log g ­
J=1..M fJ

log f

J

If true intensity is FJ, redefine S as S(fJ,FJ) = J [(fJ-FJ) - (fJlogfJ-fJlogFJ)]
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Maximize entropy while minimizing І Q = S - І + J fJІ fJ = FJ exp[-І/fJ - ]


Deconvolution: LIRA

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https://github.com/astrostat/LIRA An Image Restoration Technique with Error Estimates, Esch, D.N., Connors, A., Karovska, M., & van Dyk, D.A., 2004, ApJ 610, 1213 Detecting Unspecified Structure in Low-Count Images, Stein, N.M., van Dyk, D.A., Kashyap, V.L., Siemiginowska, A., 2015, ApJ, accepted(?)

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