Lecture 24. Introduction to Error Analysis. Experimental Nuclear Physics PHYS 741

Transcription

1 Lecture 24 Introduction to Error Analysis Experimental Nuclear Physics PHYS 741 References and Figures from: - Bevington Data Reduction and Error Analysis 1

2 Seminar Announcement Friday, December 5th Susan Gardner, University of Kentucky Shedding Light on Dark Matter: How Faraday Rotation Can Limit a Dark Magnetic Moment 2:30 pm; 5280 Chamberlin; 2

3 Next Week No lectures next week Use next week to prepare your course presentations 3

4 Course Project All talks are to be posted by Friday, December 12, 2008, 5pm CST Please post talks (in PDF or PPT) + references (in PDF) if you have them on a website or them to me so that I can download them and review them before Dec UW provides MyWebSpace. You can post files there. Final presentations will be on December 15, pm and December 16, 9am - noon. 4

5 Course Project Tuesday, Dec 15 afternoon/evening pm Quark-Gluon Plasma - Tobias Binning Strange Quarks in Protons and Neutron Charge Distribution - Michael McFarlane Reaccelerated Beams - Greg Severin Origin of the Matter/Antimatter Asymmetry - Larissa Ejzak Tuesday, Dec 16 morning am Underground Physics - Bryce Littlejohn Solar Neutrino Problem - Gabriel Mengin Dark Matter - Adam Dally Relic Supernova Neutrino Background - Annie Malkus theta13 and Long Baseline Neutrino Oscillation Experiments - Christine Lewis 5

6 Course Project Evaluations Some criteria to keep in mind: are key experimental and theoretical concepts introduced and understood (at graduate level) why is the topic relevant? what new physics can we learn? critical discussion of experiment results: what are the limitations? what are the assumptions? explain the figures and data you show clarity of presentation 6

7 Histogram of Measurements 7

8 Gaussian/Normal Distribution /2 8

9 Covariance Covariance provides a measure of the strength of the correlation between two or more sets of random variates. The covariance for two random variates X and Y, each with sample size N, is defined by the expectation value If the variables are correlated in some way, then their covariance will be nonzero. If cov(x,y)>0 then Y tends to increase as X increases, and if cov(x,y)<0, then Y tends to decrease as X increases. Note that while statistically independent variables are always uncorrelated, 9

10 A Radioactive Source Experiment 10

11 Measurements and Fit: Good or Bad? 11

12 Histogram of Data & Poisson Distribution in Each Bin histogram=sample data line= parent distribution Poisson based on sample distribution 12

13 Histogram of Data & Poisson Distribution in Each Bin histogram=sample data line= parent distribution Poisson based on parent distribution 13

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15 Chisquare Distribution k= number of degrees of freedom 15

16 Asymmetric Sample Distribution solid curve = two Gaussian curves 16

17 One-parameter Confidence Level 17

18 Multiparameter Confidence Levels allow all parameters to vary hold some parameters fixed, vary only subset 18

19 Feldman&Cousin what are the issues they deal with? what processes is it applicable to? how is it done? 19

20 Feldman&Cousin what are the issues they deal with? measurements & limits coverage of confidence intervals how to construct confidence intervals what processes is it applicable to? Poisson process with background Gaussian errors with bounded physical region how is it done? 20

21 Confidence Intervals: Measurements and Limits rate or flux or # of events x confidence interval (CL=68.3%) confidence interval (CL=99%)

22 Issue of Coverage Confidence intervals undercover Measurement pretends to be more accurate than it actually is Correct coverage Confidence intervals overcover (i.e. are too conservative) Reduced power to reject wrong hypotheses Proper coverage can be tested by Monte Carlo simulations

23 Flip-Flopping The flip-flopping attitude (example): We will state a measurement with a 1σ error (i.e. CL=68.3%) if the measurement result is above mσ, and an 99% CL upper limit otherwise. Flip-flopping between measurements and upper limits with different confidence levels spoils the coverage of the stated confidence intervals Easy to show with a toy Monte Carlo

24 Classical Confidence Intervals 24

25 Bayesian Interval 25

26 Example 26

27 Feldman & Cousins Approach Provides confidence intervals that change smoothly from upper limits to measurements User just needs to decide for a confidence level Flip-flopping problem is solved Uses Neyman s construction and a Likelihood Ratio to decide what values are included into confidence intervals

28 (Frequentist) Definition of the confidence interval for the measurement of a quantity x: If the experiment were repeated and in each attempt a confidence interval is calculated, then a fraction α of the confidence intervals will contain the true value of x (called µ). A fraction 1-α of the confidence intervals will not contain µ. Note: Experiments must not be identical 28

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