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Material Type: Notes; Professor: Zimmerman; Class: EXPERIMENTAL PHYSICS; Subject: Physics; University: University of Colorado - Boulder; Term: Unknown 1989;

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Download Covariance and Correlation - Slides | PHYS 2150 and more Study notes Experimental Physics in PDF only on Docsity! PHYSICS 2150 LABORATORY Prof. Eric D. Zimmerman Lecture 5 February 12, 2008 ANNOUNCEMENT • We will have a sixth lecture next week. • The problem set will be assigned then, and due a week later. COVARIANCE AND CORRELATION • Take a desired measurement q(x,y) • Error propagation says • Making two assumptions here: errors are gaussian and x and y are uncorrelated. • If there is a correlation, the error on q can be either larger or smaller than our estimate • Sometimes (epidemiology, other complex systems) the correlation itself is something interesting to learn Covariance and correlation Take a generic function q as a function of x and y: q(x, y) Ou error ropag tion currently states !q = ! " !q !x!x #2 + $ !q !y!y %2 This was justified under the assumption of x and y uncorrelated What about if x and y are correlated? Example from K0 mass lab February 7, 2006 Physics 2150 Lecture 4 – p. 11 AN EXAMPLE OF CORRELATED VARIABLES: THE K0 MASS • Incoming K+ hits stationary neutron, producing proton and K0 • The K0 travels a short distance and decays to π+π− • Need to measure angle θT between pions • Also need to measure θ±, angles between pions and K0 • Measuring θT is easy, but K0 direction can be hard if its path is short AN EXAMPLE OF CORRELATED VARIABLES: THE K0 MASS • If direction of the blue line is wrong, then θ+ and θ− will be wrong by equal amounts but in opposite directions • θT = θ+ + θ− will still be OK • Thus we can say that our measurements of θ+ and θ− will be correlated (actually anticorrelated). COVARIANCE VS. CORRELATION • The covariance σxy can be normalized to create a correlation coefficient r=σxy / σxσy • r can vary between −1 and 1. r=0 indicates that the variables are uncorrelated; |r|=1 means the variables are completely correlated (i.e. knowing the value of x completely determines the value of y). • Sign indicates direction of covariance: positive means that x>xmean indicates likely y>ymean; negative means x>xmean indicates likely y<ymean. CORRELATION AS A MEASUREMENT • Sometimes the correlation itself is interesting. • In order to establish the significance of the correlation, need to ask what r is, as well as how many measurements were taken to establish it. • For given r and N, look in table (Taylor Appendix C) to find out probability of randomly measuring a particular correlation value • Random probabilities <5% or <1% are generally considered significant depending on the application. • Caution: when you look at 20 correlations and their random probabilities, you expect one to be <5% just by chance! CORRELATION: A NON-
PHYSICS EXAMPLE
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COUNTING EXPERIMENTS: THE POISSON DISTRIBUTION • Start with a large amount of radioactive material with a very long half-life (compared to the experiment) • Measure the decays with a detector at a fixed distance • Measure the number of decays in five 100-second intervals • Should the five intervals all have the same number of decays? • Should there be a consistent trend? COUNTING EXPERIMENTS • Should there be a consistent trend? • NO! The long half-life assures that over the time of the experiment, the decay rate isn’t changing significantly. • Should the five intervals all have the same number of decays? • NO! The decay is a random process. A POISSON PROCESS • Assume a decay rate of 0.01 decays/sec • On average, will get 1 decay/100 seconds. • Sometimes you will get zero • Sometimes you will get two • Rarely you will get three or more • What is the probability of getting exactly n decays in 100 seconds? • Answer: Poisson distribution. Applies to any discrete process where events occur randomly at a constant rate.