correlation stuff

Correlation Stuff PhL 3.22.13 Note that page numbering is turned on in this template. I have now added a version of this text block near the start of Section 35. Expectation, correlation and covariance. In the field of statistics, one can imagine a "random variable" Am associated with position m in the pulse train, and this variable can take certain values ak (perhaps 0 and 1). If all pulse trains in our ensemble of pulse trains count the same in our average (they do), then one can write <aman> as E(AmAn) = (1/I) !Syntax Error, Iaimain in the sense of E(XY) = (1/I) !Syntax Error, Ixiyi where E(q) means the expectation value of quantity q. This particular expectation value has a special name: it is called the correlation of X and Y. So in fact we have <aman> = E(AmAn) = (1/I) !Syntax Error, Iaimain ≡ corr(AmAn) // correlation of Am and An Two other expectation values of interest are these, the mean and the covariance, E(Am) = (1/I) !Syntax Error, Iaim = <am> // mean of Am E( [Am-<am>][ [An-<an>]) = (1/I) !Syntax Error, I[aim - <am>] [ain- <an>] = cov(AmAn) // covariance of Am and An If it happens that <am> = 0 for a certain pulse train, then covariance and correlation are the same. ___________________________________________________________________________________ The following path did not seem to work out, it is replaced with cross-correlation discussion in Spectral. But I will keep it alive here. Autocorrelation. Consider again this equation from above, E(XY) = (1/I) !Syntax Error, Ixiyi = corr(XY). Suppose there are I = 2N+1 pulse trains in our ensemble, and we choose to enumerate them with index i in the range (-N,N) so then E(XY) = !Syntax Error, Ixiyi = corr(XY). We are summing over a set of (2N+1) random "events" (pulse trains in example above). Random variable X takes value xi in the ith random event. If x can take a continuous set of values, then we can enumerate them by some continuous parameter s which by fiat we require to lie in a range (-B,B). Now we have a function x(s). Then think of xi = x(si). Divide the s parameter space into (2N+1) pieces of width Δs so that si = iΔs where Δs = (2B)/(2N+1). The above then becomes E(XY) = !Syntax Error, IΔs x(i Δs) y(i Δs) = corr(XY). Now let N→∞ so Δs → 0 and we get E(XY) = !Syntax Error, Ids x(s) y(s) = corr(XY). Now define EA(XY) ≡ 2A E(XY) and corrA(XY) = 2A corr(XY) to get EA(XY) = !Syntax Error, Ids x(s) y(s) = corrA(XY) Now take the limit A→∞ to get E(XY) = !Syntax Error, I ds x(s) y(s) = corr(XY) where E(XY) ≡ E∞(XY) and the same for corr(XY). What does this mean? We have an infinite number of independent random events labeled by continuous variable s (instead of discrete i). In event s, random variable X takes some value x(s) and random variable Y takes value y(s). The integral is then the correlation between X and Y. Now imagine that the value xi = x(ti) where x(t) is a continuous function. And let Δt ≡ (2N+1)-1. Then E(XY) = !Syntax Error, IΔt x(ti)y(ti) = corr(XY). Now leet Suppose xi could take a continuous set of values in range (-N,N), so we would then call it x(t). Then ignoring the factor (1/I) we might write the above as E(XY) = !Syntax Error, Idt x(t)y(t) = corr(XY). Letting I go to infinity, we could write this as (see Appendix A regarding δ(0)), [2πδ(0)] E(XY) = !Syntax Error, Idt x(t)y(t) = [2πδ(0)]corr(XY). We could then redefine our notation so this reads E(XY) = !Syntax Error, Idt x(t)y(t) = corr(XY) So this integral gives the correlation between X which takes values x(t) and Y which takes values y(t). Now suppose y(t) = x(t+a) = xa(t). Then we have E(XXa) = !Syntax Error, Idt x(t) xa(t) = corr(XXa) = !Syntax Error, Idt x(t) x(t+a) This then is the correlation between x(t) and a time-shifted copy of itself. Since the same function appears twice in the integral, this integral is called the auto-correlation function of x(t) associated with a shift by amount a. Looking back now at (32.1) and changing its variable rx(a) ≡ !Syntax Error, I dt x(t) x(t + a) (32.1) we see that our autocorrelation function rx(a) = E(XXa) = corr(XXa) .