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## Correlation

Correlation" is a concept similar to cosine. A cosine measures the angle between two vectors. It is given by the dot product of the two vectors divided by their magnitudes:
 (48)
This is the sample normalized correlation we first encountered on page  as a quality measure of fitting one image to another.

Formally, the normalized correlation is defined using x and y as zero-mean, scalar, random variables instead of sample vectors. The summation is thus an expectation instead of a dot product:
 (49)

A practical difficulty arises when the ensemble averaging is simulated over a sample. The problem occurs with small samples and is most dramatically illustrated when we deal with a sample of only one element. Then the sample correlation is
 (50)
regardless of what value the random number x or the random number y should take. For any n, the sample correlation scatters away from zero. Such scatter is called bias." The topic of bias and variance of coherency estimates is a complicated one, but a rule of thumb seems to be to expect bias and variance of of about for samples of size n. Bias, no doubt, accounts for many false discoveries,'' since cause-and-effect is often inferred from correlation.

Next: Coherency Up: CROSSCORRELATION AND COHERENCY Previous: CROSSCORRELATION AND COHERENCY
Stanford Exploration Project
10/21/1998