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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:
\begin{displaymath}
c \eq { ({\bf x} \cdot {\bf y})
\over
\sqrt{ ({\bf x} \cdot {\bf x}) ({\bf y} \cdot {\bf y}) } }\end{displaymath} (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:
\begin{displaymath}
c \eq {\E(xy) \over \sqrt{ \E(x^2)\, \E(y^2) } }\end{displaymath} (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
\begin{displaymath}
\hat{c} \eq {xy \over \vert x\vert \, \vert y\vert} \eq \pm 1\end{displaymath} (50)
regardless of what value the random number x or the random number y should take. For any n, the sample correlation $\hat{c}$ 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 $\hat{c}$ of about $1/\sqrt{n}$ for samples of size n. Bias, no doubt, accounts for many false ``discoveries,'' since cause-and-effect is often inferred from correlation.


next up previous print clean
Next: Coherency Up: CROSSCORRELATION AND COHERENCY Previous: CROSSCORRELATION AND COHERENCY
Stanford Exploration Project
10/21/1998