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``**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