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The concept of ``**coherency**'' in time-series analysis
is analogous to correlation.
Taking *x*_{t} and *y*_{t} to be time series,
we find that they may have a mutual relationship
which could depend on time delay, scaling, or even filtering.
For example, perhaps *Y*(*Z*) = *F*(*Z*) *X*(*Z*) + *N*(*Z*),
where *F*(*Z*) is a filter and
*n*_{t} is unrelated noise.
The generalization of the correlation concept is to define
coherency by
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(51) |

*Correlation* is a real scalar.
*Coherency* is a complex function of frequency;
it expresses the frequency dependence of correlation.
In forming an estimate of coherency, it is
always essential to simulate ensemble averaging.
Note that if the ensemble averaging were to be omitted,
the coherency (squared) calculation would give

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(52) |

which states that the coherency squared is unity, independent of the data.
Because correlation scatters away from zero,
we find that coherency squared is
biased away from zero.

** Next:** The covariance matrix of
** Up:** CROSSCORRELATION AND COHERENCY
** Previous:** Correlation
Stanford Exploration Project

10/21/1998