Coherency

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

$$C \eq {\E\left[ X \left( {1 \over Z}\right) \, Y(Z) \right] \over \sqrt{\E(\overline {X} X)\, \E(\overline {Y} Y) } }$$ (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

$$\vert C\vert^2 \eq \overline{C} C \eq {(\overline{\overline{... ...(\overline{X} Y) \over (\overline{X} X) (\overline{Y} Y)} \eq 1$$ (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.