Coherence Function from Spectral Analysis - Signal Processing of Random Physiological Signals - page 208

the “Goodness Fit.” The correlation coefficient squared shows the percentage of variation
in the output or dependent variable that is accounted for from the input or independent
variable. One word of caution when using this comparison is the correlation coefficient
squared r 2 between one dependent and multiple independent variables is that the r 2 does
not represent the linear relationship. There may be linear, quadratic, or cubic relationship
contained within the correlation coefficient squared r 2 term.
The coherence function uses almost identical calculations when comparing two
spectra, where

G xx (of autospectrum 1) is analogous to (18.1),
G yy (of autospectrum 2) is analogous to (18.2), and
G xy (of both cross-spectrums) is analgous to (18.3).

2 , is equivalent to the correlation coefficient squared,
r 2 . It should be noted that the correlation coefficient, r 2 is a single value measure of
“Goodness,” the coherence is a function with values related to frequency analysis of
continuous and/or sampled random data. Therefore, the coherence function is evaluated
at specific analysis frequencies. From (18.6), it can be noted that it is necessary to calculate
the autospectra of both the input and output signals, as well as, the cross-spectra of the
input-output signals.

The coherence function , γ

G xy ( f )

2

2

γ

xy ( f )

= (18.6)

G xx ( f ) G yy ( f )

In Fig. 18.1, a single input-output system is shown with transfer function H ( f ).
The coherence will show causality (unlike the correlation coefficient) between the input
and the output. Another advantage over the correlation or cross correlation, which is
used in signal processing, is that the coherence function is a function of frequency. The
maximum value of one (1) for the coherence function occurs when all of the output
(response) is directly attributed to the input (stimulus), whereas at lowest value, which
is equal to 0, of the coherence function occurs at frequencies where little or no transfer
of energy from input to output is taking place. If an output signal is interfered with by