# Multi-signal coherence test

Is there an accepted measure for multi-signal coherence?

The signals are time-domain EEG voltage readings from various parts of the scalp. I was working with magnitude-squared coherence, but it only tests for coherence between two signals.

What I would like is some sort of measure for a general set of signals, if that makes sense.

• The problem is that coherence is a measure of distance, and there's no single "best" way to generalize a distance to describe the similarity between a set of points. Which one is best suited for you depends entirely on your requirements. What properties would your ideal measure have? When would it be small and when would it be large? When would it be negative? – Andy Jones Dec 23 '14 at 12:05
• Are you looking for a scalar value? Here's a few metrics that look at 'map strength', but they aren't coherence based and they obviously lose a lot of information. I haven't seen any in common practice except for Global Field Power. It wouldn't suffice to choose a seed channel, and show a topographic plot of coherence? It might be useful to show what you're trying to do. Also there is some debate over using raw coherence as an EEG metric, as shown here. – Malz Dec 28 '14 at 20:12

What you're looking for is a cross spectral analysis. These notes show how it works for a bivariate case, but it's easy to apply this to multivariate series.

You start with multivariate time series:$x_t=(x_{1t},\dots,x_{nt})$. You define the matrix of autocovariate functions

$\Gamma(j)=\gamma_{mk}(j)\equiv cov(x_{mt},x_{k,t-j})$,

then the cross spectrum matrix is

$f(\omega)=f_{mk}(\omega)\equiv\frac{1}{2\pi}\sum_{j=-\infty}^\infty e^{-i\omega j}\gamma_{mk}(j)$.

From here you go on and define cross periodogram, cross spectral densities and all the usual stuff. For instance, the squared coherence is:

$\rho_{mk}=\frac{|f_{mk}(\omega)|^2}{f_{mm}(\omega)f_{kk}(\omega)}$

Similarly, you can get cross phases.