## Momentary ARMA coherence

Purpose
Description
Tips
Macro Synopsis
Modules
Related Functions
References

### Purpose

Momentary ARMA coherence

### Description

"Momentary ARMA coherence" computes the time-dependent ARMA coherence of two signals x and y within a given frequency interval estimated by an adaptive bivariate ARMA model. In contrast to Coherence, "Momentary ARMA coherence" is more suitable for the analysis of the stochastic coupling of instationary signals.

This function yields a pseudo 2D output signal of the momentary ARMA coherence displaying time (x-axis) versus frequency (y-axis) with the coherence values given on the z-axis.

The coherence function is given by

using the coherence matrix obtained by the estimation of the bivariate ARMA model parameters A and B.

The estimation is computed by a Least Mean Squares (LMS) approach according to
and
with being the composed signal vector.

Parameters of "Momentary ARMA coherence" are:

• the lower frequency f0 in reciprocal x-axis units,
• the upper frequency f1 in reciprocal x-axis units,
• the order P the of autoregressive (AR) process,
• the order Q of the moving average (MA) process,
• the adaptation time ta for the variance estimator (in units of the x-axis),
• the adaptation factor f for parameter estimation (see above),
• the number of points timeres for the plot resolution along the time axis and
• the number of points freqres for the plot resolution along the frequency axis.

Warning: Note that if the input signals have different x-axis scales (sampling periods), the signal with the largest scale will be adapted automatically by interpolating between successive sample points. The type of interpolation can be set in the Basic Options menu.

### Tips

The adaptation factor f usually is around 0.01. With the choice of a relatively high f (say, 0.03), a quick parameter adaptation is achieved enabling the model to react more quickly after rapid structure changes in the input signals, but this will also lead to an estimation sequence that is less smooth and tends to be less robust on the other hand.

### Macro Synopsis

z = MomARMAcoherence([x,y],f0,f1,P,Q,ta,f,timeres,freqres,N);
signal x, y, z;
float f0, f1;
int P, Q;
float ta;
int timeres, freqres, N;

Note that z is a 2D signal.

Spectral

### Related Functions

Coherence, Momentary ARMA bandpower, ARMA dependence, Momentary ARMA spectrum, Momentary bandpower, Momentary frequency, Momentary mean, Momentary power.

### References

Haykin [41], Schack et al. [40]