wcoher - Wavelet coherence

Syntax

WCOH = wcoher(Sig1,Sig2,Scales,wname) WCOH = wcoher(...,Name,Value) [WCOH,WCS] = wcoher(...) [WCOH,WCS,CWT_S1,CWT_S2] = wcoher(...) [...] = wcoh(...,'plot')

Description

WCOH = wcoher(Sig1,Sig2,Scales,wname) returns the wavelet coherence for the input signals Sig1 and Sig2 using the wavelet specified in wname at the scales in Scales. The input signals must be real-valued and equal in length.

WCOH = wcoher(...,Name,Value) returns the wavelet coherence with additional options specified by one or more Name,Value pair arguments.

[WCOH,WCS] = wcoher(...) returns the wavelet cross spectrum.

[WCOH,WCS,CWT_S1,CWT_S2] = wcoher(...) returns the continuous wavelet transforms of Sig1 and Sig2.

[...] = wcoh(...,'plot') displays the modulus and phase of the wavelet cross spectrum.

Input Arguments

`Sig1`	A real-valued one-dimensional input signal. `Sig1` is a row or column vector.
`Sig2`	A real-valued one-dimensional input signal. `Sig2` is a row or column vector.
`Scales`	`Scales` is a vector of real-valued, positive scales at which to compute the wavelet coherence.
`wname`	Wavelet used in the wavelet coherence. `wname` is any valid wavelet name.

Name-Value Pair Arguments

`'asc'`	Scale factor for arrows in quiver plot. `wcoher` represents the phase using `quiver`. `asc` corresponds to the `scale` input argument in `quiver`. Default: 1
`'nas'`	Number of arrows in scale. Together with the number of scales, `nas` determines the spacing between the `y` coordinates in the input to `quiver`. The `y` input to `quiver` is `1:length(Scales)/(nas-1):Scales(end)` Default: 20
`'nsw'`	Length of smoothing window in scale. `nsw` is a positive integer that specifies the length of a moving average filter in scale. Default: 1
`'ntw'`	Length of smoothing window in time. `ntw` is a positive integer that specifies the length of a moving average filter in time. Default: `min[20,0.05*length(Sig1)]`
`'plot'`	Type of plot. `plot` is one of the following strings: `'cwt'` Displays the continuous wavelet transforms of signals 1 and 2. `'wcs'` Displays the wavelet cross spectrum. `'wcoh'` Displays the phase of the wavelet cross spectrum. `'all'` Displays all plots in separate figures.

Output Arguments

`WCOH`	Wavelet coherence.
`WCS`	Wavelet cross spectrum.
`CWT_S1`	Continuous wavelet transform of signal 1.
`CWT_S2`	Continuous wavelet transform of signal 2.

Definitions

Wavelet Cross Spectrum

The wavelet cross spectrum of two time series, x and y is:

where C_x(a,b) and C_y(a,b) denote the continuous wavelet transforms of x and y at scales a and positions b. The superscript * is the complex conjugate and S is a smoothing operator in time and scale.

For real-valued time series, the wavelet cross spectrum is real-valued if you use a real-valued analyzing wavelet, and complex-valued if you use a complex-valued analyzing wavelet.

Wavelet Coherence

The wavelet coherence of two time series x and y is:

For real-valued time series, the wavelet coherence is real-valued if you use a real-valued analyzing wavelet, and complex-valued if you use a complex-valued analyzing wavelet.

Examples

Wavelet coherence of sine waves in noise with delay:

t  = linspace(0,1,2048);
x = sin(16*pi*t)+0.5*randn(1,2048);
y = sin(16*pi*t+pi/4)+0.5*randn(1,2048);
wname  = 'cgau3';
scales = 1:512;
ntw = 21;	% smoothing parameter
% Display the modulus and phased of the wavelet cross spectrum.
wcoher(x,y,scales,wname,'ntw',ntw,'plot');

Sine wave and Doppler signal:

t  = linspace(0,1,1024);
x = -sin(8*pi*t) + 0.4*randn(1,1024);
x = x/max(abs(x));
y = wnoise('doppler',10);
wname  = 'cgau3';
scales = 1:512;
ntw = 21;	% smoothing parameter
% Display of the CWT of the two signals.
wcoher(x,y,scales,wname,'ntw',ntw,'plot','cwt');
% Display of the wavelet cross spectrum.
wcoher(x,y,scales,wname,'ntw',ntw,'nsw',1,'plot','wcs');
% Display of the modulus and phased of the wavelet cross spectrum.
wcoher(x,y,scales,wname,'ntw',ntw,'plot');

References

Grinsted, A, J.C. Moore, and S. Jevrejeva. "Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics. 11, 2004, pp. 561-566.

Torrence. C., and G. Compo. "A Practical Guide to Wavelet Analysis". Bulletin of the American Meteorological Society, 79, pp. 61-78.