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wcoher

(Not recommended) Wavelet coherence

wcoher in not recommended. Use wcoherence instead.

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 character vectors: '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.

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');

collapse all

Wavelet Cross Spectrum

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

${C}_{xy}\left(a,b\right)=S\left({C}_{x}^{*}\left(a,b\right){C}_{y}\left(a,b\right)\right)$

where Cx(a,b) and Cy(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:

$\frac{{|S\left({C}_{x}^{*}\left(a,b\right){C}_{y}\left(a,b\right)\right)|}^{2}}{S\left(|{C}_{x}\left(a,b\right){|}^{2}S\left(|{C}_{y}\left(a,b\right){|}^{2}}$

where Cx(a,b) and Cy(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 coherence is real-valued if you use a real-valued analyzing wavelet, and complex-valued if you use a complex-valued analyzing wavelet.

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.