How to calculate the covariance of two vectors using wcoherence?

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Jakob Sievers on 4 Nov 2016
Hi there
How does one calculate the covariance of two vectors from the cross-spectrum outputs of the wcoherence function?
Below is a code example of my problem. My first thought was to simply perform a double integral over the wavelet cross-spectrum, wcs, using trapz (see cc_wav in the code) but that does not give an answer similar to the covariance output (see resulting figure title).
What am I missing here?
/Jakob
% make data (similar to "doc wcoherence")
rng default;
t = 0:0.001:2;
x = cos(2*pi*10*t).*(t>=0.5 & t<1.1)+cos(2*pi*50*t).*(t>= 0.2 & t< 1.4)+0.25*randn(size(t));
y = sin(2*pi*10*t).*(t>=0.6 & t<1.2)+sin(2*pi*50*t).*(t>= 0.4 & t<1.6)+ 0.35*randn(size(t));
% calculate cross-spectrum
[~,wcs,f,coi] = wcoherence(x,y,1/0.001);
wcs=real(wcs);
% covariances
cc_cov=cov(x,y);cc_cov=cc_cov(2);
cc_wav=trapz(t,trapz(f,wcs)); %????!
% plot results
figure
h = pcolor(t,log10(f),wcs);
h.EdgeColor = 'none';
ax = gca;
hold on;
plot(ax,t,log10(coi),'k--','linewidth',2);
colorbar;
ax.XLabel.String='Time (s)';
ax.YLabel.String='Logarithmic frequency';
ax.Title.String = {'Wavelet Cospectrum';['cov(x,y)=',num2str(cc_cov),' | cov_{wavelet}=',num2str(cc_wav)]};