Solved – Cross-correlation in Matlab

cross correlationeconometricsMATLABself-studytime series

What is the difference between:

1) xcov(x,y,10,'unbiased')/sqrt(xcov(x,x,0,'unbiased')*xcov(y,y,0,'unbiased'));
2) xcorr(x,y,10,'unbiased');
3) [A, B] = crosscorr(x,y,10);

I think (but I am not sure) that "crosscorr" removes the means and "xcorr" doesn't, but i don't get why the first one gives a completely different answer compared to the others. Am I using a wrong formula in 1)?

Thank you very much

Best Answer

(For binaural hearing research), I have never used crosscorr(), only xcorr() with the number of lags (i.e. the amount you want to "shift" one of the signals in each direction to calculate the cross correlation coefficient at each point -- the measure of similarity) equal to the duration of the signal (in seconds) times the sampling rate of your signal.

however, it is a bit more difficult to eyeball an interpretation of xcorr(), so I often use a normalized xcorr() function called ncorr(). In GNU Octave normalized xcorr can be called by passing the "coeff" option to xcorr2() in the signal package or for Matlab the variety of normalization options for the xcorr function: http://www.mathworks.com/help/signal/ref/xcorr.html

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Solved – Cross Correlation (xcorr in MATLAB) vs Pearson Correlation (corr in MATLAB)

The function xcorr() is used for calculating the time cross correlation of time series (Or any indexed signal).
The corr() function is all about calculating the correlation of so supposedly 2 random variables.

While the 2 coincide if used correctly and wisely pay attention that they basically normalize differently the operation (Hence you don't get what you expect).

In code:

vA = randn([10, 1]);
vB = randn([10, 1]);
xcorr(vA, vB, 0. 'coeff')
ans =

    0.3070
A.' * vB / sqrt((vA.' * vA) * (vB.' * vB))

ans =

    0.3070
corr(vA, vB)

ans =

    0.3425
vAA = vA - mean(vA);
vBB = vB - mean(vB);
xcorr(vAA, vBB, 0, 'coeff')

ans =

    0.3425
corr(vA, vB)

ans =

    0.3425

As you can see from the code above what's you missing is that corr() removes the mean of the data while xcorr() doesn't.

So corr(vA, vB) is equivalent of xcorr(vA - mean(vA), vB - mean(vB), 0, 'coeff').

Solved – How to calculate the product of Gaussian variables in Matlab

You made a mistake in the derivation. Let's start from this integral: $$ p(x) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \delta(ab-x) p(a) p(b) da ~ db $$ To remove the $\delta$, you need to make a change of variable from $b$ to $c=ab$, for which $dc = a ~ db$. The integral becomes: $$ p(x) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \delta(c-x) p(a) p(c/a) \frac{1}{|a|} da ~ dc \\ = \int_{-\infty}^{\infty} p(a) p(x/a) \frac{1}{|a|} da $$ When $p(a)$ is standard normal, $p(x) = \frac{1}{\pi} {\rm BesselK}(0,|x|)$. You can compute this in Matlab with the code besselk(0,abs(x))/pi.

Best Answer

Related Solutions

Solved – Cross Correlation (xcorr in MATLAB) vs Pearson Correlation (corr in MATLAB)

Solved – How to calculate the product of Gaussian variables in Matlab

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