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

correlationcross correlationMATLAB

Are the values of xcorr(x,y) in MATLAB correlation values or not? I'm asking this because in MATLAB xcorr(x,y,'coeff') normalizes values. Is it normalizing covariance values to get correlations?

I'm confused if cross correlation values are necessarily between -1 and 1 like Pearson correlation values.

Also, I see from running an example that xcorr(x,y,0,'coeff') != corr(x,y). Could someone explain this?

Best Answer

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').

Related Solutions

Solved – Coefficient for a zero lag cross-correlation different from Pearson correlation r

A simple simulation without the NA values will show that ccf and cor give the same results. So the issue is with how ccf and cor treat NA values.

In your code you use na.pass for ccf and pairwise.complete.obs for cor. The documentation for ccf gives the following warning when using na.pass:

This means that the estimate computed may well not be a valid autocorrelation sequence, and may contain missing values.

Now the option pairwise.complete.obs for cor omits the observations where either of series are NA. So if you want the same behaviour as cor you should do the same. Simple simulation shows that if you use na.omit instead of na.pass the ccf and cor give the same result.

Now the question is how exactly ccf calculates the covariances when na.action=na.pass? This can be seen by inspecting the code of acf, which ccf calls. Here is the offending line:

if (demean) 
        x <- sweep(x, 2, colMeans(x, na.rm = TRUE), check.margin = FALSE)

The default option for demean is TRUE, if it is FALSE it is assumed that the means are zero. It is clear then that for na.pass the mean for the series without the NA values will be calculated from all of the observations, rather than from observations where both of the series are not NA.

So in the end the issue was the same as in the question mentione by @lekshmi dharmarajan. The scaling is different when NA values are treated differently.

Solved – Explaining the difference between Pearson correlation and distance correlation

Distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal dimension.

The population distance correlation coefficient is zero if and only if the random vectors are independent. Thus, distance correlation measures both linear and nonlinear association between two random variables or random vectors.

This is in contrast to Pearson's correlation, which can only detect linear association between two random variables.

Best Answer

Related Solutions

Solved – Coefficient for a zero lag cross-correlation different from Pearson correlation r

Solved – Explaining the difference between Pearson correlation and distance correlation

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