Interpreting xcorr results compared to corrcoef

I am having trouble understanding why I am getting different outputs when using coeff and a normalized xcorr, specifically at 0 lags.

Shouldn't 0 lags produce the same r value when using an xcorr compared to coeff since the time series is matched up in both cases?

Note - I am using "coeff" because the two sets of data are equal in length, but the ampltitudes are much different.

Below illustrates the example:

I have two sets of data (A and B) that are 1000 datapoints each.

I use corrcoef(A,B) and then I use xcorr(A,B,500,'coeff").

corrcoef outputs a single value that is rather low such as r = 0.15.

xcorr outputs r values at lags -500 to 500, and they are all higher than r = 0.15.

However my question is, shouldn't r at 0 lag be = 0.15 since the data is already lined up in time?

Answer 1

[David Goodmanson]

Hi Jonathan,

For the input column vectors, corrcoeff subtracts the mean off of each one and normalizes each to be a unit vector. xcorr with the 'coeff' option normalizes, but doesn't subtract off the mean.

a = (1:1000)';
b = mod((a.^(4/3)),1000);

corrcoef([a b])
ans =
    1.0000    0.1745
    0.1745    1.0000

xcorr(a,b,0,'coeff')    % zero lag only
ans =
    0.7865

aa = a-mean(a);
bb = b-mean(b);

xcorr(aa,bb,0,'coeff')
ans =
    0.1745              % agrees

[JM]

Hey David, thank you for the explanation this is very helpful.

[Adam Danz]

^^ Agreed (+1)

[Shuya Zhong]

Super helpful and solved my doubts! Thanks!

[David Goodmanson]

Hello SZ, thanks for the comment. it's good to hear that a blast from the past still might be useful.

[zmkal]

Hi @David Goodmanson, still helpful but has raised a few new questions for me.

Maybe someone or you can help?

First, I calculated an autocorrelation function using xcorr:

[akf, lags] = xcorr(A-mean(A), 'coeff');

and secondly via corrcoef:

B = A;

for i = 1:length(A)-1

B = circshift(B,1);

R = corrcoef(A,B);

corr(i,1) = R(2,1);

end

Of course, the first variant also shifts the function in the negative direction and the second variant only in the positive direction in the way I implemented it. However, from what I read above, I would have expected the functions to look the same in the range of the positive shift?

But they do not ....