How to estimate the auto-correlation function value r(k)=E[X(n)X(n-k)]

If X(n) is an available time series, I want to estimate the auto-correlation function value of r(k). The definition of r(k) is r(k)=E[X(n)X(n-k)]. Should I use the command "xcorr(x)"? But I will obtain a sequence, how could I get a value? Is this way right? I define y(n)=x(n-k), and use the xcorr(x,y)? But still I need a value and how to do the time delay process y(n)=x(n-k)?

Answer 1

[Nirmal Gunaseelan]

XCORR will return the autocorrelation values and the associated lags at which these values were calculated. The examples section has one:

ww = randn(1000,1);
[c_ww,lags] = xcorr(ww,10,'coeff');
stem(lags,c_ww)

What do you mean by just one value representing ACF? The sequence of values represent correlation between values at different lag indices (time) and a single value does not make sense.

[Jiazeng Shan]

Yes! The corresponding value in c_ww to the value of lags being 1 is the r(1)=E[X(n)X(n-1)], right? So if I want to find the r(k), I just output the lags containing k and search the corresponding value in c_ww?

[Nirmal Gunaseelan]

That should give you the ACF at just that lag (time index)

[Jiazeng Shan]

Yes, that is what I need. I want the ACF value when lag are 0, 1 and so on. If the definition r(1)=E[X(n)X(n-1)],should I use xcorr(ww,10,'biased') instead of xcorr(ww,10,'coeff')? How to determine the options? Thanks.

[Nirmal Gunaseelan]

Option depends on what you want - the document clearly gives the various options and default (none) should work for most cases.