On Mon, Mar 18, 2013 at 1:00 PM, Pierre Haessig <pierre.haessig@crans.org> wrote:
Hi Sudheer,

Le 14/03/2013 10:18, Sudheer Joseph a écrit :
Dear Numpy/Scipy experts,
                                              Attached is a script which I made to test the numpy.correlate ( which is called py plt.xcorr) to see how the cross correlation is calculated. From this it appears the if i call plt.xcorr(x,y)
Y is slided back in time compared to x. ie if y is a process that causes a delayed response in x after 5 timesteps then there should be a high correlation at Lag 5. However in attached plot the response is seen in only -ve side of the lags.
Can any one advice me on how to see which way exactly the 2 series are slided back or forth.? and understand the cause result relation better?( I understand merely by correlation one cannot assume cause and result relation, but it is important to know which series is older in time at a given lag.
You indeed pointed out a lack of documentation of in matplotlib.xcorr function because the definition of covariance can be ambiguous.

The way I would try to get an interpretation of xcorr function (& its friends) is to go back to the theoretical definition of cross-correlation, which is a normalized version of the covariance.

In your example you've created a time series X(k) and a lagged one : Y(k) = X(k-5)

Now, the covariance function of X and Y is commonly defined as :
 Cov_{X,Y}(h) = E(X(k+h) * Y(k))   where E is the expectation
 (assuming that X and Y are centered for the sake of clarity).

If I plug in the definition of Y, I get Cov(h) = E(X(k+h) * X(k-5)). This yields naturally the fact that the covariance is indeed maximal at h=-5 and not h=+5.

Note that this reasoning does yield the opposite result with a different definition of the covariance, ie. Cov_{X,Y}(h) = E(X(k) * Y(k+h))  (and that's what I first did !).

Therefore, I think there should be a definition in of cross correlation in matplotlib xcorr docstring. In R's acf doc, there is this mention : "The lag k value returned by ccf(x, y) estimates the correlation between x[t+k] and y[t]. "
(see http://stat.ethz.ch/R-manual/R-devel/library/stats/html/acf.html)

Now I believe, this upper discussion really belongs to matplotlib ML. I'll put an issue on github (I just spotted a mistake the definition of normalization anyway)

You might be interested in the statsmodels implementation which should be similar to the R functionality.