<html><body><div style="color:#000; background-color:#fff; font-family:times new roman, new york, times, serif;font-size:12pt"><div></div><div style="font-family: 'times new roman', 'new york', times, serif; font-size: 12pt;"><div style="font-family: 'times new roman', 'new york', times, serif; font-size: 12pt;"><div dir="ltr"><font size="2" face="Arial"><b><span style="font-weight:bold;">Thank you Pierre,</span></b></font></div><div dir="ltr"><font size="2" face="Arial"><span><span style="font-weight: bold;"> </span>It appears the numpy.correlate uses the frequency domain method for getting the ccf. I would like to know how serious or exactly what is the issue with normalization?. I have computed cross correlation using the function and interpreting the results based on it. It will be helpful if you could tell me if there is a significant bug in the
function</span></font></div><div dir="ltr"><font size="2" face="Arial"><span>with best regards,</span></font></div><div dir="ltr"><font size="2" face="Arial"><span>Sudheer</span></font></div><div dir="ltr"><font size="2" face="Arial"> <b><span style="font-weight:bold;">From:</span></b> Pierre Haessig <pierre.haessig@crans.org><br> <b><span style="font-weight: bold;">To:</span></b> numpy-discussion@scipy.org <br> <b><span style="font-weight: bold;">Sent:</span></b> Monday, 18 March 2013 10:30 PM<br> <b><span style="font-weight: bold;">Subject:</span></b> Re: [Numpy-discussion] Numpy correlate<br> </font> </div> <br><div id="yiv47526519">
<div>
Hi Sudheer,<br>
<br>
Le 14/03/2013 10:18, Sudheer Joseph a écrit :
<blockquote type="cite">
<div style="font-family: 'times new roman', 'new york', times, serif; font-size: 12pt;"><span>Dear Numpy/Scipy experts,</span></div>
<div style="font-family: 'times new roman', 'new york', times, serif; font-size: 16px; color: rgb(0, 0, 0); background-color: transparent; font-style: normal;"><span>
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)</span></div>
<div style="background-color:transparent;"><span>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.</span></div>
<div style="background-color:transparent;"><span>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.</span></div>
</blockquote>
You indeed pointed out a lack of documentation of in
matplotlib.xcorr function because the definition of covariance can
be ambiguous.<br>
<br>
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.<br>
<br>
In your example you've created a time series X(k) and a lagged one :
Y(k) = X(k-5)<br>
<br>
Now, the covariance function of X and Y is commonly defined as :<br>
Cov_{X,Y}(h) = E(X(k+h) * Y(k)) where E is the expectation<br>
(assuming that X and Y are centered for the sake of clarity).<br>
<br>
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.<br>
<br>
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 !).<br>
<br>
<br>
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]. "<br>
(see
<a rel="nofollow" class="yiv47526519moz-txt-link-freetext" target="_blank" href="http://stat.ethz.ch/R-manual/R-devel/library/stats/html/acf.html">http://stat.ethz.ch/R-manual/R-devel/library/stats/html/acf.html</a>)<br>
<br>
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)<br>
<br>
<br>
<br>
Coming back to numpy :<br>
There's a strange thing, the definition of numpy.correlate seems to
give the other definition "z[k] = sum_n a[n] * conj(v[n+k])" (
<a rel="nofollow" class="yiv47526519moz-txt-link-freetext" target="_blank" href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.correlate.html">http://docs.scipy.org/doc/numpy/reference/generated/numpy.correlate.html</a>)
although its usage prooves otherwise. What did I miss ?<br>
<br>
best,<br>
Pierre<br>
</div>
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