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I noticed that in signal processing they have this term called cross-covariance. The cross covariance function produces covariances of two functions with different lags. At the center of the vector there are no lags and thus it's just covariance. With one exception though, the cross covariance function is not divided by $N$. The naming is thus very misleading indeed as we are not talking about the same covariance anymore.

This kind of "covariance" rears it's head time and time again in different places, so I was just wondering what is it called?

Here are the exact formulas: \begin{align} {\rm Covariance}(X,Y) &= \frac{Σ(X_n - X_{\rm avg})(Y_n - Y_{\rm avg})}{N} \\ \text{"Cross covariance" covariance}(X,Y), \text{no lags} &= Σ(X_n - X_{\rm avg})(Y_n - Y_{\rm avg}) \end{align}

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  • $\begingroup$ Please deabbreviate DSP. $\endgroup$ – amoeba May 11 '15 at 21:01
  • $\begingroup$ @amoeba Digital signal processing, but I use it as an abbreviation for signal processing in general. $\endgroup$ – Dole May 11 '15 at 21:34
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This is just the sum of cross-products, which is what it should be called.

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    $\begingroup$ Do you have any idea why some people are calling it covariance, and in some cases even correlation? (Correlation is usually covariance / Standard deviation of X * standard deviation. of Y). This seems like a really incorrect/inconsistent terminology. Check wikipedia articles on cross-correlation, cross-covariance and autocorrelation for example. They are also saying that correlation is "Normalized cross-correlation", and using this terminology to define autocorrelation (which is not the usual normalized form). $\endgroup$ – Dole May 11 '15 at 18:29
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    $\begingroup$ On the Wikipedia page for cross-covariance, the formula I see is ${\rm Cov}(X_t, Y_s) = E[(X_t-\mu_t)(Y_s-\nu_s)]$. That notation is for the expected value, which does imply averaging / dividing by N. Outside of that, I don't know why people would use different terms. It would be incorrect, but in context they may want to stress the conceptual continuity. $\endgroup$ – gung May 11 '15 at 18:36
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    $\begingroup$ Below it: " In signal processing, the cross-covariance is often called cross-correlation and is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative time between the signals, is sometimes called the sliding dot product, and has applications in pattern recognition and cryptanalysis." And in the cross-correlation article the autocorrelation is defined to be the cross-product strictly... I think this is just how they do it in DSP. Statistics defenitions are consistent, DSP aren't. $\endgroup$ – Dole May 11 '15 at 20:04

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