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I am new here. My question is simple. Why doesn't the definition of cross-correlation between two time series include a mean centering? Wikipedia defines the cross correlation of two functions as the following,

$$ \left( f\star g \right)\left[ n \right]=\sum\limits_{m=-\infty }^{\infty }{{{f}^{*}}\left[ m \right]g\left[ m+n \right]} $$

On the other hand if we look at the definition of the auto-correlation function then we see that it is given as:

$$ {{R}_{XX}}\left( s,t \right)=\frac{\mathsf{\mathbb{E}}\left[ \left( {{X}_{t}}-{{\mu }_{t}} \right)\left( {{X}_{s}}-{{\mu }_{s}} \right) \right]}{{{\sigma }_{t}}{{\sigma }_{s}}} $$

Okay one is continuous time and the other is discrete time … but let's assume we can go between the two easily enough by substituting integrals for summations. Now seeing as how one is a generalization of the other, it beats me why the first definition does not include a normalization or mean removal.

My questions are, and forgive me if they are naive (I just can't find a definitive reference, nor are the terms and definitions standard across fields),

  1. Should we mean center each time series individually when we try to find lagged cross correlations between them (i.e. perform time delay analysis)?
  2. What are the differences between the two approaches? From running a few simulations it seems if you look at non-negative time series, if you don't mean center then by the first equation the values obtained will be huge, whereas if you mean-center you have possibility of cancellations and smaller values.
  3. What about normalization by the standard deviation? This is normally required when you want to compare two different time series that take values in different range intervals. But again the Wikipedia definition doesn't include this (though it later defines zero-normalized cross correlation and normalized cross-correlation … without references). When to use one or the other? When to standardize and when not to?

I understand that when the serie are auto-correlated then doing lagged cross correlations is more involved because we need to take care of spurious correlations … does this have to do with the definitions? For example is this assumed? I don't see it being stated. In more detail, are the definitions with and without standardization equivalent for strongly stationary series but essentially different for stationary series?

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In Wikipedia the cross-correlation in time series analysis is defined exactly like you wanted, i.e. mean centered and normalized. You're referring to the definition in signal processing.

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  • $\begingroup$ Right, I just want to understand the difference between the two; because it seems to me over finite intervals of summation and integration it really amounts to different things. Here is a well cited article: pnas.org/content/112/17/E2235 that seems to use lagged cross-correlations without standardization; where the entire argument rests on lagged cross-correlations exhibiting a single peak. From my investigations they exhibit multiple peaks if you standardize them first. $\endgroup$ – ITA Dec 18 '18 at 16:35
  • $\begingroup$ @ITA, it's basically the same thing ignoring the constant shift when the mean is not zero. $\endgroup$ – Aksakal Dec 18 '18 at 16:38
  • $\begingroup$ I don't think so over discrete time. Assume $f$ and $g$ were standardized and consider the equation where we add back their means: $$ \int\limits_{-\infty }^{\infty }{\left( f+{{\mu }_{f}} \right)}\left[ t+\tau \right]\left( g+{{\mu }_{g}} \right)\left[ t \right]dt $$ When expanding an integrating over the real line the shifts don't change the mean, so you are right, there is just a constant shift. But over discrete time and finite summation intervals, this is not true. The "constants" depend on the time lag. $\endgroup$ – ITA Dec 18 '18 at 16:47

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