In optics¹, we have a notion of coherence, that's defined as a normalised correlation/cross-correlation/autocorrelation function: Simplifying notation from the linked Wiki page we can write the (un-normalised, first order) correlation function $$G^{(1)} = \left\langle XY\right\rangle,$$ where I'm using $\left\langle\ldots\right\rangle$ to represent the expectation value, which can then be normalised as $$g^{(1)} = \frac{\left\langle XY\right\rangle}{\sqrt{\left\langle X^2\rangle\langle Y^2\right\rangle}},$$ where $X$ and $Y$ are some matrices/operators/variables etc.

All seems well and good... Except the problem was already mentioned in the first sentence - the correlation between $X$ and $Y$ is defined as $$g = \frac{\left\langle XY\right\rangle - \left\langle X\rangle\langle Y\right\rangle}{\sqrt{\left\langle X^2\right\rangle - \left\langle X\right\rangle^2}\sqrt{\left\langle Y^2\right\rangle - \left\langle Y\right\rangle^2}},$$ which is the normalised covariance $$G = \left\langle XY\right\rangle - \left\langle X\rangle\langle Y\right\rangle.$$

So, how do we reconcile these similar yet different definitions? More importantly, how do you know which is the right one to use? In order to tell how correlated two sets of data taken from the expectation of two different measurements of a signal are.

Initially, it seemed like the former might apply to operators and matrices in physics, while the latter applies to variables in statistics. However, there are a couple of issues with this - in taking the expectation of an operator, it instead becomes a continuous variable, which can be treated statistically. The second is that I sometimes see the first definition of correlation $G^{(1)}$ used alongside the definition of covariance $G$, which if nothing else, is a confusing way of mixing different definitions.

posted about 4 years ago

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Mithrandir24601‭ staff

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