Covariance and correlation

The mathematical description of covariance and correlation are very similar. Both are used to describe the degree of similarity between two random variables, or sets of random variables. There is a certain amount of disagreement as to the naming conventions used. The conventions used in this article will be those used by Oppenheim & Shafer (1975). For two sets of random variates $\{x_{i}\}$ and $\{y_{i}\}$ we have:

correlation matrix	$\phi _{xy}(n,m)=E[x_{n}\,y_{m}]$
covariance matrix	$\gamma _{xy}(n,m)=E[(x_{n}-E[x_{n}])\,(y_{m}-E[y_{m}])]$
autocorrelation matrix	$\phi _{xx}(n,m)=E[x_{n}\,x_{m}]$
autocovariance matrix	$\gamma _{xx}(n,m)=E[(x_{n}-E[x_{n}])\,(x_{m}-E[x_{m}])]$

In the case of stationarity, the means are constant and the covariance or correlation are functions only of the difference in the indices:

cross correlation	$\phi _{xy}(m)=E[x_{n}\,y_{n+m}]$
cross covariance	$\gamma _{xy}(m)=E[(x_{n}-E[x])\,(y_{n+m}-E[y])]$
autocorrelation	$\phi _{xx}(m)=E[x_{n}\,x_{n+m}]$
autocovariance	$\gamma _{xx}(m)=E[(x_{n}-E[x])\,(x_{n+m}-E[x])]$

Each of these statistics may be normalized by dividing by the respective standard deviations. For example, the normalized cross correlation is written:

${\frac {E[x_{n}\,y_{n+m}]}{\sigma _{x}\sigma _{y}}}$

where $\sigma _{x}$ and </math>\sigma_y</math> are the standard deviations of the $\{x_{i}\}$ and $\{y_{i}\}$ respectively.

These definitions are easily extended to the case of continuous random variables.

References

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