Covariance and correlation

Main articles: Covariance and Correlation

In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in similar ways.

correlation	$\rho _{XY}=E[(X-E[X])(Y-E[Y])]/(\sigma _{X}\sigma _{Y})\;$
covariance	$\sigma _{XY}=E[(X-E[X])\,(Y-E[Y])]$

where E is the expected value operator and $\sigma _{X}$ and $\sigma _{Y}$ are the standard deviations of X and Y, respectively. Notably, correlation is dimensionless while covariance is in units obtained by multiplying the units of the two variables. The covariance of a variable with itself (i.e. $\sigma _{XX}$) is called the variance and is more commonly denoted as $\sigma _{X}^{2},$ the square of the standard deviation. The correlation of a variable with itself is always 1 (except in the degenerate case where the two variances are zero, in which case the correlation does not exist).

In the case of a stationary time series, both the means and variances are constant over time and the covariance and correlation are functions of the time difference when the two variables are measured:

cross correlation	$\rho _{XY}(m)=E[(X_{n}-E[X])\,(Y_{n+m}-E[Y])]/(\sigma _{X}\sigma _{Y}),$
cross covariance	$\sigma _{XY}(m)=E[(X_{n}-E[X])\,(Y_{n+m}-E[Y])].$

Although the values of the theoretical covariances and correlations are linked in the above way, the probability distributions of sample estimates of these quantities are not linked in any simple way and they generally need to be treated separately. These distributions depend on the joint distribution of the pair of random quantities (X,Y) when the values are assumed independent across different pairs. In the case of a time series, the distributions depend on the joint distributions of the whole time-series.

See also

• correlation and dependence