Kovariation

Kovariation, commonly spelled covariation or covariance in English, is a statistical concept that describes how two random variables vary together. When the variables tend to increase or decrease together, their covariation is positive; when one tends to increase as the other decreases, it is negative. A covariation of zero indicates no linear association between the pair, though nonlinear relationships may still exist.

Mathematically, the population covariation of random variables X and Y is defined as Cov(X,Y) = E[(X − μX)(Y − μY)], where μX and μY are the respective means. If X and Y have zero means, Cov(X,Y) = E[XY]. For sample data, the unbiased estimator is the sample covariance SXY = Σ (xi − x̄)(yi − ȳ) / (n − 1). Covariance matrices extend the idea to multiple variables, with entries Cov(Xi, Xj). The covariance matrix is symmetric and positive semidefinite.

Several properties hold: Cov(aX + b, Y) = a Cov(X,Y); Cov(X, aY + b) = a Cov(X,Y); Cov(X, X) = Var(X). The covariance is related to the correlation coefficient by Corr(X,Y) = Cov(X,Y) / (σX σY), where σX and σY are the standard deviations of X and Y. Unlike correlation, covariance is not scale-invariant, so its magnitude depends on the units of X and Y.

Applications include regression analysis, principal components analysis, portfolio theory, and any study of how variables co-move. Caution is warranted: non-zero covariation does not imply causation, and observed covariation can be affected by confounding factors.