Kovariansen

Kovariansen is a statistical measure that describes how two random variables change together. If they tend to increase together, the covariance is positive; if one tends to increase while the other decreases, it is negative. A zero covariance indicates that the variables are uncorrelated, though not necessarily independent.

Mathematically, the population covariance of random variables X and Y is defined as Cov(X,Y) = E[(X − μX)(Y − μY)] = E[XY] − μX μY, where μX and μY are the means of X and Y. For a sample of n paired observations (x1,y1),...,(xn,yn), the sample covariance is cov_hat = (1/(n−1)) Σ (xi − x̄)(yi − ȳ). The covariance has the same units as the product of the two variables.

Key properties include symmetry (Cov(X,Y) = Cov(Y,X)) and the fact that Cov(X,X) = Var(X). Covariance is not standardized, so its magnitude depends on the scales of the variables.

Relation to correlation: the correlation coefficient, ρ(X,Y) = Cov(X,Y) / (σX σY), standardizes covariance to a dimensionless value between −1 and 1. Correlation measures linear association independent of scale, whereas covariance alone does not.

Applications and extensions: covariance is fundamental in multivariate statistics, regression analysis, and portfolio theory. The covariance matrix, Σ, contains Cov(Xi,Xj) for a set of variables and is symmetric and positive semi-definite. In time series, the autocovariance function γ(h) = Cov(Xt, Xt+h) describes dependence across lags.

Estimation caveats: zero covariance does not imply independence in general; sample covariance estimates require adequate data and, for unbiased estimation, the n−1 normalization is commonly used.