CORRELATION COEFFICIENT AND COVARIANCE

It gives the degree of correlation between two random variables. It is given as Pij = COJ(j’j ; — 1 — Pij — 1 • Pij = 0 for independent variables xt and xj, and for definitely correlated processes p = 1j. If a variable d is dependent on many x;, then the correlation coefficient for each of xi can be utilized to determine the degree of this correlation with d as

E (d(k) — d)(xi(k) — xi) k=1_

NN

4 E (d(k) — d)ME (xi(k) — x,)2 У k=1 У k=1

The ‘‘under bar’’ represents the mean of the variable. If |p(d, x;)| approaches unity, then d can be considered linearly related to particular x,. The covariance between two variables is defined as

Cov(xi, xj) = E{ [x, — E(x,)] [xj — E(xj)] }

By definition, the covariance matrix should be symmetric and positive semidefinite. It gives theoretical prediction of the state-error variance. If the parameters are used as variables in the definition, it gives the parameter estimation error covariance matrix. The square roots of the diagonal elements of this matrix give standard deviations of the errors in estimation of states or parameters, as the case may be. It is emphasized that the inverse of the covariance matrix is the indication of the information content
in the signals about the parameters or states. Large covariance signifies higher uncertainty and low information and low confidence in the estimation results.