) + (

- = const, hence: = var(P) + (

- )^2 = var(P) + square( bias(P wrt ) ) Dec6: Let P and A be both r.v.'s , eg P is a prediction, A is actual value to be predicted, and it usually becomes known later. MSE(P - A) = E[ P - A ]^2 = E[ A - P ]^2 = MSE(A - P) = E[ (P -

- ) ]^2 = var(P) + var(A) - 2.cov(A,P) + (

- )^2 = var(P - A) + ( bias(P wrt A) )^2 = var(A - P) + ( bias(A wrt P) )^2 = bias^2 + [ sd(P) - sd(A) ]^2 + 2.(1 - corr).sd(A).sd(P) { Dec6a } = bias^2 + [ sd(P) - sd(A).corr ]^2 + (1 - corr^2).var(A) { Dec6b } Note the symmetry (wrt r.v.'s P, A ) of Dec6a, and asymmetry of Dec6b . Theil's interpretations of the components of both decompositions of MSE(P-A) : Dec6a: bias ie errors of central tendency + [ sd(P) - sd(A) ]^2 = errors of unequal variations + errors due to incomplete covariation (would be 0 if corr = 1 ) Dec6b: bias ie errors of central tendency + [ sd(P) - sd(A).corr ]^2 = errors due to regression ie [ sd(P)^2 - 2.corr.sd(A).sd(P) + (corr.sd(A))^2 ] ie sd(P).[ 1 - corr.sd(A)/sd(P) ]^2 ie sd(P).[ 1 - beta( A on P ) ]^2 where beta is for A=beta.P + alfa would be 0 if beta( A on P ) == slope(of A on P) = 1. + (1 - corr^2).var(A) = errors due to unexplained variance, ie not explained by the regression of A on P, ie the minimal errors wrt the linear estimator (which is the regression line). This errors cannot be eliminated by linear corrections of the predictions P. These errors would be 0 if corr = 1. The meaning of 'linear' depends on the context. It may mean that : - either the principle of superposition holds - or a proportionality, like Y - E[Y] = C*(X - E[X]) -.- # References : Grewal M.S.: Kalman filters and observers ; in Wiley Encyclopedia of Electrical and Electronics Engineering, vol.11, 1999, p.83 Jensen Roderick V.: Classical chaos ; American Scientist, 75/3, March 1987 Jones, Ch.P.: Investments : Analysis and Management, 2nd ed., 1988; chap.19 = Portfolio theory (of Markowitz ) Markowitz H.: Portfolio Selection: Efficient Diversification of Investments, 1959 Theil Henry: Applied Economic Forecasting, 1966, 1971, ch.2 google: "Millman's theorem" Thevenin { see the 1st hit } -.- the end.