Random Fields Conditional on Dynamic Measurements

1.1. Formulation

As can be seen from Equation (35), the random eigenvalues can be expressed as a linear combination of the random variables Rj with coefficients Yij. Constructing a new set of random variables Q by the transformation

Q = [y 1, у 2,…, Y n]T R = TT R, (38)

whereby the transformation vectors yi are defined as yi = [Y’l, Yi2, …, Yin], the mean and covari­ance of Q are

E [Q] = rTE [R] (39)

and

Cov [Q, Q] = rTCov [R, R] Г, (40)

respectively. Given a set of m measurements s, these measurements can be interpreted as a realiza­tion of the random vector S = [Q1, Q2, …, Qm]. To include the measurement information in the statistical description of the random field, a conditional random field is defined with mean

E[Qj|s] = E[Qj] + Cov[Qj, S](Cov[S, S])-1(s – E[S]) (41)

and covariance

Cov[Qj, Qk|s] = Cov[Qj, Qk] – Cov[Qj, S](Cov[S, S])-1Cov[S, Qk] (42)

that possesses the desired properties (Ditlevsen, 1991). In other words, the random field described by Equations (41) and (42) is conditional on complete agreement with the measurements s made. The conditional mean and covariance of the original random variables R – as required by Equations (36)

and (37) – are given by

E [R|s] = T-TE [Q|s]

(43)

and

Cov [R, R|s] = r-TCov [Q, Q|s] Г-1,

(44)

respectively.