Overview

In Chapters 4 and 5, parameter estimation applying the maximum likelihood method required minimization of the likelihood cost function given by

where ?( t _k) = z( t _k) ? y( t _k) represents the residuals at the discrete time point t _k.

For a specified model structure and the data being analyzed, the number of observations n _y and the data points N to be evaluated are defined and fixed. Accordingly, the last term in the above equation is a constant and, hence, neglected without affecting minimization.

The first and the second terms on the right-hand side of Eq. (E.1) contain R ^?1 and R, respectively. To facilitate the derivation, we rewrite the right-hand side such that both terms contain R ^{? 1}. Using the matrix expression

and after minor simplifications, the above equation can be rewritten as

Now, writing Eq. (E.3) in a component form, and denoting the elements of R ^{? 1} as a _ij leads to

where we used the matrix algebra result

in which A _ij denote the cofactor of the element a _ij of a matrix A.

Partial differentiation of Eq. (E.4) and making use of the results

leads to

From Eq. (E.5) we know that the quantity in the denominator of the second term on the right-hand side of Eq. (E.7) is nothing but A. For the minimum of L with...