Variances and covariances of the least-squares parameter estimates

4 Variances and Covariances of the Least-Squares Parameter Estimates 

The matrix M describing the variances and covariances in the best values of the unknowns X, is written as (Hamilton 1964) 

The estimate e F — F°)(F — F°)T is the variance-covariance matrix of the observations My defined earlier. Thus, 

In other words, the variance-covariance matrix of the unknowns is the inverse of the matrix B. As described in the preceding section, the elements of B are the sums over the products of the derivatives of the observational equations with respect to the unknowns. Expression (4.21b) shows what we might have anticipated intuitively. As the elements of B-1 will tend to be inversely proportional to those of B, an element of B 1 mainly related to small derivatives will be large. Consequently, when the observations are not sensitive to an unknown parameter, the errors in the unknown parameter are large, and vice versa. 

It is quite common that only the relative values of the variances and covariances in the observations can be estimated. We may then write 

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