## VARIANCE-COVARIANCE MATRIX OF THE PARAMETERS

Earlier, the weight matrix was defined as the inverse of the variance-covariance matrix of the observations. This latter matrix, when fully populated, contains the variances of the observations, in order on the leading diagonal, and the covariances between them, where they exist, as the off-diagonal terms. Similar variancecovariance matrices can also be set up for all the other vector terms that appear in, or can be derived from, the least squares solution. The derivations of these variance-covariance matrices all make use of the Gauss propagation of error law, which may be interpreted like this. If two vectors s and t are related in the equation:

where K is a matrix, then their variance-covariance matrices are related by the equation:

The least squares solution for the parameters is:

where (ATWA)−1ATWis the counterpart of the matrix K above. To find σ(x), apply the Gauss propagation of error law to the least squares solution for x.

But σ(b) is W−1 and so can be replaced by it. When the terms in the brackets { } are transposed the expression becomes:

Since (ATWA)−1 is a symmetrical matrix, it is the same as its own transpose. By combining terms that are multiplied by their own inverse, this expression reduces to:

The variance-covariance matrix of the parameters is therefore the inverse of the normal equations matrix and this is often written as N−1.

The variance-covariance matrix of the parameters is a symmetrical matrix of the form:

Matrices, which are symmetrical, such as this one, are often written for convenience as upper triangular matrices, omitting the lower terms as understood, as below:

The terms on the leading diagonal are the variances of the parameters, and the off-diagonal terms are the covariances between them. Covariances are difficult to visualize.Amore helpful statistic is the coefficient of correlation r12, which is defined as:

A value of 1 indicates that any error in the two parameters will be in the same sense by a proportional amount. −1 indicates that it will be proportional, but in the opposite sense. 0 indicates that there is no relationship between the errors in the two parameters. Matrices of coefficients of correlation are useful for descriptive purposes but do not have any place in these computations.