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of that element. Thus, for the correlation coefficient matrix of Example 5.1, the logs of
parameters “ro2” and “h2” show medium to high correlation, as is indicated by the value of
elements (1,3) and (3,1) of the correlation coefficient matrix, viz. -0.8756. This explains why,
individually, these parameters are determined with a high degree of uncertainty in the
parameter estimation process, as evinced by their wide confidence intervals.
5.2.11 The Normalised Eigenvector Matrix and the Eigenvalues
The eigenvector matrix is composed of as many columns as there are adjustable parameters,
each column containing a normalised eigenvector. Because the covariance matrix is positive
definite, these eigenvectors are real and orthogonal; they represent the directions of the axes
of the probability “ellipsoid” in the n-dimensional space occupied by the n adjustable
parameters.
In the eigenvector matrix the eigenvectors are arranged from left to right in increasing order
of their respective eigenvalues; the eigenvalues are listed beneath the eigenvector matrix. The
square root of each eigenvalue is the length of the corresponding semiaxis of the probability
ellipsoid in n-dimensional adjustable parameter space.
If the ratio of a particular eigenvalue to the lowest eigenvalue pertaining to the parameter
estimation problem is particularly large, then the respective eigenvector defines a direction of
relative insensitivity in parameter space. The eigenvector pertaining to the highest eigenvalue
is worthy of attention in most parameter estimation problems, for this defines the direction of
maximum insensitivity, and hence of greatest elongation of the probability ellipsoid in
adjustable parameter space. If this eigenvector is dominated by a single element, then the
parameter associated with that element may be quite insensitive, the “magnitude of its
insensitivity” being defined by the square root of the magnitude of the corresponding
eigenvalue. However if this eigenvector contains a number of significant components rather
than just one, then this is an indication of insensitivity associated with a group of parameters
(ie. parameter correlation). The correlated parameters are those whose eigenvector
components are significantly non-zero.
The ratio of the highest to lowest eigenvalue constitutes another significant item of
information that is forthcoming as a by-product of the parameter estimation process. The
square root of this ratio is related to the “condition number” of the matrix that PEST must
invert when solving for the parameter upgrade vector - see equation 2.23. If the condition
number of a matrix is too high, then inversion of this matrix becomes numerically difficult or
even impossible. In the present instance this is an outcome of the fact that solution of the
inverse problem approaches nonuniqueness as elongation of the probability ellipsoid
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increases. In general, if the ratio of the highest to lowest eigenvalue is greater than about 10 ,
there is a strong possibility that PEST is having difficulty in calculating the parameter
upgrade vector because of parameter insensitivity and/or correlation. Its performance may be
seriously degraded as a result.
5.3 Other PEST Output Files
5.3.1 The Parameter Value File