The Shape Design Criterion

In certain occasions the volume criterion is not appropriate. Fn particular when we have an ill-conditioned problem, use of the volume criterion results in an elongated ellipsoid (like a cucumber) for the joint confidence region that has a small volume however, the variance of the individual parameters can be very high. We can determine the shape of the joint confidence region by examining the cond( ) which is equal to and represents the ratio of the principal axes of 

In this case, it is best to choose the experimental conditions which will yield the minimum length for the largest principal axis of the ellipsoid. This is equivalent to 

we can determine the condition number and X,nin of matrix Anew using any eigenvalue decomposition routine that computes the eigenvalues of a real symmetric matrix and use the conditions (xN+0 that correspond to a maximum of 

When the parameters differ by several orders of magnitude between them, the joint confidence region will have a long and narrow shape even if the parameter estimation problem is well-posed. To avoid unnecessary use of the shape criterion, instead of investigating the properties of matrix A given by Equation 12.2, it is better to use the normalized form of matrix A given below (Kalogerakis and Luus, 1984) as AR. 

Essentially this is equivalent to using (Sf/dk kj instead of ( 3f/ 3k,) for the sensitivity coefficients. By this transformation the sensitivity coefficients are normalized with respect to the parameters and hence, the covariance matrix calculated using Equation 12.4 yields the standard deviation of each parameter as a percentage of its current value. 

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