Uncorrelated linear combinations of variables

When the parameters are strongly correlated, it is still possible to define a set of mutually uncorrelated combinations of the parameters. This can be shown as follows. If T represents the matrix of the eigenvectors of the variance-covariance matrix Mx, then Mx is diagonalized by the transformation 

Because of Eq. (4.31), A is the variance-covariance matrix of the set of unknowns X, which we will refer to as the eigenparameter. The eigenparameters X are, by the definition of the variance-covariance matrix, not correlated. 

Some of the linear combinations will be well defined and others poorly defined. The latter may be eliminated in a filtering procedure, referred to in the literature under the names characteristic value filtering, eigenvalue filtering, and principal component analysis. If the parameter set is not homogeneous, but includes different types, relative scaling is important. Watkin (1994) recommends that the unit be scaled such that similar shifts in all parameters lead to similar changes in the error function S. 

Diamond (1966) has applied a filtering procedure in the refinement of protein structures, in which poorly determined linear combinations are not varied. In charge density analysis, the principal component analysis has been tested in a refinement of theoretical structure factors on diborane, B2H6, with a formalism including both one-center and two-center overlap terms (Jones et al. 1972). Not unexpectedly, it was found that the sum of the populations of the 2s and spherically averaged 2p shells on the boron atoms constitutes a well-determined eigenparameter, while the difference is very poorly determined. Correlation between one- and two-center terms was also evident in the analysis. 

The value of Eq. (4.32) is that it shows exactly which features of the structure are well determined and which are poorly determined in the fitting procedure. For the two-dimensional example of Fig. 4.1, the eigenparameters correspond to the principal axes of the variance-covariance ellipsoid in the figure. In general, they define the principal axes of the hyper-ellipsoid in w-dimensional parameter space which represents the variance-covariance matrix. 

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