Applying eigenvalue analysis to experimental design

whose columns contain the normalized eigenvectors of X X. The covariance matrix of 9 is 

We see that X X is singular, as a result of the lack of any experiments with X2. But, let us say that this deficiency of the data set was not so immediately obvious. We could still diagnose the simation using the eigenvector decomposition (8.27), 

The eigenvector corresponding to the zero eigenvalue is of the form [o + c — c], suggesting that we need to add an experiment that varies xi and X2 in opposite directions. Therefore, we add an experiment whose predictor variables equal those in the second experiment plus 

The zero eigenvalue has been removed, and we now are able to estimate all parameters to finite accuracy. 

This analysis is useful for designing a set of experiments to yield the desired accuracy. Given an a priori estimate of a, we estimate the corresponding width of the confidence intervals. If this accuracy is insufficient, we add more experiments, until the expected accuracy is deemed sufficient. 

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