Relationships between the definition of the hyperplane width to the fidelity of principal component loading patterns

When applying eigenanalysis, one decision analysts make is the determinatio n of what magnitude an eigen vector coefficient (e.g., principal component (PC) loading) must achieve to be considered as physically important. Such c oefficients can be displayed on maps or in a time series or tables to gain a fuller understanding of a large array of multivariate data. Previously, s uch a decision on what value of loading designates a useful signal (hereaft er called the loading "cutoff") for each eigenvector has been purely subjec tive. The importance of selecting such a cutoff is apparent since those loa ding elements in the range of zero to the cutoff are ignored in the interpr etation and naming of PCs since only the absolute values of loadings greate r than the cutoff are physically analyzed. This research sets out to object ify the problem of best identifying the cutoff by application of matching b etween known correlation/covariance structures and their corresponding eige npatterns, as this cutoff point (known as the hyperplane width) is varied. A Monte Carlo framework is used to resample at five sample sizes. Fourteen different hyperplane cutoff widths are tested, bootstrap resampled 50 times to obtain stable results. The key findings are that the location of an opt imal hyperplane cutoff width (one which maximized the information content m atch between the eigenvector and the parent dispersion matrix from which it was derived) is a well-behaved unimodal function. On an individual eigenve ctor. this enables the unique determination of a hyperplane cutoff value to be used to separate those loadings that best reflect the relationships fro m those that do not. The effects of sample size on the matching accuracy ar e dramatic as the values for all solutions (i.e., unrotated, rotated) rose steadily from 25 through 250 observations and then weakly thereafter. The s pecific matching coefficients are useful to assess the penalties incurred w hen one analyzes eigenvector coefficients of a lower absolute value than th e cutoff (termed coefficient in the hyperplane) or, alternatively, chooses not to analyze coefficients that contain useful physical signal outside of the hyperplane. Therefore, this study enables the analyst to make the best use of the information available in their PCs to shed light on complicated data structures.