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Keywords:

  • Principal factor analysis;
  • Factor analysis;
  • Eigenvalue analysis;
  • Multivariate analysis;
  • Weighted factor analysis;
  • Procrustean analysis

Abstract

Two approximate methods for weighted principal components analysis (WPCA) were devised and tested in numerical experiments using either empirical variances (obtained from replicated data) or assumed variances (derived from unreplicated data). In the first (‘spherical’) approximation each data vector was assigned a weight proportional to the geometrical mean of its variances in all dimensions. The arithmetical mean of variances was used instead in the other approximation. Both the numerical experiments with artificial data containing random errors of various kinds (constant, proportional, constant plus proportional, Poisson) and the analysis of two sets of Raman spectra clearly indicated the necessity of introducing statistical weights. The spherical approximation was found to be slightly better than the arithmetical one. The application of statistical weighting was found to improve the performance of PCA in estimation problems.