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Theory of the distribution of error eigenvalues resulting from principal component analysis with applications to spectroscopic data
Author(s) -
Malinowski Edmund R.
Publication year - 1987
Publication title -
journal of chemometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.47
H-Index - 92
eISSN - 1099-128X
pISSN - 0886-9383
DOI - 10.1002/cem.1180010106
Subject(s) - eigenvalues and eigenvectors , normalization (sociology) , principal component analysis , mathematics , function (biology) , matrix (chemical analysis) , mathematical analysis , statistics , physics , chemistry , quantum mechanics , chromatography , evolutionary biology , sociology , anthropology , biology
The distribution of error eigenvalues resulting from principal component analysis is deduced by considering the decomposition of an error matrix in which the errors are uniformly distributed. The derived probability function is\documentclass{article}\pagestyle{empty}\begin{document}$$ P(\lambda ^0 _j) = N(r - j + 1)(c - j + 1) $$\end{document}Where λ 0 j is the j th error eigenvalue, r and c are the numbers of rows and columns in the data matrix, and N is the normalization constant. This expression is tested and validated by investigations involving model data. The distribution function is used to determine the number of factors responsible for various sets of spectroscopic data taken from the chemical literature (including nuclear magnetic resonance, infrared and mass spectra).