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Assessment of the effective rank of a (co)variance matrix: A non‐parametric goodness‐of‐fit test
Author(s) -
Tomis̆ić Vladislav,
Simeon Vladimir
Publication year - 1993
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.1180070505
Subject(s) - goodness of fit , principal component analysis , mathematics , rank (graph theory) , eigenvalues and eigenvectors , parametric statistics , matrix (chemical analysis) , statistics , combinatorics , chemistry , physics , chromatography , quantum mechanics
Each eigenvector of the dispersion matrix [ X ] T [ X ] was shown to be a partial predictor of the original data matrix [ X ], the sum of the predictions from the individual principal components being equal to the expectance of [ X ]. By comparing the distributions of the members of two neighbouring predicted matrices, [ X̃ ] 1… i and [ X̃ ] 1… i +1 (i.e. the sums of the first i and i + 1 individual predictions respectively), it was shown that they should be indistinguishable provided that i is equal to or greater than the effective rank of [ X ], and significantly different otherwise. This was confirmed by analysing the visible absorption spectra of methyl orange and methyl red solutions as well as the Raman spectra of Na 2 SO 4 and MgSO 4 solutions. On the grounds of these findings, a non‐parametric goodness‐of‐fit test for assessing the effective rank of [ X ] was proposed which proved to be comparatively conservative and more robust than most currently used tests.