Hi Matt, thank you for your answer!. The references I have about Malinowski´s work and some applications are:Malinowski, E.R., Theory of error in factor analysis. Analytical Chemistry, 1977. 49(4): p. 606-612.
Malinowski, E.R., Theory of the distribution of error eigenvalues resulting from principal component analysis with applications to spectroscopic data. Journal of Chemometrics, 1987. 1(1): p. 33-40.
Malinowski, E.R., Statistical F-tests for abstract factor analysis and target testing. Journal of Chemometrics, 1989. 3(1): p. 49-60.
Malinowski, E.R., Adaptation of the Vogt–Mizaikoff F-test to determine the number of principal factors responsible for a data matrix and comparison with other popular methods. Journal of Chemometrics, 2004. 18(9): p. 387-392.
McCue, M. and E.R. Malinowski, Target Factor Analysis of the Ultraviolet Spectra of Unresolved Liquid Chromatographic Fractions. Applied Spectroscopy, 1983. 37(5): p. 463-469.
Beauchemin, S., D. Hesterberg, and M. Beauchemin, Principal Component Analysis Approach for Modeling Sulfur K-XANES Spectra of Humic Acids. Soils Science Society of America Journal, 2002. 66: p. 83-91.
Wasserman, S.R., et al., EXAFS and principal component analysis: a new shell game. Journal of Synchrotron Radiation, 1999. 6: p. 284-286.
I think I don't understand what you mean by "eigenvalues explain exactly the amount of variance ... but they are different". Can you clarify? Giving an actual example might help.
Here are results obtained by ATHENA, the factional variance explained by each eigenvalue is calulated by dividing the eigenvalue between the sum of them all, right?:
# Eignevalues Variance Cumulative variance 1 8,864394 0,80585 0,805854 2 1,227578 0,11160 0,917452 3 0,708334 0,06439 0,981846 4 0,129478 0,01177 0,993617 5 0,045127 0,00410 0,997719 6 0,012229 0,00111 0,998831 7 0,009464 0,00086 0,999692 8 0,001489 0,00014 0,999827 9 0,000989 0,00009 0,999917 10 0,000617 0,00006 0,999973 11 0,000298 0,00003 1 Here are results obtained with matrix calculator for the same data:
Eigenvalue Explained Variance Cumulative variance 1418,3057 0,80586 0,80586 196,4118 0,11160 0,917453 113,3331 0,06439 0,981846 20,7162 0,01177 0,993617 7,2205 0,00410 0,997719 1,9566 0,00111 0,998831 1,5144 0,00086 0,999692 0,2381 0,00014 0,999827 0,1583 0,00009 0,999917 0,0987 0,00006 0,999973 0,0477 0,00003 1,000000
The eigenvalues are used then to evaluate the function IND and F test, and depending on the values of eigenvalues, function IND reach a minimum value when the set of primary components are separated from the secondary ones that just explained experimental errors (in the equations lambda are the eigenvalues, r the numbers of rows, c columns, n the number of primary components):The results obtained with the two sets of eigenvalues are diferent but they reach the minimum in the same n. The F test also gives me similar levels of significance for the two sets, but I do not undestand why I´m not hable to find the same eigenvalues that ATHENA does.
By the way, I tested the possibility you told to not divide the data by the standard deviation and still couldn´t find the same
eigenvalues.