STANDARD ERRORS IN THE EIGENVALUES OF A CROSS-PRODUCT MATRIX:THEORY AND APPLICATIONS

New expressions are derived for the standard errors in the eigenvalues of a cross-product matrix by themethod of error propagation.Cross-product matrices frequently arise in multivariate data analysis,especially in principal component analysis (PCA).The derived standard errors account for the variabilityin the data as a result of measurement noise and are therefore essentially different from the standarderrors developed in multivariate statistics.Those standard errors were derived in order to account for thefinite number of observations on a fixed number of variables,the so-called sampling error.They can beused for making inferences about the population eigenvalues.Making inferences about the populationeigenvalues is often not the purposes of PCA in physical sciences,This is particularly true if themeasurements are performed on an analytical instrument that produces two-dimensional arrays for onechemical sample:the rows and columns of such a data matrix cannot be identified with observations onvariables at all.However,PCA can still be used as a general data reduction technique,but now the effectof measurement noise on the standard errors in the eigenvalues has to be considered.The consequencesfor significance testing of the eigenvalues as well as the usefulness for error estimates for scores andloadings of PCA,multiple linear regression (MLR) and the generalized rank annihilation method(GRAM) are discussed.The adequacy of the derived expressions is tested by Monte Carlo simulations.

STANDARD ERRORS IN THE EIGENVALUES OF A CROSS-PRODUCT MATRIX:THEORY AND APPLICATIONS

New expressions are derived for the standard errors in the eigenvalues of a cross-product matrix by the method of error propagation.Cross-product matrices frequently arise in multivariate data analysis, especially in principal component analysis (PCA).The derived standard errors account for the variability in the data as a result of measurement noise and are therefore essentially different from the standard errors developed in multivariate statistics.Those standard errors were derived in order to account for the finite number of observations on a fixed number of variables,the so-called sampling error.They can be used for making inferences about the population eigenvalues.Making inferences about the population eigenvalues is often not the purposes of PCA in physical sciences,This is particularly true if the measurements are performed on an analytical instrument that produces two-dimensional arrays for one chemical sample:the rows and columns of such a data matrix cannot be identified with observations on variables at all.However,PCA can still be used as a general data reduction technique,but now the effect of measurement noise on the standard errors in the eigenvalues has to be considered.The consequences for significance testing of the eigenvalues as well as the usefulness for error estimates for scores and loadings of PCA,multiple linear regression (MLR) and the generalized rank annihilation method (GRAM) are discussed.The adequacy of the derived expressions is tested by Monte Carlo simulations.