Standard errors in the eigenvalues of a cross‐product matrix: Theory and applications

Abstract

SUMMARYNew 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.

Uncertainty estimation and figures of merit for multivariate calibration (IUPAC Technical Report)

“…for standard error of prediction), although some work deals with alternative methods such as PCR [22,56,87,90,105,106], ILS [22], ANNs [107], nonlinear PLSR [108], CLS [18,27,109], and GSAM [110,111]. Considerable progress has been achieved for some higher-order methods, namely, rank annihilation factor analysis (RAFA) [112][113][114], GRAM [109,[115][116][117][118][119][120], BLLS with calibration using pure standards [50] and mixtures (as well as some alternatives) [44,45], N-PLS [56,121,122], and PARAFAC [123,124].…”

“…Considerable progress has been achieved for some higher-order methods, namely, rank annihilation factor analysis (RAFA) [112][113][114], GRAM [109,[115][116][117][118][119][120], BLLS with calibration using pure standards [50] and mixtures (as well as some alternatives) [44,45], N-PLS [56,121,122], and PARAFAC [123,124]. It is noted that the expressions proposed for N-PLS and PARAFAC are rather crude approximations that can likely be refined using the parameter results derived in [125,126].…”

“…When second-and higher-order multivariate calibrations exploiting the second-order advantage are employed, interferences that are not present in the calibration standards can in principle be modeled, and prediction residuals will in general reflect the instrumental noise level [44,45,50,56,109,[115][116][117][118][119][120][121][122][123][124][125][126]. This is true provided the data follow the required bi-or tri-linear structure.…”

Generalized rank annihilation method. I: Derivation of eigenvalue problems

“…M = UBVT, and making use of the orthogonality properties of U and V leads to the standard eigenvalue problem --_ ( U T N V 6 -' ) Z * = Z*II (10) where Z" = O v T Z . The concentration ratio of the analyte of interest is found as the only nonzero eigenvalue.…”

Generalized rank annihilation method: standard errors in the estimated eigenvalues if the instrumental errors are heteroscedastic and correlated

“…Much of the present confusion in previous derivations of standard errors in the estimated eigenvalues 13,14 might have arisen by not making this distinction explicit in the notation.…”