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

Abstract

SUMMARYThe generalized rank annihilation method (GRAM) is a method for curve resolution and calibration that uses two data matrices simultaneously, i.e. one for the unknown and one for the calibration sample. The method is known to become an eigenvalue problem for which the eigenvalues are the ratios of the concentrations for the samples under scrutiny. Previously derived standard errors in the estimated eigenvalues of GRAM have very recently been shown to be based on unrealistic assumptions about the measurement errors. In this paper a systematic notation is introduced that enables the propagation of errors that are based on realistic assumptions concerning the datagenerating process. The error propagation will be performed in detail for the case that one data order modulates the second one. Extensions to more complicated error models are indicated.

Precision of prediction in second‐order calibration, with focus on bilinear regression methods

“…With other calibration/prediction methods such as PARAFAC, TLD [8] and PLS-type methods, precision formulae are lacking. GRAM [2] is an exception, for which Faber et al [9] derived standard errors. Indirect quantification of type cross-validation, which is the standard tool in first order, is not a good idea when the calibration involves only few specimens.…”

Detection, Identification, and Quantification of Nonpolar High Molecular Weight Contaminants in Jet and Diesel Fuels by Liquid Chromatography

“…PLS-based data modeling is intended to correlate the underlying linear variance that can be found within multivariate data sets (in the present case, the HPLC-ELSD data collected from training fuel samples with known contamination levels) to vectors of calibration data (in the present case, the known contamination levels) for the purposes of producing predictive models. It should be noted that alternative data modeling techniques such as multivariate curve resolution (MCR), the generalized rank annihilation method (GRAM), and target factor analysis (TFA) exist that forego this indirect approach to more directly focus upon the modeling of underlying chemical variance via the explicit calibration of models toward the presence of pure chemical components. Such data modeling methodologies would have distinct advantages in scenarios in which the detection and quantification of pure chemical components would be both reliable and desirable.…”

Enhanced Chemical Analysis Using Parallel Column Gas Chromatography with Single-Detector Time-of-Flight Mass Spectrometry and Chemometric Analysis

“…Equation 5 closely predicts the concentration prediction error when noise in the data is uniformly distributed and uncorrelated. A modified form of this equation has been developed when additional noise terms become significant . However, the trend seen in eq 5 still holds: the increased NAS of a component in a parallel- versus single-column system increases its second-order sensitivity, which in turn reduces the error in its quantitation.…”