Basis sets for multivariate regression

• Published in: Analytica chimica acta Vol. 428; no. 1; pp. 31 - 40
• Main Author: Kalivas, John H.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.02.2001; Elsevier
• Subjects: Analytical chemistry; Basis set; Chemistry; Exact sciences and technology; Multivariate calibration; Near infrared spectroscopy; Regression model; Spectrometric and optical methods; Near infrared spectroscopy; Basis set; Regression model; Multivariate calibration; Near infrared spectrometry; Data analysis; Chemical analysis; Multicomponent analysis; Regression analysis; Multivariate analysis; Chemometrics; Principal component analysis; Quantitative analysis
• ISSN: 0003-2670; 1873-4324
• DOI: 10.1016/S0003-2670(00)01225-3

Estimates of regression coefficients for a multivariate linear model have been the subject of considerable discussion in the literature. A purpose of this paper is to discuss biased estimators using common basis sets. Estimators of focus are least squares, principal component regression, partial least squares, ridge regression, generalized ridge regression, continuum regression, and cyclic subspace regression. Variations of these methods are also proposed. It is shown that it is not the common basis set used to span the calibration space or the number of vectors from the common basis set used to form respective calibration models that are important, i.e. a parsimony emphasis. Instead, it is suggested that the size and direction of the calibration subspace used to form the models is essential, i.e. a harmony consideration. The approach of the paper is based on representing estimated regression vectors as weighted sums of basis vectors.