SHORT COMMUNICATION THE RELATIONSHIP OF CLOSURE,MEAN CENTERING AND MATRIX RANK INTERPRETATION

This paper describes an investigation into the relationship of closure,a baseline offset and mean centeringto the interpretation of matrix rank.The equivalence of a certain type of closure to a constant baseline(i.e.a simple numerical offset which may vary between response channels but is constant over all samples)is demonstrated.A systematic approach to the interpretation of the rank of a matrix is given.     

THE EFFECT OF MEAN CENTERING ON PREDICTION IN MULTIVARIATE CALIBRATION

Traditionally,one form of preprocessing in multivariate calibration methods such as principal componentregression and partial least squares is mean centering the independent variables(responses)and thedependent variables(concentrations).However,upon examination of the statistical issue of errorpropagation in multivariate calibration,it was found that mean centering is not advised for some datastructures.In this paper it is shown that for response data which(i)vary linearly with concentration,(ii)have no baseline(when there is a component with a non-zero response that does not change inconcentration)and(iii)have no closure in the concentrations(for each sample the concentrations of allcomponents add to a constant,e.g.100%)it is better not to mean center the calibration data.That is,the prediction errors as evaluated by a root mean square error statistic will be smaller for a model madewith the raw data than a model made with mean-centered data.With simulated data relativeimprovements ranging from 1% to 13% were observed depending on the amount of error in thecalibration concentrations and responses.     

SHORT COMMUNICATION COMMENTS ON THE POWER METHOD

In a recent short communication,Miyashita et al.have commented on the weakness of the NIPALSalgorithm(equivalently the power method)for calculating the eigenvalues out of order.They offer adiagnostic to ascertain when this may have occurred and suggested a modification to the NIPALSalgorithm to avoid this situation.Further comments regarding the use of the power method andMiyashita's presentation of its weakness are warranted.The general inadequacies of methods fordecomposing a matrix with degenerate eigenvalues and their relationship to the orthogonal design ofexperiments are discussed.