R=D-CA; ssq=∑∑Rij2
A very prominent decomposition of D is the singular value decomposition,SVD
D=USV
SVD is usually unique, but unfortunately, this decomposition is chemically not useful; the matrices U and V are abstract eigenvectors and contain no directly chemically meaningful information. However, the matrices U and V are related to the useful matrices C and A.
The soft-modeling analysis or soft modeling of multivariate data has been a core area of research in chemometrics for many years.
Its principles are astonishingly simple: the analysis is based on the decomposition of a matrix of data into the product of two smaller matrices which are interpretable in chemical terms.
Generally, one resulting matrix is related to the concentrations of the species involved in the process and the other matrix contains their response vectors.
(2)Intensity ambiguities refer to scale indeterminacy of the matrix factorization described by the bilinear model.
(3) Rotation ambiguities are the more important and more difficult source of ambiguities in MCR methods and they are the main reason for non-unique solutions. Rotation ambiguities are found in all mixture analysis problems.
D=CA=USV CA=UST-1TV with C=UST-1 and A=TV
Therefore, in absence of any constraint, an infinite number of rotations and solutions are possible. This intrinsic indeterminacy of MCR methods is usually referred in the literature as rotation ambiguity.
1.Rotation ambiguities
• There are three types of ambiguities in MCR methods, permutation, intensity and rotation ambiguities.
(1)Permutation ambiguities refer to the exchangeable order of the components in the rows and columns of the factor matrices resolved by the bilinear model.
For instance: D=CA (multivariate spectroscopic data)
There muBiblioteka t be a solution where the product CA represents D as well as possible, as defined by a minimal sum of squares of the residuals R,
化学计量学
Resolution of Rotational Ambiguity for ThreeComponent Systems
• Abstract: Soft modeling of multivariate data is a
powerful method for the analysis of processes that cannot be described quantitatively by a chemical model. Soft modeling usually does not result in unique solutions. Thus, the determination of the range of feasible solutions is important. For two-component systems the determination of that range is well-understood; for threecomponent systems the task is remarkably more complex. We present a novel method that can be applied to any multivariate data set, irrespective of overlap or realistic noise level.