Latent variable decomposition

This book contains several different NIR applications in food analysis, and many of them use multivariate data handling. Our aim in this chapter is to discuss the aspects of latent variable decomposition in principal component analysis and partial least squares regression and to illustrate their use by an application in the NIR region. 

PCR and PLR are useful when the matrix does not contain the full model representation. The first step of PCR is the decomposition of the data matrix into latent variables through PCA and the dependent variable is then regressed onto the decomposed independent variables. PLS performs, however, a simultaneous and interdependent PCA decomposition in a way that makes that PLS sometimes handles dependent variables better than does PCR. 

For the first latent variable, PLS decomposition is started by selecting yj, an arbitrary column of Y as the initial estimate for ui. Usually, the... 

As shown by Kvalheim (6), approaches to decomposition of multivariate data in terms of latent variables can be developed within the frame of a generalized NIPAL algorithm (7), for instance, decomposition into principal components (8) and decomposition using the PLS approach (9,10). 

Partial least squares regression [16-18] has been employed since the early 1980s and is closely related to PCR and MLR [18]. In fact, PLS can be viewed as a compromise midway between PCR and MLR [19]. In determining the decomposition of R (and consequently removing unwanted random variance), PCR is not influenced by knowledge of the estimated property in the calibration set, c. Only the variance in R is employed to determine the latent variables. Conversely, MLR does not factor R prior to regression all variance correlated to c is employed for estimation. PLS determines each latent variable to simultaneously optimize variance described in R and correlation with c, p. Technically, PLS latent variables are not principal components. The PLS factors are rotations of the PCA PCs for a slightly different optimization criterion. 

PCA is a bilinear decomposition/projection technique capable of condensing large amounts of data into few parameters, called principal components (PCs) or latent variables/factors, which capture the levels, differences and similarities among the samples and variables constituting the modelled data. This task is achieved by a linear transformation under the constraints of preserving data variance and imposing orthogonality of the latent variables. 

PCR method is a two-step process, in which the projection stage is separated and independent from the regression one. As discussed in Section 3.3, this can lead to the drawback that the components that are extracted in the decomposition step, based only on the information about the X-matrix, can be poorly predictive for the Y-block. Starting from these considerations, another method was proposed, PLS regression [2,18,19], in which information in Y is actively used also for the definition of the latent variable space. Indeed, PLS looks for components which compromise between explaining the variation in the X-block and predicting the responses in Y. This corresponds to a bilinear model, which can be summarized mathematically as ... 

Physical and Chemical Properties - Physical State at 15 X and 1 atm. Liquid Molecular Weight Variable — 200 to 2000 Boiling Point at 1 atm. Not pertinent (decomposes) Freezing Point -22 to -58, -30 to -50, -243 to 223 Critical Temperature Not pertinent Critical Pressure Not pertinent Specific Gravity 1.012 at 20 °C (liquid) Vqjor (Gas) Specific Gravity Not pertinent Ratio of Specific Heats of Vapor (Gas) Not pertinent Latent Heat of Vaporization Not pertinent Heat [Pg.322]

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