Analysis of multiblock and hierarchical PCA and PLS models

Abstract

Multiblock and hierarchical PCA and PLS methods have been proposed in the recent literature in order to improve the interpretability of multivariate models. They have been used in cases where the number of variables is large and additional information is available for blocking the variables into conceptually meaningful blocks. In this paper we compare these methods from a theoretical or algorithmic viewpoint using a common notation and illustrate their differences with several case studies. Undesirable properties of some of these methods, such as convergence problems or loss of data information due to deflation procedures, are pointed out and corrected where possible. It is shown that the objective function of the hierarchical PCA and hierarchical PLS methods is not clear and the corresponding algorithms may converge to different solutions depending on the initial guess of the super score. It is also shown that the results of consensus PCA (CPCA) and multiblock PLS (MBPLS) can be calculated from the standard PCA and PLS methods when the same variable scalings are applied for these methods. The standard PCA and PLS methods require less computation and give better estimation of the scores in the case of missing data. It is therefore recommended that in cases where the variables can be separated into meaningful blocks, the standard PCA and PLS methods be used to build the models and then the weights and loadings of the individual blocks and super block and the percentage variation explained in each block be calculated from the results. © 1998 John Wiley & Sons, Ltd.

Relating the sensory sensation ‘creamy mouthfeel’ in custards to rheological measurements

“…To obtain a feeling for the correlation between the sensory measurements and the reduced set of rheological measurements, the sensory and rheological data sets were combined and a consensus principal component analysis (consensus PCA) [21] was performed, resulting in the loading plot shown in Figure 4. Creamy, airy and fatty mouthfeel are related and they clearly relate negatively to rheological variables describing break-up of the structure (ssr-fd , ssrtmax, dss-critstrainA); thus, when break-up of the structure is at low deformation, ratings for these three attributes are high.…”

Comparison of Timber Consumption in U.S. and China

“…Since the initial set of independent variables is usually quite large selected subjectively and a number of coefficients will most likely have been estimated by Ordinary Least Squares methods (Forward, Backward or Stepwise approaches provided by most standard statistical programs). Although partial least squares regression (PLS) and principal component regression (PCR) provide good results, the problems still cannot be solved efficiently (Johan et al, 1998). More recently, several modified PCR and PLS methods have been proposed, for example a hierarchical PCA (HPCA), a consensus PCA (CPCA，a hierarchical PLS (HPLS) and a multi-block PLS (MBPLS) (Svante et al,1996).These methods divide the set of independent variables into multiple blocks according their physical context and try to explain the dependent variables from different dimensions, in order to solve regression by multi-block independent variables.…”

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