Two-block PLS

Fig. 37.6. PLS biplot obtained from the pharmacological data in Table 37.9, after log double-centering and analysis by two-block PLS [56]. Circles represent 17 reference neuroleptic compounds, squares denote tests. Areas of circles and squares are proportional to the potencies of the compounds and the sensitivities of the tests, respectively. Reproduced with permission of E.J. Kaijalainen.

Figure 1 is the summary of the two block PLS algorithm using the equation numbers. 

Figure 1. Two block PLS (numbers correspond to the equation numbers).

Are the standardization samples representative of the calibration samples The second important point is the nature of the standardization samples used. For some standardization algorithms, successful standardization results can only be obtained if the representativity of the standardization samples is satisfactory. This is the case for multivariate algorithms such as the two-block PLS [43,44], the direct and PDS [22] or the algorithms based on neural networks [45,46]. Therefore, these algorithms can only be applied when the experimental calibration domain is well covered by the standardization samples [24]. 

How many standardization samples are available for standardization An important point to consider is the number of standardization samples necessary to estimate the transfer parameters. Some methods based either on univariate local corrections (Shenk-Westerhaus algorithm) or on multivariate local correction (PDS, standardization in the wavelet domain, etc.) can be performed with a reduced number of standardization samples. However, multivariate methods based on the whole spectral range (direct standardization, two-block PLS) usually require more standardization samples to obtain a good and stable estimation of the standardization parameters [21]. 

This standardization approach consists of transferring the calibration model from the calibration step to the prediction step. This transferred model can be applied to new spectra collected in the prediction step in order to compute reliable predictions. An important remark is that the standardization parameters used to transfer calibration models are exactly the same as the ones used to transfer NIR spectra. Some standardization methods based on transferring spectra yield a set of transfer parameters. For instance, the two-block PLS algorithm yields a transfer matrix, and each new spectrum collected in the prediction step is transferred by simply multiplying it by the transfer matrix. For these standardization methods, the calibration model can be transferred from the calibration step to the prediction step using the same transfer matrix. It should be pointed out that all standardization methods yielding a transfer matrix (direct standardization, PDS, etc.) could be used in order to transfer the model from the calibration to the prediction step. For instrument standardization, the transfer of a calibration model from the master instrument to the slave instruments enables each slave instrument to compute its own predictions without systematically transferring the data back to the master instrument. 

PLS is a linear regression extension of PCA which is used to connect the information in two blocks of variables X and Yto each other. It can be applied even if the features are highly correlated. 

A drawback of the method is that highly correlating canonical variables may contribute little to the variance in the data. A similar remark has been made with respect to linear discriminant analysis. Furthermore, CCA does not possess a direction of prediction as it is symmetrical with respect to X and Y. For these reasons it is now replaced by two-block or multi-block partial least squares analysis (PLS), which bears some similarity with CCA without having its shortcomings. 

B program, PLS-2, uses the partial least squares (PLS) method. This method has been proposed by H. Wold (37) and was discussed by S. Wold (25). In such a problem there are two blocks of data, T and X. It is assumed that T is related to X by latent variables u and t is derived from the X block and u is derived from the Y block. 

Fig. 13.3 Schematic diagram of two different membrane units, (a) Membrane separator unit composed of two machined blocks of PTFE or Ti (A), and PTFE membrane (B), impregnated with stationary liquid, (b) Membrane unit. The PTFE membrane is placed between the two blocks made of titanium. The two channels (donar and acceptor) that are formed have a nominal volume of 12 pL.

Any regional Influence on rainwater composition would be expected to affect all three sites reported here. A PLS two block model 9) was used to predict the variance In rainwater composition at one site from the variance In rainwater composition at an upwind site. 

The method which satisfies these conditions is partial least squares (PLS) regression analysis, a relatively recent statistical technique (18, 19). The basis of tiie PLS method is that given k objects, characterised by i descriptor variables, which form the X-matrix, and j response variables which form the Y-matrix, it is possible to relate the two blocks (or data matrices) by means of the respective latent variables u and 1 in such a way that the two data sets are linearly dependent ... 

With the molecular descriptors as the X-block, and the senso scores for sweet as the Y-block, PLS was used to calculate a predictive model using the Unscrambler program version 3.1 (CAMO A/S, Jarleveien 4, N-7041 Trondheim, Norway). When the full set of 17 phenols was us, optimal prediction of sweet odour was shown with 1 factor. Loadings of variables and scores of compounds on the first two factors are shown in Fig es 1 and 2 respectively. Figure 3 shows predicted sweet odour score plotted against that provid by the sensory panel. Vanillin, with a sensory score of 3.3, was an obvious outlier in this set, and so the model was recalculated without it. Again 1 factor was r uired for optimal prediction, shown in Figure 4. 

PLS (Projections to Latent Stmctures) is a method by which it is possible to obtain quantitative relations between a matrix of descriptors (independent variables), called the X block, and a matrix of one or more response variables, called the Y block. The method is based on projections, similar to principal components analysis, and the quantitative relation between the two blocks is obtained by a correlation between the components of the respective block. 

The principles behind PLS are simple and easily understood from a geometrical illustration. The method is based upon projections, similar to principal components analysis. The two blocks of variables are given by the matrices X and Y. The following notations will be used ... 

The PLS algorithm is one ofthe standard methods used for two-block modeling, for example, for multivariate calibration as given as follows. 

This two-block predictive PLS regression has been found very satisfactory for multivariate calibration and many other types of practical multivariate data analysis. This evaluation is based on a composite quality criterion that includes parsimony, interpretability, and flexibility of the data model lack of unwarranted assumptions wide range of applicability good predictive ability in the mean square error sense computational speed good outlier warnings and an intuitively appealing estimation principle. See, for example. Reference 6, Reference 7, and References 15-17. 

Around 1980 the simplest PLS model with two blocks (X and Y) was slightly modified by Svante Wold and Harald Martens in order to suit better the data from science and technology, and was shown to be useful for dealing with complicated data sets where ordinary regression was difficult or impossible to apply. To give PLS a more descriptive meaning. Wold et al. have also started to interpret PLS as a projection to latent structures . 

The idea behind multiveuiate hierarchical modeling is very simple. Take one model dimension (component) of an existing projection method, say PLS (two-block), and substitute each variable by a score vector from a block of variables. We call these score vectors superveuiables . On the upper level of the model, a simple relationship, a supermodel , between rather few supervariables is developed. In the lower layer of the model, the details of the blocks are modeled by block models as block scores time block loadings. Conceptually this corresponds to seeing each block as an entity, and then developing PLS models between the superblocks . The lower level provides the variables (block scores) for these block relationships. 

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