Partial least-squares in latent variables

Partial least-squares in latent variables (PLS) is sometimes called partial least-squares regression, or PLSR. As we are about to see, PLS is a logical, easy to understand, variation of PCR. 

The most promising new approach in multivariate statistical methods is the PLS (partial least squares in latent variables) method [26, 27, 38, 607 — 610]. Many, even hundreds or thousands of independent variables (the X block) can be correlated with one or several dependent variables (the Y block). PLS analysis is a principal component-like method, with the main difference that the vectors are not indepen-... 

Another problem is to determine the optimal number of descriptors for the objects (patterns), such as for the structure of the molecule. A widespread observation is that one has to keep the number of descriptors as low as 20 % of the number of the objects in the dataset. However, this is correct only in case of ordinary Multilinear Regression Analysis. Some more advanced methods, such as Projection of Latent Structures (or. Partial Least Squares, PLS), use so-called latent variables to achieve both modeling and predictions. 

Partial least-squares path modeling with latent variables (PLS), a newer, general method of handling regression problems, is finding wide apphcation in chemometrics. This method allows the relations between many blocks of data ie, data matrices, to be characterized (32—36). Linear and multiple regression techniques can be considered special cases of the PLS method. 

Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

We are about to enter what is, to many, a mysterious world—the world of factor spaces and the factor based techniques, Principal Component Analysis (PCA, sometimes known as Factor Analysis) and Partial Least-Squares (PLS) in latent variables. Our goal here is to thoroughly explore these topics using a data-centric approach to dispell the mysteries. When you complete this chapter, neither factor spaces nor the rhyme at the top of this page will be mysterious any longer. As we will see, it s all in your point of view. 

Partial least squares regression (PLS). Partial least squares regression applies to the simultaneous analysis of two sets of variables on the same objects. It allows for the modeling of inter- and intra-block relationships from an X-block and Y-block of variables in terms of a lower-dimensional table of latent variables [4]. The main purpose of regression is to build a predictive model enabling the prediction of wanted characteristics (y) from measured spectra (X). In matrix notation we have the linear model with regression coefficients b ... 

H. Wold, Soft modelling by latent variables the non-linear iterative partial least squares (NIPALS) algorithm. In Perspectives in Probability and Statistics, J. Gani (Ed.). Academic Press, London, 1975, pp. 117-142. 

Partial least squares (PLS) projections to latent structures [40] is a multivariate data analysis tool that has gained much attention during past decade, especially after introduction of the 3D-QSAR method CoMFA [41]. PLS is a projection technique that uses latent variables (linear combinations of the original variables) to construct multidimensional projections while focusing on explaining as much as possible of the information in the dependent variable (in this case intestinal absorption) and not among the descriptors used to describe the compounds under investigation (the independent variables). PLS differs from MLR in a number of ways (apart from point 1 in Section 16.5.1) ... 

Multivariate calibration has the aim to develop mathematical models (latent variables) for an optimal prediction of a property y from the variables xi,..., jcm. Most used method in chemometrics is partial least squares regression, PLS (Section 4.7). An important application is for instance the development of quantitative structure—property/activity relationships (QSPR/QSAR). 

Regression can be performed directly with the values of the variables (ordinary least-squares regression, OLS) but in the most powerful methods, such as principal component regression (PCR) and partial least-squares regression (PLS), it is done via a small set of intermediate linear latent variables (the components). This approach has important advantages ... 

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. 

M. Sjostrom, S. Wold, W. Lindberg, J.A. Persson and H. Martens, A multivariate calibration problem in analytical chemistry solved by partial least squares models in latent variables. Anal. Chim. Acta, 150, 61-70 (1983). 

Spatial Interrelationships In the chemical composition among two or more blocks (sites) can be calculated by partial least squares (PLS) (9 ). PLS calculates latent variables slmlllar to PG factors except that the PLS latent variables describe the correlated (variance common to both sites) variance of features between sites. Regional Influences on rainwater composition are thus Identified from the composition of latent variables extracted from the measurements made at several sites. Gomparlson of the results... 

A further better solution is offered by partial least squares (PLS), whose acronym may signify also projections onto latent structures. These latent structures, more frequently called latent variables (TVs) or PLS components, are directions in the space of the predictors with a connotation... 

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 ... 

Sjoestroem, M., Wold, S., Lindberg, W., Persson, J.A. and Martens, H., A Multivariate Calibration Problem in Analytical Chemistry Solved by Partial Least Squares Models in Latent Variables Anal. Chim. Acta 1983, 150, 61-70. 

Partial least squares regression (PLS) [WOLD et al., 1984] is a generalized method of least squares regression. This method uses latent variables i, 2,. .., i.e. matrix U, for separately modeling the objects in the matrix of dependent data Y, and t, t2,. .., i.e. matrix T, for separately modeling the objects in the matrix of independent data X. These latent variables U and T are the basis of the regression model. The starting points are the centered matrices X and Y ... 

Partial Least Squares Regression (PLS) is a multivariate calibration technique, based on the principles of Latent Variable Regression. Originated in a slightly different form in the field of econometrics, PLS has entered the spectroscopic scene.46,47,48 It is mostly employed for quantitative analysis of mixtures with overlapping bands (e.g. mixture of glucose, fructose and sucrose).49,50... 

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