Chemometrical latent variable

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). 

In addition to the measured values and the analytical values (e.g. content, concentration), latent variables are included in the scheme. Latent variables can be obtained from measured values or from analytical values by means of mathematical operations (e.g. addition, subtraction, eigenanal-ysis). By means of latent variables and their typical pattern (represented in chemometric displays) special information can be obtained, e.g. on quality, genuineness, authenticity, homogeneity, origin of products, and health of patients. 

An essential concept in multivariate data analysis is the mathematical combination of several variables into a new variable that has a certain desired property (Figure 2.14). In chemometrics such a new variable is often called a latent variable, other names are component or factor. A latent variable can be defined as a formal combination (a mathematical function or a more general algorithm) of the variables a latent variable summarizes the variables in an appropriate way to obtain acertain property. The value of a latent variable is called score. Most often linear latent variables are used given by... 

Also nonlinear methods can be applied to represent the high-dimensional variable space in a smaller dimensional space (eventually in a two-dimensional plane) in general such data transformation is called a mapping. Widely used in chemometrics are Kohonen maps (Section 3.8.3) as well as latent variables based on artificial neural networks (Section 4.8.3.4). These methods may be necessary if linear methods fail, however, are more delicate to use properly and are less strictly defined than linear methods. 

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). 

Oveilitting Oceifitting occurs when the model used to describe a data set is overly convex. An example of this in regression analysis is the use of a sccond-onfcr polynomial to describe the relationship between two variables when the true relationship is a straight line. In chemometrics, the most common exan le of overfitting is the use of too many latent variables in a... 

Often, relationships between measured process parameters and desired product attributes are not directly measurable, but must rather be inferred from measurements that are made. This is the case with several spectroscopic measurements including that of octane number or polymer viscosity by NIR. When this is the case, these latent properties can be related to the spectroscopic measurement by using chemometric tools such as PLS and PCA. The property of interest can be inferred through a defined mathematical relation.39 Latent variables allow a multidimensional data set to be reduced to a data set of fewer variables which describe the majority of the variance related to the property of interest. This data compression using the most relevant data also removes the irrelevant or noisy data from the model used to measure properties. Latent variables are used to extract features from data, and can result in better accuracy of measurement and a reduced measurement time.4... 

V-M Taavitsainen and P Korhonen. Nonlinear data analysis with latent variables. Chemometrics Intell. Lab. Sys., 14 185-194, 1992. 

Burnham AJ, Viveros R, MacGregor JF, Frameworks for latent variable multivariate regression, Journal of Chemometrics, 1996,10, 31 -5. 

In the second case, we took into account over 400 quinolones reported in the literature. Again, a chemometric approach based on multivariate characterization and design in the resulting latent variables permitted us to select a set of 32 molecules with a well-balanced structural variation on which to derive the QSAR models. Linear PLS modeling allowed ranking of the relative importance of individual structural features, and, by CARSO analysis, a new class of compounds was predicted, the lead of which was tested and shown to be as active as expected. This preliminary lead, after a proper modification, is presently being tested for further development. 

M. Reis, P. Saraiva, Heteroscedastic latent variable modelling with applications to multivariate statistical process control, Chemometrics and Intelligent Laboratoy Systems 80 (2006) 57-66. 

A FTIR spectrum is complex, containing many variables per sample and making visual analysis very difficult. Hence, to extract extra useful information, i.e., latent variables, from the whole spectra chemometric analysis was performed considering the whole FTIR data set using principal components analysis (PCA) for an exploratory overview of data. This method could reveal similarity/dissimilarity patterns among propolis samples, simplifying... 

Kvalheim, O.M. Karstang, T.V. (1989). Interpretation of latent-variable regression models. Chemometrics and Intelligent Laboratory Systems, Vol. 7, No.1-2, (December 1989), pp. 39-51, ISSN 0169-7439... 

Before leaving stage 1 of the two-stage regression with latent variables, we note that ridge regression can be formulated in a similar fashion to that for OLS, and the interested reader is referred to Frank and Friedman s (re)view of regression tools in chemometrics. ... 

Conceptually similar to PCA, PLS is more useful for QSAR studies, because it relates chemical structure (e.g., CoMFA fields) to the target property (stored in the Y matrix). The pioneering work in PLS was done by H. Wold in the discipline of econometrics and was implemented by S. Wold, Martens, and Kowalski in chemometrics." PLS was first implemented in 3D-QSAR by Cramer and co-workers and is increasingly referred to as projection to latent structures. PLS is based on two PCA models (in the X and Y matrices), with the difference that the resulting PCA models are rotated to maximize the fit between the X and Y latent variables. As in PCA, the X and Y data matrices of the training set (e.g., CoMFA fields and biological activities, respectively) include a model and noise that is, they are a combination of score and loadings matrices, and the noise matrices E and F ... 

Chemometrics mainly focuses on the chemical model, rather than on random effects as is typical for statistics. A chemometric approach does not exclude theory or human experience from problem solving they tell us how to define the problem, what to measure and how to pre-process measurement data. The basic hypothesis suggests that complicated chemical systems can be described (characterized) by a set of (measured) variables and that models (latent variables) can help to find the essential information. Selection or creation of appropriate problem-relevant features is often more important than the method which is applied for data interpretation. 

Camacho J. Observation-based missing data methods for exploratory data analysis to unveil the connection between observations and variables in latent subspace models Journal of Chemometrics 25 (2011) 592 - 600. 

Partial least squares regression (PLS) is more important in chemometrics than in other fields of applied statistics (see Partial Least Squares Projections to Latent Structures (PLS) in Chemistry). PLS can be considered as an alternative method to PCR and LDA. The aim of data interpretation is to build a linear model for the prediction of a response y from the independent variables (regressors, features)x],X2- - Xp as given in equation (27) ... 

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