Multivariate linear models

The relationship between a criterion variable and two or more predictor variables is given by a linear multivariate model ... 

The methodology used to develop linear multivariable models is described in the following section. 

At the beginning of Section 6.3, the possibilities of transformations of variables for modeling nonlinear relationships were discussed. In the ACE method, these transformations need not be predefined, but are found by the algorithm. To understand the ACE method, we start from the linear multivariate model for a single dependent variable y and p independent variables Xj (cf Eq.(6.13)) ... 

In its simplest, and in chemistry the most used, form, PLS is a method for relating two data matrices X and Y to each other by a linear multivariate model. PLS stands for projection to latent structures by means of partial least squares. It derives its usefulness from its ability to analyze data with many noisy, collinear, and even incomplete variables in both X and Y. PLS has the desirable property that for the parameters regarding the observations (samples, compounds, objects, items), the precision improves with the increasing number of relevant X variables. This corresponds to the intuitive belief of most chemists that many variables provide more information about the observations than do just a few variables. 

Much effort has been devoted to the development of reliable calculation methods for the prediction of the retention behaviour of analyses with well-known chemical structure and physicochemical parameters. Calculations can facilitate the rapid optimization of the separation process, reducing the number of preliminary experiments required for optimization. It has been earlier recognized that only one physicochemical parameter is not sufficient for the prediction of the retention of analyte in any RP-HPLC system. One of the most popular multivariate models for the calculation of the retention parameters of analyte is the linear solvation energy relationship (LSER) ... 

This chapter is organized in the following way. First, the general model of the CSTR process, based on first principles, is derived. A linearized approximate model of the reactor around the equilibrium points is then obtained. The analysis of this model will provide some hints about the appropriate control structures. Decentralized control as well as multivariable (MIMO) control systems can be designed according to the requirements. 

Multivariate models have been successful in identifying source contributions in urban areas. They are not independent of Information on source composition since the chemical component associations they reveal must be verified by source emissions data. Linear regressions can produce the typical ratio of chemical components in a source but only under fairly restrictive conditions. Factor and principal components analysis require source composition vectors, though it is possible to refine these source composition estimates from the results of the analysis (6.17). 

The PLS approach to multivariate linear regression modeling is relatively new and not yet fully investigated from a theoretical point of view. The results with calibrating complex samples in food analysis 122,123) j y jnfj-ared reflectance spectroscopy, suggest that PLS could solve the general calibration problem in analytical chemistry. 

In traditional method validation, assessment of the calibration has been discussed in terms of linear calibration models for univariate systems, with an emphasis on the range of concentrations that conform to a linear model (linearity and the linear range). With modern methods of analysis that may use nonlinear models or may be multivariate, it is better to look at the wider picture of calibration and decide what needs to be validated. Of course, if the analysis uses a method that does conform to a linear calibration model and is univariate, then describing the linearity and linear range is entirely appropriate. Below I describe the linear case, as this is still the most prevalent mode of calibration, but where different approaches are required this is indicated. 

Linear Regression with Several Response Variables 6.4.2.1 The Multivariate Linear Regression Model... 

In the resolution of any multicomponent system, the main goal is to transform the raw experimental measurements into useful information. By doing so, we aim to obtain a clear description of the contribution of each of the components present in the mixture or the process from the overall measured variation in our chemical data. Despite the diverse nature of multicomponent systems, the variation in then-related experimental measurements can, in many cases, be expressed as a simple composition-weighted linear additive model of pure responses, with a single term per component contribution. Although such a model is often known to be followed because of the nature of the instrumental responses measured (e.g., in the case of spectroscopic measurements), the information related to the individual contributions involved cannot be derived in a straightforward way from the raw measurements. The common purpose of all multivariate resolution methods is to fill in this gap and provide a linear model of individual component contributions using solely the raw experimental measurements. Resolution methods are powerful approaches that do not require a lot of prior information because neither the number nor the nature of the pure components in a system need to be known beforehand. Any information available about the system may be used, but it is not required. Actually, the only mandatory prerequisite is the inner linear structure of the data set. The mild requirements needed have promoted the use of resolution methods to tackle many chemical problems that could not be solved otherwise. 

Principal component analysis (PCA) and multivariate curve resolution-alternating least squares (MCR-ALS) were applied to the augmented columnwise data matrix D1"1", as shown in Figure 11.16. In both cases, a linear mixture model was assumed to explain the observed data variance using a reduced number of contamination sources. The bilinear data matrix decomposition used in both cases can be written by Equation 11.19 ... 

Jalali-Heravi and coworkers used the three descriptors of their multivariable model, that is, Offord charge-to-mass parameter, corrected steric constant, and molar refractivity, as the input parameters for generating the network (Fig. 14.2). In fact, they proposed an MLR-ANN model for the prediction of the electrophoretic mobility of peptides. The purpose for choosing the MLR parameters as inputs for the ANN mode was to compare the abilities of linear and nonlinear models in predicting the electrophoretic mobilities of peptides (9). 

From a data analytical point of view, data can be categorised according to structure, as exemplified in Table 1. Depending on the kind of data acquired, appropriate data analytical tools must be selected. In the simplest case, only one variable/number is acquired for each sample in which case the data are commonly referred to as zeroth-order data. If several variables are collected for each sample, this is referred to as first-order data. A typical example could be a ID spectrum acquired for each sample. Several ID spectra from different samples may be organised in a two-way table or a matrix. For such a matrix of data, multivariate data analysis is commonly employed. It is clearly not possible to analyse zeroth-order data by multivariate techniques and one is restricted to traditional statistics and linear regression models. When first- or second-order data are available, multivariate data analysis may be used and several advantages may be exploited,... 

Equations (2.21) have been written in the order and manner above to bring out the dynamic interdependence of the states that will normally emerge as a feature of models of typical industrial processes. While the derivative of one state may depend only on the current value of that state, as in the case of the valve travels, x and Xi, others will depend not only on their own state but also on a number of others. This latter situation arises above in the cases of control valve travel, Xi, and the liquid mass in the tank, m. The dependence may be linear in some cases, but in any normal process model, there will be a large number of nonlinear dependencies, as exhibited above by the derivative for tank liquid mass, which is dependent on a term multiplying the square of one state by the square-root of another. This is an important point to grasp for those more accustomed to thinking of linear, multivariable control systems such systems are idealizations only of a nonlinear world. 

Although a majority of the published ADMET models are based on linear multivariate methods as discussed in Section 16.3.3.1, other nonlinear methods have also been employed. The most commonly used nonlinear method in ADMET modeling is neural networks (NNs). Backpropagation NNs have been used to model absorption, permeation, as well as solubility and toxicological effects. A particular problem for many NNs is the tendency for these networks to overtrain (see further discussions on model validation in Section 16.3.3.4), which needs to be closely monitored to avoid the situation where the derived model becomes an encyclopedia , that is, the model can perfectly explain the variance of the investigated property of the compounds used to derive the model but have quite poor predictive ability with respect to new compounds. 

In the first case, the structural description of over 100 thienyl- and furyl-benzimidazoles and benzoxazoles was multivariately characterized to identify three latent variables. A set of 16 informative molecules was derived thereafter on applying a central composite design criterion in these latent variables to all the available structures. The data were analyzed by a linear PLS model, which permitted the optimization of three structural features out of four. The fourth one, the substituent linked to the homocyclic ring of the bicyclic system was finally optimized by the CARSO procedure in terms of the substituents PPs, predicting two new compounds as possible optimal structures. Indeed, later analysis revealed the accuracy of these predictions. 

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. 

The peak parameter representation (PPR) [21] uses only the set of non-linear basis function parameters in the multivariate modelling. The PPR method can be written as follows ... 

The parameters of the model were estimated from the experimental data using a non linear multivariate curve fitting technique. In this process the temperature dependence of the diffusion coefficient for glucose was assumed to be small in the range of temperatures studied. The equilibrium constant was assumed to be given by ... 

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