Multivariate analysis construction

Fig. 9 Steps for automated determination of metastable zone using ATR-FTIR and FBRM. While automatically collecting the IR spectra for calibration, the metastable limit is determined using FBRM. Then the model for relating the IR spectra to solution concentration is constructed using multivariate analysis such as principal component regression (PCR) or partial least squares (PLS). Using this model, the solubility curve can be obtained from the IR spectra of saturated slurry.

The g construct has been criticized by researchers who are committed to the multivariate analysis of psychometric indices and by researchers who believe that existing psychometric indices fail to encompass important dimensions of intelligence. I shall briefly consider both kinds of criticisms. 

This type of multivariate analysis is concerned with the inter-relationships within a set of variables. The procedure involves construction of a number of factors to explain variation in the measured variables. Data reduction is the result, as the number of factors created is less that the number of variables. 

Other methods consist of algorithms based on multivariate classification techniques or neural networks they are constructed for automatic recognition of structural properties from spectral data, or for simulation of spectra from structural properties [83]. Multivariate data analysis for spectrum interpretation is based on the characterization of spectra by a set of spectral features. A spectrum can be considered as a point in a multidimensional space with the coordinates defined by spectral features. Exploratory data analysis and cluster analysis are used to investigate the multidimensional space and to evaluate rules to distinguish structure classes. 

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

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) ... 

Discriminant Analysis (DA) is a multivariate statistical method that generates a set of classification functions that can be used to predict into which of two or more categories an observation is most likely to fall, based on a certain combination of input variables. DA may be more effective than regression for relating groundwater age to major ion hydrochemistry and well construction because it can account for complex, non-continuous relationships between age and each individual variable used in the algorithm while inherently coping with uncertainty in the age values used for... 

It is worth remarking that a gas sensor array is a mere mathematical construction where the sensor outputs are arranged as components of a vector. Arrays can also be utilized to investigate the properties of chemical sensors, or even better, the peculiar behaviour of a sensor as a component of an array. In this chapter, the more common sensor array methodologies are critically reviewed, including the most general steps of a multivariate data analysis. The application of such methods to the study of sensor properties is also illustrated through a practical example. 

Despite the broad definition of chemometrics, the most important part of it is the application of multivariate data analysis to chemistry-relevant data. Chemistry deals with compounds, their properties, and their transformations into other compounds. Major tasks of chemists are the analysis of complex mixtures, the synthesis of compounds with desired properties, and the construction and operation of chemical technological plants. However, chemical/physical systems of practical interest are often very complicated and cannot be described sufficiently by theory. Actually, a typical chemometrics approach is not based on first principles—that means scientific laws and mles of nature—but is data driven. Multivariate statistical data analysis is a powerful tool for analyzing and structuring data sets that have been obtained from such systems, and for making empirical mathematical models that are for instance capable to predict the values of important properties not directly measurable (Figure 1.1). 

It may be possible to use an array of electrodes containing various enzymes in combination with multivariate statistical analyses (principal component analysis, discriminant analysis, partial least-squares regression) to determine which pesticide(s) the SPCE has been exposed to and possibly even how much, provided sufficient training sets of standards have been measured. The construction methods for such arrays would be the same as described in this protocol, with variations in the amounts of enzyme depending on the inhibition constants of other cholinesterases for the various pesticides of interest. 

A limiting factor in noninvasive optical technology is variations in the optical properties of samples under investigation that result in spectral distortions44 8 and sampling volume (effective optical path length) variability 49-54 These variations will impact a noninvasive optical technique not only in interpretation of spectral features, but also in the construction and application of a multivariate calibration model if such variations are not accounted for. As a result, correction methods need to be developed and applied before further quantitative analysis. For Raman spectroscopy, relatively few correction methods appear in the literature, and most of them are not readily applicable to biological tissue.55-59... 

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