Multivariable data sets

Canonical Correlation Analysis (CCA) is perhaps the oldest truly multivariate method for studying the relation between two measurement tables X and Y [5]. It generalizes the concept of squared multiple correlation or coefficient of determination, R. In Chapter 10 on multiple linear regression we found that is a measure for the linear association between a univeiriate y and a multivariate X. This R tells how much of the variance of y is explained by X = y y/yV = IlylP/llylP. Now, we extend this notion to a set of response variables collected in the multivariate data set Y. 

Matrix formed by a set of correlation coefficients related to m variables in multivariate data sets, R = (rXi,Xj). It is relevant in multicomponent analysis. 

We continue considering multivariate data sets, e.g. a series of spectra measured as a function of time, reagent addition etc. In short, a matrix of... 

The basic theory of Kohonen maps—and only this will be treated here—is mathematically simple. A typical Kohonen map consists of a rectangular (often quadratic) array of fields (squares, cells, nodes, neurons) with a typical size of 5 x 5 (25 fields) to 100 x 100 (10,000 fields). Each field k is characterized by a vector wk, containing the weights wki, wia, , with in being the number of variables of a multivariate data set X (Figure 3.18) the lengths of the weight vectors are, for instance,... 

An overview of the multivariate data set can first be obtained by PCA. Since the objects summed up to a constant value (compositional data, see Section 2.2.4), the data were first transformed with the isometric logratio (ILR) transformation. 

On the other hand, factor analysis involves other manipulations of the eigen vectors and aims to gain insight into the structure of a multidimensional data set. The use of this technique was first proposed in biological structure-activity relationship (i. e., SAR) and illustrated with an analysis of the activities of 21 di-phenylaminopropanol derivatives in 11 biological tests [116-119, 289]. This method has been more commonly used to determine the intrinsic dimensionality of certain experimentally determined chemical properties which are the number of fundamental factors required to account for the variance. One of the best FA techniques is the Q-mode, which is based on grouping a multivariate data set based on the data structure defined by the similarity between samples [1, 313-316]. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or covariance) relationships explored with R-mode factor analysis. The measure of similarity used is the cosine theta matrix, i. e., the matrix whose elements are the cosine of the angles between all sample pairs [1,313-316]. 

The goal of Q-mode FA is to determine the absolute abundance of the dominant components (i.e., physical or chemical properties) for environmental contaminants. It provides a description of the multivariate data set in terms of a few end members (associations or factors, usually orthogonal) that account for the variance within the data set. A factor score represents the importance of each variable in each end member. The set of scores for all factors makes up the factor score matrix. The importance of each variable in each end member is represented by a factor score, which is a unit vector in n (number of variables) dimensional space, with each element having a value between -1 and 1 and the... 

Chemometric evaluation methods can be applied to the signal from a single sensor by feeding the whole data set into an evaluation program [133,135]. Both principle component analysis (PCA) and partial least square (PLS) models were used to evaluate the data. These are chemometric methods that may be used for extracting information from a multivariate data set (e.g., from sensor arrays) [135]. The PCA analysis shows that the MISiC-FET sensor differentiates very well between different lambda values in both lean gas mixtures (excess air) and rich gas mixtures (excess fuel). The MISiC-FET sensor is seen to behave as a linear lambda sensor [133]. It... 

Partial Least Squares Regression is a valuable tool in FTIR-spectroscopy, not only for (routine) quantitative analysis of mixtures, but also as a research application. Due to its ability to expose correlations in complex, multivariate data sets, PLS is gaining importance rapidly in spectroscopy-assisted-research. 

FIGURE 3.4 Arrangement of a multivariate data set in matrix form. 

This chapter deals with multivariate data sets. In the present context, this means that complete spectra are observed as a function of reaction time, e.g., with a diode-array detector. As we will demonstrate, the more commonly performed single-wavelength measurements can be regarded as a special case of multiwavelength measurements. 

The multivariate methods of data analysis, like discriminant analysis, factor analysis and principal component analysis, are often employed in chemometrics if the multiple regression method fails. Most popular in QSRR studies is the technique of principal component analysis (PCA). By PCA one reduces the number of variables in a data set by finding linear combinations of these variables which explain most of the variability [28]. Normally, 2-3 calculated abstract variables (principal components) condense most (but not all) of the information dispersed within the original multivariable data set. 

The third application of GSA shows a simple classification problem between two groups of objects. liie two variable data set is shown in Figure 6. The "o" and "x" symbols represent classes 0 and 1, respectively. The two variables, xl and x2, could represent actual measurements of the objects to be classified. In real life classification problems, they would more likely be composite variables such as the first two principal components of a multivariate data set containing several measurements (e.g. pH, concentrations of various trace elements, near infra-red reflectance signals at multiple wavelengths, etc.) on each object in the set. 

PLS is a method by which blocks of multivariate data sets (tables) can be quantitatively related to each other. PLS is an acronym Partial Least Squares correlation in latent variables, or Projections to Latent Structures. The PLS method is described in detail in Chapter 17. 

Advances in science often come as a result of advances in technology, and this has been especially tme in the fields of genomics, proteomics, and metabolomics. The development of microarrays, advancements in MS and NMR instmmentation, and computing power to analyze large multivariate data sets have provided scientists with unprecedented ability to measure biological response at the... 

Theory. PCA is a frequently used variable reduction technique, which can be used to visualize the objects of a multivariate data set in a lowerdimensional space. This technique calculates new latent variables, called principal components (PCs), which are hnear combinations of the original manifest... 

The simplest multivariate data set is a data set consisting of measurements (or calculated properties) of J variables on I objects. Such a data set can be arranged in an / x. / matrix X. This matrix X contains variation which is supposed to be relevant for the (chemical) problem at hand. Several types of methods are available to investigate this variation depending on the purpose of the research and the problem definition. 

The quantitative data analysis protocols described herein may be generally applied to multivariant data sets and promise to be of increasing utility for the analysis of RSSF spectral data in the near future. It is notable that many of the commercially available RSSF units incorporate data analysis software based upon both global analysis and/or SVD into their data processing software packages. 

We conclude this section on multivariate methods by mentioning a number of additional techniques for obtaining graphic representations of multidimensional data sets. A good general review may be found in Everitt [82]. Amongst the techniques described there is the Andrews plot [83]. For the rth observation in a multivariate data set, the Andrews function is defined as ... 

In principle both the classical and the inverse approach use a multivariate data set. But in the classical approach the variance is minimised, whereas in the inverse approach one tries to find an equilibrium between bias and variance. Therefore the bias is reduced and by the procedure of predictive receivable error sum of squares either via a singular value decomposition or the bidiagonalisation method estimated values, either according to principle component regression or partial least squares, are found. The multilinear regression on the other hand will find the best linear unbiased estimation as an approach to a true concentration. Besides applications in absorption spectroscopy, fluorescence spectra can also be evaluated [74]. 

Eigen analyis A mathematical procedure for identifying the principal axes of a multivariate data set. 

Factor analysis has recently been used in source partitioning modeling of molecular marker investigations [1-4,296-300]. Q-mode factor analysis is based on grouping a multivariate data set based on the data structure defined by the similarity between samples. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or co-variance) relationships e q)lored with R-mode factor analysis. 

The residual terms associated with each system of equations represent the difference between the linear programming estimate and the actual concentration of each organic contaminant in the sample. The optimum solution for each system of equations is that for which the residual terms are minimized. Since a perfect modeling solution would accoimt for 100% of the measured concentration for each organic contaminant, the validity of the present environmental forensic MM model can be evaluated by calculating a mean residual percent of each contaminant (the mean residual for each contaminant divided by the mean contaminant concentration) [Ij. Therefore, the use of linear programming technique partitioning helps correct the initial end-member compositions of SWMs and/or their leachates, and their abundances, to better fit the observed multivariate data set, as well as to specify and select the compositions of the end-members. 

Szydlo, R. M., Ford, M. G., Greenwood, R. and Salt, D. W. (1985) The use of multivariate data sets in the study of structure-activity relationships of synthetic pyrethroid insecticides Part II. The relationships between pharmacokinetics and toxicity, in QSAR and Strategies in the Design of Bioactive Compounds (ed. J. K. Seydel), VCH, Weinheim. 

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