Variables multivariate methods

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. 

Sets of spectroscopic data (IR, MS, NMR, UV-Vis) or other data are often subjected to one of the multivariate methods discussed in this book. One of the issues in this type of calculations is the reduction of the number variables by selecting a set of variables to be included in the data analysis. The opinion is gaining support that a selection of variables prior to the data analysis improves the results. For instance, variables which are little or not correlated to the property to be modeled are disregarded. Another approach is to compress all variables in a few features, e.g. by a principal components analysis (see Section 31.1). This is called... 

As introduced earlier, inputs can be transformed to reduce their dimensionality and extract more meaningful features by a variety of methods. These methods perform a numeric-numeric transformation of the measured input variables. Interpretation of the transformed inputs requires determination of their mapping to the symbolic outputs. The inputs can be transformed with or without taking the behavior of the outputs into account by univariate and multivariate methods. The transformed features or latent variables extracted by input or input-output analysis methods are given by Eq. (5) and can be used as input to the interpretation step. 

The traditional acute, subchronic, and chronic toxicity studies performed in rodents and other species also can be considered to constitute multiple endpoint screens. Although the numerically measured continuous variables (body weight, food consumption, hematology values) generally can be statistically evaluated individually by traditional means, the same concerns of loss of information present in the interrelationship of such variables apply. Generally, traditional multivariate methods are not available, efficient, sensitive, or practical (Young, 1985). 

A simple strategy for variable selection is based on the information of other multivariate methods like PCA (Chapter 3) or PLS regression (Section 4.7). These methods form new latent variables by using linear combinations of the regressor... 

Multivariate methods, on the other hand, resolve the major sources by analyzing the entire ambient data matrix. Factor analysis, for example, examines elemental and sample correlations in the ambient data matrix. This analysis yields the minimum number of factors required to reproduce the ambient data matrix, their relative chemical composition and their contribution to the mass variability. A major limitation in common and principal component factor analysis is the abstract nature of the factors and the difficulty these methods have in relating these factors to real world sources. Hopke, et al. (13.14) have improved the methods ability to associate these abstract factors with controllable sources by combining source data from the F matrix, with Malinowski s target transformation factor analysis program. (15) Hopke, et al. (13,14) as well as Klelnman, et al. (10) have used the results of factor analysis along with multiple regression to quantify the source contributions. Their approach is similar to the chemical mass balance approach except they use a least squares fit of the total mass on different filters Instead of a least squares fit of the chemicals on an individual filter. 

Above mentioned examples clearly show that if multivariate data processing methods are applicable, analytical information can be derived with a minimal amount of pre-information and a foreseeing of a maximum of problems. When the sampled object is homogenous, multivariate methods are only applicable when the analytical method itself produces multivariate signals. This is the case when several signals (e.g. spectra) are obtained for the sample as a function of another variable (e.g. time, excitation wavelength). For e mple in GC-MS, a mass spectrum is m sured of the eluents every. 1 a 1 second. In excitation-emission spectroscopy, spectra are measured at several excitation-wavelengths. The potentials of the application of multivariate... 

Liu, Y. Multivariate Methods for Relating Different Sets of Variables with Applications in Food Research. PhD Thesis, UnivCTsity of Bristol, UK, 1990. 

The term factor is a catch-all for the concept of an identifiable property of a system whose quantity value might have some effect on the response. Factor tends to be used synonymously with the terms variable and parameter, although each of these terms has a special meaning in some branches of science. In factor analysis, a multivariate method that decomposes a data matrix to identify independent variables that can reconstitute the observed data, the term latent variable or latent factor is used to identify factors of the model that are composites of input variables. A latent factor may not exist outside the mathematical model, and it might not therefore influence... 

It should be noted that there are other multivariate variable selection methods that one could consider for their application. For example, the interactive variable selection (IVS) method71 is an actual modification of the PLS method itself, where different sets of X-variables are removed from the PLS weights (W, see Equation 8.37) of each latent variable in order to assess the usefulness at each X-variable in the final PLS model. 

How can multivariate methods be used to avoid the problems associated with the OVAT approach In general, multivariate methods use the information contained in the relation between the variables (correlations or covariances) and therefore data like those in Figure 6.3 present no problem. The risk of type I errors is kept under control in multivariate analysis by considering all variables simultaneously. To consider all variables simultaneously involves a... 

The strategy that we have therefore sought to follow is to exploit multivariate methods which can measure many variables simultaneously. The resulting data floods necessitate the use of robust, multivariate chemometric methods. These too are now available in many flavours, with different strengths and weaknesses. 

Variable selection is particularly important in LC-MS and GC-MS. Raw data form what is sometimes called a sparse data matrix, in which the majority of data points are zero or represent noise. In fact, only a small percentage (perhaps 5% or less) of die measurements are of any interest. The trouble with this is that if multivariate methods are applied to the raw data, often the results are nonsense, dominated by noise. Consider the case of performing LC-MS on two closely eluting isomers, whose fragment ions are of principal interest. The most intense peak might be the molecular... 

Among the multivariate methods the most important are principal components analysis (PCA), factor analysis, cluster analysis and the pattern recognition method, from which only PCA will be briefly described below. PCA is used to find such a system of new variables, called principal components (PC), which explains the variation of a given data set in a more convenient way than the original system of variables, e.g. xl9...,Xj,...,xm. The greater convenience of PC consists mainly in a reduction of dimensions, m, in which the data were originally described, because the PC variables are chosen so that only two or three of them should be sufficient to characterize the variation of the data. The PC are linear combinations of the original variables, xj9 used to characterize the set of objects,... 

Therefore it is necessary to use methods which allow all variables (substituents) to be investigated simultaneously, i.e. multivariate methods [26], This can be achieved by using design (e.g., factorial design [28] or D-optimal design [29]) in principal properties and evaluating the result by multivariate analysis as shown below. 

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. 

Geladi, P. and Tosato, M.L. (1990). Multivariate Latent Variable Projection Methods SIMCA and PLS. In Practical Applications of Quantitative Structure-Activity Relationships (QSAR) in Environmental Chemistry and Toxicology (Karcher, W. and Devillers, J., eds.), Kluwer, Dordrecht (The Netherlands), pp. 145-152. 

The remainder of this section details the potential application of multivariate methods in the selection of endpoints and in the evaluation of exposure and effects of stressors in ecosystems. Particular reference is made to the application of these methods to the current framework for ecological risk assessment. Examples of the use of multivariate methods in detecting effects and in selecting important measurement variables are covered using both field surveys and multispecies toxicity tests. 

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