Multivariate techniques association analysis

The following examples demonstrate the usefulness of multivariate methods in the evaluation of field ecological data and laboratory multispecies toxicity tests. In each of the examples, several multivariate techniques were used — generally Euclidean and cosine distances (Figure 11.29), principal components, and nonmetric clustering and association analysis. 

In addition to univariate statistical analysis, the data were also examined by means of multivariate statistical techniques. In particular, R-mode factor analysis was used, which is a very effective tool to interpret anomalies and to help identify their sources. Factor analysis allows grouping of anomalies by compatible geochemical associations from a geologic-mineralogical point of view, the presence of mineralizing processes, or processes connected to the surface environment. Based on this analysis, six meaningful chemical associations were identified (Fig. 15.8). 

Thousands of chemical compounds have been identified in oils and fats, although only a few hundred are used in authentication. This means that each object (food sample) may have a unique position in an abstract n-dimensional hyperspace. A concept that is difficult to interpret by analysts as a data matrix exceeding three features already poses a problem. The art of extracting chemically relevant information from data produced in chemical experiments by means of statistical and mathematical tools is called chemometrics. It is an indirect approach to the study of the effects of multivariate factors (or variables) and hidden patterns in complex sets of data. Chemometrics is routinely used for (a) exploring patterns of association in data, and (b) preparing and using multivariate classification models. The arrival of chemometrics techniques has allowed the quantitative as well as qualitative analysis of multivariate data and, in consequence, it has allowed the analysis and modelling of many different types of experiments. 

Consideration of the results from a simple multi-element analysis will serve to illustrate terms and parameters associated with the techniques used. This example will also introduce some features of matrix operators basic to handling multivariate data. In the scientific literature, matrix representation of multi-... 

According to Ennis (1988), the application of the various multivariate analysis techniques (factor, cluster, discriminant analysis, multidimensional scaling) to classification in sensory analysis has been very valuable but is of little help for understanding the modes of perception. Mathematical models are proposed for predicting human sensory responses and the author concludes that they need development before they are able to improve the understanding of the complex perceptions associated with foods and beverages . 

To this point our discussions have largely focused on the application of matrices to linear problems associated with simultaneous equations, applications that commonly arise in least-square, multiple regression techniques. One further important function that occurs in multivariate analysis and the analysis of variance is the quadratic form. 

Cheng, M. D., Lioy, R J., and Opperman, A. J. (1988) Resolving PMi0 data collected in New Jersey by various multivariate analysis techniques, in PMl0 Implementation of Standards, C. V. Mathai and D. H. Stonefield, eds., Air Pollution Control Association, Pittsburgh, PA, pp. 472-483. 

Chapter 4 retrieves the basic ideas of classical univariate calibration as the standpoint from which the natural and intuitive extension of multiple linear regression (MLR), arises. Unfortunately, this generalization is not suited to many laboratory tasks and, therefore, the problems associated with its use are explained in some detail. Such problems justify the use of other more advanced techniques. The explanation of what the multivariate space looks like and how principal components analysis can tackle it is the next step forward. This constitutes the root of the regression methodology presented in the following chapter. 

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