Multivariate statistical methods

MacGregor, J. F., Marlin, T. E., Kresta, J. V., and Skagerberg, B., Multivariate statistics methods in process analysis and control. In Chemical Process Control, CPCIV, (Y. Arkun and W.H. Ray, eds.). CACHE, AIChE Publishers, New York, 1991. 

With regard to linear projection based methods, the latent variables or scores determined by linear multivariate statistical methods such as... 

Piovoso, M. J., and Kosanovich, K. A., Applications of multivariate statistical methods to process monitoring and controller design, Int. J. Control 59(3), 743-765 (1994). 

PCA [12, 16] is a multivariate statistics method frequently applied for the analysis of data tables obtained from environmental monitoring studies. It starts from the hypothesis that in the group of original data, there is a set of reduced factors or dominant components (sources of variation) which influence the observed data variance in an important way, and that these factors or components cannot be directly measured (they are hidden factors), since no specific sensors exist for them or, in other words, they cannot be experimentally observed. 

Keil, D. et al., Evaluation of multivariate statistical methods for analysis and modeling of immunotoxicology data, Toxicol. Sci. 51, 245, 1999. 

Hierarchical Cluster Analysis (HCA) is a multivariate statistical method that can be used assign groundwater samples or monitoring sites to distinct categories (hydrochemical facies). HCA offers several advantages over other methods of... 

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... 

Bakraji, E. H., Othman, I., Sarhil, A., and Al-Somel, N. (2002). Application of instrumental neutron activation analysis and multivariate statistical methods to archaeological Syrian ceramics. Journal of Trace and Microprobe Techniques 20 57-68. 

Solvent selection based on cohesion parameters, like those of Hansen [12], and by multivariate statistical methods like principal components analysis are two potential methods that can be used for solvent selection. These effects will be examined further in section 3.6.1. 

B. Berente, D.D.C. Garcia, M. Reichenbacher and K. Danzer, Method development for the determination of anthocyanins in red wines by high-performance liquid chromatography and classification of German red wines by means of multivariate statistical methods. J. ChromatogrA 871 (2000) 95-103. 

This book is the result of a cooperation between a chemometrician and a statistician. Usually, both sides have quite a different approach to describing statistical methods and applications—the former having a more practical approach and the latter being more formally oriented. The compromise as reflected in this book is hopefully useful for chemometricians, but it may also be useful for scientists and practitioners working in other disciplines—even for statisticians. The principles of multivariate statistical methods are valid, independent of the subject where the data come from. Of course, the focus here is on methods typically used in chemometrics, including techniques that can deal with a large number of variables. Since this book is an introduction, it was necessary to make a selection of the methods and applications that are used nowadays in chemometrics. 

The focus is on multivariate statistical methods typically needed in chemo-metrics. In addition to classical statistical methods, also robust alternatives are introduced which are important for dealing with noisy data or with data including outliers. Practical examples are used to demonstrate how the methods can be applied and results can be interpreted however, in general the methodical part is separated from application examples. 

Chapter 3 starts with the first and probably most important multivariate statistical method, with principal component analysis (PC A). PC A is mainly used for mapping or summarizing the data information. Many ideas presented in this chapter, like the selection of the number of principal components (PCs), or the robustification of PCA, apply in a similar way to other methods. Section 3.8 discusses briefly related methods for summarizing and mapping multivariate data. The interested reader may consult extended literature for a more detailed description of these methods. 

Manly, B. F. J. Multivariate Statistical Methods A Primer. Chapman Hall, London, United Kingdom, 2000. 

Chemometrics, as defined by Kowalski (1), includes the application of multivariate statistical methods to the study of chemical problems. SIMCA (Soft Independent Method of Class Analogy) and other multivariate statistical methods have been used as tools in chemometric investigations. SIMCA, based on principal components, is a multivariate chemometric method that has been applied to a variety of chemical problems of varying complexity. The SIMCA-3B program is suitable for use with 8- and 16-bit microcomputers. 

Morrison, D. F. "Multivariate Statistical Methods" McGraw-Hill, New York, 1967, p. 338. 

Table 1 lists a number of studies carried out in urban sites of the Eastern Mediterranean Basin that have used multivariate statistical methods to quantify the mass contribution of sources of particulate matter. In most of these studies, four or five major source categories have been detected. These categories include road/soil dust, traffic emissions, marine aerosol, fuel oil combustion emissions, biomass... 

In another very recent paper (Son et al., 2009b), the fermentative performances of yeast strains used for grape must fermentation were monitored by NMR and multivariate statistical methods. Characterization of the properties of wine yeasts is important because they affect wine quality. In this paper, the changes of metabolites in must during alcoholic... 

However, it should be emphasized that the statistical methods presented here are no cures for poor data. Irrelevant or erroneous measurement and poorly planned experiments will still be irrelevant, erroneous and poorly planned in spite of any statistical analysis. There are, however, many examples of excellent data that have been seriously mutilated by poor statistical analysis. The aim of this chapter is to present multivariate statistical methods for design and... 

There are apparently many multivariate statistical methods partly overlapping in scope [11]. For most problems occurring in practice, we have found the use of two methods sufficient, as discussed below. The first method is called principal component analysis (PCA) and the second is the partial least-squares projection to latent structures (PLS). A detailed description of the methods is given in Appendix A. In the following, a brief description is presented. 

Because data analysis is of central interest, particularly in the application of chemometric methods in the field of environmental research, a rough list of important multivariate statistical methods is given below (Tab. 1-1). 

Manly, B.F.J. Multivariate Statistical Methods A Primer, Chapman and Hall, London, 1986 Massart, D.L., Vandeginste, B.G.M., Deming, S.N., Michotte, Y., Kaufman, L. Chemometrics A Textbook, Elsevier, Amsterdam, 1988... 

Identification of the pollutant pattern Application of multivariate statistical methods (see, for example, Section 9.2) for the detection of emitters or origins. 

Relationships between variables, which can be expressed as correlations, or relationships between the groups cannot be detected. Multivariate statistical methods must therefore be applied. 

The visual estimation of differences between groups of data has to be proved using multivariate statistical methods, as for example with multivariate analysis of variance and discriminant analysis (see Section 5.6). 

In environmental chemistry in particular it cannot be the last step of the investigation to produce data tables on pollutant concentrations. Multivariate statistical methods offer a tool for the investigation of the multifactorial and complex events in the environment. 

To demonstrate the accuracy, two dust and two soil reference materials were analyzed with the described method. The mean value of the correlation coefficients between the certified and the analyzed amounts of the 16 elements in the samples is r = 0.94. By application of factor analysis (see Section 5.4) the square root of the mean value of the communahties of these elements was computed to be approximately 0.84. As frequently happens in the analytical chemistry of dusts several types of distribution occur [KOM-MISSION FUR UMWELTSCHUTZ, 1985] these can change considerably in proportion to the observed sample size. In the example described the major components are distributed normally and most of the trace components are distributed log-normally. The relative ruggedness of multivariate statistical methods against deviations from the normal distribution is known [WEBER, 1986 AHRENS and LAUTER, 1981] and will be tested using this example by application of factor analysis. 

Commonly the compromising conditions of routine environmental monitoring lead to restrictions on the accuracy and the precision of sampling and analysis. The purpose of this section is to show that under these conditions multivariate statistical methods are a useful tool for qualitative extraction of new information about the degree of stress of the investigated areas, and for identification of emission sources and their seasonal variations. The results represented from investigation of the impact of particulate emissions can, in principle, be transferred to other environmental analytical problems, as described in the following case studies. 

Multivariate statistical methods should be preferred for evaluating such multidimensional data sets since interactions and resulting correlations between the water compounds have to be considered. Fig. 8-1, which shows the univariate fluctuations in the concentrations of the analyzed compounds, illustrates the large temporal and local variability. Therefore in univariate terms objective assessment of the state of pollutant loading is hardly possible. 

These multivariate statistical methods consider relative pollution changes or relationships of variances, because the basis of the computations is the matrix of the correlation coefficients. The absolute values of the concentration changes are not considered. Therefore conclusions regarding the actual state of pollution can only be drawn with respect to the actual data. 

---