Methods of multivariate statistics

The application of methods of multivariate statistics (here demonstrated with examples of cluster analysis, multivariate analysis of variance and discriminant analysis, and principal components analysis) enables clarification of the lateral structure of the types of feature change within a test area. 

The following two examples [EINAX et al., 1990 KRIEG and EINAX, 1994] demonstrate not only the power, but also the limits of multivariate statistical methods applied to the description of polluted soils loaded with heavy metals from different origins. Case studies with chemometric description of soil pollution by organic compounds are also discussed in the literature. DING et al. [1992], for example, evaluated local sources of chlorobenzene congeners in soil samples by using different methods of multivariate statistical analysis. 

In this section we shall consider the rather general case where for a series of chemical compounds measurements are made in a number of parallel biological tests and where a set of descriptor variables is believed to be related to the biological potencies observed. In order to imderstand the data in their entirety and to deal adequately with the mathematical properties of such data, methods of multivariate statistics are required. A variety of such methods is available as, for example, multivariate regression, canonical correlation, principal component analysis, principal component regression, partial least squares analysis, and factor analysis, which have all been applied to biological or chemical problems (for reviews, see [1-11]). Which method to choose depends on the ultimate objective of an analysis and the property of the data. We have found principal component and factor analysis particularly useful. For this reason and also since many multivariate methods make use of components for factors we will start with these methods in some detail, while the discussion of other approaches will be less extensive. 

These sensors can be used as simple single sensors or they can form a sensor array for multiple analytes detection. The sensor array can also be formed by unselective sensors that can generate a response pattern that can be analyzed by powerful methods of multivariate statistics. In principle, appropriate sensors can be placed along all the food chain as shown in Figure 15.1. 

Throughout the 1970s, appHcations of pattern recognition were found in the chemical sciences. Other methods of multivariate mathematics and statistics were borrowed or invented, and a new discipline called chemometrics arose. In 1974, the Chemometrics Society was formed, and the first Chemometrics newsletter came out in 1976 (12). 

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). 

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

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. 

Peter Filzmoser was bom in 1968 in Weis, Austria. He studied applied mathematics at the Vienna University of Technology, Austria, where he wrote his doctoral thesis and habilitation, devoted to the field of multivariate statistics. His research led him to the area of robust statistics, resulting in many international collaborations and various scientific papers in this area. His interest in applications of robust methods resulted in the development of R software packages. J ( He was and is involved in the organization of several y scientific events devoted to robust statistics. Since... 

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. 

Many applications of GC-MS and HPLC-UV-VIS prove the powerful capabilities of these methods to analyze complex mixtures. The principal limiting factor is the obtainable separation, which can be optimized as described in section 3.1. When the physico-chemical separation is incomplete, a mathematical improvement of the resolution can be considered by the application of multivariate statistics. 

The arrival of computers in every chemical laboratory has made possible the use of multivariate statistical analysis and mathematics in the analysis of measured chemical data. Sometimes, the methods were inadequate or only partially suitable for a particular chemical problem, so handling methods were modified or new ones developed to fit the chemical problem. On the basis of these elements, common to every field of chemistry, in 1974 a new chemical science was identified chemometrics, the science of chemical information. In the same year, Bruce Kowalski and Svante Wold founded the Chemometrics Society, which since then has been spreading information on multivariates in chemistry all over the world. 

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

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. 

By application of multivariate statistical methods for consideration of the overall environmental situation of a river it is possible to optimize the sampling strategy. 

Sediment analyses are useful for characterization of pollution over a long period [MULLER, 1981]. Assessment of the state of a river and of the interactions between the components can be made by application of multivariate statistical methods only, because the strongly scattering territorial and temporal courses [FORSTNER and MULLER, 1974 FORSTNER and WITTMANN, 1983] are not compatible with many univariate techniques. FA shall serve as a tool for the recognition of variable structures and for the differentiated evaluation of the pollution of both river water and sediment [GEISS and EINAX, 1991 1992],... 

Modeling of the connections and the binding forms requires the application of multivariate statistical methods. A regression model with latent variables, e.g. the PLS method (see Section 5.7.2 for mathematical details), seems to be useful. The PLS method models not only the influences of the plan matrix on the responses, but also the interactions between the metals in the response matrix. In the matrix of the dependent variables the electroche-mically available parts of the heavy metals cea were filled in (Tab. 8-12) ... 

The purpose of this case study is to investigate the following two questions by means of multivariate statistical methods ... 

In soil science, the empirical description of soil horizons predominates. Only a few applications of statistical methods in this scientific field are described. SCHEFFER and SCHACHTSCHABEL [1992] give an example for the classification of different soils into soil groups using cluster analysis. They claim the objectivity of the results to be one advantage of multivariate statistical methods. 

After xenobiotic/toxin exposure, differentially expressed proteins are identified by the comparison of SELDI spectra from control and treated samples. By combining groupwise statistics with N-fold regulations, single biomarkers (m/z) can be selected. As to be expected from the complexity of the proteome, in many cases no single marker will be able to discriminate between the groups. Rather, a complex pattern of multiple markers will be acquired (Figure 8). Discovery of such markers/pattems can be successful by application of multivariate statistics methods on the data set. However, for the identification of specific protein expression patterns bioinformatics tools are... 

Finally it is important to note that modern analytical equipment frequently offers opportunities for measuring several or many characteristics of a material more or less simultaneously. This has encouraged the development of multivariate statistics methods, which in principle permit the simultaneous analysis of several components of the material. Partial least squares methods and principal component regression are examples of such techniques that are now finding extensive uses in several areas of analytical science. ... 

Martm-Alvarez, P.J., Herranz, A. (1991). Application of multivariate statistical methods to the differentiation of gin brands. J. Sci. Food Agric., 57, 263-272. 

These differ in the exact way in which variations in the data (responses) are used to predict the concentration. Software for accomplishing multivariate calibration is available from several companies. The use of multivariate statistical methods for quantitative analysis is part of the subdiscipline of chemistry called chemometrics. 

The choice of properties has a major influence on pattern recognition methods (unsupervised multivariate statistical or neural network methods) in particular and different property sets can resnlt in qnite different patterns of similarity between compounds. Several methods are available to make selections of subsets of uncorrelated properties which can be used... 

The detection and diagnosis tasks can be carried out on the process measurements to obtain critical insights into the performance of not only the process itself but also the automatic control system that is deployed to assure normal operation. Today, the integration of such tasks into the process control software associated with Distributed Control Systems (D-CS) is in progress. The technologies continue to advance, especially in the incorporation of multivariate statistics as well as recent developments in signal processing methods such as wavelets and hidden Markov models. 

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