Basic multivariate methods

In Chapter 2, we approach multivariate data analysis. This chapter will be helpful for getting familiar with the matrix notation used throughout the book. The art of statistical data analysis starts with an appropriate data preprocessing, and Section 2.2 mentions some basic transformation methods. The multivariate data information is contained in the covariance and distance matrix, respectively. Therefore, Sections... 

The main goal of this chapter is to present the theoretical background of some basic chemometric methods as a tool for the assessment of surface water quality described by numerous chemical and physicochemical parameters. As a case study, long-term monitoring results from the watershed of the Struma River, Bulgaria, are used to illustrate the options offered by multivariate statistical methods such as CA, principal components analysis, principal components regression (models of source apportionment), and Kohonen s SOMs. 

This Add-in provides a basic functionality for many of the multivariate methods described in Chapters 4-6 and can be used when solving the problems. 

All the wavelet coefficients above resolution level k are set to zero and do not contribute. For each combination i the chosen multivariate method is applied to the masked data. Fig. 5 illustrates the basic idea behind the method. 

With respect to the applied mathematical algorithms, quantitative evaluation of spectra can be subdivided into univariate and multivariate methods (cf. Fig. 22.2). The independent variables x and respectively, are denoted regressors, whereas the dependent variables y and y , respectively, are denoted regressands. The basic sequence of a quantitative evaluation is always the same ... 

Lohninger, H. 1999. Teach Me Data Analysis, Springer, Berlin. (This package contains a book and a CD-ROM, and most of the examples are focused on analytical chemistry. Covers basic statistics and some multivariate methods.)... 

Chatfield and Collins (1980), in the introduction to their chapter on cluster analysis, quote the first sentence of a review article on cluster analysis by Cormack (1971) The availability of computer packages of classification techniques has led to the waste of more valuable scientific time than any other statistical innovation (with the possible exception of multiple-regression techniques). This is perhaps a little hard on cluster analysis and, for that matter, multiple regression but it serves as a note of warning. The aim of this book is to explain the basic principles of the more popular and useful multivariate methods so that readers will be able to understand the results obtained from the techniques and, if interested, apply the methods to their own data. This is not a substitute for a formal training in statistics the best way to avoid wasting one s own valuable scientific time is to seek professional help at an early stage. 

J. N. Miller, Basic statistical methods for analytical chemistry. Part 2 Calibration and regression methods. A Review, Analyst, 1991, 116, 3-13. R. G. Brereton, Introduction to multivariate calibration in analytical chemistry. Analyst, 2000, 125, 2125-2154. 

Some methods that paitly cope with the above mentioned problem have been proposed in the literature. The subject has been treated in areas like Cheraometrics, Econometrics etc, giving rise for example to the methods Partial Least Squares, PLS, Ridge Regression, RR, and Principal Component Regression, PCR [2]. In this work we have chosen to illustrate the multivariable approach using PCR as our regression tool, mainly because it has a relatively easy interpretation. The basic idea of PCR is described below. 

This variation on Newton s method usually requires more iterations than the pure version, but it takes much less work per iteration, especially when there are two or more basic variables. In the multivariable case the matrix Vg(x) (called the basis matrix, as in linear programming) replaces dg/dx in the Newton equation (8.85), and g(Xo) is the vector of active constraint values at x0. 

The FORTRAN program ARTHUR (Harper et al. 1977)—running on main frame computers at this time—comprised all basic procedures of multivariate data analysis and made these methods available to many chemists in the late 1970s. 

Basic understanding and efficient use of multivariate data analysis methods require some familiarity with matrix notation. The user of such methods, however, needs only elementary experience it is for instance not necessary to know computational details about matrix inversion or eigenvector calculation but the prerequisites and the meaning of such procedures should be evident. Important is a good understanding of matrix multiplication. A very short summary of basic matrix operations is presented in this section. Introductions to matrix algebra have been published elsewhere (Healy 2000 Manly 2000 Searle 2006). 

In this work, two methods for the determination of caffeine in energy drinks by derivative spectrophotometry and by partial least-squares multivariate spectropho-tometric calibration (PLS-1) are described. Proposed methods involve background correction methods that interferences from vitamins, taurin and food colours were minimized by treating the sample with basic lead acetate and NaHC03 for the arralysis of caffeine in energy drinks. 

The multivariate tools typically used for the NIR-CI analysis of pharmaceutical products fall into two main categories pattern recognition techniques and factor-based chemometric analysis methods. Pattern recognition algorithms such as spectral correlation or Euclidian distance calculations basically determine the similarity of a sample spectrum to a reference spectrum. These tools are especially useful for images where the individual pixels yield relatively unmixed spectra. These techniques can be used to quickly define spatial distributions of known materials based on external reference spectra. Alternatively, they can be used with internal references, to locate and classify regions with similar spectral response. 

SIMCA (each class described by a PC model). The basic idea of the SIMCA method is that multivariate data measured on a group of similar objects, a proper class are well approximated by a simple PC model. 

Among the multivariate statistical techniques that have been used as source-receptor models, factor analysis is the most widely employed. The basic objective of factor analysis is to allow the variation within a set of data to determine the number of independent causalities, i.e. sources of particles. It also permits the combination of the measured variables into new axes for the system that can be related to specific particle sources. The principles of factor analysis are reviewed and the principal components method is illustrated by the reanalysis of aerosol composition results from Charleston, West Virginia. An alternative approach to factor analysis. Target Transformation Factor Analysis, is introduced and its application to a subset of particle composition data from the Regional Air Pollution Study (RAPS) of St. Louis, Missouri is presented. 

If we do have, and include, a priori knowledge in addition to the measurements or numerical data we should use methods from the second family of basic data analysis methods. Here the data are considered to be grouped in respect of the objects, or maybe in respect of the features. Within this family we may further distinguish between non-causally and causally determined data, or by analogy with correlation and regression, we may distinguish between multivariate relationships and dependencies. 

There are many excellent articles and books on multivariate calibration which provide greater details about the algorithms.7 14 This article will compare the basic methods, illustrated by case studies, and will also discuss more recent... 

Data preprocessing is important in multivariate calibration. Indeed, the relationship between even basic procedures such as centring the columns is not always clear, most investigators following conventional methods, that have been developed for some popular application but are not always appropriately transferable. Variable selection and standardisation can have a significant influence on the performance of calibration models. 

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