Multiple discriminant analysis

Figure 4. Multiple discriminant analysis for bread making quality of wheat flour.

Discriminant analysis (DA) performs samples classification with an a priori hypothesis. This hypothesis is based on a previously determined TCA or other CA protocols. DA is also called "discriminant function analysis" and its natural extension is called MDA (multiple discriminant analysis), which sometimes is named "discriminant factor analysis" or CD A (canonical discriminant analysis). Among these type of analyses, linear discriminant analysis (LDA) has been largely used to enforce differences among samples classes. Another classification method is known as QDA (quadratic discriminant analysis) (Frank and Friedman, 1989) an extension of LDA and RDA (regularized discriminant analysis), which works better with various class distribution and in the case of high-dimensional data, being a compromise between LDA and QDA (Friedman, 1989). 

Scarponi, G., Moret, I., Capodaglio, G. and Cescon, P. (1982) Multiple discriminant analysis in the analytical differentiation of Venetian wines. 3. A reelaboration with addition of data from samples of 1979 vintage Prosecco wine. J. Agric. Food Chem., 30, 1135-1140. 

Alternatives to Multiple Linear Regression Discriminant Analysis, Neural Networks and Classification Methods... 

We will explore the two major families of chemometric quantitative calibration techniques that are most commonly employed the Multiple Linear Regression (MLR) techniques, and the Factor-Based Techniques. Within each family, we will review the various methods commonly employed, learn how to develop and test calibrations, and how to use the calibrations to estimate, or predict, the properties of unknown samples. We will consider the advantages and limitations of each method as well as some of the tricks and pitfalls associated with their use. While our emphasis will be on quantitative analysis, we will also touch on how these techniques are used for qualitative analysis, classification, and discriminative analysis. 

He, X., Fung, W. K. J. Multivariate Anal. 72, 2000, 151-162. High breakdown estimation for multiple populations with applications to discriminant analysis. 

Multiple Linear Regression Discriminant Analysis Partial Least Squares (PLS)... 

Meisel P, Schroeder C, Wulff K, Siegmund W (1997) Relationship between human genotype and phenotype of N-acetyltransferase (NAT2) as estimated by discriminant analysis and multiple linear regression 1.Genotype and N-acetylation in vivo. Pharmacogenetics 7 241-246... 

Discriminant analysis evaluates the distance between individual points and several centroids hypothesized to exist in the hyperspace defined by elemental concentrations. Davis (28) provides a clear and concise description of the algebra involved in two-group and multiple-group discriminant analysis, showing that discriminant functions are equivalent to the eigenvectors of W-1B, where W1 is the inverse of the within-group sums of products matrix, and B is the between-group sums of products matrix. The Mahalanobis distances from an unknown point to each of the alternative centroids... 

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. 

Equations (25) are linear with respect to x and this classification technique is referred to as /inear discriminant analysis, with the discriminant function obtained by least squares analysis, analogous to multiple regression analysis. 

Linear discriminant analysis is closely related to multiple regression analysis. Whereas in multiple regression, the dependent variable is assumed to be a continuous function of the independent variables, in discriminant analysis the dependent variable, e.g. Group A or Group B, is nominal and discrete. Given this similarity, it is not surprising that the selection of appropriate variables to perform a discriminant analysis should follow a similar scheme to that employed in multiple regression (see Chapter 6). 

As with multiple regression analysis, the most commonly used selection procedures involve stepwise methods with the F-test being applied at each stage to provide a measure of the value of the variable to be added, or removed, in the discriminant function. The procedure is discussed in detail in Chapter 6. 

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