Multivariate discriminant function

Schulz, J., Mollmann, C., Hirche, H. J., 2007. Vertical zonation of the zooplankton community in the Central Baltic Sea in relation to hydrographic stratification as revealed by multivariate discriminant function and canonical analysis. Journal of Marine Systems, 67, 47-58. 

A different approach to unmixing involves the use of multivariate discriminant functions. Collins et al. (1996, 1997) successfully apply this technique to sediment source discrimination in suspended river particulates and river over-bank deposits. 

D. Coomans, I. Broeckaert, M. Jonckheer and D.L. Massart, Comparison of multivariate discrimination techniques for clinical data—Application to the thyroid functional state. Meth. Inform. Med., 22 (1983) 93-101. 

Distance (in multivariate data) Discriminant function Discriminant variable... 

Multivariate statistical analysis using classes of variables and calculating discriminant functions as linear combinations of the variables that maximize the inter-class variance and minimize the intra-class variance. Volume 2(2). 

The MANOVA enables significant class separation with a multivariate scaled separation measure of 330.9. The sampling times 5 a.m. and 11 p.m. are well separable from the times 11 a.m. and 5 p.m. by the optimum separation set which consists in the features suspended material, iron, magnesium, nickel, and copper. The result of discriminant analysis is shown in the plane of the two strongest discriminant functions (Fig. 8-3). 

The principle of multivariate analysis of variance and discriminant analysis (MVDA) consists in testing the differences between a priori classes (MANOVA) and their maximum separation by modeling (MDA). The variance between the classes will be maximized and the variance within the classes will be minimized by simultaneous consideration of all observed features. The classification of new objects into the a priori classes, i.e. the reclassification of the learning data set of the objects, takes place according to the values of discriminant functions. These discriminant functions are linear combinations of the optimum set of the original features for class separation. The mathematical fundamentals of the MVDA are explained in Section 5.6. 

Fig. 9-9 demonstrates the results of MVDA for the three investigated territories in the plane of the computed two discriminant functions. The separation line corresponds to the limits of discrimination for the highest probability. The results prove that good separation of the three territories with a similar geological background is possible by means of discriminant analysis. The misclassification rate amounts to 13.0%. The scattering radii of the 5% risk of error of the multivariate analysis of variance overlap considerably. They demonstrate also that the differences in the multivariate data structure of the three territories are only small. 

There are a large number of mediods for supervised pattern recognition, mostly aimed at classification. Multivariate statisticians have developed many discriminant functions, some of direct relevance to chemists. A classical application is the detection of forgery of banknotes. Can physical measurements such as width and height of a series of banknotes be used to identify forgeries Often one measurement is not enough, so several parameters are required before an adequate mathematical model is available. 

If the assumption is made that the covariance matrices for the two classes are the same, then the discriminant function developed using the Bayes theorem and the multivariate normal assumption simplifies from that above to the following ... 

The linear discriminant function is a most commonly used classification technique and it is available with all the most popular statistical software packages. It should be borne in mind, however, that it is only a simplification of the Bayes classifier and assumes that the variates are obtained from a multivariate normal distribution and that the groups have similar covariance matrices. If these conditions do not hold then the linear discriminant function should be used with care and the results obtained subject to careful analysis. 

The adaptive least squares (ALS) method [396, 585 — 588] is a modification of discriminant analysis which separates several activity classes e.g. data ordered by a rating score) by a single discriminant function. The method has been compared with ordinary regression analysis, linear discriminant analysis, and other multivariate statistical approaches in most cases the ALS approach was found to be superior to categorize any numbers of classes of ordered data. ORMUCS (ordered multicate-gorial classification using simplex technique) [589] is an ALS-related approach which... 

We also make a distinction between parametric and non-parametric techniques. In the parametric techniques such as linear discriminant analysis, UNEQ and SIMCA, statistical parameters of the distribution of the objects are used in the derivation of the decision function (almost always a multivariate normal distribution... 

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... 

The approach of Fisher (1938) was originally proposed for discriminating two populations (binary classification), and later on extended to the case of more than two groups (Rao 1948). Here we will first describe the case of two groups, and then extend to the more general case. Although this method also leads to linear functions for classification, it does not explicitly require multivariate normal distributions of the groups with equal covariance matrices. However, if these assumptions are not... 

This supervised classification method, which is the most used, accepts a normal multivariate distribution for the variables in each population ((Ai,..., A ) Xi) ), and calculates the classification functions minimising the possibility of incorrect classification of the observations of the training group (Bayesian type rule). If multivariate normality is accepted and equality of the k covariance matrices ((Ai,..., Xp) NCfti, X)), Linear Discriminant Analysis (LDA) calculates... 

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