Discriminant variable

In feature selection one selects from the m variables a subset of variables that seem to be the most discriminating. Feature selection therefore constitutes a means of choosing sets of optimally discriminating variables and, if these variables are the results of analytical tests, this consists, in fact, of the selection of an optimal combination of analytical tests or procedures. 

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

Fig. 8.5. Discrimination of object quality (+/—) by means of one variable Xi (a), of two variables X and x2 (b) which are represented as biplot in (c) from which the discriminant variable dv (discriminant function df) (d) can be derived...

Some tests that are widely used in univariate statistics are listed here together with hints for their use within R and the necessary requirements, but without any mathematical treatment. In multivariate statistics these tests are rarely applied to single variables but often to latent variables for instance, a discriminant variable can be defined via a two-sample f-test. 

In multivariate classification, the latent variable is a discriminant variable possessing optimum capability to separate two object classes. 

Appropriate latent variables (discriminant variables defined by regression models, Section 5.2)... 

In case of two groups, the Fisher method transforms the multivariate data to a univariate discriminant variable such that the transformed groups are separated as much as possible. For this transformation, a linear combination of the original x-variables is used, in other words a latent variable. 

Figure 5.6 visualizes the idea of Fisher discriminant analysis for two groups in the two-dimensional case. The group centers (filled symbols) are projected on the discriminant variable, giving yi and y2 the mean of both is the classification threshold y0. 

Figure 5.30 shows the density functions of the estimated y-values (discriminant variable) of the calibration set (left) and the test set (right). The solid lines refer to a PLS model with two components, the dashed lines to a PLS model with one component, and the dashed-dotted lines to a PLS model with three components. The maxima of the density functions are approximately at the values —1 and 1, corresponding to the group codes. The overlap of the density functions indicates wrong group assignments for the calibration set, the overlap is smallest for the model with two components, for the test set two or three components are better than one component. 

FIGURE 5.30 D-PLS is applied to the phenyl data using a model with one, two, and three components calculated from the calibration set. The density functions of the estimated y-values (discriminant variable) are shown for the calibration set (left) and for the test set (right). 

Two groups of objects can be separated by a decision surface (defined by a discriminant variable). Methods using a decision plane and thus a linear discriminant variable (corresponding to a linear latent variable as described in Section 2.6) are LDA, PLS, and LR (Section 5.2.3). Only if linear classification methods have an insufficient prediction performance, nonlinear methods should be applied, such as classification trees (CART, Section 5.4), SVMs (Section 5.6), or ANNs (Section 5.5). 

Dudewicz, E.J. Statistical Analysis of Magnetic Resonance Imaging Data in The Normal Brain, Part I Data, Screening, Normality, Discrimination, Variability" unpublished report, 1985. 

Differences in the phenolic composition of these types of sherries enable their differentiation, even during their earliest stages of production. In fact, discriminate variables, obtained by Linear Discriminant... 

The modelling power varies between 1 (excellent) and 0 (no discrimination). Variables with M below 0.5 are of little use. In this example, it can be seen that variables 4 and 6 are not very helpful, whereas variables 2 and 5 are extremely useful. This information can be used to reduce the number of measurements. 

Cerrato Oliveros et al. (2002) selected array of 12 metal oxide sensors to detected adulteration in virgin olive oils samples and to quantify the percentage of adulteration by electronic nose. Multivariate chemometric techniques such as PCA were applied to choose a set of optimally discriminant variables. Excellent results were obtained in the differentiation of adulterated and non-adulterated olive oils, by application of LDA, QDA. The models provide very satisfactory results, with prediction percentages >95%, and in some cases almost 100%. The results with ANN are slightly worse, although the classification criterion used here was very strict. To determine the percentage of adulteration in olive oil samples multivariate calibration techniques based on partial least squares and ANN were employed. Not so good results were carried out, even if there are exceptions. Finally, classification techniques can be used to determine the amount of adulterant oil added with excellent results. 

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