Class variables

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods. 

In Section 33.2.2 we showed how LDA classification can be described as a regression problem with class variables. As a regression model, LDA is subject to the problems described in Chapter 10. For instance, the number of variables should not exceed the number of objects. One solution is to apply feature selection or... 

A more recently introduced technique, at least in the field of chemometrics, is the use of neural networks. The methodology will be described in detail in Chapter 44. In this chapter, we will only give a short and very introductory description to be able to contrast the technique with the others described earlier. A typical artificial neuron is shown in Fig. 33.19. The isolated neuron of this figure performs a two-stage process to transform a set of inputs in a response or output. In a pattern recognition context, these inputs would be the values for the variables (in this example, limited to only 2, X and x- and the response would be a class variable, for instance y = 1 for class K and y = 0 for class L. 

Corresponding to final static class variables in Java (const static pointers in C++). 

Class variables. Data stored once per class and shared by all its instances. 

Invariants over class variables, and pre- and postconditions attached to message signatures, are treated as for joining types They are anded. 

LDA uses a space that is defined by a basis set of vectors, called linear discriminants (LDs) that are similar to the PCs obtained from PCA analysis. Like PCs, LDs are linear combinations of the original M variables in the jc data that are also orthogonal to one another. However, they are determined using a quite different criterion where the ratio of between-class variability and within-class variability in the calibration data is maximized. 

Although the SIMCA method is very versatile, and a properly optimized model can be very effective, one must keep in mind that this method does not use, or even calculate, between-class variability. This can be problematic in special cases where there is strong natural clustering of samples that is not relevant to the problem. In such cases, the inherent interclass distance can be rather low compared to the mtraclass variation, thus rendering the classification problem very difficult. Furthermore, from a practical viewpoint, the SIMCA method requires that one must obtain sufficient calibration samples to fully represent each of the J classes. Also, the on-line deployment of a SIMCA model requires a fair amount of overhead, due to the relatively large number of parameters and somewhat complex data processing instructions required. However, there are several current software products that facilitate SIMCA deployment. 

The PE class has the most within class variability this can also be seen in the plot of preprocessed data (Figure 4.20). 

Regression analysis with the dependent class variable y where y = 1 or y = 2 should yield results similar to those from discriminant analysis [LACHENBRUCH, 1975]. Therefore let us first try to predict class memberships by the two variables, x, and x2, without using an intercept in the regression model. 

The test of the four lots is an example of a one-way ANOVA. The one-way comes from the fact that there is only one category (lot) into which the data is classified. Often, we have more than one category (class variable) in which we need to classify data. Although our interest may be to determine only whether a particular class variable has meaning, it is important to include other class variables that may influence the variability of the data. ANOVA involves a null hypothesis for each classification variable that proposes that the means at each different level of the class (category) are all equal. If we reject the null hypothesis we conclude in favor of the alternative hypothesis, that at least one mean in the class differs from at least one other mean in the class. This is also a conclusion that... 

Table 9 shows the construction of the ANOVA table. If the variance estimate of a class variable MS variabie deviates significantly from that obtained by that for random error MSettot, then the null hypothesis that the means at the different levels for that variable are equal is rejected. In other words, the classification of data by that variable is explanatory of the variation observed in the data. We conduct the test by using the variance ratio test F = MSvariabie/AfSError, with... 

Class variable is a subclass of the basic class generic-variable. From the class variable emanates the tree of subclasses, a partial view of which is shown in Fig. 19. Unlike other modeling approaches, MODEL.LA. does not represent variables through their values alone, but it provides an extensive structure that includes many additional attributes in the description of a variable. The additional attributes allow MODEL. LA. to reason about these variables and not just acquire their values. Thus, a set of methods in the class variable allow any of the subclasses to monitor their values, react with predefined procedures when the value of the variable changes, invoke values from external databases, and so on. 

Tarko, L. and Ivanciuc, O. (2001) QSAR modeling of the anticonvulsant activity of phylacetanilides with PRECLAV (property evaluation by class variables). MATCH Commun. Math. Comput. Chem., 44, 201-214. 

Up until now it has been assumed that x consists of continuous variables. OLS is not predicated on x being continuous, although this makes it convenient to explain the model. An extremely important data type is a categorical variable where the variable of interest takes on discrete values. These variables are also called factors or class variables. For instance, whether a person is considered a smoker can be coded as either yes or no. The variable race may take on the values White, Black, Asian, or Hispanic. These variables must enter the model through what are called dummy variables or indicator variables which are themselves categorical variables that take on the value of either 0 or 1. If there are k levels in the categorical variable, then k—1... 

On the other hand, there may be cases where a random effect is included in the model but not in the mean model. One case would be in a designed experiment where subjects were first randomized into blocks to control variability before assignments to treatments. The blocks the subjects were assigned to are not necessarily of interest—they are nuisance variables. In this case, blocks could be treated as a random effect and not be included in the mean model. When the subject level covariates are categorical (class) variables, such as race, treating random effects beyond random intercepts, which allows each subject to have their own unique baseline, is not usually done. 

Property Chemical class Variables Method Citation Ref. No. 

Finally, a last consideration about problems with the data set representativeness. As it has been claimed in a pniblished report a LDA was applied to differentiate 12 classes of oils on the basis of the chromatographic data, where some classes contained two or three members only (and besides, the model was not validated). There is no need of being an expertise chemometrician to be aware of two or three samples are insufficient to draw any relevant conclusion about the class to which they belong. There are more sources of possible data variance than the number of samples used to estimate class variability (Daszykowski Walczak, 2006). The requirements of a sufficient number of samples for every class could be envisaged according to a class modelling technique to extract the class dimensionality and consider, for instance, a number of members within three to ten times this dimensionality. 

The probabUistic model for a naive Bayes classifier is a conditional model P(T Xi, X2,..., X ) over a dependent class variable F, conditional on features Xi, X2, X. Using Bayes s theorem, F(F Xj,..., X ) oc P(F) 7(Xi,..., X F). The prior probability F(F = j) can be calculated based on the ratio of the class j samples such as P(F = 7) = (number of class j samples)/(total number of samples). Having formulated the prior probabihties, the likelihood function p(Xi, X2,..., X F) can be written as ]/[ j p(Xi F) under the naive conditional independence assumptions of the feature X, with the feamre Xj for j i. A new sample is classified to a class with maximum posterior probability, which is argmaxr erF (r7)nr ( i 1 /)- If the independence assumption is correct, it is the Bayes optimal classifier for a problem. Extensions of the naive Bayes classifier can be found in Demichelis et al. (2006). 

Weighting fpr a supervised learning uses the ratio of an "inter-class variability" and an "intraclass variability" C148/ 1531. 

This is called classification. The predicted values /(Xj) should agree with the known class variable value for as many observations as possible. To express this quantitatively, we introduce a cost function... 

The descriptors were calculated here, for each complex, using the computer programs MOPAC/PM6, PRoperty Evaluation by CLAss Variables (PRECLAV) (Tarko 2005), and DESCRIPT (Tarko 2008a). The LogP descriptor was calculated using the KowWin algorithm of EPISuite software (EPISuite Meylan and Howard 1995). The type of chemical bonds was defined according to Table 4.1. 

PV Is a class variable, which is associate with the equipment class. 

POS The position of the NAS is related to the running equipment, affected fluid, and the process variable. POS can be represented as class variable associated with equipment, fluid, or as a sub-class associated with the equipment or fluid. 

---