CART model

Although CART modeling of the classes might be perfect as in Example 5.13, the prediction of new observations is sometimes less perfect. This might be reasoned by overfitting and some further treatment, especially pruning of a tree, should be apphed. 

In CART models, no assumptions are necessary regarding the distribution of the input variables as made in many other multivariate methods. Another advantage is the treatment of missing values. In Section 5.1, we learned about column means or random numbers to deal with missing values. CART provides more sophisticated methods for this purpose, for example, by... 

Here we consider again the 4-class problem used in Example 5.12. CART modeling of this data set by a single tree reveals... 

Application of ensemble methods implies variation of several parameters in order to optimize the final model. Typical parameters are minimum observations per leaf or per branch node, pruning, and split criterions as well as surrogate decision splits. In this example, the maximum number of decision splits (branch nodes) per layer - a set of nodes that are equidistant from the root node - has been chosen for improving the CART model by the ensemble methods bagging and boosting. 

Figure 5.36a demonstrates the fractions of misclassifica-tions in dependence on the maximum number of splits for both resubstitution and cross-validation in a bagged CART model. Although the resubstitution error might be close to zero, a more realistic model is based on the maximum number of splits for which a minimal classification error for cross-validation is observed. The smallest error for the bagged trees is found at 16 splits where the cross-validated fraction of misclassification is 12.0% (cf. Figure 5.36b for the decision boundaries). 

Figure 5.37 Dependence of the fraction of misclassifications in boosted CART models on the maximum number of splits (a) and the decision boundaries for 14 splits per layer (b).

However, the instability of tree-based methods implies also here a much higher error in case of cross-validation by the simple leave-one-out method, that is, the cross-validated fraction of misclassified objects for CART is with 2.25%, 10 times higher than the resubstitution error. This error can be only reduced if ensemble methods are included in the model budding step. A bagged CART model revealed a cross-validation error of only 1.0% (Figure 5.38e). The fraction of misclassifica-tions for the cross-validated models increases for QDA, SVM, and A-NN to 5.5%, 5.0%, and 4.75%, respectively. The cross-validated classifications by LDA reveal 58.8% of misclassified objects as expected from the type of data. 

Israelachivil et al. [25] proposed a phenomenal model for describing the interrelations between friction and adhesion. Consider the system shown in Fig. 28, where a spherical molecule slides over a corrugated solid surface. The scenario is somehow like pushing the wheel of a cart over a road paved with cobblestones, so it is also known as the cobblestone model. 

The basis functions in a CART or inductive decision tree model are given by... 

Frank IE (1989) Classification models discriminant analysis, SIMCA, CART. Chemom Intell Lab Syst 5 247... 

The SunRay two-seat model has a 5.5 hp, 48V motor and uses 6 lead/ acid golf cart batteries. It has a 180W solar panel and a 55 mile range with a maximum speed of 22 mph. 

Models of the form y =f(x) or v =/(x1, x2,..., xm) can be linear or nonlinear they can be formulated as a relatively simple equation or can be implemented as a less evident algorithmic structure, for instance in artificial neural networks (ANN), tree-based methods (CART), local estimations of y by radial basis functions (RBF), k-NN like methods, or splines. This book focuses on linear models of the form... 

In connection with the SHEBA project, the U.S. Department of Energy s Atmosphere Radiation Measurement (ARM) program indicated its intention to develop a Cloud and Radiation Testbed (CART) facility on the North Slope of Alaska. The principal focus of this program will be on atmospheric radiative transport, especially as modified by clouds (such transport impacts the growth and decay of sea ice), as well as testing, validation, and comparison of radiation transfer models in both the ice pack and Arctic coastal environment. 

A new carbon-cArbtyi bond fortm and the entire Cart)ox/k Acid group -s added to the model. 

These sprayers are mounted on a two-wheel cart with handles for pushing. Trailer hitches are available for towing the units. Spray material is hydraulically agitated. Some models have 15- to 30-gallon tanks. Pumps deliver 1.5 to 3 gallons per minute at pressures up to 250 psi. 

The most popular classification methods are Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), Regularized Discriminant Analysis (RDA), K th Nearest Neighbours (KNN), classification tree methods (such as CART), Soft-Independent Modeling of Class Analogy (SIMCA), potential function classifiers (PFC), Nearest Mean Classifier (NMC) and Weighted Nearest Mean Classifier (WNMC). Moreover, several classification methods can be found among the artificial neural networks. 

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