Guide Definable additive categories: purity and model theory

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It is an essential step for generating useful for prediction tree models, and because it can be computationally difficult to do, this method is often not found in tree classification or regression software. Another general issue that arises when applying tree classification or regression methods is that the final trees can become very large. In practice, when the input data are complex and, for example, contain many different categories for classification problems and many possible predictors for performing the classification, then the resulting trees can become very large.

This is not so much a computational problem as it is a problem of presenting the trees in a manner that is easily accessible to the data analyst, or for presentation to the "consumers" of the research. The classic Breiman et. However, it is easy to see how the use of coded predictor designs expands these powerful classification and regression techniques to the analysis of data from experimental designs e. The process of computing classification and regression trees can be characterized as involving four basic steps:.

These steps are very similar to those discussed in the context of Classification Trees Analysis see also Breiman et al. See also, Computational Formulas. Operationally, the most accurate prediction is defined as the prediction with the minimum costs. The notion of costs was developed as a way to generalize, to a broader range of prediction situations, the idea that the best prediction has the lowest misclassification rate. In most applications, the cost is measured in terms of proportion of misclassified cases, or variance.

In this context, it follows, therefore, that a prediction would be considered best if it has the lowest misclassification rate or the smallest variance. The need for minimizing costs, rather than just the proportion of misclassified cases, arises when some predictions that fail are more catastrophic than others, or when some predictions that fail occur more frequently than others. In the case of a categorical response classification problem , minimizing costs amounts to minimizing the proportion of misclassified cases when priors are taken to be proportional to the class sizes and when misclassification costs are taken to be equal for every class.

The a priori probabilities used in minimizing costs can greatly affect the classification of cases or objects. Therefore, care has to be taken while using the priors. Misclassification costs. Sometimes more accurate classification of the response is desired for some classes than others for reasons not related to the relative class sizes. If the criterion for predictive accuracy is Misclassification costs, then minimizing costs would amount to minimizing the proportion of misclassified cases when priors are considered proportional to the class sizes and misclassification costs are taken to be equal for every class.

Case weights. Case weights are treated strictly as case multipliers. For example, the misclassification rates from an analysis of an aggregated data set using case weights will be identical to the misclassification rates from the same analysis where the cases are replicated the specified number of times in the data file.

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However, note that the use of case weights for aggregated data sets in classification problems is related to the issue of minimizing costs. Suppose that in an aggregated data set with two classes having an equal number of cases, there are case weights of 2 for all cases in the first class, and case weights of 3 for all cases in the second class. The second basic step in classification and regression trees is to select the splits on the predictor variables that are used to predict membership in classes of the categorical dependent variables, or to predict values of the continuous dependent response variable.

In general terms, the split at each node will be found that will generate the greatest improvement in predictive accuracy. This is usually measured with some type of node impurity measure, which provides an indication of the relative homogeneity the inverse of impurity of cases in the terminal nodes. If all cases in each terminal node show identical values, then node impurity is minimal, homogeneity is maximal, and prediction is perfect at least for the cases used in the computations; predictive validity for new cases is of course a different matter The Gini index of node impurity is the measure most commonly chosen for classification-type problems.

As an impurity measure, it reaches a value of zero when only one class is present at a node. With priors estimated from class sizes and equal misclassification costs, the Gini measure is computed as the sum of products of all pairs of class proportions for classes present at the node; it reaches its maximum value when class sizes at the node are equal; the Gini index is equal to zero if all cases in a node belong to the same class.

The Chi-square measure is similar to the standard Chi-square value computed for the expected and observed classifications with priors adjusted for misclassification cost , and the G-square measure is similar to the maximum-likelihood Chi-square as for example computed in the Log-Linear module.

Definable Additive Catagories: Purity and Model Theory

For regression-type problems, a least-squares deviation criterion similar to what is computed in least squares regression is automatically used. Computational Formulas provides further computational details. As discussed in Basic Ideas, in principal, splitting could continue until all cases are perfectly classified or predicted. What is required is some reasonable stopping rule.

Minimum n. One way to control splitting is to allow splitting to continue until all terminal nodes are pure or contain no more than a specified minimum number of cases or objects.

This option can be used when Prune on misclassification error , Prune on deviance , or Prune on variance is active as the Stopping rule for the analysis. Fraction of objects. Another way to control splitting is to allow splitting to continue until all terminal nodes are pure or contain no more cases than a specified minimum fraction of the sizes of one or more classes in the case of classification problems, or all cases in regression problems.

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This option can be used when FACT -style direct stopping has been selected as the Stopping rule for the analysis. For classification problems, if the priors used in the analysis are equal and class sizes are equal as well, then splitting will stop when all terminal nodes containing more than one class have no more cases than the specified fraction of the class sizes for one or more classes. Alternatively, if the priors used in the analysis are not equal, splitting will stop when all terminal nodes containing more than one class have no more cases than the specified fraction for one or more classes.

See Loh and Vanichestakul, for details. The size of a tree in the classification and regression trees analysis is an important issue, since an unreasonably big tree can only make the interpretation of results more difficult. Some generalizations can be offered about what constitutes the "right-sized" tree. It should be sufficiently complex to account for the known facts, but at the same time it should be as simple as possible. It should exploit information that increases predictive accuracy and ignore information that does not.


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It should, if possible, lead to greater understanding of the phenomena it describes. One strategy is to grow the tree to just the right size, where the right size is determined by the user, based on the knowledge from previous research, diagnostic information from previous analyses, or even intuition.

The other strategy is to use a set of well-documented, structured procedures developed by Breiman et al. These procedures are not foolproof, as Breiman et al. FACT-style direct stopping. We will begin by describing the first strategy, in which the user specifies the size to grow the tree. Specifically, three options are available for performing cross-validation of the selected tree; namely Test sample, V-fold, and Minimal cost-complexity. Test sample cross-validation.

The first, and most preferred type of cross-validation is the test sample cross-validation. In this type of cross-validation, the tree is computed from the learning sample, and its predictive accuracy is tested by applying it to predict the class membership in the test sample.


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If the costs for the test sample exceed the costs for the learning sample, then this is an indication of poor cross-validation. In that case, a different sized tree might cross-validate better. The test and learning samples can be formed by collecting two independent data sets, or if a large learning sample is available, by reserving a randomly selected proportion of the cases, say a third or a half, for use as the test sample. V-fold cross-validation.

This type of cross-validation is useful when no test sample is available and the learning sample is too small to have the test sample taken from it. The user-specified 'v' value for v-fold cross-validation its default value is 3 determines the number of random subsamples, as equal in size as possible, that are formed from the learning sample.


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  • A tree of the specified size is computed 'v' times, each time leaving out one of the subsamples from the computations, and using that subsample as a test sample for cross-validation, so that each subsample is used v - 1 times in the learning sample and just once as the test sample. The CV costs cross-validation cost computed for each of the 'v' test samples are then averaged to give the v-fold estimate of the CV costs.

    Minimal cost-complexity cross-validation pruning.

    On the other hand, if Prune on deviance has been selected as the Stopping rule , then minimal deviance-complexity cross-validation pruning is performed. The only difference in the two options is the measure of prediction error that is used. Prune on misclassification error uses the costs that equals the misclassification rate when priors are estimated and misclassification costs are equal, while Prune on deviance uses a measure, based on maximum-likelihood principles, called the deviance see Ripley, The sequence of trees obtained by this algorithm have a number of interesting properties.

    They are nested, because the successively pruned trees contain all the nodes of the next smaller tree in the sequence. Initially, many nodes are often pruned going from one tree to the next smaller tree in the sequence, but fewer nodes tend to be pruned as the root node is approached. The sequence of largest trees is also optimally pruned, because for every size of tree in the sequence, there is no other tree of the same size with lower costs.

    Tree selection after pruning. The pruning, as discussed above, often results in a sequence of optimally pruned trees.