PROCEDIMIENTO CONTENCIOSO - TRIBUTARIO CAPÍTULO I

Stochastic discrimination is a methodology for combining weak classifiers (in this case, random rules) to produce a complex classifier that can generalise to new data. Kleinberg’s algorithm for stochastic discrimination requires that the weak classifiers be ‘enriched’, all covering a greater proportion of the target class than the other class, to differentiate between classes, and also that the set of classifiers be ‘uniform’, covering the training data as evenly as possible. These requirements are discussed in more detail below.

As the weak classifiers all cover a greater proportion of the target class (due to enrichment) than they do of the other class, instances of the target class

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will, on average, have a higher coverage than those of the other class. Due to uniformity, instances that the randomly generated weak classifiers would tend to cover less frequently will instead receive coverage closer to the average. The result of this process as the number of weak classifiers increases is thus that the coverages for instances of both the target and non-target classes tend to approach normal distributions, with the mean for the target class greater than that for the non-target class, and decreasing variances for both means.

The threshold for class prediction for test instances is therefore the mid- point between the mean coverages for the two classes. More specifically, test instances are classified according to Equations 2.5-2.6 (for a two-class problem, classes P (the target class, for which the set is enriched) and N). The coverage on the test instance is compared to the midpoint between the overall mean coverages for each class, and if it is greater than that midpoint, it is classified as being of the target class. The mean coverage on the target class is expected to be higher than that on the other class, due to the enrichment requirement.

Predict P if prop(Instance) > meanCoverage(P) + meanCoverage(N)

2 (2.5)

Where:

Instance = the test instance to be classified

prop(Instance) = the proportion of classifiers in the set that cover Instance meanCoverage(Class) = the mean coverage of training instances of that class by the ruleset meanCoverage(P) = size(P) X i=1 prop(Pi) size(P) (2.6) Where:

size(P) = the number of instances of class P Pi = the ith instance of class P

Enrichment

Enrichment is a rule-level quality: a rule is enriched for a particular class, if it covers a greater proportion of the instances of that class than it does of the instances of the other class.

A rule is enriched if #coveredP #totalP

> #coveredN #totalN

(2.7) For example, given a simple dataset containing 10 instances of class A and 20 instances of class B and a rule that covered 6 instances of class A and 11 instances of class B, the calculation would be as follows:

The rule is enriched for class A if 6 10 >

11

20 (2.8)

As 0.6 is greater than 0.55, the rule is enriched for class A. In addition, to avoid overly specific rules, rules are required to cover more than one instance of the class for which they are enriched.

Uniformity

Uniformity is a ruleset-level quality - a uniform ruleset covers the training instances of a given class as evenly as possible. The current coverage of a ruleset affects the selection of new rules to be added, in a similar fashion to Boosting. Rrr-sd defines uniformity as the standard deviation of the coverages of each instance of the target class in the training set, so that the best theoretically possible uniformity is 0, at which point each instance of the target class would be covered by exactly the same number of rules.

While determining if a particular rule is enriched is a simple mathematical calculation, determining whether adding that rule will satisfy the uniformity constraint is less straightforward. Several approaches to this problem were investigated before a satisfactory solution was found for Rrr-sd.

• Fixed threshold – Setting a fixed threshold for the standard deviation, and rejecting rules that would bring the standard deviation over this threshold, either has very little effect on the uniformity (if the threshold is too high) or rejects a high proportion of rules, requiring large numbers of rules to be generated, and can result in non-termination, with no possible rule that can keep the uniformity under the threshold (if the threshold is too low).

• Annealing – An ‘annealing’ approach, where the threshold is initially set high and then decreased as rules are added, initially has the draw- back of a high threshold, and later displays the high rejection rate (and concomitant requirement for many rules to be generated) and potential non-termination of a low threshold.

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• Decreasing – Requiring each rule to keep the uniformity value the same or decrease it in order to be added has a similar drawback to a low threshold, in that the number of rules that will satisfy such a constraint grows smaller as more rules are added, and eventually non-termination results.

• Increasing Threshold – The reverse of the ‘annealing’ method, start- ing with a low threshold and raising the threshold gradually as rules fail, resetting it to the low value when a rule is added, avoids the non- termination problem, but is strongly affected by the size of increments and frequency of threshold raising. Similarly to the simple threshold method, if the threshold goes up quickly, uniformity is hardly affected, but if it goes up slowly, although the uniformity is improved, the number of rules generated to add a single rule is prohibitive.

All of the above methods for ensuring uniformity share the property that the number of rules that will be generated and discarded before one is added to the ruleset is unknown. Therefore, Rrr-sd uses an alternative approach to ensure uniformity that guarantees progress within a fixed number of rules. A “maximum batch size” is selected, and as enriched rules are generated, they are added to the batch. When the batch reaches its maximum size, a subset of the rules in the batch will be added to the ruleset. This subset is determined by evaluating the resulting uniformity for adding each possible non-zero subset of rules, and selecting the subset resulting in the best value for uniformity. The remaining rules in the batch are discarded. This ensures that at least one rule in every batch will be added to the ruleset, while also maintaining an acceptable level of uniformity.