Propuesta de mejora

Wang et al. [2003] proposed a fuzzy learning algorithm, FAQR, which is also based on the AQR crisp inductive learning strategy. Since FUZZYSEEDSEARCH and FAQR both take ideas from AQR, we briefly review the FAQR algorithm (for a detailed review of FAQR refer to Section 2.3) and then discuss the similarities and differences between FAQR and FUZZYSEEDSEARCH.

FAQR consists of two layers. The top layer starts with an empty rule R, and a set of positive and negative fuzzy instances, P and N respectively. The following steps are then iterated until the rule covers all positive instances. The instance that has the highest concept membership and is notα-covered byR is selected as positive seed. An instance i is α-covered if µR(i) ≥ α. The function GenComplex is

used to generate a set of candidate conjunctions thatα-cover the positive seed and no negative instances. The best conjunctionCbestis then added to the rule by formingR = R ∨ Cbest. The best conjunction

has the highest evaluationE(c),

E(c) =µinclude(c) ρ µexclude(c) (9.1)

µinclude(c) = P

i∈P(µP(i) τ µc(i)) P

i∈PµP(i)

(9.2) µexclude(c) =

P

i∈N(µN(i) τ (1 − µN(i)) P

i∈NµN(i)

(9.3) where ρ is an addition or a union operator, such as maximum, and τ is a t-norm operator, such as minimum. WhenR covers all positive instances it is returned as result.

The procedure GenComplex generates a set of candidate conjunctions. It initialises a setCsetof candi-

date solutions with the set of all single term descriptions, such as[T emperature = hot] that α-cover the positive seed. It then repeats the following procedure while any description inCset α-covers a neg-

ative instance. The conjunction c ∈ Cset with the smallest valueµexclude(c) is selected and then the

negative instance inXN(c) that has the highest membership to N is chosen as the negative seed. All

conjunctions inCset are then refined not toα-cover the negative seed. This refinement process is per-

formed as follows. LetS be the set of terms that α-cover the positive seed and not the negative seed. Then the newCset is obtained by forming conjunctions of descriptions in the oldCsetwith terms inS.

All descriptions in the newCset that are subsumed by other descriptions are removed, and then the set Csetis pruned by removing the worst complexes until its size is less than a specified threshold. Once all

complexes inCsetcover no negative instances,Csetis returned as result.

Clearly FUZZYSEEDSEARCHand FAQR have many similarities. Both are inductive learning strategies, and both induce incomplete fuzzy rules. An incomplete fuzzy rule is a rule of which the antecedent does not necessarily contain all linguistic terms. FAQR and FUZZYSEEDSEARCH share the heuristic to use positive and negative seeds to guide the search for rule descriptions. Both methods employ a generate- and-test strategy of generating a set of candidate rule descriptions and then selecting the best based on an evaluation function. Both methods also employ a beam search. However, there are also many fundamental differences between the two algorithms’ search strategies, which we enumerate shortly. (1) FAQR does not implement the fuzzy set covering methodology. Set covering algorithms iteratively induce rules that cover the set of positive instances, but not the set of negative instances, and after the induction of each individual rule, the positive instances covered by the rule are removed from the training set while the negative instances are retained (see Section 3.2 and Def. 3.2.1, Point 1). FAQR does not remove covered instances from the training set, but keeps adding more rules until all positive instances are covered. However, at each step FAQR will cover at least one positive instance that was not covered before. FUZZYSEEDSEARCH, on the other hand, fits with in the FCF framework, and thus implements

the fuzzy set covering methodology. At each rule induction step the positive instances covered by the induced rule are removed. Thus, FUZZYSEEDSEARCH follows a separate-and-conquer strategy, and each rule is biased to cover as many positive instances in the set of “still not covered” positive instances. FAQR does not have this bias. FAQR will thus prefer rule one over rule two if rule one covers more positive instances than rule two, regardless of whether the positive instances are already covered by other rules. For example, if rule one covers 20 positive instances of which one is the positive seed and the others are already covered, and rule two covers 15 positive instances of which one is the positive seed and the others were not covered before, FAQR will prefer rule one. This can result in much more complex rule sets, containing many rules that differ in only a few terms and covers overlapping sets of instances.

(2) FAQR’s description language only allows conjunctions of linguistic terms, i.e. it does not allow internal disjunction. In contrast, FUZZYSEEDSEARCH induces a rule set where the antecedent of each rule is a description in FuzzyAL.

(3) FUZZYSEEDSEARCH uses exclusion as specialization operator whereas FAQR uses append as spe- cialization operator. This has important implications. FUZZYSEEDSEARCH performs a systematic search from top to bottom in its description lattice. The top layer contains the mgc that covers all instances. By excluding terms from descriptions FUZZYSEEDSEARCH restricts the descriptions more and more such that they cover fewer and fewer instances. The search is then guided to cover progres- sively fewer negative instances while still covering as many positive instances as possible. FAQR starts its search with a set of conjunctions, each consisting of only one term, such that the conjunctions cover at least one positive instance. The conjunctions are then restricted to cover fewer and fewer instances by adding more and more terms to them. The search is guided to result in a set of conjunctions that cover none of the negative instances while still covering the positive seeds. Thus, during the search pro- cess FUZZYSEEDSEARCH is biased toward high positive coverage and low negative coverage, whereas FAQR is guided only toward low negative coverage.

(4) Due to the difference in FUZZYSEEDSEARCH and FAQR’s specialization operators they choose seeds in different ways. The terms chosen for exclusion by FUZZYSEEDSEARCH should cover the negative seed and not the positive seed. After the exclusion of a term, the specialization will still cover the positive seed and may now not cover the negative seed any longer. It will (not) cover the negative seed if (no more) terms that have the negative seed in their extensions remain in the internal disjunction. The terms chosen by FAQR to add to the current conjunctions should cover the positive seed and not the negative seed. Thus, the specializations still cover the positive seed, but will definitely not cover the negative seed.

(5) FUZZYSEEDSEARCH as specialization model fits in the FCF framework. As such, FUZZYSEED- SEARCH inherits all the beneficial characteristics of FCF’s top layers. Some of the more important inherited characteristics are search efficiency measures, early stopping and rule pre-pruning criteria, beam search, and an easily interchangeable description evaluation function. We showed that the evalua- tion function is very important for the learning process, and it is also very important to note that different evaluation functions are suited to different learning problems. The Accuracy evaluation function, for ex-

ample, is well suited to deal with noisy data and incomplete domain knowledge, but is not the optimal choice when dealing with the problem of small disjuncts. The Fuzzy Laplace andls-Content evaluation functions, on the other hand, are better positioned to deal with the problem of small disjuncts. However, they do not perform as well as the Accuracy function in the presence of noise or incomplete domain knowledge, especially with respect to the size of the rule sets induced.

(6) FAQR and FUZZYSEEDSEARCH both perform a beam search. However, FUZZYSEEDSEARCH performs a systematic top-to-bottom general-to-specific search of the description lattice, moving down one layer at a time. During each step the next layer of the description lattice is considered, and up to a user adjustable number of conjunctions in this layer are generated and tested. The best conjunction found during the whole search is stored in the parameter bestconj. FAQR maintains a set of current conjunctions which may be specialized. This set may grow up to a user adjustable size—if it grows bigger than this size, the worst conjunctions are removed. If more specialized conjunctions are worse than more general conjunctions these are removed from the set, and are in fact again specialized during the next iteration of the search process. Thus, FAQR does not in general perform a systematic top-down search, and may again jump back up to more general conjunctions. The beam search is also not done in a systematic way, and may include conjunctions at different levels of generality.

(7) FAQR has two pruning steps. Conjunctions that are subsumed by other conjunctions and all con- junctions worse than the user defined number of best conjunctions are pruned from the search. FUZZY- SEEDSEARCH also employs a fixed beam width, but also has further pruning criteria. Since a general- to-specific search is performed, FUZZYBEXA can determine whether further specialization can improve a conjunction to such a degree that it can replace the current best conjunction. If this is not the case, the conjunction is removed from the search. For example, if the best conjunction found thus far covers 20 positive and no negative instances, all conjunctions that cover less than 20 positive instances can be removed from the search process. Conjunctions are also not overspecialised. During specialization a conjunction is not simply specialized by excluding from it all the terms that cover the negative seed and not the positive seed. If excluding a term has no benefit, it is not excluded, resulting in FUZZYSEED- SEARCH’s bias towards maximal generality and maximal classification accuracy. Duplicate specializa- tions are removed from the search for efficiency reasons.

(8) The final difference we discuss here is not about the search method but about the semantic interpre- tation of a rule. The interpretation of Wang et al states that the membership degree of an instance to the rule consequent can be set equal to the membership degree of the instance to the rule antecedent. Thus, the degree to which an instance matches the antecedent can be used to predict the class membership of the instance. Our interpretation differs in that we take the membership degree of an instance to a rule as the certainty or confidence that the rule fires. The certainty or degree to which a rule fires does not predict the membership of the instance to the rule consequent, but specifies that the instance membership to the rule consequent lies within a certain range—the range[αc, 1]. If αc was not set, then the instance

membership to the concept is simply greater than zero. For example, ifαc = 0.8 and we have the rule

IFX = x1∨ x2THENY , and an instance i matches the antecedent to degree 0.75, our interpretation is

Table 9.8:The FUZZCONRI specialization model.

INPUT: sets of positive instancesP , negative instances N , and conjunctions to specializeC

OUTPUT: set of specializationsS S = ∅

FOR each conjunctionc ∈ C DO

remove uninteresting terms fromc.usable FOR each termL ∈ c.usable DO

cnew = specialize (c)

IFcnew ∈ S THEN CONTINUE cnew.XP = computeXP(cnew)

IFcnew.XP = ∅ THEN CONTINUE cnew.XN = computeXN(cnew) S = S ∪ {cnew}

END FOR END FOR RETURNS

concept with membership in the range[0.8, 1]. We do not believe that the membership of an instance to the antecedent can be used to predict the membership of the instance to the concept with an acceptable degree of accuracy without any form of (likely non-linear) transformation of the input domain to the output domain, e.g. like that performed by a neural network.