Redundancy Detection and Removal Tool for Transparent Mamdani Systems

In Mamdani systems, redundancy of fuzzy rule bases that derives from extensive sharing of a limited number of output membership functions among the rules, is often an overlooked property. In current study, means for detection and removal of such kind redundancy have been developed. Our experiments with case studies collected from literature and Mackey-Glass time series prediction models show error-free rule base reduction by 30-60% that partially cures the curse of dimensionality problem characteristic to fuzzy systems. 1 Motivation One very acute problem that is marring the large scale applications of fuzzy logic is the combinatorial explosion of rules (curse of dimensionality). As the number of membership functions (MFs) and/or input variables increases, the upper bound on the count of fuzzy rules grows exponentially: Rmax = N ∏ i=1 Si, (1) where Si is the number of MFs per i-th input variable (i = 1, ...,N). Andri Riid Laboratory of Proactive Technologies, Tallinn University of Technology, Ehitajate tee 5, 19086, Tallinn, Estonia e-mail: [email protected] Kalle Saastamoinen Department of Military Technology, National Defence University, P.O. Box 7 FI-00861, Helsinki, Finland e-mail: [email protected] Ennu Rüstern Department of Computer Control, Tallinn University of Technology, Ehitajate tee 5, 19086, Tallinn, Estonia e-mail: [email protected] V. Sgurev et al. (Eds.): Intelligent Systems: From Theory to Practice, SCI 299, pp. 397–415. springerlink.com c © Springer-Verlag Berlin Heidelberg 2010 398 A. Riid, K. Saastamoinen, and E. Rüstern In the ideal fuzzy system the number of fuzzy rules R = Rmax, meaning that the rule base of the system is fully defined and contains all possible antecedent combinations. Situation R > Rmax indicates a failure in fuzzy system design either redundant or contradictory rules are present, both of which are the signs of sloppy system design. In real life applications, however, the number of rules often remains well below Rmax for several reasons. First of all, commonly there is not enough material (data) or immaterial (knowledge) evidence to cover the input space universally, not only because it would be too time consuming to collect exhaustive evidence in large scale applications but also because of potential inconsistency that certain antecedent combinations may present (an antecedent “IF sun is bright AND rain is heavy” could be one such example). Moreover, it is common practice that for the sake of compactness, the rules with little relevance are excluded from the model (for all we know they may be based on few noisy samples). The exclusion decision of a given rule may be based on its contribution to approximation properties (using singular value decomposition, orthogonal transforms, etc. [1]) or on how often or to what degree a given rule is contributing to the output (this, for example, can be easily evaluated by computing cumulative rule activation degrees on available data). On the whole, rule base reduction can be fitted under two categories error-free reduction or degrading reduction. Error-free reduction searches for existing redundancies in the model. In other words, if error-free reduction is effective, it is actually an indicator that initial system design was not up to the standard. With degrading simplification, the model is made less complex by removing nonredundant system parameters. Incidentally, this is achieved at the expense of system universality, accuracy etc. Typically, reduction is carried out on initial complex model. However, with certain design methodologies unnecessary complexity is avoided by model design procedure. A typical example is the application of tree partitioning of the input space (instead of more common grid partitioning) but the most common constructive compactness-friendly approach these days (related primarily to 1st order TakagiSugeno systems [2]) is fuzzy clustering. With clustering, the rules are created in product space only in regions where data concentration is high. Interestingly enough, the side effect of that is the redundancy of cluster projections that are used as the prototypes for MFs of the model. The projections that become fuzzy sets may be highly similar to each other, similar to the universal set or reduced to singleton sets, which calls for adequate methods to deal with that [3]. Another feature of product space clustering is that R is always a lot smaller than Rmax (in fact R = Si prior to simplification). For this reason and also from interpolational aspect, product space clustering is not very well suited for Mamdani modeling. In Mamdani systems, a relatively small set of output MFs is typically shared among rules. This creates substantial redundancy potential, which can exploited for rule base reduction. For a special class of Mamdani systems (transparent Mamdani systems, more closely observed in Sect. 2) this reduction can actually be error-free, i.e. without any performance loss. In Sect. 3, practical redundancy detection and Redundancy Detection and Removal Tool for Transparent Mamdani Systems 399 removal scenarios have been investigated. Sect. 4 considers the typical implementation issues when designing a computer program for reduction of Mamdani systems. The remainder of the paper presents application examples and performance analysis. Note that the current implementation of the reduction tool can be freely downloaded from http://www.dcc.ttu.ee/andri/rdart”. 2 Transparent Mamdani Systems Generally, fuzzy rules in Mamdani-type fuzzy systems are based on the disjunctive rule format IF x1 is A1r AND x2 is A2r AND ... ... AND xN is ANr THEN y is Br OR ... (2) where Air denote the linguistic labels of the i-th input variable associated with the r-th rule (i = 1, ...,N), and Br is the linguistic label of the output variable, associated with the r-th rule. Each Air has its representation in the numerical domain the membership function μir (the same applies to Br that is represented by γr) and in general case the inference function that computes the fuzzy output F(y) of the system (2) has the following form F(y) = R ⋃ r=1 (( N ⋂ i=1 μir(xi) ) ∩ γr ) , (3) where ∪r denotes the aggregation operator (corresponds to OR in (2), ∩ is the implication operator (THEN) and ∩i is the conjunction operator (AND). In order to obtain crisp output, (3) is generally defuzzified with center-of-gravity method y = Ycog(F(y)) = ∫ Y yF(y)dy ∫ Y F(y)dy . (4) The results obtained in current paper are valid for a class of Mamdani systems that satisfy the following requirements: • The inference operators used here are product and sum. With product-sum inference (4) reduces to y = ∑ R r=1 τrcrsr ∑r=1 τrsr , (5) where τr is the activation degree of r-th rule (computed with the conjunction operator (product)) and cr and sr are the center-of-gravity and area of γr, respectively (see [4]). • The input MFs (s = 1, ...,Si) are given by the following definition: 400 A. Riid, K. Saastamoinen, and E. Rüstern μ s i (xi) = ⎧ ⎪⎪⎨ ⎪⎩ xi−as−1 i asi−a i , as−1 i < xi < a s i as+1 i −xi as+1 i −ai , ai < xi < a s+1 i 0, as+1 i ≤ xi ≤ as+1 i , (6) Such definition of input MFs satisfies input transparency condition assumed for correct interpretation of Mamdani rules (see [5] for further details), however, in current paper we are more interested in its other property, namely Si ∑ s=1 μ s i = 1. (7) • The number of output MFs is relatively small and they are shared among rules (this is the usual case in Mamdani systems). 3 Error-Free Rule Base Reduction Principles Consider a pair of fuzzy rules that share the same output MF Bξ IF x1 is A s1 1 AND ... AND xi is A s i ...... AND xN is A sN N THEN y is Bξ IF x1 is A s1 1 AND ... AND xi is A s+1 i ...... AND xN is A sN N THEN y is Bξ (8) It is possible to replace these two rules by a single one: IF x1 is A s1 1 AND ... AND xi is (A s i OR A s+1 i ) ... ... AND xN is A sN N THEN y is Bξ (9) This replacement can be validated very easily, as it derives from (5) that numerically, (8) is represented by (10). μ s i N ∏ j=1 j =i μ s j j + μ s+1 i N ∏ j=1 j =i μ s j j . (10) Obviously (10), is equivalent to (11) (μ s i + μ s+1 i ) N ∏ j=1 j =i μ s j j , (11) which is nothing else than a representation of (9), assuming that the OR operand is implemented through sum. This line of logic, while hardly practical for the reduction of fuzzy systems (fuzzy logic software does not usually have any support for such constructions as (9) and numerically, (11) is not really an improvement over (10)), however, has three offsprings (or special cases) that can be really useful as evidenced below. Redundancy Detection and Removal Tool for Transparent Mamdani Systems 401 Lemma 1. Consider not a pair but a subset of fuzzy rules consisting of Si rules that share the same output MF Bξ so that IF x1 is A s1 1 AND ... AND xi is A s i ...... AND xN is A sN N THEN y is Bξ s = 1, ...,Si (12) Apparently, this would be equivalent to a rule IF x1 is A s1 1 AND ... AND xi is (A 1 i OR A 2 i OR ... OR A Si i ) ... ... AND xN is A sN N THEN y is Bξ (13) We proceed by showing that (13) is equivalent to (14). IF x1 is A s1 1 AND ... AND xi−1 is A si−1 i−1 AND xi+1 is A si+1 i+1 ... ... AND xN is A sN N THEN y is Bξ (14) Proof. For proof we need to show that Si ∑ s=1 μ s i N ∏ j=1 j =i μ s j j = N ∏ j=1 j =i μ s j j , (15) which is valid when Si ∑ s=1 μ s i = 1, (16) which is ensured by (6) that concludes the proof. Example 1. Consider three rules of a two-input tipping system: IF f ood is bad AND service is bad THEN tip is zero IF f ood is OK AND service is bad THEN tip is zero IF f ood is good AND service is bad THEN tip is zero (17) If there are no more linguistic labels of food quality as Fig. 1 clearly implies, it is indeed the case that if service is good, output of the system (the amount of tip) is independent from food quality that can be expressed by the following single rule IF service is bad AND f ood is whatever THEN tip is zero, (18) where “whatever” (or “don’t care”) describes the situation that food quality may have any value in its domain without a slightest effect to the output and can thus be removed from the rule, resulting in a nice compressed formulation IF service is bad THEN tip is zero (19) Lemma 2: If a subset of fuzzy rules consisting of Si−1 rules share the same output MF 402 A. Riid, K. Saastamoinen, and E. Rüstern zero zero OK good bad