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Tikk, D. et al.

Paper:

Fuzzy Rule Interpolation and Extrapolation Techniques: Criteria and Evaluation Guidelines Domonkos Tikk∗1 , Zsolt Csaba Johanyák∗2 , Szilveszter Kovács∗3 , and Kok Wai Wong∗4 ∗1 Dep.

of Telecommunications and Media Informatics, Budapest University of Technology and Economics Magyar tudósok krt. 2, H-1117 Budapest, Hungary E-mail: [email protected] ∗2 Institute of Information Technology, Kecskem´ et College Izsáki u´ t 10, H-6000 Kecskemét, Hungary E-mail: [email protected] ∗3 Department of Information Technology, University of Miskolc H-3515 Miskolc-Egyetemváros, Hungary E-mail: [email protected] ∗4 School of Information Technology, Murdoch University South St., Murdoch, Western Australia 6150, Australia E-mail: [email protected] [Received December 22, 2010; accepted March 18, 2011]

This paper comprehensively analyzes Fuzzy Rule Interpolation and extrapolation Techniques (FRITs). Because extrapolation techniques are usually extensions of fuzzy rule interpolation, we treat them both as approximation techniques designed to be applied where sparse or incomplete fuzzy rule bases are used, i.e., when classical inference fails. FRITs have been investigated in the literature from aspects such as applicability to control problems, usefulness regarding complexity reduction and logic. Our objectives are to create an overall FRIT standard with a general set of criteria and to set a framework for guiding their classification and comparison. This paper is our initial investigation of FRITs. We plan to analyze details in later papers on how individual techniques satisfy the groups of criteria we propose. For analysis, MATLAB FRI Toolbox provides an easy-to-use testbed, as shown in experiments.

Keywords: fuzzy rule interpolation (FRI), fuzzy rule extrapolation (FRE), criteria of FRITs, evaluation guidelines of FRI methods

1. Introduction Fuzzy-Rule-Based Systems (FRBSs) have been applied in applications such as control engineering, expert systems, pattern recognition, operation research, and decision support systems [1–7]. FRBS output is generated by an inference mechanism based on a knowledge base of IF–THEN rules. Originally based on Zadeh’s initial linguistic variable concept, rule bases were assembled using expert knowledge. This was replaced from the 90s – particularly in engineering applications – by automatic rule base construction extracting rules from numerical sample 254

data. Classical inference – Zadeh’s CRI [8], MamdaniLarsen [9, 10], Takagi-Sugeno [11, 12], etc. – determine output by rule matching, i.e., matching observed input to rule premises and calculating conclusions as weighted combinations of rule consequents with nonzero matching in which weights depend on the degree of matching. Due to incomplete knowledge, rule-base construction may produce sparse or incomplete rule bases that may be due to missing or insufficient expertise or available sample data for covering all possible input configurations. Such rule bases thus do not completely (or sufficiently) cover the range of possible input, as defined in Section 2. In sparse or incomplete rule bases, classical fuzzy inference approaches do not always generate meaningful output because actual input is not guaranteed to fire any rule. Such situations can, however, be treated by approximative approaches. The approximative approaches most commonly applied are Fuzzy Rule Interpolation (FRI) and Fuzzy Rule Extrapolation (FRE). FRE is usually an extension of FRI. In the literature, FRI attracted more interest and FRE was mostly neglected [13–17], even though the latter significantly broadens FRI applicability. When we refer here to FRITs, we mean an approximative technique dealing with sparse rule bases including both FRI and FRE; when approximation is concerned, we explicitly specify either interpolation or extrapolation. The application of FRIs can also be justified on other grounds. The main motivation of the first proposed FRIT [18] originated in fuzzy system complexity [19] – the “curse of dimensionality” problem, i.e., the rule base size and inference algorithm complexity grow exponentially with input space dimensions. This issue is settled, in part, by omitting redundant or insignificant rules. The original rule base’s completeness cannot, however, be guaranteed after rule base reduction, i.e., the reduced rule base is often sparse. Despite the numerous FRITs proposed since 1991,

Journal of Advanced Computational Intelligence and Intelligent Informatics

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Evaluation of Fuzzy Rule Interpolation Techniques

a general set of criteria for FRIT consisting of both mathematical- and application-originated requirements does not exist. Our goal here is to correct this deficiency. The several initiatives in the literature to set up FRIT criteria are divided into two groups – approach-oriented and summarization-oriented. The approach-oriented papers focus on a new FRI motivated by critiques of other methods or new analytical aspects, e.g., [15, 20, 21], using conditions in FRIT analysis that are, in our opinion, motivated to justify the proposed approach or aspect, i.e., are not comprehensive enough. Summarization-oriented papers include brief summaries mostly focusing on selected FRI aspects, e.g., [22–25], criticizable on the same ground. [21], for example, analyzes and compares proposed MACI’s general applicability, complexity, approximative power, and fuzziness of conclusion. Jenei [15] axiomatically treated FRITs by setting eight conditions on rule interpolation/extrapolation, focusing on applicability, consistency, functionality, and logic. Baranyi et al. [20] investigated the general methodology for applicability, consistency, and shape-preservation – all of which appear in the second, which are also summarizations. As this paper’s main contribution, we define the set of criteria and properties an ideal FRIT must meet. These properties may serve as a standard and be used to guide FRIT classification and comparison. To facilitate comparison and evaluation, we recently created the MATLAB FRI Toolbox [26], which implements several important FRITs and can be extended by other contributors. This paper is organized as follows. Section 2 defines the notion of a sparse rule base and characterizes FRIT applicability. Section 3 gives FRIT criteria and properties investigated in Section 4 by two examples using our MATLAB FRI Toolbox. Section 5 presents conclusions.

2. Sparse Rule Bases

2.2. Rule Base Coverage Let us characterize the applicability of rule-matchingbased fuzzy inference mechanisms regarding rule base properties. We first define the activation degree of a particular rule and rule base, using the degree of matching. Definition 1. The activation degree (degree of matching) of rule Ri for n-dimensional observation A∗ is ωh,t (Ri ) = h t(Ai1 , A∗1 ), . . . ,t(Ain , A∗n ) , . . . (2) where t denotes an arbitrary t-norm, and h can be an arbitrary t-conorm or aggregation operator. In Eq. (2) the most typical h is the min function. Rules with nonzero activation degree are called activating, firing, or matching rules. Definition 2. The activation degree (degree of matching) of rule base R = {Ri |i = 1, ..., r } for observation A∗ is equal to the highest rule activation degree in the rule base r

We assume readers to be familiar with basic fuzzy set theory, e.g., [27].

2.1. Fuzzy Rule Base The knowledge base for approaching fuzzy paradigm reasoning consists of fuzzy IF–THEN rules. The fuzzy rule ensemble partially maps areas between regions of input and output space, formalized in the model of available but possibly uncertain knowledge. Uncertainty is encoded in fuzzy set membership functions describing input and output space. Let X j ( j = 1, . . ., n) be input dimensions and Y output space, denoting the Cartesian product of input dimensions by X = ×nj=1 X j . A fuzzy IF–THEN rule is given as Ri : if Ai1 ∧ Ai2 ∧ · · · ∧ Ain then Bi

. . . . . (1)

where antecedents Ai j ∈ F (X j ), consequents Bi ∈ F (Y ), and F (Z) denote the entirety of all fuzzy subsets of Z. We denote the (n-dimensional) Cartesian product of antecedents Ai j , ( j = 1, . . . , n) of rule Ri by A(i) . Based on Vol.15 No.3, 2011

the concept dominant in literature, we limit our investigation to rules of form Eq. (1), i.e., when fuzzy sets in the antecedent are connected by a conjunction. If other logical connectives or unary operators, such as disjunction and negation, are allowed, conflicts may occur in the rule base [28], which must then be resolved by inference mechanisms. Multiple output rule bases – SIMO or MIMO – can be decomposed into single output rule bases thus, without loss of generality, only MISO rule bases are investigated. We require that X j , j = 1, . . ., n and Y be bounded and gradual [19, 29]. This guarantees a total ordering for each by which a partial ordering can be introduced among F (X j ) and F (Y ) elements. In practice, X and Y are typically compact subsets of Rn and R.

ω (R) = max ω (Ri ). . . . . . . . . . . . (3) i=1

Rule-matching-based fuzzy inference is applicable if it generates a conclusion. This condition holds if the rule base activation degree is nonzero for arbitrary observation. Practically speaking, it is reasonable to specify an ε > 0 threshold for the minimal rule base activation degree. ε can also be interpreted as a prescribed minimal confidence value of the conclusion, so the applicability of such methods depend on the rule base, or more precisely, how the entirety of rule antecedents covers X, characterized by the next definition. Definition 3. Let R = {Ri |i = 1, ..., r } be a rule base with rules of form Eq. (1). If ∀A∗ ∈ F (x) : ω (R) ≥ ε > 0

. . . . . . . (4)

then R forms an ε -cover of X. The ε -coverage of a rule base guarantees that the rulebase activation degree is at least ε for arbitrary observation, hence rule-matching-based models apply. We call such rule bases ε -dense.


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Note that Definition 3 requires ε -coverage of multidimensional X, which is not necessarily equivalent to ε coverage of each dimension of X. An ε -cover rule base is obtained by multidimensional grid partition. Each input dimension is partitioned by membership functions of fuzzy sets, where their kernels are typical values of xi , and, between these kernels, overlapping membership functions cover the space to at least level ε . The number of overlapping membership functions forming the partition of a dimension should be minimized to have distinguishable membership functions that maintain the linguistic interpretability of fuzzy rules [30]. Only neighboring membership functions are usually allowed to overlap. In the next step, a rule antecedent is created from every possible combination of one-dimensional fuzzy sets and rules are made by assigning consequents to multidimensional antecedents. The set of rules thus obtained of form Eq. (1) with pairwise overlapping membership functions divide the input domain into a grid of fuzzy hyperboxes, parallel to the axes. These hyperboxes are Cartesian product-space intersections of corresponding univariate fuzzy sets. If input dimensions are covered according to the Ruspini partition (∀x j ∈ X j , ∑ ri=1 Ai j (x j ) = 1), then the rule set forms a 0.5-cover of X. The required number of rules to get the ε -cover is then n

r = ∏ N j . . . . . . . . . . . . . . . (5) j=1

where N j is the number of fuzzy sets in the j-th dimension. This quantity grows exponentially with the number of dimensions – usually termed the “curse of dimensionality.” If the rule base does not cover the input domain sufficiently, its minimal activation degree cannot be guaranteed. Thus exist inputs for which no conclusion can be determined by rule-matching-based inference. When antecedents of a rule base R do not form an ε -cover for a given value of ε , the rule base is called ε -sparse. Definition 4. Let R = {Ri |i = 1, ..., r } be a rule base with rules of form Eq. (1). If ∃A∗ ∈ F (x) : ω (R) ≤ ε ,

(ε > 0) . . . . . (6)

for a given ε , then R is ε -sparse. If R is ε -sparse for all ε > 0, then it is sparse. Figure 1 shows a sparse rule base with twodimensional input space. Grey rectangles and the hatched rectangle show rule antecedents and the two-dimensional observation. In (ε -)sparse rule bases, fuzzy rule interpolation or extrapolation is required to generate the conclusion by approximate reasoning.

2.3. Interpolation or Extrapolation? The application of interpolation or extrapolation depends on the location of the observation for rule antecedents. Specifying the relative position of two fuzzy 256

Fig. 1. Sparse rule base.

sets requires an ordering relation and a distance function defined for fuzzy sets. FRITs approach the determination of the distance of fuzzy sets two ways – using α -cuts or reference points. Kóczy and Hirota [31] defined a partial ordering for Convex and Normal Fuzzy (CNF) sets with the help of α -cuts. Let A, B ∈ F (X). We say that A precedes B, denoted by A ≺ B, if ∀α ∈ (0, 1] inf {[A]α } ≤ inf {[B]α } and . . . . . . . . (7) sup {[A]α } ≤ sup {[B]α } . Using Eq. (7), the family of upper and lower – superscripts U and L – distances of ≺-comparable fuzzy sets are defined for each α : dαL (A, B) = inf([A]α ) − inf([B]α ) and dαU (A, B) = sup([A]α ) − sup([B]α ). . . . (8) The applicability of this approach is limited due to the CNF requirement and partial ordering. These limitations are waived using the Hausdorff-distance of α -cuts in defining the distance of fuzzy sets [15]: DH (A, B) = sup dH ([A]α , [B]α ).

. . . . (9)

α ∈(0,1]

The second approach, e.g., [15, 20, 21, 32, 33], uses the reference point concept [14, 20, 34], rp, to determine the position of fuzzy sets. By definition, A ≺ B if and only if rp (A) < rp (B), which is a total ordering, in which case the distance of fuzzy sets is determined as: d (A, B) = |rp (A) − rp (B)| .

. . . . . . . (10)

Using such an ordering enables us to formulate conditions for applying FRI and FRE easily. A onedimensional case is investigated, multidimensional extensions are given by aggregation. If observation A∗ is located so that Ai1 and Ai2 exists such that Ai1 ≺ A∗ ≺ Ai2 , then FRI is applied based on rules Ri1 and Ri2 to obtain the conclusion. Otherwise, when all rule antecedents Ai (i = 1, . . . , r) either precede or are preceded by A∗ : ∀i ∈ [1, r] : A∗ ≺ Ai or Ai ≺ A∗ . . . . . . . (11) then FRE is applied based on the two (or more) rules closest to A∗ .


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3. FRIT Criteria and Properties The general set of criteria for FRIT we introduce, incorporating mathematical- and application-oriented requirements, assists in future FRIT classification and comparison and lays down the foundation for applying FRITs suitable for different applications. We consider our requirements as FRIT properties – not as an axiomatic characterization. In an axiomatic characterization, we would exclude numerous – mostly early – methods from the arena of FRITs meeting even the basic properties only under weakened conditions. We define rule interpolation as a mapping: Definition 5. Let R = {Ri |i = 1, ..., r} be a rule base with multidimensional rules of form Eq. (1). We call mapping I: F (X) → F (Y ) rule interpolation, if it assigns to each observation A∗ ∈ F (X) an (interpolating) conclusion I(A∗ ) = B∗ ∈ F (Y ) that satisfies at least Properties 1, 5, and 7. Several additional conditions and properties can be expected from an FRI in all types of applications or in particular cases. To facilitate classification and comparison, we divide them into four groups presented in the next sections.

3.1. Applicability and Extendibility Fuzzy set A ∈ F (Z) is valid if its membership function is valid. The validity of a fuzzy set is characterized by α -cuts as follows: ∀α , α1 < α2 ∈ (0, 1] : inf {[A]α } ≤ sup {[A]α } and inf {[A]α1 } ≤ inf {[A]α2 } and sup {[A]α2 } ≤ sup {[A]α1 } . . . . . . . . . . . (12) Property 1 Mapping validity: For each A∗ ∈ F (X) with a valid fuzzy membership function, the conclusion generated by mapping I, B∗ = I(A∗ ) ∈ F (Y ) should also be a valid fuzzy set in the sense of Eq. (12) for an arbitrary set of rules. Although Property 1 is a straightforward requirement, several methods do not meet it. As pointed out, e.g., in [20, 21, 35], the Kóczy-Hirota (KH) method [18] and its modification VKK interpolation [36] do not always generate valid fuzzy set as a conclusion, e.g., Figs. 6–7. For KH, [35] gave necessary and sufficient conditions on the shape and the location of the observation and the rule base to meet Property 1, and similar restrictions apply to VKK interpolation. Property 2 General applicability: Mapping I must be applicable to an arbitrary rule base and observation, without constraints regarding fuzzy set shape.

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In generally applying rule interpolation/extrapolation mapping, Property 2 is a natural requirement, but few FRITs meet it without constraint, e.g., [20, 33]. FRI and FRE impose different conditions on the membership functions of the rule base such as normality, upper semicontinuity [15], convexity [18, 36], or singleton observation [37]. Some discrepancies have been eliminated, e.g., the extension of [37], renamed FIVE, handles fuzzy observations [38]. Practically speaking, it is reasonable to weaken Property 2 to control the shape of input because mostly piece-wise linear and Gaussian shaped fuzzy sets are encountered in applications, and complicated, irregularly shaped input sets raise computational requirements. Property 3 Applicability in multidimensional input space: Mapping I must be applicable to arbitrary finite dimensions of input space. FRITs were originally motivated in attempts to reduce complexity, which is meaningful only in the case of many input dimensions, so FRITs working only with a onedimensional rule base have limited applicability. Surprisingly, researchers often neglect discussing this case, as pointed out in [39], although, it is not always straightforward to extend a one-dimensional method to a multidimensional case. Property 4 Extrapolation capability of the method: A method with mapping I applies to extrapolation if it generates a conclusion when the observation is located in an extrapolative position, defined in Eq. (11). As stated, only certain FRI methods are extrapolatively extensible [13–15, 17, 33, 37, 38, 40, 41].

3.2. Location and Interpretation of Conclusions Property 5 Gradual semantic interpretability: Let sZ : F (Z) × F (Z) → R denote the similarity function defined in the fuzzy sets of Z. Then, for A∗ , A(1) , A(2) ∈ F (X) , if sX (A∗ , Ai1 ) ≥ sX (A∗ , Ai2 ) then sY (I(A∗ ), Bi1 ) ≥ sY (I(A∗ ), Bi2 ), where Ri j : Ai j → Bi j ( j = 1, 2) are two rules of rule base R. This property was interpreted as “the more similar the observation to an antecedent, the more similar the conclusion should be to the corresponding consequent of the given antecedent” in [29]. Many researchers only consider the extreme case of Property 5 – when the observation coincides with a rule antecedent referred to as compatibility with the rule base [15, 20, 22]. In logic, this property is called Modus Ponens. Note that it is also called continuity of the model characterized by the fuzzy relation of the rule base [42]. Property 6 Monotonicity: Let A∗ , A+ be two observations such that A∗ is more specific in all dimension than A+ , that is for all ∀ j = 1, . . . , n, A∗j ⊆ A+j , then I(A∗ ) ⊆ I(A+ ), i.e., the same relation should be held for the two conclu-


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observation A∗ are singleton, then I(A∗ ) should be singleton.

sions. This property originated in [15], where it is shown that only the method proposed in that paper satisfies it. Property 7 Preserving “in between”: In a linear interpolation, if Ai1 j ≺ A∗j ≺ Ai2 j for all j = 1, . . . , n, then Bi1 ≺ I(A∗ ) ≺ Bi2 , where Ri j : Ai j → Bi j ( j = 1, 2) are two rules of rule base R. Property 7 states [15, 22] that if the antecedent sets of two neighboring rules surround an observation in all antecedent dimensions, the approximated conclusion should also be surrounded by the consequent sets of these rules.

3.3. Shape Invariance As shortly discussed about Property 2, FRITs often restrict the shape of fuzzy sets in rules for practical reasons. Fuzzy sets in rules are commonly of a certain type, i.e., the shape of all membership functions is, e.g., equally singleton, triangular, trapezoid, piece-wise linear, or Gaussian bell shape. Property 8 Shape invariance of mapping: Let all fuzzy sets in rules be of the same type T. Mapping I is shapeinvariant if I(A∗ ) is also type T. This property ensures the closedness of mapping I regarding the given membership function shape. When the validity property (Property 1) is not harmed, α -cut-based methods [18, 21, 36, 40] satisfy Property 8 for piecewise linear type because the conclusion is determined for breakpoint set levels and, in between them, flanks are obtained by linear interpolation. It is interesting to investigate how the fuzziness of the generated conclusion depends on the fuzziness of input – an issue raising two opposite approaches in the literature. In the first case, the less uncertain the observation, the less fuzziness the approximated consequent has [15, 20, 22]. In the most extreme case, a crisp observation should produce a crisp consequence. This concept is strongly related to the monotonicity condition in Property 6. In the second approach, the fuzziness of the estimated consequent originates in the nature of the fuzzy rule base [21, 39], i.e., a crisp conclusion can be expected only if all consequents of rules considered during interpolation are crisp, i.e., the knowledge base produces precise information from fuzzy input data. This approach is a special case of shape invariance. These two approaches can be formalized as follows. Property 9 The fuzziness of the conclusion: Mapping I of a FRIT is allowed to generate a singleton conclusion in two cases: a) Observation A∗ is singleton, then conclusion I(A∗ ) should be singleton and b) All BI , where I denotes the indices of rules that contribute to the calculation of conclusion I(A∗ ), and 258

IMUL [39], for example, satisfies case a) if consequents BI has a single core, i.e., triangle sets. MACI [21] and LESFRI [40] fall into case b).

3.4. Global Mapping Properties Property 10 Mapping continuity (smoothness): For arbitrary ε > 0 there exists a δ > 0 such that if dX (A∗ , A+ ) ≤ δ then dY (I(A∗ ), I(A+ )) ≤ ε , where dZ (·, ·) is a distance function defined in domain Z. This condition prescribes that similar observations should lead to similar results [15, 20, 22]. From an approximation theory view, mapping I approximates an input-output function. Many FRITs consider only the two closest rules from the rule base when calculating the conclusion, although a better approximation of the input-output function can often be obtained if the conclusion is generated using more rules. The number of rules has an analog role in rule interpolation as the number of measurement points in function interpolation. The approximation of mapping I can be characterized based on this analogy. Property 11 Universal approximation property: Let f (x): Rn → R be the continuous approximated function and Ik ( f , x) the mapping of fuzzy rule interpolation/extrapolation, where k is the number of rules considered for the approximation of f (x). Ik ( f , x) has a universal approximation property, if f (x) = lim Ik ( f , x) k→∞

holds independently from the position of the rules. This means that function f (x) can be approximated arbitrarily well if the number of rules is not limited. This property has been proven for some α -cut based FRITs, the generalized KH interpolation [43], and for its improved version, MACI [34], exploiting the analogy between these methods and Shepard interpolation [44]. A FRIT’s computational complexity directly determines its time demand and often influences its memory consumption. Computational complexity is thus a determinative factor in the method’s applicability in real time or embedded systems. Property 12 Computational complexity of the method: A FRIT is applied by rule omission for rule base reduction if its time complexity is inferior to O(T n ), where T is the maximum number of fuzzy sets in original rule base R . Mapping decomposability is described by two dual conditions [15]. The first states that the interpolated conclusion corresponding to an observation obtained by the t-norm (fuzzy intersection) of two observations should be more specific than the t-norm of interpolated conclusions


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Fig. 3. Input space with triangular (left) and singleton (right) observations. Fig. 2. Surface showing the relationship between nonlinear function input and output in Eq. (15).

corresponding to the original observations in Eq. (13). The second states that the interpolated conclusion corresponding to an observation obtained by the s-norm (fuzzy union) of two observations should be less specific than the s-norm of interpolated conclusions corresponding to original observations in Eq. (14). Property 13 Decomposability of mapping: Let A∗ and A+ be two observations. Mapping I is decomposable if it satisfies . . . . . (13) I t(A∗ , A+ ) ⊆ t I(A∗ ), I(A+ ) ∗ ∗ + + . . . . (14) I s(A , A ) ⊇ s I(A ), I(A ) .

Fig. 4. Partition of input dimensions with triangular observations.

Fig. 5. Partition of input dimensions with singleton observations.

4. Experiments To demonstrate the properties described so far, we created two rule bases modeling the nonlinear function in Fig. 2. sin(x) , 4

x, y ∈ [−2, 2] × [−2, 2] . . . . . . . . (15)

f (x, y) = y · e−x

2 −y2

+

The two fuzzy systems are sparse, with six rules in each. Four of the six rules correspond to the extrema of input space mapped on output space. The remaining two rules determine the output of the modeled function for two central regions of input space. The first fuzzy rule base (FRB-1) has only trapezoidal fuzzy sets both on antecedent and consequent sides. The second fuzzy rule base (FRB-2) has singleton fuzzy sets on the consequent side, while input fuzzy sets remain the same. With the two fuzzy rule bases, we created four experimental settings, two for each, by testing both with two observations. We randomly picked a triangle and a singleton observation. Rule bases are not optimized for FRITs. Fig. 3 shows antecedent space for the two fuzzy rule bases and observations. Figs. 4 and 5 show antecedent partitions of the two fuzzy rule bases and observations, in thick lines. Experiments were conducted using our MATLAB FRI Toolbox [26]. Settings are given on the toolbox home page (http://fri.gamf.hu/examples/). More tests are easily executed by varying test setting parameters in the toolbox. In experimentation, we tested the following 10 FRITs: KH linear interpolation [18], generalized KH interpolaVol.15 No.3, 2011

tion [43], MACI [21], Baranyi’s solid cutting method with fixed point law (SCM + FPL) [20], Conservation of Relative Fuzziness (CRF) interpolation [45] (Figs. 6 and 7), Fuzzy Interpolation based on Vague Environment (FIVE) [37, 38], VKK interpolation [36], IMUL [39], Least Squares-based Fuzzy Rule Interpolation (LESFRI) [40], and Fuzzy Rule Interpolation based on Polar Cuts (FRIPOC) [33] (Figs. 6 and 7) In addition to demonstrating the methods, experiments can show by counterexamples whether a method fails to meet properties defined in Sections 3. These few examples are, however, not appropriate for proving the meeting of properties; that requires analytical proofs. This future work can, however, be supported by empirical observation drawn from such experiments.

4.1. Experiments with Trapezoidal Consequents Figure 6 shows conclusions generated by the above methods using FRB-1 and both observations. Experiments show that Property 1 is violated by KH, generalized KH, and VKK. Note also that Property 8 is violated by FRIPROC. Properties 5 and 7 are harmed in singleton observation by CRF, VKK, and IMUL both for FRB-1 and FRB-2. Some of these problems are alleviated in SISO, implying that affected methods do not meet Property 3. 4.2. Experiments with Singleton Consequents Figure 7 shows conclusions generated by the above methods using FRB-2 and both observations. Due to


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Fig. 7. Conclusions generated by using FRB-2 and triangle observation (left) and singleton observation (right).

Table 1. Summary of experiments on properties. The plus sign (+) indicates that the method passed both tests, while a minus sign (−) shows that a counterexample was found. In Property 9, we denote the subcase the method satisfying (a or b).

Fig. 6. Conclusions generated by using FRB-1 and triangle observation (left) and singleton observation (right).

space constraints, results obtained by generalized KH, FIVE, and FRIPROC are omitted here, but available on the toolbox web site. Note that although the rp of the IMUL conclusion is within range of a triangle observation, its support becomes wide at both FRB-1 and FRB-2. Tests suggest that SCM + FPL, CRF, FIVE, and IMUL satisfy case a) while 260

MACI and LESFRI satisfy case b) of Property 9. We summarize findings on experiments in Figs. 6 and 7 and in Table 1.


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5. Conclusions Having investigated fuzzy rule interpolation and extrapolation, we have characterized sparse rule bases where FRITs substitute conventional rule-matching-based fuzzy inference, because the latters are unable to generate sensible output when no fuzzy rule is fired. In the last 20 years, this fact motivated researchers to propose FRITs. Having created an overall standard for FRITs with a general set of criteria, we show these properties to be useful for guiding FRIT classification and comparison. We have conducted experiments with ten selected methods. This paper is our initial work in the investigation of FRITs, and subsequent papers will analyze in detail how individual techniques meet each group of the criteria proposed here. Acknowledgements This research was supported in part by Hungarian National Scientific Research Fund grant no. OTKA K77809.

References:

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Name: Domonkos Tikk

Name: Zsolt Csaba Johanyák

Affiliation: Institute of Information Technology, Kecskemét College

Address: Izsáki u´ t 10, H-6000 Kecskemét, Hungary

Brief Biographical History: 1990 Received M.Sc. in Mechanical Engineering from Technical University of Cluj Napoca, Romania 1991-1996 Assistant Professor, Department of Information Technologies, Kecskemét College, Hungary 1996-2008 Senior Lecturer, Department of Information Technologies, Kecskemét College, Hungary 2005 Received M.Sc. in Information Engineering from University of Miskolc, Hungary 2008 Received Ph.D. in Information Science and Technology from University of Miskolc, Hungary 2008- Associate Professor, Institute of Information Technologies, Kecskemét College, Hungary

Main Works:

Affiliation: Budapest University of Technology and Economics

• FRI MATLAB ToolBox, SFMI MATLAB ToolBox • fuzzy systems, software engineering, and quality management

Membership in Academic Societies:

• John von Neumann Computer Society • Hungarian Fuzzy Association • Scientific Association for Infocommunications Hungary (HTE) • European Organization for Quality (EOQ)

Address: Magyar Tudósok krt. 2, H-1117 Budapest, Hungary

Brief Biographical History: 1995 Received M.Sc. in Computer Science from E¨tvös University, Budapest, Hungary 2000 Received Ph.D. in Information Science from Budapest University of Technology and Economics (BUTE), Hungary 2000-2008 Researcher, Project Leader, Department of Telecommunications and Media Informatics at BUTE 2009-2010 Humboldt Fellowship Holder, Humboldt University in Berlin, Germany 2010- Chief Scientific Officer, Gravity R&D Ltd., Budapest, Hungary

Main Works:

• text mining, recommender systems, and fuzzy systems


• Association for Computing Machinery (ACM) • John von Neumann Computer Society • Hungarian Fuzzy Association

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Name:

Name:

Szilveszter Kovács

Kok Wai Wong

Affiliation:

Affiliation:

Associate Professor, Department of Information Technology, University of Miskolc

Murdoch University, School of Information Technology

Address:

Address:

Miskolc-Egyetemváros, H-3515, Miskolc, Hungary

South St., Murdoch, Western Australia 6150, Australia

Brief Biographical History:

Brief Biographical History:

1989 Received M.Sc. in Electrical Engineering from Technical University of Budapest, Hungary 1993 Received M.Sc. in Computer Engineering from Technical University of Budapest, Hungary 1998 Received Ph.D. in Engineering from University of Miskolc, Hungary 1989-1998 Network Manager, Computer Centre, University of Miskolc, Hungary 1998-2005 Senior Lecturer, Department of Information Technology, University of Miskolc, Hungary 1999-2001 Research Fellow, Gifu Prefectural Research Institute of Manufacturing Information Technology, Japan 2005-2010 Senior Research Fellow, Institute of Information Technologies, Kecskemét College, Hungary 2009-2010 Research Fellow, Department of Cybernetics and Artificial Intelligence, Faculty of Electrical Engineering and Informatics, Technical University of Kosice, Slovakia 2009-2010 Head of the Department, Department of Automation, University of Miskolc, Hungary 2005-Associate Professor, Department of Information Technology, University of Miskolc, Hungary

2007- Associate Professor, Murdoch University 2003- Assistant Professor, Nanyang Technological University

Main Works:

• Y. S. Ong, M. H. Lim, N. Zhu, and K. W. Wong, “Classification of Adaptive Memetic Algorithms: A Comparative Study,” IEEE Trans. of Systems, Man, and Cybernetics, Part B: Cybernetics, Vol.36, No.1, pp. 141-152, February 2006. • Z. C. Johanyak, D. Tikk, S. Kovacs, and K. W. Wong, “Fuzzy Rule Interpolation Matlab Toolbox – FRI Toolbox,” Proc. of IEEE Int. Conf. on Fuzzy Systems 2006, Vancouver, Canada, pp. 1427-1433, July 2006. • K. W. Wong, D. Tikk, T. D. Gedeon, and L. T. Koczy, “Fuzzy Rule Interpolation for Multidimensional Input Spaces with Applications,” IEEE Trans. of Fuzzy Systems, Vol.13, No.6, pp. 809-819, December 2005.


• IEEE, IEEE Computational Intelligence Society, IEEE Computer Society • Association for Computing Machinery (ACM) • Australian Computer Society (ACS) • Asia Pacific Neural Network Assembly (APNNA)

Main Works:

• fuzzy systems, fuzzy rule interpolation and its embedded applications, behaviour-based control and reinforcement learning


• EURO (The Association of European Operational Research Societies) Working Group on Fuzzy Sets (EUROFUSE) • Integrated Intelligent Systems, Japanese-Hungarian Joint Laboratory (IISL) • John von Neumann Computer Society • Hungarian Fuzzy Association

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