Fuzzy Rule Interpolation Based on Polar Cuts - Semantic Scholar

Johanyák, Zs. Cs., Kovács Sz.: Fuzzy Rule Interpolation Based on Polar Cuts, Computational Intelligence, Theory and Applications, Springer Berlin Heidelberg, 2006, ISBN 978-3-540-34780-4, pp. 499-511 draft paper

Fuzzy Rule Interpolation Based on Polar Cuts Zsolt Csaba Johany´ ak1 and Szilveszter Kovács2 1

2

Department of Information Technology, GAMF Faculty, Kecskemét College, Kecskemét, H-6001 Pf. 91, Hungary, [email protected] Department of Information Technology, University of Miskolc, Miskolc-Egyetemv´ aros, Miskolc, H-3515, Hungary, [email protected]

Systems applying fuzzy logic are rule based ones. The collection of the rules the so called rule base can be characterized as dense or sparse depending on whether there exist rules for all the possible observations. In the sparse case for some observations there are no rules whose antecedent part would overlap the observation at least partially. Therefore the classical compositional reasoning methods can not produce an acceptable conclusion. The inference techniques based on fuzzy rule interpolation are developed for especially this purpose. This paper proposes a new fuzzy rule interpolation based inference technique applying the concept of linguistic term shifting and polar cut. It is called FRIPOC (Fuzzy Rule Interpolation based in POlar Cuts) and it is applicable in the case of sparse and dense rule bases, too. Its main advantages are its comprehensibility, extrapolation capability and its applicability even if the height of one or more fuzzy sets is smaller than one. The rest of this paper is organized as follows. Section 1 gives a brief overview on the relevant fuzzy rule interpolation techniques grouping them depending on the main steps they are following. Section 2 presents the main structure and the steps and stages that characterize the method FRIPOC. Section 3 introduces the concept of the polar cut and a fuzzy set interpolation technique called FEAT-p based on it as a possible implementation for the first and third stage of the first step. In section 4 the authors propose a technique for the determination of the position of the consequent sets that is an extension and adaptation of the Shepard 2D interpolation [15]. Section 5 introduces a new polar cut based single rule inference method for the determination of the conclusion. In section 6 some relevant features of the new method are outlined through some numerical examples.

1 A brief overview of fuzzy rule interpolation methods The fuzzy rule interpolation (FRI) based inference techniques have been used for several years in order to alleviate the problems arising from the information


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Zsolt Csaba Johany´ ak and Szilveszter Kov´ acs

gaps in sparse rule bases. They can be divided into two groups depending on whether they are producing the approximated conclusion directly or a new intermediate rule is interpolated first. Relevant members of the first group are among others the α-cut based interpolation (KH) [10] proposed by Kóczy and Hirota, which was the first developed technique, the modified α-cut based interpolation (MACI) [17] introduced by Tikk and Baranyi, the fuzzy interpolation based on vague environment (FIVE) [12] developed by Kovács and Kóczy, the improved fuzzy interpolation technique for multi-dimensional input spaces (IMUL) [20] proposed by Wong, Gedeon and Tikk, the interpolative reasoning based on graduality (IRG) [2] introduced by Bouchon-Meunier, Marsala and Rifqi, the interpolation by the conservation of fuzziness (GK) [4] developed by Gedeon and K´ oczy, the method based on the conservation of the relative fuzziness (CRF) proposed by Hirota, K´ oczy and Gedeon, and the VKK method [19] introduced by Vass, Kalm´ ar and K´ oczy. The structure of the methods belonging to the second group can be described best by the generalized methodology of the fuzzy rule interpolation introduced by Baranyi, K´ oczy and Gedeon in [1]. As other typical members of this group can be mentioned the ST method [22] introduced by Yan, Mizumoto and Qiao, the interpolation with generalized representative values (IGRV) [5] developed by Huang and Shen, the technique proposed by Jenei in [6], and the method being presented in this paper. The solvability and approximate solvability of fuzzy relation equations and the approximation quality of approximate solutions was studied by Perfilieva and Gottwald in [13].

2 The structure of the proposed method The method FRIPOC (Fuzzy Rule Interpolation based on POlar Cuts) essentially follows the concepts of the generalized methodology of fuzzy rule interpolation (GM) introduced by Baranyi et al. in [1]. The position of the

Fig. 1. Options for the reference point and the related set distances


Fuzzy Rule Interpolation Based on Polar Cuts

3

fuzzy sets is characterized by a reference point during the calculations. For example the centre of the core, the centre of gravity, the centre of the support or the projection of the centre of the core to the horizontal axis can play this role (Fig. 1.). In the case of the polar cut based set interpolation and single rule reasoning methods the latter choice offers the most advantages. Besides its information content about the middle one from the most relevant (having the maximal membership value) elements of the set it also reduces the need for calculations due to the fact that its ordinate value is always zero. Further on this type of reference point is used during the calculations. The distance of the fuzzy sets is measured by the horizontal distance between the reference points of the sets. The method consist of two steps. First a new intermediate rule is interpolated, of which antecedent part contains fuzzy sets whose position is identical with the position of the sets describing the observation in each dimension. This task is done in three stages. First the antecedent part of the new rule is determined through a set interpolation method. The application of the technique FEAT-p introduced in section 3 is proposed by the authors for this purpose. Next the position of the fuzzy sets belonging to the consequent part of the new rule is calculated. The method suggested by the authors for this task is presented in section 4. Thirdly the shape of the consequent sets is determined using the same technique as in the case of the antecedent sets. The conclusion is determined in the second step by firing the interpolated rule. A special single rule reasoning technique called SURE-p, which is based also on polar cuts, is introduced for this task in section 5.

3 Fuzzy set interpolation based on linguistic term shifting and polar cuts The task of the fuzzy set interpolation is to determine the antecedent and consequent sets that belong to the new rule. The method is the same in the case of each linguistic term regardless of it belongs to an antecedent or consequent universe of discourse. The calculations are done separately for each input and output dimension. The starting point is a fuzzy partition with the reference points of the sets determined in advance and the reference point of the observation (conclusion) in the actual dimension/partition. All the sets in the partition belong to the antecedent (consequent) part of one or more rules. The reference point of the new set is identical with the the reference point of the observation (conclusion) in the actual dimension. The method goes out from the assumption that a better set approximation can be attained by taking into consideration not only the two sets flanking the observation/conclusion but all the available linguistic terms in the partition. First all sets are shifted horizontally in order to reach the coincidence of the horizontal position of their reference points with the position of the interpolation (see fig. 2). This idea is similar to the concept in [2], but that method uses and translates only


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Fig. 2. The original partition and the result of the shifting

the two flanking sets into the location of the observation. Next the shape of the new set is determined from the collection of the overlapped sets. There are several solutions for this task. Similar to the choice of the reference point the selection of the calculation mode of the shape is also a tuning point. In [8] the authors present a solution with low computational complexity called FEAT-α (Fuzzy SEt interpolAtion Technique based on α-cuts). It is based on α-cuts and its application area is however, restricted to the most popular case of the convex and normal fuzzy (CNF) sets. Further on the concept

Fig. 3. Polar cut

of the polar cut is introduced and next based on it a solution called FEAT-p (Fuzzy SEt interpolAtion Technique based on polar cuts) is proposed. Its main advantage is that it can also be applied in cases when the normality condition is not satisfied for all the sets participating in the interpolation process, i.e. the height of one or more sets is smaller than 1. The concept of the polar cut is strong related to the application of a polar co-ordinate system whose origin coincides with the abscissa of the reference point of the observation. A polar cut is defined by a value pair {ρ, θ} that determines a point on the shape of the linguistic term. The value ρ denotes the polar distance at the angle θ (Fig. 3). The authors are going out from the



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assumption that an extension and a resolution principle of the fuzzy sets can be defined for polar cuts, too. This extension principle states that the solution of a problem for fuzzy sets can be found in the form of solving it first for its polar cuts and then extending the solution to the fuzzy case. The resolution principle states in this case that a fuzzy set can be decomposed into polar cuts. The shape calculation technique FEAT-p is based on the above defined extension principle. For each polar cut of the interpolated set the value ρ is calculated as weighted average of the polar distances ρ of the shifted sets for the same θ angle using the formula (1).  nj P   wjk ·ρ(Ajkθ )    k=1  d A∗j , Ajk > 0  nj P wjk ρ Aijθ = (1)  k=1       ρ (Ajkθ ) d A∗j , Ajk = 0, k = 1..nj

where ρ denotes the length of a polar cut, j is the actual antecedent (consequent) dimension, θ is the angle of the actual cut, nj is the number of the sets in the partition, Ajkθ is the polar cut of the k th set, wjk is the weighting factor of the k th set, Aijθ is the interpolated polar cut and the superscript i denotes that the set is an interpolated one. The collection of the angles , the so called polar levels, for which the calculations are done, should be set-up in such mode to include the values 0, π/2 and π. It seems to be natural that the sets whose original position were in the neighbourhood of the reference point of the observation to exercise higher influence as those ones situated in farther regions of the universe of discourse. Therefore the weighting factor should be dependent on distance. The simplest weighting factor is the reciprocal value of the distance, which can be expressed by the formula (2) with p = 1, but there are several recommendations in the literature for more or less analogue cases. For example in [10] the square of the reciprocal value of the distance is suggested (p = 2). The authors of [18] propose the use of the reciprocal value of the distance on the mth power (p = m), where m is the number of the antecedent dimensions. wjk =

1 p d A∗j , Ajk

(2)

The formula (1) separates the case when the position of the interpolation coincides with the actual set of the partition (d A∗j , Ajk = 0). Its reason is that the weighting factor (2) contains the distance in the denominator. Thus if the reference point of the observation (conclusion) is the same as one of the original sets of the partition the interpolated set will be the same as that linguistic term. This feature ensures the fulfilment of the condition 4 from [7], namely the compatibility with the rule base, for the rule interpolation method based on the above mentioned method.


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4 The position of the consequent sets The position of the fuzzy sets belonging to the consequent part of the new rule is determined independently in each output dimension. The task can be defined as a problem of finding a point on a hyper-surface defined by the reference points of the antecedent sets (sets belonging to the antecedent parts of the existing rules) and the consequent sets in the actual output dimension. Due to the sparse character of the rule base an na dimensional interpolation has to be done for irregularly spaced data, where na is the number of the output dimensions. It can be expressed in general by the formula (3). RP Bli = f RP Ai1 , RP Ai2 , ..., RP Aij , ..., RP Aina (3) where RP Bli is the reference point of the interpolated consequent set in the lth output dimension and RP Aij is the reference point of the interpolated antecedent set in the j th input dimension. The function should pass through the known points of the hyper-surface and it should be smooth, i.e. continuous and once differentiable. There are several applicable linear or non-linear functions that take into consideration either only the points (rules) situated in the closest neighbourhood of the interpolation or all the known points. The authors suggest the use of an interpolation function that is an extension and adaptation of the Shepard interpolator [15] for the case of arbitrary number of antecedent dimensions. The antecedent part of each rule can be thought of as a point in the antecedent hyper-space. Its co-ordinates are given by the reference points of the sets belonging to it. The point corresponding to the antecedent of the interpolated rule is at the same time also the representing point of the observation. Further on the Euclidean distance between these points is used as the measure of the closeness of the antecedents and by this means also the closeness of the rules. The proposed interpolation function (4) determines the reference point of the conclusion as a weighted average of the reference points of the consequent sets of the known rules in the actual output dimension.

RP

Bli

=

N P

RP (Blj ) · sj

j=1 N P

(4) sj

j=1

where RP Bli is the reference point of the interpolated consequent set in the lth dimension, N is the number of the rules, j denotes the actual rule, sj is the weight attached to the j th rule. The rules whose antecedent part is in the closer neighbourhood of this point should exercise higher influence than those situated farther. Therefore the weighting factor is a distance function. Shepard proposed in [15] several variants of the weighting factors for its interpolation function. The first of



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them, which applies the inverse of the square of the distance was chosen by the authors to be applied considering it as the one having the lowest computational complexity. Its adapted version, the formula (5) is the inverse of the square of the distance between the antecedent of the interpolated rule and the antecedent of the j th rule. It is actually the sum of the squares of distances measured along each antecedent dimension. sj =

1 2 d (RAi , RAj )

=

na P

k=1

1 2 RP Aik − RP (Ajk )

(5)

where RAi is the antecedent of the interpolated rule, RAj is the antecedent of the j th rule,RP Aik is the reference point of the interpolated antecedent in the k th dimension (identical with the reference point of the observation in the k th dimension), RP (Ajk ) is the reference point of the reference point of the antecedent set of the j th rule in the k th dimension and na is the number of the antecedent dimensions. Generally the fuzzy sets are identified in the different formulas by two indexes (e.g. in (1) and (2)) the first indicating the dimension and the second indicating the ordinal number of the set. Contrary to this in the last two formulas ((4) and (5)) the second subscript gives the number of the rule of which antecedent part the set belongs to. It is because this notation mode simplifies the formulas. Shepard suggested in [15] for the 2D case the use of maximum 10 closest points in order to reduce the computational needs. However, when the number of the dimensions is much more than two and the rule base is sparse it seems to be easier to take into consideration all the rules than to seek those ones that are in a special proximity of the observation.

5 Single rule reasoning based on polar cuts In the second step of the inference the conclusion is generated by firing the new rule. The reference point of the interpolated conclusion in the current dimension will be the same as the reference point of the consequent set of the new rule in the current dimension. Usually the antecedent part of the rule does not fit perfectly the observation. Therefore a special single rule reasoning technique is needed. There are several methods for this task in the literature, but their common drawback is that their applicability is restricted to some regular cases. For example the similarity transfer method introduced in [16] requires the normality of the sets. Beside this the revision principle based FPL and SRM techniques presented in [14] also demand the coincidence between the support of the antecedent set and the support of the observation. Generally these conditions are not fulfilled. Therefore some transformations of the fuzzy relation are needed when one decides for their application. The technique SURE-p (Single rUle REasoning based on polar cuts) being presented alleviates this problem. In addition its advantage is its applicabil-


8


Fig. 4.

ity in multi-dimensional cases. SURE-p is based on the concept of polar cut. Although it determines the conclusion sets in each consequent dimension independently, there are some common calculations that have to be done only once at the beginning. Thus for each polar cut firstly the difference between the polar distance of the antecedent set and the polar distance of the observation in each dimension is calculated, and the result is divided by the range of the linguistic variable (6). ρ Aijθ − ρ A∗j rjθ = (6) rangeaj th where rjθ is di the relative difference at the θ level in the j antecedent i ∗ mension, ρ Ajθ the polar distance of the antecedent set, ρ Aj is the polar distance of the observation and rangeaj is the range of the antecedent linguistic variable in the j th dimension. Next an average relative difference is calculated taking into consideration the relative differences in all antecedent dimensions (7). na P rjθ

rθ =

j=1

na

(7)

where rθ is the average relative difference at the θ level, na is the number of the antecedent dimensions. In each consequent dimension the corresponding polar cut is calculated supposing that the relative difference at θ level between the polar distances of the rule consequent and the conclusion is equal to rθ as expressed in formula (8). i ∗ ρ Blθ − ρ (Blθ ) = rθ (8) rangecl i ∗ where ρ Blθ the polar distance of the interpolated consequent set, ρ (Blθ ) is the polar distance of the conclusion, rangecl is the range of the consequent



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linguistic variable in the lth output dimension and θ is the polar angle. Due to the nature of the fuzzy sets the resulting height of the conclusion has to be maximized to 1. Thus arises the formula (9).  h i 1 i  M in ρ B − r · range , sin (θ) > 0  θ cl sin(θ) lθ   ∗ ρ (Blθ )= (9)     i rho Blθ − rθ · rangecl sin (θ) = 0

Fig. 5. Non-convex conclusions obtained by the formula (9)

Due to the revision of the interpolated consequent sets based on the average relative antecedent difference the formula (9) can easily lead to a nonconvex fuzzy set. As an example figure 5 presents the consequent partitions of a system with two output dimensions (output1 and output2). The interpolated conclusion sets (B1∗ and B2∗ ) obtained by the formula (9) are drawn with bold lines. In order to alleviate this problem the calculations should start at polar level π/2 in top-down direction, they should be done separately for the right and left flanks of the linguistic terms as well a control and correction algorithm should be included. Further on the basic ideas and the steps that have to be done are presented only for the case of the right flank of the set. The calculation of the left flank is similar. The convexity requirement is satisfied if and only if the horizontal distance to the centre of the polar co-ordinate system of each point is not smaller than the same distance calculated for the previous point and if the vertical distance to the centre of the polar co-ordinate system of each point is not greater than the same distance calculated for the previous point. This condition can be expressed by the formula (10).


10


ρ

∗c Blθ(k)

=

 ∗  ρ Blθ(k)          ∗  ρ B   lθ(k−1) ·           ∗  ρ Blθ(k−1) ·                   ρ B∗ lθ(k)

k=1 cos(θ(k−1)) cos(θ(k))

k > 1 and ∗ ρ(Blθ(k−1) )·cos(θ(k−1))

>1

k > 1 and ∗ ρ(Blθ(k) ·sin(θ(k)) )

>1

∗ ρ Blθ(k) ·cos(θ(k))

sin(θ(k−1)) sin(θ(k))

∗ ρ Blθ(k−1) ·sin(θ(k−1))

(10)

otherwise

where θ is an array containing the polar angles necessary for the calculation ∗ of the right flank from π/2 to 0 in descending order, ρ Blθ(k) is the polar ∗c distance calculated by the formula (9) and ρ Blθ(k) is the corrected polar distance.

6 Numerical examples In the followings the sensitivity of the method FRIPOC to the value of its parameter p will be studied through some numerical examples. Figure 6 presents

Fig. 6. FRIPOC applied with p = 0.001 to a system with 2 input and 1 output dimension and 2 rules

a fuzzy system having two antecedent (input1 and input2) dimensions and one consequent dimension (output1). There are triangular, trapezoidal and rectangular (crisp) set shapes and four of the sets are subnormal. For the sake of simplicity each original antecedent partition contains only two sets that are surrounding the observation drawn by bold line. Based on the same consideration the original consequent partition also contains two fuzzy sets. The rule



11

base consist of two rules according to (11). R1 : if A∗1 = A11 and A∗2 = A21 then B1∗ = B11 R2 : if A∗1 = A12 and A∗2 = A22 then B2∗ = B12

(11)

The observation is trapezoid shaped in both antecedent dimensions (A∗1 and A∗2 ). The third axes (frame) contains the final interpolated conclusion marked by bold line and obtained for the value p = 0.001. Figure 7 contains

Fig. 7. Conclusions obtained for p = 1, p = 2 and p = 10

three further results obtained for the values 1, 2 and 10 of the parameter p. One can clearly observe that increasing the value of p the second rule (R2 ), which is visibly the nearest one to the observation keeps getting more dominant and the corrected interpolated conclusion (B1∗c ) becomes more and more similar to the set B12 .

7 Conclusions The interpolation based fuzzy reasoning methods ensure an acceptable conclusion even in cases when there are no rules whose antecedent part would overlap the observation. In this paper a new technique called FRIPOC is presented that introduces the concept of polar cuts and linguistic term shifting for fuzzy rule interpolation. It determines the conclusion in two steps following the concept of GM [1]. First an intermediate rule is interpolated whose antecedent part is in the same position as the observation in each antecedent dimension and next the result is determined by firing the new rule. The authors suggest the application of a new method called FEAT-p for the set interpolation tasks and the use of an adapted version of the Shepard interpolation for the determination of the position of the consequent part of the rule in the first step. A new technique called SURE-p is suggested as single rule reasoning method for the second step. The main advantages of the method FRIPOC are its comprehensibility,


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extrapolation capability and its applicability even in subnormal cases. The sensitivity of the method to the value of the parameter p is outlined through some numerical examples. The method is implemented in Matlab and can be downloaded from [3]. This website is dedicated to a fuzzy rule interpolation Matlab toolbox development project (introduced in [9]) aiming the implementation of various FRI techniques.

8 Acknowledgements This research was partly supported by the Kecskemét College, GAMF Faculty grant no: 1N076/2006.

References 1. Baranyi P, K´ oczy L T, Gedeon T D (2004) A Generalized Concept for Fuzzy Rule Interpolation. In: IEEE Transaction On Fuzzy Systems, ISSN 1063-6706, Vol. 12, No. 6: 820–837 2. Bouchon-Meunier B, Marsala C, Rifqi M (2000) Interpolative reasoning based on graduality. In: Proc. FUZZ-IEEE’2000: 483–487 3. Fuzzy Rule Interpolation Matlab Toolbox website. http://fri.gamf.hu 4. Gedeon T, K´ oczy L T (1996) Conservation of fuzziness in the rule interpolation, Intelligent Technologies, International Symposium on New Trends in Control of Large Scale Systems, Vol. 1, Herl’any: 13–19 5. Huang Z, Shen Q (2004) Fuzzy interpolation with generalized representative values, in Proceedings of the UK Workshop on Computational Intelligence: 161–171 6. Jenei S (2001) Interpolation and Extrapolation of Fuzzy Quantities revisited (I). An Axiomatic Approach. In: Soft Computing, 5: 179–193 7. Johany´ ak Zs Cs, Kov´ acs Sz (2006) A brief survey and comparison on various interpolation based fuzzy reasoning methods”, Acta Politechnica Hungarica, Journal of Applied Sciences at Budapest Tech Hungary, Vol. 3, No. 1: 91–105 8. Johany´ ak Zs Cs, Kov´ acs Sz (2006) Fuzzy set approximation based on linguistic term shifting, MicroCad 2006, Miskolc, March 16-17, 2006, Section N: Applied Information Engineering: 123–128 9. Johany´ ak Zs Cs, Tikk D, Kov´ acs Sz, Wong K W (2006) Fuzzy Rule Interpolation Matlab Toolbox - FRI Toolbox, IEEE WCCI 2006, (accepted for publication) 10. K´ oczy L T, Hirota K (1991) Rule interpolation by α-level sets in fuzzy approximate reasoning. In: J. BUSEFAL, Automne, URA-CNRS. Vol. 46. Toulouse, France: 115–123 11. K´ oczy L T,Hirota K, Gedeon T D(1997) Fuzzy rule interpolation by the conservation of relative fuzziness. In: Technical Report TR 97/2. Hirota Lab, Dept. of Comp. Int. and Sys. Sci., Tokyo Inst. of Techn., Yokohama 12. Kov´ acs, K´ oczy L T (1999) Application of an approximate fuzzy logic controller in an AGV steering system, path tracking and collision avoidance strategy. Fuzzy Set Theory and Applications. In: Tatra Mountains Mathematical



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Publications, Mathematical Institute Slovak Academy of Sciences, Vol. 16, Bratislava, Slovakia: 456–467 Perfilieva I, Gottwald S (2003) Solvability and Approximate Solvability of Fuzzy Relation Equations, Int. J. of General Systems, Vol.32 (4): 361–372 Shen Z, Ding L, Mukaidono M (1993) Methods of revision principle. In: Proc. 5th IFSA World Congr.: 246–249 Shepard D (1968) A two dimensional interpolation function for irregularly spaced data. In: Proc. 23rd ACM Internat. Conf.: 517–524 Shi Y, Mizumoto M, Wu Z Q (1995) Reasoning conditions on K´ oczy’s interpolative reasoning method in sparse fuzzy rule bases. In: Fuzzy Sets Syst., Vol. 75: 63–71 Tikk D, Baranyi P (2000) Comprehensive analysis of a new fuzzy rule interpolation method. In IEEE Trans. Fuzzy Syst., Vol. 8: 281–296 Tikk D, Jo´ o I, K´ oczy L T, V´ arlaki P, Moser B, Gedeon T D (2002) Stability of interpolative fuzzy KH-controllers. Fuzzy Sets and Systems, 125(1): 105–119. Vass G, Kalm´ ar L, K´ oczy L T (1992) Extension of the fuzzy rule interpolation method. In: Proc. Int. Conf. Fuzzy Sets Theory Applications (FSTA ’92), Liptovsky M., Czechoslovakia: 1–6 Wong K W, Gedeon T D, and Tikk D (2000) An improved multidimensional α-cut based fuzzy interpolation technique. In: Proc. Int. Conf Artificial Intelligence in Science and Technology (AISAT’2000), Hobart, Australia: 29–32 Yam Y, K´ oczy L T (1997) Representing membership functions as points in high dimensional spaces for fuzzy interpolation and extrapolation. In: Technical Report CUHK-MAE-97-03, Dept. Mech. Automat. Eng., The Chinese Univ. Hong Kong, Hong Kong Yan S, Mizumoto M, Qiao W Z (1996) An Improvement to K´ oczy and Hirota’s Interpolative Reasoning in Sparse Fuzzy Rule Bases. In: International Journal of Approximate Reasoning, Vol. 15: 185-201

Fuzzy Rule Interpolation Based on Polar Cuts - Semantic Scholar

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