The Collapsing Method of Defuzzification for Discretised Interval Type-2 Fuzzy Sets

Transcription

1 The Collapsing Method of Defuzzification for Discretised Interval Tpe- Fuzz Sets Sarah Greenfield, Francisco Chiclana, Robert John, Simon Coupland Centre for Computational Intelligence De Montfort niversit Leicester LE 9BH, K {sarahg, chiclana, rij, simonc}@dmuacuk Abstract Defuzzification of tpe- fuzz sets is a computationall intense problem This paper proposes a new approach for defuzzification of interval tpe- fuzz sets The collapsing method converts an interval tpe- fuzz set into a representative embedded set RES which, being a tpe- set, can then be defuzzified straightforwardl The novel Representative Embedded Set Theorem REST, with which the method is inetricabl linked, is epounded, stated and proved within this paper Additionall the Pseudo Representative Embedded Set PRES, a useful and easil calculated approimation to the RES, is discussed Introduction Tpe- Fuzz Inferencing Sstems Tpe- fuzz sets were originall proposed b Zadeh [6]; their advantage over tpe- fuzz sets is their abilit to model second-order uncertainties A fuzz inferencing sstem FIS is a computerised aid to decision making, which uses fuzz sets It works b appling fuzz logic operators to common-sense linguistic rules An FIS can be of an tpe; here we are concerned with the tpe- FIS, ie one which emplos tpe- sets An FIS of an tpe usuall starts with a crisp number, and passes through the five stages of fuzzification, antecedent computation, implication, aggregation/combination of consequents, and defuzzification When implementing a tpe- FIS researchers discretise the membership functions This work is concerned onl with discretised tpe- fuzz sets Defuzzification The output of the fourth stage of a tpe- FIS is a tpe- fuzz set, which requires defuzzification to convert it into a crisp number, the answer to the problem presented to the FIS It is this final defuzzification stage that is the focus of this paper The defuzzification techniques available for discretised tpe- sets are: Full Defuzzification In a conventional generalised tpe- FIS, the defuzzification stage consists of two parts, tpe-reduction and defuzzification proper Tpe-reduction [5], pages 48-5 in which the tpe- fuzz set is converted to a tpe- fuzz set, the tpe-reduced set TRS, involves the processing of all the embedded sets [5], definition 3-, page 98 within the original discretised tpe- set These sets are ver numerous For instance, when a prototpe tpe- FIS performed an inference using sets which had been discretised into 5 slices across both the and -aes, the number of embedded sets in the aggregated set was calculated to be in the order of 9 63 Though individuall easil processed, embedded sets under full defuzzification give rise to a processing bottleneck simpl b virtue of their high cardinalit Sampling Method The sampling method of defuzzification [] is an efficient, cut-down alternative to full defuzzification In this technique, onl a relativel small random sample of the totalit of embedded sets is processed The resultant defuzzified value, though surprisingl accurate, is nonetheless an approimation The Karnik-Mendel Iterative Procedure This procedure [] is an efficient method of defuzzification for interval tpe- sets but produces onl a ver good approimation to full defuzzification since it works b finding the mid-point of the TRS interval without taking account of the distribution of the values along the interval Table contrasts these three methods, comparing them in relation to efficienc, eactness and applicabl None of these methods is both efficient and eact

2 Table : Comparison of Full, Sampling, and Iterative Methods Method Efficienc Eactness sage Full poor perfect generalised Sampling ecellent approimate generalised Iterative ecellent approimate interval 3 Reversal of Blurring Mendel and John [4], page 8 describe the transformation from a tpe- to a tpe- membership function b a process of blurring: Imagine blurring the tpe- membership function b shifting the points either to the left or the right, and not necessaril b the same amounts, Then, at a specific value of, sa, there no longer is a single value for the membership function u ; instead the membership function takes on values wherever the vertical line [ = ] intersects the blur These values need not all be weighted the same; hence, we can assign an amplitude distribution to all of these points Doing this for all X, we create a three-dimensional membership function a tpe- membership function that characterizes a tpe- fuzz set The collapsing method of defuzzification is a response to the challenge of reversing blurring Further motivation for this research was to devise an efficient and eact tpe- defuzzification technique 4 Embedded Sets An interesting feature of the FIS is that embedded sets onl appear during the final defuzzification stage Theoreticall the could be emploed in the earlier stages, but to do so would be impractical A defuzzification technique that reversed blurring would obviate the need for processing embedded sets, which would be an enormous practical advantage, as well as being satisfing from a theoretical perspective Though the collapsing algorithm does not require processing of embedded sets, it involves the creation, and subsequent defuzzification, of a single tpe- set which is embedded Moreover, the embedded set concept is used in the proof of the theorem section upon which the method is based 5 Overview Our research sees developing generalised tpe- sstems as a challenge for the research communit However in this communit the interval case is often eplored before the generalised case This paper reports the first results on the interval case with a view to future etension of the research to the generalised case We propose a straightforward, eact, iterative defuzzification technique for a discretised interval tpe- fuzz set 6 Preliminaries The discussion in this article relies on certain assumptions and definitions Discretisation Technique It is assumed that the domain is discretised into an arbitrar number m of vertical slices, and the codomain into slices, which are the end-points of the secondar domain Defuzzification Method The analsis presented in this paper presupposes that the centroid method of defuzzification is adopted Scalar Cardinalit The concept of scalar cardinalit is frequentl encountered in the following analsis For tpe- fuzz sets, Klir and Folger [3], p7 define scalar cardinalit as follows: The scalar cardinalit of a fuzz set A defined on a finite universal set X is the summation of the membership grades of all the elements of X in A Thus, A = µ A X The smbol as used here represents sum To distinguish scalar cardinalit from cardinalit in the classical sense of the number of members of a set, we adopt the smbol for scalar cardinalit, ie A represents the scalar cardinalit of A The Representative Embedded Set In this section we introduce the idea of the representative embedded set, a concept which fundamentall underpins the collapsing method of defuzzification that is the subject of this paper An interval tpe- fuzz set is a tpe- fuzz set in which ever secondar membership grade takes the value Such a set is completel specified b its footprint of uncertaint [4], definition 5, page 9, since all its secondar grades are b definition

3 equal to In the analsis which follows, to speak in terms of the footprint of uncertaint FO of an interval tpe- fuzz set is equivalent to referring to the interval set itself The Collapsing Technique It is helpful to think of the interval FO as a blurred tpe- membership function [4], page 8 The collapsing technique is the reversal of this process to create a tpe- fuzz set from an interval FO The tpe- set s membership function is derived so that its defuzzified value is equal to that of the interval set It is a simple matter to defuzzif a tpe- set, and to do so would be to find the defuzzified value of the original interval set Hence the collapsing process reduces the computational compleit of interval tpe- defuzzification We term this special tpe- set the representative embedded set It is a representative set because it has the same defuzzified value as the original interval tpe- fuzz set It is an embedded set because it lies within the FO of the interval tpe- fuzz set Before defining the representative embedded set, we first define the representative set Definition Representative Set A tpe- fuzz set is a representative set RS of an interval tpe- fuzz set if it has the same defuzzified value as the interval set Definition Representative Embedded Set Let F be an interval tpe- fuzz set with defuzzified value X F Then tpe- fuzz set R is a representative embedded set RES of F if its defuzzified value X R is equal to that of F, ie X R = X F, and its membership function lies within the FO of F Clearl an RS ma be embedded, ie an RES, or non-embedded 3 Solitar Collapsed Slice Lemma SCSL In this section we concentrate on the derivation of an RES in the special case of an interval FO formed b upwardl blurring the membership function of a tpe- fuzz set at a single domain value I figure, to create a vertical slice which is an interval as opposed to a point This interval is the secondar domain at I The FO formed b this blurring is depicted in figure, and consists of the shaded triangular region plus the line L We derive a formula for the RES of this somewhat unusual interval FO in terms of the original tpe- membership function and the amount of blurring Tpe- Set with a Single Blurred Grade Let A be a tpe- fuzz set that has been discretised into m vertical slices We calculate the defuzzified value of A, X A, b finding the centroid of A: X A = i=m i= µ A i i i=m i= µ A i = i=m i= µ A i i A To avoid division b, it is assumed throughout this paper that A > bi B A Derivation of an RES We know that an RES will lie within the FO of the interval tpe- fuzz set it represents The objective of this analsis is to derive an epression for the membership function of an RES in terms of the upper and lower membership functions of the interval set to be defuzzified Our strateg is two-stage: We derive a formula for the special case of the interval FO which has onl one blurred vertical slice We generalise this formula to the tpical interval FO in which ever point of the original tpe- membership function has been blurred, ie one for which the upper membership function is greater than the lower membership function for ever domain value I Figure : At = I the membership function of tpe- fuzz set A has been blurred, increasing the membership grade b the amount b I, creating a new tpe- fuzz set B Now suppose the membership function of A is blurred upwards at domain value I, so that I, instead of corresponding to the point µ A I, corresponds to the co-domain range [µ A I,µ A I + b I ] Let B figure be a tpe- fuzz set whose membership function is the same as that of A apart from

4 are known values ~ FO of F X F = X A + X B = X A + X A + b I I X A A + b I = X A + b I I X A A + b I I Figure : FO of interval tpe- fuzz set F, which consists of the original line of tpe- set A plus the triangular region at the point I, for which µ B I = µ A I + b I X B, the defuzzified value of B, ma be calculated: Tpe- Defuzzification of Set R Let R be an RES of F such that the membership function of R is the same as that of A for all domain values i apart from I At this point the membership function deviates from that of A so that µ R I takes the value µ A I + r I Figure 3 depicts the membership function of R Following the same chain of reasoning as in the derivation of X B, we work out an epression for X R in terms of A, X A, I and r I, where A, X A, I are known values: X R = X A + r I I X A A + r I X B = µ B i i µ B i = µ A i i + b I I µ A i + b I = A X A + b I I A + b I = X A + A X A + b I I A + b I = X A + b I I X A A + b I X A A R ri B Interval Set with One Blurred Slice Let F figure be an interval tpe- fuzz set whose lower membership function is A and upper membership function B The membership function of A is identical to that of B apart from the blurring at the point I which makes µ B I greater than µ A I b the amount b I Full Defuzzification of Set F Full defuzzification requires that all the embedded sets of a tpe- set be processed to form the tpereduced set F contains onl two embedded sets, namel A and B Therefore the tpe-reduced set of F consists of the two pairs of co-ordinates X A, and X B, We find the defuzzified value of F b calculating the mean of X A and X B, ie X A + X B Let X F be the defuzzified value of F X F will be epressed in terms of A, X A, I and b I, all of which I Figure 3: R, the representative embedded set of F, is indicated b the undashed line Equating X R and X F to Derive an Epression for r I The defuzzified values X R and X F are b definition equal section, and b equating these values we are able to obtain a formula for r I in terms of A and b I : X R = X F r I = b I A A + b I We have arrived at the membership function of R, and in so doing proved the solitar collapsed slice lemma:

5 Lemma Solitar Collapsed Slice Lemma Let A be a discretised tpe- fuzz set which has been blurred b amount b I at a single point I to form the FO of interval tpe- fuzz set F Then R, the RES of F, has a membership function such that µ R i = µ A i + A b I A + b I if i = I, µ A i otherwise b 4 Representative Embedded Set Theorem REST L We generalise the solitar collapsed slice lemma to the tpical situation in which ever point of the tpe- membership function has been blurred First we present the concept behind the theorem: How an interval tpe- set ma be collapsed to create a representative embedded set We then state the theorem, and go on to prove it inductivel Collapsing an Interval Set to Form an RES We have considered an etremel atpical interval FO whose membership function follows the course of a tpe- fuzz set apart from at one slice I, at which its membership grade opens up into a secondar co-domain [µ I,µ I + b I ] We have done this to provide a simple et illustrative eample of the collapsing process, as a basis for generalisation to the tpical interval FO The SCSL section 3 tells us how to calculate the RES for this special case of an interval tpe- set Now we proceed to look at the tpical interval FO, in which the upper membership grade is greater than the lower membership grade at ever point The difference between the lower and upper membership grades at an given point is the amount of blur b i at that point, ie µ i µ L i = b i The solitar collapsed slice lemma does not appl in this situation However, this lemma ma be applied repeatedl to FOs assembled in stages using slices taken from the interval set Collapsing the st FO to Form RES R The first interval FO to be collapsed figure 4 comprises the slice at =, plus the rest of the lower membership function L, represented b the shaded triangular region plus the line L The lower membership function of the FO is the line L, and the upper membership function starts at = at the line, but immediatel descends to L slice, after which it follows the course of L slices m The SCSL tells us that this interval set ma be collapsed into its RES R, depicted in figure 5 The collapse increases the membership grade µ L b r to µ R Figure 4: The first slice in interval tpe- fuzz set F r Figure 5: The first slice collapsed, creating an RES R for the interval tpe- fuzz set F Collapsing the nd FO to Form RES R We now move on to the second FO Figure 6 shows this FO before it is collapsed The SCSL is reapplied, but instead of the lower membership function being L, it is now R The RES of the second FO is R, which is depicted in figure 7 Collapsing the k + th FO to Form RES R k+ Suppose FOs,,k have been collapsed in turn, with R k being the most recentl formed RES Then it is the turn of the k + th FO to be collapsed The lower membership function is R k This situation prior to the k + th FO s collapse is represented in figure 8; the situation after the collapse in figure 9 Collapsing the m th FO to Form RES R Suppose FOs,,m have been collapsed R

6 Bk+ b R Rk 3 k+ Figure 6: For the interval tpe- fuzz set F, the first slice is collapsed, and the second slice is shown Figure 8: Slices to k collapsed, slice k + about to be collapsed, for interval tpe- fuzz set F rk+ r R Rk+ 3 k+ Figure 7: Slices and collapsed, creating RES R for the interval tpe- fuzz set F Figure 9: Slices to k + collapsed, creating RES R k+ for interval tpe- fuzz set F in turn, with R m being the most recentl formed RES Then it is the turn of the m th FO to be collapsed The lower membership function is R m, and the slice to be collapsed is slice m at = After the collapse the new lower membership function is R m As the m th slice is the final slice, then R m is the RES of the entire interval tpe- fuzz set, and is equivalent to the RES R We now state and prove the representative embedded set theorem Theorem Representative Embedded Set Theorem A discretised tpe- interval fuzz set F, having defuzzified value X F, lower membership function L, and upper membership function, ma be collapsed into a tpe- fuzz set R, known as the repre- sentative embedded set, whose defuzzified value X R is equal to X F, with membership function such that: i j= b i µ R i = µ L i +, i j= where b i = µ i µ L i Proof Let X F be an interval tpe- fuzz set, and R,R m be tpe- fuzz sets formed b collapsing vertical slices,m inclusive of X F, ie R is formed b collapsing slice, R b collapsing slices and, and R i b collapsing slices to i R m is a representative set of X F, equivalent to the R of the previous discussion

7 Proof b induction on the number of collapsing vertical slices will be used: Basis Collapsing the st slice to form R : Figures 4 and 5 depict the collapse of the first slice The resultant RES is R For R, i =, and i = We need to prove that µ R = µ L + L b b, but this is actuall what we have when we appl the solitar collapsed slice lemma section 3 for i = Induction hpothesis: Assume the theorem is true for R k, ie that slices,,k have been collapsed to form tpe- fuzz set R k, and that µ Rk i = µ L i + In this formula, for i > k, b i = i j= i j= b i Induction Step: Now we collapse slice k +, which is a single slice Appling the SCSL section 3 to R k we have: r k+ = R k b k+ R k + b k+ We need to prove that for all i µ Rk+ i = µ L i + i j= i j= b i The proof will be split up into three cases Case : i k µ Rk+ i = µ Rk i Case : i = k + = µ L i + µ Rk+ i = µ L i + r k+ i j= i j= b i = µ L i + R k b i R k We know that R k = = m j= k j= = µ Rk j µ L j + + k j=, and therefore we conclude that µ Rk+ i = µ L i + Case 3: i > k + µ Rk+ i = µ Rk i = µ L i + m j=k+ k j= k j= i j= i j= µ L j b i b i Conclusion: Drawing the three cases together, we conclude that for all i, µ Rk+ i = µ L i + i j= i j= b i 5 Collapsing as Tpe-Reduction Conventional tpe-reduction [5], pages 48-5, whether of interval or generalised tpe- fuzz sets, creates the tpe-reduced set, which is a tpe- set whose domain values are the centroids of all the embedded sets of the original tpe- fuzz set The significant point, however, is that a tpe- fuzz set is produced from a tpe- fuzz set, which is what happens when an interval set is collapsed Both methods of tpe-reduction result in a tpe- fuzz set, which is easil defuzzified However conventional tpe-reduction is in itself so computationall comple that the fact that the process creates an easil defuzzified tpe- set is spurious Moreover, it is far less computationall comple to defuzzif an RES than a TRS, as an RES has far fewer points to process than its equivalent TRS

8 6 Pseudo Representative Embedded Set The SCSL section 3 tells us that: r I = A b I A + b I We have assumed section 3 that A > Therefore, b dividing the numerator and denominator b A, we can deduce that r I = b I + b I A < b I As A increases, r I increases and approaches b I b I can therefore be considered an approimation for r I This approimation makes sense intuitivel The REST section states that: i j= b i µ R i = µ L i + i j= The epression j=i j= replaces A in the SCSL The same line of reasoning that was applied to the blurred single point case applies equall here As each slice is collapsed, L + j=i j= increases, which means that as the collapse progresses, the r i for each collapsed slice is a closer approimation to b i Definition 3 Pseudo Representative Embedded Set For an interval tpe- fuzz set, the pseudo representative embedded set PRES is the tpe- fuzz set whose membership function at ever point takes the mean value of the lower membership function and the upper membership function of the interval set Smbolicall µ P i = µl i + µ i The approimation of the PRES to a RES is equivalent to the approimation involved in equating r i to b i, given that for all vertical slices, r i < b i 7 Conclusions and Further Work The collapsing method of defuzzification generates a tpe- embedded set from an interval tpe- fuzz set that is representative in that it has the same defuzzified value as the original tpe- set The representative embedded set ma be thought of as a tpe-reduced set, but the collapsing form of tpereduction is vastl more efficient than the conventional, full tpe-reduction The collapsing method promises to be efficient and accurate in its implementation Future work will consider the following: Testing the Collapsing Method This method requires testing, to quantif both the time saving it makes possible and confirm its accurac Tests ma be performed for either defuzzification in isolation, or defuzzification as part of an interval tpe- FIS Discretisation with More Than Co-Domain Slices The research presented in this paper concerns the situation where the co-domain is sliced at points for ever domain value, at the lower and upper membership grades for that domain value It would be desirable to etend the collapsing method to cover discretisation in which the co-domain is sliced more than twice Other Methods of Defuzzification There is no obvious reason wh this technique ma not be etended to other methods of defuzzification besides the centroid method Generalised Tpe- Fuzz Sets The technique as described in this paper applies to interval tpe- fuzz sets; we plan to etend it to generalised sets The PRES as a Substitute for an RES It is far easier to calculate the PRES membership function than that of an RES, but of course the defuzzified value of the PRES is onl an approimation It would be useful to investigate both mathematicall and eperimentall the etent to which accurac is lost in replacing an RES b the PRES References [] S Greenfield, R John, S Coupland, A Novel Sampling Method for Tpe- Defuzzification, Proc KCI 5, London, 5, pp 7 [] N N Karnik and J M Mendel, Centroid of a Tpe- Fuzz Set, Information Sciences, vol 3,, pp 95 [3] G J Klir and T A Folger, Fuzz Sets, ncertaint, and Information, Prentice-Hall International, 99 [4] J M Mendel and R I John, Tpe- Fuzz Sets Made Simple, IEEE Transactions on Fuzz Sstems, vol, number,, pp 7 7 [5] J M Mendel, ncertain Rule-Based Fuzz Logic Sstems: Introduction and New Directions, Prentice-Hall PTR, [6] Lotfi A Zadeh, The Concept of a Linguistic Variable and its Application to Approimate Reasoning II, Information Sciences, vol 8, part 4, 975, pp 3 357