8/13/2019 00839119

1/5

SVD Based Reduction for SubdividedRule BasesPeter Barany?, Yeung Yamb,Chi-Tin Yangband Annamiria Virkonyi-K6czyckesemch Group for Mechanics of the Hungarian Academy of Science andDept. Telecommunication and Telematics, Technical University ofBudapest, Sztoczek u.2., Budapest, H-I1I 1 ,

(baranyi8 lektro,get,bme.hu)bDeparttnentof Mechanical and Automation Engineering, Chinesc University of H a ng Ko ng , S h a h , N.T., Hong Kong,yyamdmae.cuhk.edu.hk)

Dept. M easurement and Information Systems, Technical University of Budapest, Hungary, koczy @mic.bme.hu)

Abstarct-Thls p a p e r is motivated by t h e fact that thoughfuzzy and B-spline techniques are popu l ar engineeringtools, their utilisation is being restricted by theirexponential complexity property. As a resul t SVD basedreduction techniques have emerged. These methods applysingular value decomposition to the characteristicmatrixof the ru le base. The m a x i m u m size o f the rule base takeninto considerat ion is limitcd by size o f operat ion memorynvailablc for singular value decomposition.The method proposed in this paper is capable ofapplying singular value decomposit ion step by s t ep to thepartitions of the rule base, Therefore, usidg the proposedextension, there i s no limit, theoretically, for the size of therulc bases.1 INTRODUCHON

The advantage of fuzzy control lies in its ability to mimicand implement the actions of expert operator(s) without theneed of accuratc mathematical models, The drawback,however, i s that there is no standardised framework regardingthe desig n, optimality, redu cibility, and partitioning of a fuzzyrule set. A fuzzy rule base, b e it generated from expertoperators or by some learning or identification schemes, maycontain redundant, weakly contributing, or outrightinconsistetit components. Moreover, in pursuit of goodapproximation, onc may be tempted to overly assign thenumber of antecedent sets thereby resulting in large fuzzyrule bases and much problems in computation time andstorage space. A formal approach to extracting the morepcrtinent elements of a givcn rule set is, hencc, highlyde sirab1e.The quest to minimise the complexity of fuzzy ule-basesfor reduced computation time and the number of rules is notnew, n 1974,Mumdani the first to modify the CRI algorithmto obtain considerable calculation time reduction. Hisinference algorithm is based on the cylindrical approximationof fuzz y relations. O ther algorithms, such ns Sugeno-. i uhgi-Sugeno-, h r s e n - , Product-Sum-Gravity, etc., arcsubsequently developed following the same key idea, bysimply operating the fuzzy sets along cach input dimension.Howevcr, they still sulker from the exponential calculation-thne and storage-spacc complexity that has been

demonstrated by Kdczy and Hiruaa through the uniformcotnplexity expression of [ l ] , namely, the number o f rulesgrows exponentially with the number of antecedent fizzyterms and the numbcr of input variables ( Kd c z y and Hirota[lo)).This is the reason why the number of inputs i s usuallynot more than five i n fuzzy applications, In this regards,Muser, Klement and Tikk have shown that most of the fiizzyalgorithms do not have universal approximation property ifthe number of antecedeni sets j s limited (Klement, Moser [31,Tikk [4]),A fuzzy rule-base dcsign, hence, has two imporlanlobjectives. One is to achicvc a good approximation. The otheris to reduce the number O rules. The main difficulty is thatthese two objectives are contradictory. Complexity reductionis therefore becoming a pertinent research topic of fuzzytheory.Reduction methods can be classified according to lheirunderlying concept. One group uses a new or modificdinference algorithm to induce reduction. For cxamplc.inference paradigms based on uniform complexity-expressioiiof Kdczy and Hirota 111. a-cut mapping method of Stoiccr[4] , and the minimum-distance-inference method of I M andBlen [SIcan be viewed as belonging to this group. The othergroup contains rulc-basc reduction methods, which workswith the same inference algorithm, but reduces the size of h egivcn rule base. This includes the reduction method for inputvariables by Titli et nl. [GI Sugeno's hierarchical rule-structure [71, and singular valuc bascd (SVD) rcductionmethods (8.9,13,14.15,25]. SVD has also been applied byW m g et al. [241 to reduce the dimensionality nf the inputspace, after which the system is reduced via optimisation ofan objective function. It s worth pointing out that fuzzyinterpolation methods can be included in both groups. Onecan find different types of fuzzy interpolation methods[lO,ll~2,201.The present paper focuses on the SVD based reductionmethods of [8,9,13,14,15], which are based on conductingsingular value decomposition of the rule consequents and thengenerating proper linear combinations of the originalmembership functions to fonn new ones fo: rhe reduced sei.The first work is published in 1996 for rule bases usingproduct-sum-gravity inference algorithm, piece-wisc linearantecedents, and csscntially, singleton conscqiicnts [SI.Shortly after the method is extended to non-singleton

0-7803-5877-5/00/ 10.90 oo0 I E I E712

8/13/2019 00839119

2/5

conscqucnts 19,131, Takagi-Sugewo type rule bases I9,14], and applies SVD obtaining the product of two matrices throuyli~lo a spccial fuzzy rule interpolation method [W he method & t-th dimension, reiultsn matrix 4ith si;sc ifs further generalised in [22,25]. O n the other hand,

tcchniques using B-spline face the Same exponentidlygrowing computational and parameter requirement 1161. It ni x a [ , ni 2the size ofand malrices E which size is the Same as

in each dimension except the i-th one, whcrc-turns out that thc polynomial and the rational B-spline -functions have thc same form as the product-sum-gravityinference bascd algorithms [17]. Somc applications can befound Far thesc cases in [22,23].An important advantag e of thc SVD reduction techniquesis thnt thcm is a formal measurc to fi lkring out thc redundantand weakly contributing components, This implies that thedegree of reduction can be applied according to the maximumacceptable error. For various cases, output error boundhctween the original set and the rcduced set is readilyexpressible based on thc sum of discarded singular values.The methods above apply singular valucdecomposition to the characteristic matrix of the rule base.The application of this technique is practically restricted bythe given computer capacity. Namely, the maximum size ofthe decomposition taken into consideration i s limited by thesize of operation memory available for singular valucdecomposition. Th e m ethod proposed in this paper is capableor applying singular value decomposition step by stcp to thepartitions of the original decomposition. Therelorc, using thisproposed cxtcnsion, thcrc is no limit, theoretically, of theoaramcter matrices. =;i = k

where h e size of A r js ai x , Froln one obtaills- -B= [ P i , i 2 i3 in such o way, th t C = layout@, ).

the sizc is instcad. The result of SVDI holds that:

So, method 1 i s or obtaining matrix A i n cslsc ofn = bi i 2 , i 3 s donc ws follows:=

Let S_ = h you l@ , i ) Then applying SVD to S_ yields:S =ADVT = A r c ,= s =

For instance if i = 1 thc elcments of matrix are,hence:

In order to have convcnient notations first the SVDmethod proposcd i n [22,25] is reformulated by the use of twoCiinctions:S i ~ l i 2 + h - I ) f 1 2 = h &andbil ,iz,is

Definition 2: S = layout g,i),where the size of n -is 1x.-.xn,, results in matrix S , c i l , 2+ i3 1 1 9dirnettsional matrix

that is thc two-dimensional layout of lhc i-th dimension ofmatrix E . from n jxqn3) = C,, 3

For instance in t i le case of n=3, namely, E=bp , r , s ] ,nd 111 EXTENSlON OPTHE SVD RASED REDUCTION ALGORITHMi = l :n 1 w l w e3 Suppose that an extremely large multi-dimensional E isgiven and it must be partitioned into more thnn t wo parts ineach dimcnsion, because of the size of the operation memory.

Method 1,Step Let q n l X n * x n 3 ) bc parlitioned into m atrices

7 13

8/13/2019 00839119

3/5

S V D ~ ( M ~ ,j ,

where M . =

In the same way let

let us return to step 1) which results in matrices yz andU .=2,1] one yields matrices i l , i a n d Step 3) If the 11 obtained in step 1 can be stored intothe operation memory then the original SVD mcthod can be

i = 1.3 and E r . n the final steppplied to yield I.J.p+l,i,

==I = i , j , kG i,l,]~ ~ ' ~ * ~ i , m ~ l m ~ P

S V D I ~ ~ ,.

be used, where now =

matrix

is calculated.IV. REDURCTION TECHNIQUE

Let us briefly summarise the use of the SVD reductionused for a general rational form [22].here now M = G which respectively result in= k [ = = i , i , k ]

be constructed. whereStep 2) If FIlcontains linearly dependent vectors i n any

dimensions and it requires larger memory than the avajhblcoperation memory then let us repartition as followingthen return to stcp 1).

qrl,. ,tn (XI 5 1 . .,x,>C k , l l , . , , , r , ~ k ~ l , . ~ ~ , ~ ~ )k =1

wtl ,tHfil; . ,tn 1 9 xn 1

Swapping in l can always be transformed into the following form:

1here g i = gl,i.i, are the perturbation matricesfor swapping the vectors in g i . Having the new ordering inmatrix , where where V i : t[ 2 i and

714

8/13/2019 00839119

4/5

The superscript I f here denotes reduced.Lemma I is given in [9,25]. or brief summary first let USfocus on the nominator of thc rational form.Thc cxtcndcd method yields readily thati 1.. fl nd

r 1

The reduced antecedents can be obtained fromparametersof the reducedmatrices as:

wlicre ctemcnts a j , i , j consist of matrix 2;Fit t ing numerator and denominator

The numerator and denominator of rational form are thesamc type. Therefore the proposed methd can be applied toboth individually. Howcver, this may not result in the samefunctjons p ( x i ) as required in the numerator anddenom inaror.This problem can be alleviated by constructing the n-dimensioaal matrices gk k = I ..m using elements

A ri.

bk,*, ,. .J I and with an additional inatrix Emtl =[ws,, ] I

k = 1 m + 1 (whcre V i : i t [ n i ) bo generated by applyingMethod 1 to d l matrices As a result, the reduced functionp A r . x ; ) AS calculated will be common to both numeratoratid denominator. For the reinaiiiing parameters, the functionsi,i

41, . . , , in x l , - - , x n ) an be formed with elements b l t ,of the matrices E k = l . . .m and the elements- k , n j x . .xn,,W r can be given by clemcnts of the matrixt l , . . , t ,

-r I This yields all necessary parameters for- m + l , ( n ; x + ,,the reduced form.

V, ONCLUSIONThis paper proposcs a practical extension of the SVDbased reduction algorithms for such cases when the data-may, what the use of reduction is needed for, is much largerthan thc operation memory available for singular valuedecomposition. In thesc cases the problem is that the SVDmethod can not be applied for the whole array, which implicsthat the former methods can not be used, The introduccdmethod is capable to perfomi the SVD in mentioned cases aswcll.

AKNOWLODGEMENTThis rcsearch is supported by the Hungarian Ministry ofCulturc and Education MKM) FKFP 0422/1997, andNational Sciencc Research Pound OTKA) TO19671 and

F030056, nd RGC Earmarked Research Grant 413Ri97E.REFERENCES

[l] L.Tk6cl .y and K.Hirotta J h u y inference by compactrules, Proc. of Int. Con8 on FL NN IIZUKA g o ) ,

[2] E.P.Klemcnt, L.T.K6czy and B.Moser, A rc fuzzysystems universal approximators?, h t Jour. GeneralSystems, to appear.[3] D.Tikk, Thc nowhere denseness of Takagj-Sugcno-Kangtype fuzzy controllers containing grerestricted numberof rules, TutruMountainsMaihematical Publications.[4] A,Stoica, Fuzzy Processing Based on Alpha-CutMapping, 5th IFSA, 1493, pp. 1266-1269.

[SI Wonseak Yu and Zeungnam Bien, Design of FuzzyLogic Controller with Inconsistent Rule Base, Acccptcdi n Joranral of intelligent and Fuzzy Systems161 J.Bruinzee1, VLacrose, A.Titli and W.B.Verbruggcn,Real time fuzzy control of complex systems using rulc-base reduction methods, 2th World Aun. Con.WAC96) , Monpellier, France, 1996.[7] MSugcno, M.P.Griffin, A.Bastian, Puzzy hicrarchicalcontrol of an anmanned Helicopter, 5th IFSA WurldCongress, Seoul, 1993. pp. 1262-1265.[8] Ycung Yam: Fuzzy Approximation via Grid PointSampling and Singular Value Decomposition I E E E Yr.

191 Y.Yam, P.Baranyi and Chi Tin Yang Reduction o f FuzzyRuIc Base Via Singular Value Decomposition IEEE Tr.I101 L.T.K6czy and KHirata, Sizc Reductiori byInterpolation in Fuzzy Rule Bases, IEEE Tr. S.M. C.[ l l ] SzKovAcs and L,T,K6czy, The use of the concept ofvague environment in approximate fuzzy reasoning,

Iizuka, Fukuoka,1990, pp. 307-3 0.

S.M.C.Vol. 27,NO , 1997, pp, 933-951.

F. S . V O l 7, NO 1999,pp 120-132.

1997, vol. SMC-27, ~ ~ 1 4 - 2 5 .

715

8/13/2019 00839119

5/5

Fuzzy Set Theory and Applications, Tatru Moun ta i nsMarhetnarical Publications, Math, Inst. SlovakAcademy of S 1997, ~01~12,p,169-281,I121 P.Baranyi, T. D edeoii and L.T. K6czy, A GeneralInterpolation Technique in Fuzzy Rule Bases witharbitrary Membership Functions, IEEE int, Con5S.M.C.Beijing, China, 1996, pp. 510-515.

[ I 31 P.Baranyi and Y.Yam Singular Value-BasedA proximation with Non-Singleton Fuzzy Rule Base,7 lat. Fuzzy Systems Association World CongressIFSA97), Prague. 1997, pp, 127-132.[ 141 P.Raranyi and Y.Yam, Singular Value-BasedApproximation with Takngi-Sugeno Type Fuzzy RulcBase. IEEE In t Cor on Ftczzy Systems FUZZ-IEEE97). BarceIona, Spain, 1997, pp, 265-270.[ i5] P.Baranyi, Y.Yam and L.T.Kdczy, Multi VariablesSingular Value Based Rule Interpolation, ZEEE Int.[ 161 Gcrald Farin Curves and Surfaces for Computer Aided

GeometricDesign Academic press 1997[ 171 M . Mizumoto, I Fuzzy contrnls by product-sum-gravity

method Advanccment of Fuzzy Thory and Systems inChina and Japan, Eds. Liu and Mizumnto, InternationaIAcademic Publishers, cl.l.-c.l.4. 990[ 181T.Takagi and M.Sugeno/Fuzzy dentification of systcmsand its applications to modelling and control, ZEEE Tr.[20] M . P . Kawaguchi and M-Miyakoshi Fuzzy SplincInterpolation in Sparse Fuzzy Rule Bases Proc. of 5h r Cora5 on Soft Conip. and nJ /h t . Systems IIZUKA98, lizuka. Japan, 1998, pp 664-667,[ ] .Baranyi, Y.Yam and Chi Tin Yang SVD Reduction inNumcrjcal Algorithms: Specialised to B-spline and toFurzy Logic concepts 8 int. Fuzzy Systems

Associadion World Cottgress IFSA99) 1999 Taiwan.I231 Chi-Tin Yang, PBaranyi, Y.Yam and Sz.KovAcs , ,FuzzyControl Identification Using SVD Reduction fromInput-output Data in AGV Steering System JointEurofusc-Soft and Intelligent Computing 1999Confercncc EUROFUSE-SIC99), udapest, 1999,pp[ 4]L. Wang, R. angari, and 1.Yen Principalcomponents,B-splines, and fuzzy systems reduction, n Fuzzy Logicfor thc applications to Coinplex systems, W. Chiang andJ. Lce, Eds. Singapure: World Scientific, 1996, pp. 255-259.[25f P.Bnranyi and Y,Yam Fuzzy rule base reduction inFiizzy W- THEN Rules in Computational Intelligence:Tkeoly and Application$ (EMS., D R u m and E.E.Rerrc), (Kluwcr, 2000)

CUH&.M.C. USA, 1997, pp, 1598-1603.

S.M.C. 15, 1 9 8 . 5 , ~ ~ .36-132.

155-161.

716