INCOMPETE DATA IN FUZZY INFERENCE SYSTEM

Transcription

1 XI Conference "Medcal Informatcs & Technologes" - 26 mssng data, ncomplete nformaton fuzz rules, fuzz operators, classfer Slwa POŚPIECH-KURKOWSKA * INCOMPETE DATA IN FUZZY INFERENCE SYSTEM The paper descrbes a method for dealng the ncomplete data n a Mamdan fuzz sstem and the nfluence of nference nterpretaton on an effcenc the fuzz sstem operatng on ncomplete data. Two representatons of mssng nformaton are dscussed theoretcall and then presented on an example of rs data. Varous nterpretatons of fuzz rules and varous tpes of membershp functon are examned n order to fnd a soluton of fuzz sstem that s more robust to mssng data. 1. INTRODUCTION In real world we often encounter wth nput data vectors where some features are mssng. Reasons can be varous. In medcne some examnatons can be rsk for the patent, or too expensve, or a patent can delberatel wthhold some nformaton n questonnare. Such ncompleteness of nput nformaton s a problem a human beng usuall deals wth ver effcentl. On the contrar a fuzz nference sstem where the knowledge s represented b rules (and the majort of the automatc reasonng methods), t s unable to produce a result, f the nput data does not match a predefned vector of attrbutes. Shall we elaborate a specal set of rules for ever possble case of mssng data? It would be mpractcal, and n sstems wth man nputs even unachevable for explodng complext. Rather we should make a fuzz sstem wth a gven rule base able to process ncomplete nformaton. Another queston s how to choose among man nterpretatons of fuzz nference process n order to desgn a sstem that s the most robust to partall mssng data. The problem of dealng wth mssng nformaton has alread drawn the attenton of researchers. Common probablstc soluton s to replace the mssng value wth ts estmate obtaned from the condtonal probablt dstrbuton gven the known features. In a fuzz settng Berthold and Huber [1] descrbe a method for tranng a classfer wth ncomplete data. In normal operaton of the classfer a mssng value on nput s assgned a degree of membershp equal to one. Gabrs [4] presents a generalzed fuzz mn-max neural network that can deal wth ncomplete nformaton both durng the desgn and the normal operaton. The mssng values are represented as real valued ntervals spannng the whole range of possble values. The result of the classfcaton n * Zakład Elektronk Bomedcznej, Inst. Elektronk, Poltechnka Śląska ul. Akademcka 16, Glwce, e-mal: slwa.pospech@polsl.pl

2 Slwa Pośpech-Kurkowska / XI Conference "Medcal Informatcs & Technologes" ths stuaton s rather a reduced number of alternatve classfcatons than one class. Descrpton of a dagnoss support sstem used n varous felds of medcne mentons that a user s allowed to specf some elements as unknown [2]. In the sstem for assessment of skeletal development of chldren [5] the ablt to deal wth ncomplete data s acheved b dvdng the classfer nto subclassfers that can be actvated ndependentl. 2. PROCESSING OF INCOMPLETE DATA IN THEORY 2.1. FUZZY INFERENCE SYSTEM Knowledge n a fuzz nference sstem s represented b rules. Each rule n a Mamdan sstem wth a multple nput and a sngle output has a form: Where x 1,,, x k values of nput varables A 1, A 2,, A k fuzz sets for nput varables - value of output varable B - fuzz sets of output varable x 1 s A 1 s A 2 x k s A k s B (1) Fuzz set A s descrbed b ts membershp functon µ A wth values n the range [, 1]. Statement x k s A k s nterpreted as supremum of ntersecton of the fuzz sets x k s A k. Connectve (and) can be nterpreted b mnmum operator or b product operator. Value obtaned from evaluaton of the premse s called frng level of the rule. Implcaton s nterpreted b mnmum operator or b product operator. If there are several rules, then the nference result s obtaned b aggregaton of outputs of all rules. Aggregaton s usuall performed b maxmum operator or probablstc or (probor). Whenever a crsp output s necessar, a defuzzfcaton s appled. 2.2 REPRESENTATION OF MISSING DATA Consder a fuzz nference sstem wth n nput varables. Assume that a datum for k-th nput s mssng. How to execute the nference process n ths case? Note that the onl problem that needs a specal soluton s evaluaton of premses where k-th varable s mssng. Other effects for the nference process are caused b a wa a modfed frng level of the rule projects on the concluson and the aggregaton of rules. Skp approach: f the k-th nput s mssng, then smpl skp the statement x k s A k n the premse of the rule. Frng level of the rule n ths case s calculated as: sup( x A ) (2)

3 356 Slwa Pośpech-Kurkowska / XI Conference "Medcal Informatcs & Technologes" - 26 and as from defnton of the membershp functon: so from (2) and (3) t can be concluded: sup( x A ) 1 (3) sup( x A ) sup( x A ) (4) 1 n That means that the frng level of the rule wth unknown k-th nput s alwas greater or equal to the frng level of the rule wth all known nputs. Statement (4) s true ndependent of the connectve nterpretaton, however wth product as, frng of the rule s more lkel to be hgher for mssng data then for complete. Note that skp representaton s equvalent to assumng a degree of 1 for the statement x k s A k, whch s smlar to approach proposed n [1]. Null approach: f the k-th nput s mssng, then substtute t b a null fuzz set representng our unknowledge. The null fuzz set for the k-th nput varable has a membershp functon equal to constant c =.5 over the entre doman of the k-th varable. In ths case the premse s evaluated: sup( x A ) = sup( x ) n 1 n A sup( xk Ak ) (5) 1 And as from defnton of the membershp functon If connectve s nterpreted b mnmum, then we get: sup( x A ) c (6) mn mn ( sup( x A )), c c (7) That means the rule wth mssng data s not fred at hgher level then for complete data and the frng level s cut off at c =.5. If connectve s nterpreted b product, then we get: prod prod ( sup( x A )), c = c prod( sup( x A )) That means the frng level of the rule wth mssng data s reduced b a factor of c=.5 n comparson wth the rule for complete data. (7)

4 Slwa Pośpech-Kurkowska / XI Conference "Medcal Informatcs & Technologes" DEALING WITH INCOMPLETE DATA ON EXAMPLE OF IRIS DATA In ths secton we demonstrate how the proposed methods of managng mssng data work and how the nterpretaton of nference nfluences the effcenc of the fuzz classfer. Irs plant data are used for tests. Ths well known data set [2] publshed b R.A. Fsher n 1936 s often appled n tests of classfers. The set counts 15 cases, for each case four features (sepal length, sepal wdth, petal length, petal wdth n cm) and a proper classfcaton nto one class (Irs verscolor, rs setosa, rs vrgnca) APPLIED FUZZY INFERENCE SYSTEMS Tests have been conducted on fuzz sstems wth varous tpes of membershp functons and dfferent nterpretaton of the nference process. However the basc structure of all sstems s the same. Doman of each nput varable s covered b three fuzz sets representng values tpcal for ever class. Ther membershp functons are generated automatcall on the base of the data. Output space s dvded nto three fuzz sets correspondng to classes. The same tpe of the membershp functon s used as for nputs. Parameters of the membershp functons are shown n the tab.1. Smbol Tpe of mf Varable Class 1 Class 2 Class 3 Gaus Gaussan x Tr1 Trangular x Tr2 Trangular x Trap Trapezodal x Tab.1 Membershp functons (mf) and ther parameters.

5 358 Slwa Pośpech-Kurkowska / XI Conference "Medcal Informatcs & Technologes" - 26 A rule base s the same n ever sstem and conssts of the followng three rules: R1: x 1 s A 1,1 s A 2,1 s A 3,1 s A 4,1 s B 1 R2: x 1 s A 1,2 s A 2,2 s A 3,2 s A 4,2 s B 2 (8) R3: x 1 s A 1,3 s A 2,3 s A 3,3 s A 4,3 s B 3 The frst ndex at A descrbes the nput number, the second descrbes the class, and the ndex at B descrbes the class. Inference process s nterpreted b mnmum, maxmum, product and probablstc or operators as descrbed n p.2.1. Two most popular methods for defuzzfcaton are used: center of area (COA) and mean of maxma (MOM). 3.2 EXAMPLE FOR A SINGLE CASE Ths secton demonstrates n detal how mssng data nfluences the functonng of a fuzz sstem for a sngle case (x 1 = 6.2, = 2.9, =4.3, =1.3). Table 2 shows the evaluaton of each statement x s A,k, and the frng for each rule for complete data and mssng value for. Rule R 1 s not fred, rule R 2 s fred at lower level for null representaton, at equal or slghtl hgher level for skp representaton, rule R 3 s fred at hgher level for both representatons. The change n frng levels s more substantal for prod than for mn. Fg.1 shows the output fuzz sets for rules, the degree of membershp for B 2 (proper class) s smaller and the degree of membershp for B 3 hgher for both representatons of mssng value, that means the fuzzness of the classfcaton ncreases n comparson to the result for complete data, but the defuzzfed values stll pont the proper class Complete data Mssng skp representaton Mssng null representaton x s A,k Frng of R k Frng of R k Frng of R k x s A,k x s A,k Mn Prod Mn Prod Mn Prod Tab.2 The evaluaton of the fuzz sstem tr1 wth the complete data (x 1 =6.2, =2.9, =4.3, =1.3, proper class=2) and mssng value for varable. Two nterpretatons of and are used mnmum or product operator. B 2 B 3 Aggregaton Fg.1 Outputs of rules R2, R3 and aggregaton for complete data (rule R1 s not fred, see tab.2) - sold lne, for mssng : skp representaton - dotted lne, null representaton dashed lne (classfer tr1, mplcaton prod). After MOM defuzzfcaton 2 for all, COA: complete data 2.5, null 2.19, skp 2.18.

6 Slwa Pośpech-Kurkowska / XI Conference "Medcal Informatcs & Technologes" RESULTS Fuzz sstems based on product/probablstc or operators showed to be more senstve to mssng data then sstems based on mnmum/maxmum operators (tab.3). However the deteroraton of the classfer s performance depends strongl on whch varable s mssng. Features and are much more crucal for classfcaton then features x 1 and. Influence of membershp functon tpe s not consderable. The dfference n effcenc of the classfers wth null and skp representaton s not sgnfcant. Ths effect can be caused b the fact each nput varable s present n premses of all rules that s a change n one rule concluson s compensated b smlar changes n the output of the other rules. Results for MOM defuzzfcaton are smlar as for COA., mn, aggregaton: max, prod, aggregaton: Probor Skp Null Skp Null #1 #2 #3 #4 #1 #2 #3 #4 #1 #2 #3 #4 #1 #2 #3 #4 Gaus Tr Tr Trap Tab.3. The ncrease n number of wrong classfcatons respectvel to the same fuzz sstem wth complete nput data. A comparson of null and skp representatons, mn/max vs. prod/probor nterpretaton of nference. 4. CONCLUSIONS A method for dealng wth mssng nput values has been presented that enables a classcal Mamdan fuzz sstem to process ncomplete data. No sgnfcant dfference n overall classfer performance between the two proposed representatons of mssng data was observed n tests for rs plant data. The nference nterpreted b mnmum/maxmum operators has turned out to be more robust wth respect to ncomplete nformaton than the nference based on product/probablstc sum. Future nvestgatons should nclude tests for more vared rule bases and non relatonal nterpretatons of mplcaton. BIBLIOGRAPHY [1] Berthold M.R., Huber K.-P., Mssng values and learnng of fuzz rules, Internatonal Journal of Uncertant, Fuzzness and Knowledge-based Sstems vol.6 (2) (1998) [2] Boegl K., Adlassng K.-P., Haash Y. Rothenfluh T. Letch H., Knowledge acquston n the fuzz knowledge representaton framework of a medcal consultaton sstem, Artfcal Intellgence n Medcne 3 (24) 1-26 [3] Fsher R.A., Irs dataset, [4] Gabrs B., Neuro-fuzz approach to processng nputs wth mssng values n pattern recognton problems, Internatonal Journal of Approxmate Reasonng 3 (22) [5] Pośpech-Kurkowska S., Gertch A. Pętka E., Rozmt sstem do szacowana rozwoju kostnego na podstawe rentgenogramów. Proc. BIB Conf., Bocbernetka InŜnera Bomedczna, 25,