Training of Kernel Fuzzy Classifiers by Dynamic Cluster Generation
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1 Tranng of Kernel Fuzzy Classfers by Dynamc Cluster Generaton Shgeo Abe Graduate School of Scence and Technology Kobe Unversty Nada, Kobe, Japan Abstract We dscuss kernel fuzzy classfers wth hypersphere regons, whch are defned n the feature space mapped from the nput space. Startng from a hypersphere wth a small radus defned at a datum, we expand the hypersphere f the new datum s wthn the prescrbed dstance from the center. And f not, we defne a new hypersphere. After rule generaton, we resolve overlaps of dfferent classes contractng the hyperspheres. We then defne a truncated concal membershp functon for each hypersphere. We demonstrate the usefulness of the kernel verson of fuzzy classfers wth hypersphere regons wth several benchmark data sets. 1 Introducton In support vector machnes (SVMs) [1], the nput space s mapped nto a hgh dmensonal feature space, and n the space, the optmal hyperplane s determned so that two classes are separated wth the maxmum margn. Accordng to performance comparsons n a wde range of applcatons, support vector machnes have shown to have hgh generalzaton ablty. Inspred by the success of support vector machnes, to mprove generalzaton ablty and classfcaton ablty, conventonal pattern classfcaton technques are extended to ncorporate maxmzng margns and mappng to a feature space. For example, onlne perceptron algorthms, neural networks, and fuzzy systems have ncorporated maxmzng margns [2]. There are numerous conventonal technques that are extended to be used n the hgh-dmensonal feature space, e.g., kernel perceptrons, the k-means clusterng algorthm, the kernel self organzng feature map, kernel dscrmnant analyss, kernel prncpal component analyss, kernel Mahalanobs dstance, and kernel least-squares [2]. One of the problems of support vector machnes s that t s dffcult to analyze the behavor of support vector machnes because the nput space s mapped to the hgh-dmensonal feature space. There are several approaches to vsualze support vector machnes [3, 4, 5]. Another approach s to extend fuzzy classfers to kernel fuzzy classfers that are defned n the feature space [6]. Although fuzzy classfers wth hyperbox regons such as dscussed n [7, 8] are dffcult to extend to the feature space, classfers wth hyperspheres [9, 10, 11] are relatvely easly extended. In ths paper, we defne fuzzy rules n the feature space n the way smlar to fuzzy mn-max classfers [7]. Instead of generatng and contractng hyperboxes, we generate and contract hyperspheres n the feature space. To facltate effcent calculatons n the feature space, we use kernel trcks. Namely, we use kernel functons assocated wth a mappng functon to avod explct treatment of varables n the feature space. Suppose there are tranng data belongng to one of n classes. We scan the tranng data and for a tranng datum wth no assocated hyperspheres we defne the hypersphere at the tranng datum wth a predefned small radus R ε. If there s a hypersphere that ncludes the tranng datum or whose center s wthn the prescrbed maxmum radus R max, we modfy the center and the radus of the hypersphere. If not, we generate the hypersphere at the tranng datum wth radus R ε. After generatng the hyperspheres, we check whether the hyperspheres of dfferent classes overlap. If there s an overlap, we resolve the overlap contractng the hyperspheres. In Secton 2, we dscuss the rule generaton of kernel fuzzy classfers. In Secton 3, we compare the generalzaton performance of the method wth that of other classfers usng two-class and multclass data sets. 2 Dynamc Rule Generaton 2.1 Concept In the frst stage we scan the tranng data and generate hyperspheres of each class wthout resolvng overlaps be-
2 tween classes. Then n the second stage, we resolve overlaps between classes, contractng hyperspheres. We explan the dea of hypersphere generaton usng the two-dmensonal example shown n Fg. 1. In the fgure, assume that Data 1, 2, and 3 belongng to the same class are scanned n ths order. For Datum 1, we generate the crcle wth Datum 1 beng the center and wth radus R ε. Then for Datum 2, we check f the crcle s expandable. Assume that dstance between Data 1 and 2 s shorter than R max. Then we generate the dotted crcle shown n Fgure 1, whose center s at the mddle of data 1 and 2 and whose dameter s the dstance between Data 1 and 2. If Datum 3 s nsde of the dotted crcle, we update the center and the radus. But because Datum 3 s outsde of the crcle, we check f the dstance between the center and the datum s wthn R max. If so, we update the center addng Datum 3 n addton to Datum 1 and 2 and calculate the mnmum radus that ncludes Data 1, 2, and 3. If the crcle s not expandable, we generate the crcle wth Datum 3 beng the center and wth radus R ε. x 2 1 c 3 x 2 0 class class j Fgure 2. Resoluton of overlap 2.2 Rule Generaton In the followng we dscuss the procedure of rule generaton n detal. Let the tranng nputs for class be x j for j =1,..., M, where x j s the jth tranng datum for class and M s the number of data for class. And let g(x) be the mappng functon that maps the nput space nto the feature space. The procedure for generatng hyperspheres for class s as follows. x 1 0 Fgure 1. Generaton of crcles 2 x 1 1. Generate the hypersphere S 1 wth center c 1 = g(x 1 ) and wth radus R 1 = R ε. Set N s =1, X 1 = {1}, and j =2, where N s s the number of generated hyperspheres for class and X 1 s the set of ndces of data for calculatng the center c Check f x j s n a sphere. Namely f there exsts such k (1 k N s ) that satsfes g(x j ) c k R k, (1) After generatng hyperspheres for each class wthout consderng the overlap between classes, we resolve overlaps by contractng hyperspheres. In Fg. 2, two crcles belongng to classes and j are overlapped. We resolve ths overlap, contractng the crcles as shown n the fgure. For each hypersphere we defne a membershp functon for datum x, n whch the degree of membershp s 1 f x s n the sphere and the degree of membershp decreases as x moves away from the hypersphere. Fg. 3 shows an example for a two dmensonal case. The shape of the membershp functon s a truncated concal. Usually, the degree of membershp s defned between 0 and 1, but to avod unclassfable regons, we assume negatve degree of membershp. x j s n hypersphere S k, where c k = 1 X k j X k g(x j ). (2) Here, X k s the number of elements n X k. If there are more than one hypersphere that satsfy (1), we select one whose value on the left-hand sde of (1) s the smallest. Update the center gven by (2) addng x j, update R k, and go to Step 4. Otherwse, fnd hypersphere S k whose center s nearest to x j : k = arg mn g(x j ) c k. (3) k
3 x 2 1. For R j c j c ko < R j + R ko and R ko c j c ko <R j + R ko As shown n Fg. 2, two hyperspheres overlap but each center s outsde of the other hypersphere. We resolve the overlap contractng the hyperspheres by Degree = 1 r = R j + R ko c j c ko, 2 (9) R j R j r, (10) R ko R ko r. (11) 0 Degree = 0 x 1 2. For R j < c j c ko R ko One of the hypersphere centers s nsde of the other hypersphere but the other s not. We resolve the overlap contractng the hyperspheres by Fgure 3. Defnton of a membershp functon 3. Check f the hypersphere S k can be expandable for the ncluson of x j.if g(x j ) c k R max for j X k (4) s satsfed, hypersphere S k s expandable. Then, we set Otherwse, we set X k X k {x j }, (5) R k = max x j c j. j X k (6) N s N s +1, (7) X,N s = {j}, (8) and generate the hypersphere S,N s wth center c,n s = g(x j ) and wth radus R,N s = R ε. 4. If j = M, termnate the algorthm. Otherwse, j j +1and go to Step Overlap Resoluton We can resolve overlaps by contractng hyperspheres or tunng membershp functons, whch are defned n the next secton. But snce we are gong to defne truncated concal membershp functons shown n Fg. 3, we need to resolve overlaps between hyperspheres. The rato of contracton drectly nfluences the generalzaton ablty. But here, we resolve overlaps of hyperspheres S j and S ko ( k) n the followng way: R ko c j c ko R j 2, (12) R j R j 2. (13) 3. For c j c ko <R j R ko Both centers of the hyperspheres are nsde of the other hyperspheres. We resolve the overlap contractng the hyperspheres by R j c j c ko, 2 (14) R ko c j c ko. 2 (15) Contracton of hyperspheres may result n msclassfcaton of the correctly classfed tranng data, and thus may worsen the generalzaton ablty. Ths s avoded by tunng membershp functons. But here, we do not consder tunng. Instead, we optmze the value of R max by cross valdaton. 2.4 Defnton of Membershp Functons We defne the membershp functon of S j for x, m j (x), by { 1 for dj (x) R m j (x) = j, 1 d j (x)+r j for d j (x) >R j, (16) where d j (x) s the dstance between g(x) and c j and s gven by d j (x) = c j g(x). (17) The membershp functon gven by (16) takes negatve value so that any data are classfed nto a defnte class unless they are on the class boundares. Snce the slopes of the membershp functons are the same for all the hyperspheres, the class boundary on the lne segment between c j and c ko
4 s the mddle pont of the lne segment between the two hyperspheres S j and S ko. Ths s a smlar dea of the optmal separatng hyperplane of a support vector machne. But f there are msclassfed data belongng to class or j, we can tune the slopes of membershp functons [12]. But here we do not consder ths. 2.5 Kernel Methods In kernel methods, we treat the varables n the feature space mplctly usng kernels. In our proposed method, we need to calculate c j c ko and g(x) c j usng H(x, x )=g T (x) g(x). Namely, c j c ko 2 = (c j c ko ) T (c j c ko ) = 1 X j 2 H(x, x ) x,x X j g(x) c j 2 = H(x, x) 2 X j 2 H(x, x ) X j X ko x X j,x X ko + 1 X ko 2 H(x, x ), (18) x,x X ko + 1 X j 2 x X j H(x, x ) x,x X j H(x, x ). (19) In our study, we use the followng kernels: 1. lnear kernels: H(x, x )=x t x, 2. polynomal kernels: H(x, x )=(x t x +1) d, where d s a postve nteger, 3. RBF kernels: H(x, x ) = exp( γ x x 2 ), (20) where γ s a postve parameter. 4. Mahalanobs kernels: ( H(x, x ) = exp δ ) m (x x ) T Q 1 (x x ), where δ s a postve parameter and Q s the dagonal covarance matrx calculated usng the tranng data [13]. Because, except for lnear kernels, g(x ) g( x ) (21) holds, we need to calculate the second term n the rght hand sde of (19) for x but the result of the thrd term may be saved to reduce computaton tme. Lkewse, by savng the result of (18), computaton tme s reduced. 3 Performance Evaluaton We compared the generalzaton ablty of the kernel fuzzy classfer and other methods usng two groups of data sets: (1) some of the two-class data sets used n [14, 15] 1 and (2) multclass data sets used n [2, 12]. We used the lnear and Mahalanobs kernels for the kernel fuzzy classfer. Throughout the experments we set R ε =0.01 and selected the value of R max from 0.05 to 1.5 by the ncrement of 0.05 by 10-fold cross valdaton. In addton to ths we set a large value to R max to avod clusterng the class data. For Mahalanobs kernels we need to determne R max and δ. Lke Mahalanobs kernels for support vector machnes [13], we determne the values of R max and δ by lne search. Namely, frst we determne the value of R max wth δ =1by cross valdaton. Then wth the determned value of R max, we determne the value of δ by cross valdaton changng δ from 0.1 to 2 wth the ncrement of Evaluaton for Two-class Problems Table 1 lsts the specfcatons of two-class problems. Each problem has 100 or 20 tranng data sets and ther correspondng test data sets. We determned the optmal values of R max by 10-fold cross valdaton. For comparson, we used the support vector machne (SVM) wth Mahalanobs kernels and the kernel fuzzy classfer wth ellpsodal regons [6]. For the support vector machne, we determned the optmum value of the margn parameter C and δ by lne search. For the kernel fuzzy classfer wth ellpsodal regons (KFC-ER), we used RBF kernels and determned the parameters by 5-fold cross valdaton. We performed cross valdaton of the frst fve tranng data sets, and selected the medan of the best values. Then, for the optmal values, we traned the classfer for 100 or 20 tranng data sets and calculated the average recognton error and the standard devaton for the test data sets. Table 2 lsts the parameter values for the kernel fuzzy classfers determned by cross valdaton. The symbol denotes that the hghest recognton rate for the valdaton data set was obtaned when the class data were not clustered. Except for the banana and mage data sets relatvely large values were selected for R max. Table 3 lsts the average classfcaton errors and the standard devatons wth the ± symbol. The KFC-L and 1
5 Table 3. Average error rates and standard devatons of the test data sets. Data KFC-L KFC KFC-ER SVM Banana 12.6 ± ± ± ±0.5 B. Cancer 31.6 ± ± ± ±4.4 Dabetes 33.2 ± ± ± ±1.7 German 26.5 ± ± ± ±2.1 Heart 18.7 ± ± ± ±3.2 Image 2.8± ± ± ±0.5 Rngnorm 24.4± ± ± ±0.1 Table 1. Benchmark data sets for two-class problems. Data Inputs Tran. Test Sets Banana B. cancer Dabetes German Heart Image Rngnorm Table 2. Parameter values selected by cross valdaton. Data Lnear Mahalanobs R max R max δ Banana B. Cancer Dabetes German Heart 0.1 Image Rngnorm 1.2 KFC columns lst the values for the kernel fuzzy classfer wth lnear kernels and that wth Mahalanobs kernels, respectvely. The best performance n the row s shown n boldface. For the german and mage data sets, the KFC-L performed better than the KFC, but for the other four data sets, the KFC performed better. The KFC-L showed the best performance for the mage data set, but for the rngnorm data set, the KFC-L showed a large error compared to other methods. Table 4 shows the best recognton rates of the rngnorm valdaton data sets. The recognton rates of the KFC-L are very low compared to those of the KFC. Because smaller R max results n overfttng, the large value of R max was selected as optmal. And the large value caused large contracton of hyperspheres. Thus, n such a stuaton we need to optmze the membershp functons as dscussed n [12]. Except for the heart, mage, and rngnorm data sets, KFC or KFC-L performed poorly compared to KFC-ER and SVM. For these data sets to mprove the generalzaton ablty we need to optmze the membershp functons. 3.2 Evaluaton for Multclass Problems As multclass data sets, we used the data sets n [2, 12]. They were the rs data [16, 17], the numeral data for lcense plate recognton [18], the thyrod data [19], 2 the blood cell data [20], and hragana data [12, 21]. Table 5 lsts the specfcatons of the data sets. We used parwse support vector machnes. To resolve unclassfable regons, we used fuzzy support vector machnes wth mnmum operators [2]. Table 6 shows the parameter values determned by cross valdaton. Unlke two-class problems, a relatvely small value was set to R max. 2 ftp://ftp.cs.uc.edu/pub/machne-learnng-databases/
6 Table 4. Cross valdaton results (%) for the rngnorm data set. Data KFC-L KFC R max Rec. R max δ Rec Table 5. Benchmark data sets for mult-class problems. Data Inputs Classes Tran. Test Irs Numeral Thyrod Blood cell Hragana Hragana Hragana Table 6. Parameter values selected by cross valdaton. Data Lnear Mahalanobs R max R max δ Irs Numeral Thyrod Blood cell Hragana Hragana Hragana Table 7 lsts the recognton rates of the test data sets for 6 classfers. The results of the fuzzy mn-max classfer (FMM) and the k-nearest neghbor classfer (k-nn) are from [12]. They showed the best performance for the test data. For k-nn, the best k was selected from 1, 3, 5, and 7. But for other classfers, the parameters were determned by cross valdaton. The SVM used Mahalanobs kernels and the margn parameter and δ were determned by cross valdaton. For the KFC-ER, polynomal kernels were used and the polynomal degree was determned by 5-fold cross valdaton. The performance dfference between the KFC-L and KFC s small. The reason may be that because the multclass data sets are relatvely easly classfed compared to the two-class problems, the mprovement of separablty by mappng to the feature space resulted n overfttng. For the thyrod data, the KFC-L and KFC show poor recognton rates. Ths s the same tendency wth that of the leastsquares support vector machnes [2]. But for other data sets, performance s comparable. 4 Conclusons In ths paper we dscussed kernel fuzzy classfers wth hypersphere regons. We scan the tranng data and f no hyperspheres are defned for the class assocated wth a datum we defne the hypersphere at the tranng datum wth a predefned small radus. If there s a hypersphere that can nclude the tranng data wthn the prescrbed maxmum radus, we expand the hypersphere. If not, we generate the hypersphere at the tranng data wth the small radus. After generatng the hyperspheres, we resolve the overlap contractng the hyperspheres. For some two-class data sets the proposed classfers showed performance nferor to support vector machnes and kernel fuzzy classfers wth ellpsodal regons, but for multclass problems, ther performance was comparable to other methods. To further mprove the generalzaton ablty, we need to tune the membershp functons. References [1] V. N. Vapnk. Statstcal Learnng Theory. John Wley & Sons, New York, [2] S. Abe. Support Vector Machnes for Pattern Classfcaton. Sprnger-Verlag, London, [3] D. Caragea, D. Cook, and V. Honavar. Towards smple, easy-to-understand, yet accurate classfers. In Proceedngs of the Thrd IEEE Internatonal Conference on Data Mnng (ICDM 2003), pages , Melbourne, FL, 2003.
7 Table 7. Recognton rates of the test data sets. Data KFC-L KFC FMM KFC-ER k-nn SVM Irs Numeral Thyrod Blood cell Hragana Hragana Hragana [4] X. Fu, C.-J. Ong, S. Keerth, G. G. Hung, and L. Goh. Extractng the knowledge embedded n support vector machnes. In Proceedngs of Internatonal Jont Conference on Neural Networks (IJCNN 2004), volume 1, pages , Budapest, Hungary, [5] H. Núnẽz, C. Angulo, and Català. Rule extracton from support vector machnes. In Proceedngs of the Tenth European Symposum on Artfcal Neural Networks (ESANN 2002), pages , Bruges, Belgum, [6] K. Kaeda and S. Abe. KPCA-based tranng of a kernel fuzzy classfer wth ellpsodal regons. Internatonal Journal of Approxmate Reasonng, 37(3): , [7] P. K. Smpson. Fuzzy mn-max neural networks Part 1: Classfcaton. IEEE Transactons on Neural Networks, 3(5): , [8] S. Abe and M.-S. Lan. A method for fuzzy rules extracton drectly from numercal data and ts applcaton to pattern classfcaton. IEEE Transactons on Fuzzy Systems, 3(1):18 28, [9] C.-W. V. Chen and T.-Z. Lu. A compettve learnng process for hypersphercal classfers. Neurocomputng, 17:99 110, [10] P. M. Patl, U. V. Kulkarn, and T. R. Sontakke. Modular fuzzy hypersphere neural network. In Proceedngs of the Twelfth IEEE Internatonal Conference on Fuzzy Systems, volume 1, pages , [11] G. C. Anagnostopoulos and M. Georgopoulos. Hypersphere ART and ARTMAP for unsupervsed and supervsed, ncremental learnng. In Proceedngs of the IEEE-INNS-ENNS Internatonal Jont Conference on Neural Networks (IJCNN 2000), volume 6, pages 59 64, Como, Italy. [12] S. Abe. Pattern Classfcaton: Neuro-Fuzzy Methods and Ther Comparson. Sprnger-Verlag, London, [13] S. Abe. Tranng of support vector machnes wth mahalanobs kernels. In W. Duch, J. Kacprzyk, E. Oja, and S. Zadrozny, edtors, Artfcal Neural Networks: Formal Models and Ther Applcatons (ICANN 2005) Proceedngs of Internatonal Conference, Warsaw, Poland, pages Sprnger- Verlag, Berln, Germany, [14] G. Rätsch, T. Onoda, and K.-R. Müller. Soft margns for AdaBoost. Machne Learnng, 42(3): , [15] K.-R. Müller, S. Mka, G. Rätsch, K. Tsuda, and B. Schölkopf. An ntroducton to kernel-based learnng algorthms. IEEE Transactons on Neural Networks, 12(2): , [16] R. A. Fsher. The use of multple measurements n taxonomc problems. Annals of Eugencs, 7: , [17] J. C. Bezdek, J. M. Keller, R. Krshnapuram, L. I. Kuncheva, and N. R. Pal. Wll the real rs data please stand up? IEEE Transactons on Fuzzy Systems, 7(3): , [18] H. Takenaga, S. Abe, M. Takatoo, M. Kayama, T. Ktamura, and Y. Okuyama. Input layer optmzaton of neural networks by senstvty analyss and ts applcaton to recognton of numerals. Electrcal Engneerng n Japan, 111(4): , 1991.
8 [19] S. M. Wess and I. Kapouleas. An emprcal comparson of pattern recognton, neural nets, and machne learnng classfcaton methods. In Proceedngs of the Eleventh Internatonal Jont Conference on Artfcal Intellgence, pages , Detrot, [20] A. Hashzume, J. Motoke, and R. Yabe. Fully automated blood cell dfferental system and ts applcaton. In Proceedngs of the IUPAC Thrd Internatonal Congress on Automaton and New Technology n the Clncal Laboratory, pages , Kobe, Japan, [21] M.-S. Lan, H. Takenaga, and S. Abe. Character recognton usng fuzzy rules extracted from data. In Proceedngs of the Thrd IEEE Internatonal Conference on Fuzzy Systems, volume 1, pages , Orlando, 1994.
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