System identifications by SIRMs models with linear transformation of input variables

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1 ORIGINAL RESEARCH System dentfcatons by SIRMs models wth lnear transformaton of nput varables Hrofum Myama, Nortaka Shge, Hrom Myama Graduate School of Scence and Engneerng, Kagoshma Unversty, Japan Receved: November, 25 Accepted: March 3, 26 Onlne Publshed: Aprl 6, 26 DOI: 543/arv5n2p55 URL: ABSTRACT Ths paper shows the effectveness of the model proposed n the prevous paper for system dentfcatons In the frst smulaton, whch s for EX-OR, the fundamental dea of the proposed model s explaned In the second smulaton, whch s for classfcaton problems for dataset of Irs, Wne, Sonar and BCW known as benchmark problems, the capablty of the model s evaluated for nvolvng a large number of nput varables In the thrd smulaton, as one of control problems, numercal smulaton for obstacle avodance problem s performed In these smulatons, t s shown that the proposed model outperforms conventonal models n terms of system dentfcatons Key Words: SIRMs models wth LT, Classfcaton problem, Obstacle avodance problem INTRODUCTION There have been many studes on self-tunng fuzzy systems [ 5] Ther am s to construct self-tunng fuzzy systems from learnng data based on the steepest descent method (SDM) The obvous drawbacks of the method are ts large computatonal complexty and gettng stuck n a shallow local mnmum Therefore, neffectve systems wth an nference error and the number of rules are constructed In order to construct effectve systems, some novel methods have been developed whch ) create fuzzy rules one by one startng from a small number of rules, [6] 2) delete fuzzy rules one by one startng from a suffcently large number of rules, [7] 3) use a genetc algorthm to determne the structure of the fuzzy model, [5] 4) use a self-organzaton or a vector quantzaton technque to determne the ntal assgnment of fuzzy rules, [8, 9] 5) use generalzed obectve functons [] However, there are lttle effectve fuzzy nference systems Therefore, the conventonal learnng methods wth mult-obectve fuzzy modelng and fuzzy modelng wth constraned parameters of the ranges have become popular [5] On the other hand, SIRMs (Sngle-Input Rule Modules) model ams to obtan a better soluton by usng fuzzy nference system composed of sngle-nput rule modules [] Although t s easy to apply SIRMs model to the problems wth many nput varables, t s always dffcult to obtan a good performance for nonlnear problems Accordngly, SNIRMs (Small Number of Input Rule Modules) models, whch s a generalzed SIRMs, have been proposed [2 4] However, the capablty of SNIRMs models such as DIRMs (Double-Input Rule Modles) one s stll low compared wth conventonal models Therefore, we proposed SIRMs model wth lnear transformaton (LT) of nput varables and showed ts performance n prevous [5, 6] papers Correspondence: Hrom Myama; Emal: mya@eeekagoshma-uacp; Address: Graduate School of Scence and Engneerng, Kagoshma Unversty, Japan Publshed by Scedu Press 55

2 In ths paper, we nvestgate the performance of the proposed model for varous problems of system dentfcatons In the smulaton of classfcaton problems, we compare the proposed model and the conventonal model n terms of accuracy and the number of parameters Further, t s shown that the proposed model can be appled to an obstacle avodance problem, whch s an applcaton of control problems In the smulaton results, we wll show that the proposed model outperforms conventonal models n terms of accuracy and the number of parameters 2 PRELIMINARY 2 The conventonal model The conventonal model usng the SDM was prevously [, 2, 6] presented Let Z k {,, k} for the postve nteger k Let R denote the set of all real numbers Let x (x,, x m ) and y denote nput and output data, respectvely, where x, y R for Z m Then, for Z n, the -th rule of the conventonal model s expressed as R : f x s M and x m s M m then y s w () where M and w denote a membershp functon and the weght of -th rule, respectvely A membershp value µ for nput x s expressed as follows: m µ M (x ) (2) For Gaussan membershp functon, M s defned as follow: ( M (x ) exp ( ) ) 2 x c (3) 2 where c and b denote the center and wdth parameters, respectvely The output y of fuzzy nference s obtaned as follows: b Then, the followng relaton for Eq(3) hold: µ w n µ (y y r ) (7) µ c n µ (y y r ) (w y ) x c b 2 µ b n µ (y y r ) (w y ) (x c ) 2 b 3 In the followng, Gaussan functon s used as a membershp functon 22 The leanng algorthm A typcal learnng algorthm s ntroduced as the conventonal one [2] Let D {(x p,, xp m, y r p) p Z P } be a gven set of learnng data Fgure shows the learnng algorthm, whch mnmzes the followng obectve functon E P (8) (9) P (yp yp) r 2 () p y µ w µ (4) The obectve functon E, whch s the nference error between the desrable output y r and the output y, s defned as follows: E 2 (y y r ) 2 (5) In order to mnmze the obectve functon E, the procedure based on SDM updates each parameter β {c, b, w } as follows: β(t + ) β(t) K β (6) β where t s the learnng tme and K β s the learnng rate of β, whch s a constant value Fgure The flowchart of the conventonal learnng algorthm In the algorthm, θ and T max are the threshold for nference error and the maxmum number of learnng tme, respectvely 3 SIRMS AND DIRMS MODELS The SIRMs and DIRMs models are ntroduced [2 4] 56 ISSN E-ISSN

3 Each rule of SIRMs model s expressed as SIRM : {R : f x s M then y s w } n SIRM l : {R l : f x l s M l then y l s w} l n SIRM m : {R m : f x m s M m then y m s w m } n A membershp value for nput x and output for the l-th rule are expressed as follows: µ l M l (x l ) () yl µl wl (2) µl Then the output of fuzzy nference s obtaned as follow: m y h l yl (3) l Each parameter for the obectve functon E s updated as follow: h l (y y r )y l (4) w l c l b l h l µ l l µl (y y r ) (5) h l (y y r ) (wl y l ) x c l n µl (b l )2 (6) h l (y y r ) (wl y l ) (x c l )2 n µl (b l )3 (7) where h l, w l, cl and bl are parameters Lkewse, the rule of DIRMs model s expressed as DIRMs-2 {R 2 : f x s M and x 2 s M 2 then y 2 s w 2 } n and output for fuzzy nference system are shown as follows: µ ll2 M l (x l )M l2 (x l 2 ) (8) y l l 2 y µll2 µll2 l l 2 Z 2 m where Z 2 m Z m Z m, and l < l 2 w ll2 (9) h ll 2 y l l 2 (2) Each parameter for the obectve functon E s updated as follow: h ll 2 (y y r )y l l 2 (2) w ll2 c ll2 b ll2 where γ (wl l 2 parameters µ ll2 h ll 2 µll2 (y y r ) (22) h ll 2 (y y r ) γ x c ll2 (23) (b ll2 ) 2 h ll 2 (y y r ) γ (x c ll2 ) 2 (24) (b ll2 ) 3 y l l 2 ) µl l 2, and h ll 2, w ll2, c ll2 and b ll2 are DIRMs-l l 2 {R ll2 : f x l s M l DIRMs-m, m {R m,m where l < l 2 and x l2 s M l2 then y ll 2 s w ll2 } n : f x m s M m and x m s M m then y m,m s w m,m } n In ths case, a membershp value, output for the l l 2 -th rule Fgure 2 The block dagrams of three types of models for m 3 Fgure 2 shows the relaton among conventonal, SIRMs and DIRMs models for m 3 Note that each order of the numbers of rules for three models s O(h m ), O(mh) and O(m 2 h 2 ), respectvely, where h denotes the number of Publshed by Scedu Press 57

4 Artfcal Intellgence Research 26, Vol 5, No 2 parttons n a membershp functon Further, the numbers of parameters for them are (2m + )hm, (3h + )m and (5h2 + )m C2, respectvely Learnng methods for SIRMs and DIRMs models are shown as the same methods as Fgure µ y y M (z ) Pn µ w P n µ l X (26) (27) h y (28) where h for Zl s the weght for the -th module Then, the followng relaton hold: Fgure 3 The proposed model: Input x s transformed nto ntermedate z by the matrx A and SIRMs model wth varable z s formed h µ h Pn w c b Further, ak (y y r )y (29) (y y r ) (3) (w y )µ z c (y y r )h Pn (b )2 µ (3) (w y )µ (z c )2 (y y r )h Pn (32) (b )3 µ µ ak s computed as follows: h (y r y ) Pn n X µ µ (w y ) z c (b )2 (33) Fgure 4 The flowchart of the proposed learnng algorthm The flowchart for learnng algorthm of the proposed model s shown n Fgure 4 4 P ROPOSED FUZZY INFERENCE MODEL The proposed model s shown n Fgure 3[5, 6] The model conssts of two stages The frst stage performs a lnear transformaton (LT) A from nput x nto ntermedate varables z (z,, zl )T as follows: z a z a zl al ak am ak am alk alm x xk (25) xm where A (ak ) for Zl and k Zm {} and z Pm k ak xk for x The second stage s performed by SIRMs model wth z,, zl The output y s calculated as follows: 58 Fgure 5 A schematc explanaton of the proposed model: Three lnes mean the centers of three membershp functons Two nput of (; ) and (; ) are nearly on Lne 2, where and mean output and, respectvely ISSN E-ISSN

5 Artfcal Intellgence Research 26, Vol 5, No 2 In ths case, let θ and Tmax be the threshold for nference problem n the condton or the result s over 25 The MSE error and the maxmum number of learnng tme value s the average value from ten trals In Table, MSE means that the correct output s obtaned for any nput We wll determne the number of ntermedate varables l case, and MSE 25 mean that the correct output s not accordng to the threshold of nference error θ and the maxobtaned for ust 25% nput cases Therefore, the SIRMs mum number of learnng tme Tmax Note that the number model and the DIRMs one for m 3 cannot mplement the of parameters for the proposed model s (m + 3h + 2)l EX-OR problem On the other hand, the proposed model can mplement the EX-OR problem for any m Table The smulaton result for EX-OR problem wth m nput varables m 2 25 The conventonal SIRMs DIRMs The proposed (l m) Fgure 6 Smulaton on obstacle avodance, where r and θ are the dstance and the angle between the agent and the obstacle, and r2 and θ2 are the dstance and the angle between the agent and the destnaton In the followng, the dea of the proposed model s explaned Let us consder the followng fuzzy nference system for l and n 3 obtaned by learnng z 7 5x 64x2 h 93 (35) 5 N UMERICAL SIMULATIONS In ths secton, the effectveness of the proposed model s n R : f z s M then y s 6, vestgated for EX-OR, classfcaton and obstacle avodance SIRM s R : f z s M2 then y s 37, 2 problems R3 : f z s M3 then y s 6, 5 The EX-OR problems The EX-OR problem wth m nput varables s defned by the followng equaton z x x2 xm (34) M M2 M3 (z + )2 exp 2 (z 5)2 exp (z )2 exp 2 (36) (37) (38) where x,, xm, z {, } and s the Exclusve OR operator[2] Let Kc, Kb, Kw, Kh, Ka and Tmax n numercal smulatons be the learnng rate of c, b, w, h and ak and the maxmum number of learnng tme, respectvely Further, ak means each element of the matrx A (See Eq(25)) In ths smulaton, Kc, Kb, Kw 5, Kh 5, Ka and h 3 are used Further, Tmax 5,, 2 and 5 are set for the conventonal SIRMs, DIRMs and proposed methods, respectvely Furthermore, ntal parameters c, b, w, h and ak are as follows: c s set to equal ntervals Fgure 7 Learnng data for smulaton 2h (the doman of nput), and w, h and ak are randomly selected from domans [, ], [, ] and [, ], respectvely The smulaton result s shown n Table, where In Fgure 5, three lnes of Lnes, 2, and 3 mean the centers the symbol - means that t s mpossble to smulate the of three membershp functons for Eqs(36), (37) and (38), Publshed by Scedu Press 59

6 respectvely As two nput of (,) and (,) are nearly on Lne 2, the output s about Further, output of (,) and (,) for Lnes and 3 nearly equal to Therefore, output for (,) and (,) obtaned from all rules nearly equals to Lkewse, we can consder about the cases of (,) and (,) On the other hand, we have already shown that EX-OR problem wth two varables cannot be mplemented by any SIRMs model [6] 52 Classfcaton problems 53 Obstacle avodance problem As an applcaton to control problems, let us show smulatons of an obstacle avodance problem used n the prevous paper [4] See t for the detaled explanaton of the smulatons [4] Untl now, we have already performed some smulatons by the conventonal, SIRMs and DIRMs models As a result, the conventonal and DIRMs models were successful n learnng and test smulatons [4] We wll show that the proposed model s successful wth a small number of parameters For classfcaton problems, the benchmark datasets Irs, Wne, Sonar and BCW from UCI database are used [8] The numbers of data, varables and classes are 5, 4 and 3 for Irs, 78, 3 and 3 for Wne, 28, 6 and 2 for Sonar and 683, 9 and 2 for BCW, respectvely In order to evaluate the model, 5-fold cross-valdaton s used As the ntal condton, K c, K b, K w 5, K h 5, K a, h 3 and T max 5 are used Further, the ntal value of c, b, w, h and a k are as follows: c s set to equal ntervals 2h (the doman of nput), and b, w, h and a k are randomly selected from domans [, ], [, ] and [, ], respectvely Table 2 shows the smulaton result for the classfcaton problems In each box, two numbers from the top show msclassfcaton rate for tranng and test data sets, respectvely, and the bottom number shows the number of parameters The smulaton result s obtaned as the average value from twenty trals Table 2 The smulaton result for classfcaton problems The conventonal SIRMs DIRMs The proposed Irs Wne Sonar BCW 4 55 (729) 2 52 (4) 57 (276) 29 3 (45) (3) 92 (3588) 37 (72) 24 3 (6) - 25 (23) (9) 65 (656) 6 36 (6) In Table 2, the symbol - means that t s mpossble to smulate the problem Table 2 shows that the proposed method s superor to SIRMs and DIRMs models n terms of accuracy and the proposed one s superor to the conventonal model n terms of the number of parameters Fgure 8 Smulaton result for Test As shown n Fgure 6, four parameters are selected as nput varables The problem s to construct a fuzzy system by whch the moble agent avods the obstacle and arrves at the destnaton The moble agent moves wth the vector A (A x, A y ) at each step, where A x s constant and A y s only adusted as an output from the fuzzy system The learnng data wth 2 ponts are obtaned from an examnee Usng the learnng data, fuzzy nference rules are constructed for SIRMs and the proposed models As smulaton 6 ISSN E-ISSN

7 condtons for SIRMs and proposed models, K c, K b, K w 5, K h 5, K a 5, h 3 and T max 5 are selected Intal values for c and b are set to equal ntervals and 2(h ) (the doman of nput), respectvely Intal values for w, h and a k are selected randomly from [, ], [, ] and [, ], respectvely For h 3, the number of parameters of SIRMs s 4 and that of proposed model s 3, respectvely In the smulatons, only when the agent reaches the destnaton unless colldng wth the obstacle, the tral s regarded as a successful one Otherwse, the tral s regarded as a unsuccessful (faled) one There are two types of evaluatons: learnng and test For the learnng evaluaton, the postons of the agent s startng pont, the obstacle and the destnaton are same as the ones of learnng data In the learnng evaluaton, SIRMs fals n the trals but the proposed model s successful For the test evaluaton, one or more of the three postons are dfferent from the ones of learnng data The followng four types of test evaluatons are consdered n the smulatons () Test uses some dfferent agent s startng ponts (see Fgure 8) Fgure 8 shows the movements of the agent from startng ponts (, ), (, 2),, (, 8), (, 9) after learnng The test evaluaton result s shown n Fgure 8 Lke the learnng evaluaton, SIRMs fals n the trals but the proposed model s successful (2) Test 2 assumes the scenaro where the moble agent avods the obstacle placed at a dfferent place and arrves at the dfferent destnaton The proposed model s appled for the obstacle at (4, 4) and the destnaton (, 6) Every tral s successful as shown n Fgure 9 (3) Test 3 assumes the scenaro where the obstacle moves wth the fxed speed As shown n Fgure, every tral s successful Fgure Smulaton result for Test 3 Fgure The movement of the moble obstacle for Test 4 Fgure 2 Smulaton result for Test 4 Fgure 9 Smulaton result for Test 2 (4) Test 4 assumes the scenaro where the obstacle moves randomly wth a random vector B as shown n Fgure, where B s constant and the angle θ b changes randomly at each step As shown n Fgure 2, the proposed model s successful for every tral Publshed by Scedu Press 6

8 In Ref, [4] we have already shown that the smulatons are successful by DIRMs model n almost the same condton In ths paper, t s shown that they are successful by the proposed model wth half of the parameters compared to DIRMs model Lastly, we consder the nterpretaton for fuzzy rules of the proposed model Let us consder fuzzy rules of Table 3 obtaned by learnng Assume that there are three attrbutes small, mddle and large for r, r 2, θ and θ 2, and that there are two attrbutes left (A y > ) and rght (A y < ) for the drecton of A y From ths result, we can obtan the followng fuzzy rules: Z : (r s small) or (θ s large) or (r 2 s large) or (θ 2 s small) A : If Z s small then move to the rght A 2 : If Z s mddle then move to the rght A 3 : If Z s large then move to the left Z 2 : (r s large) or (θ s small) or (r 2 s small) or (θ 2 s large) A 2 : If Z 2 s small then move to the left A 22 : If Z 2 s mddle then move to the left A 23 : If Z 2 s large then move to the rght Let us consder the behavor for two cases on the places of obect (I) The case that the moble agent approaches to the obstacle: In ths case, r s small and r 2 s large are vald () If θ s large, the moble agent move to the left based on A 3 () If θ s not so large, the moble agent move to the rght based on A or A 2 Lkewse, we can also explan about the rule of Z 2 (II) The case that the moble agent approaches to the destnaton: In ths case, r s large and r 2 s small are vald () If θ 2 s large, the moble agent move to the rght based on A 23 () If θ 2 s not so large, the moble agent move to the left based on A 2 or A 22 Lkewse, we can also explan about the rule of Z That s, t s shown that the moble agent moves away from the obstacle when the moble agent approaches to the obstacle and moble agent moves to the drecton of the destnaton when the moble agent approaches to the destnaton 6 CONCLUSIONS In prevous papers, we proposed the SIRMs model wth LT of nput varables In ths paper, we showed the effectveness of the proposed method for system dentfcatons In the second smulaton of classfcaton problems, well-known benchmark datasets Irs, Wne, Sonar and BCW are used Accordng to the smulaton result, t s dffcult for the conventonal model to mplement classfcaton problems wth large number of parameters such as Wne, Sonar and BCW However, the proposed model can mplement them whle keepng hgh accuracy Further, as one of control problems, the smulaton of obstacle avodance problem s performed The smulaton result shows that the proposed model outperforms conventonal models n terms of accuracy and the number of parameters In our future work, we wll consder to refne the proposed model REFERENCES [] Kosko B Neural Networks and Fuzzy Systems: A Dynamcal Systems Approach to Machne Intellgence, Prentce Hall, Englewood Clffs, NJ 992 [2] Gupta MM, Jn L, Homma N Statc and Dynamc Neural Networks IEEE Press 23 [3] Casllas J, Cordon O, Herrera F, et al, Accuracy Improvements n Lngustc Fuzzy Modelng Studes n Fuzzness and Soft Computng 23; 29 [4] Lu B Theory and Practce of Uncertan Programmng Studes n Fuzzness and Soft Computng 29; 239 [5] Cordon O A hstorcal revew of evolutonary learnng methods for Mamdan-type fuzzy rule-based systems: Desgnng nterpretable genetc fuzzy systems Journal of Approxmate Reasonng 2; 52: [6] Arak S, Nomura H, Hayash I, et al A Fuzzy Modelng wth Iteratve Generaton Mechansm of Fuzzy Inference Rules, Journal of Japan Socety for Fuzzy Theory & Systems 992; 4(4): [7] Fukumoto S, Myama H, Kshda K, et al A Destructve Learnng Method of Fuzzy Inference Rules Proc of IEEE on Fuzzy Systems 995: [8] Kshda K, Myama H, Maeda M, et al A Self-tunng Method of Fuzzy Modelng usng Vector Quantzaton Proceedngs of FUZZ- IEEE : [9] Pedrycz W, Izakan H Cluster-Centrc Fuzzy Modelng IEEE Trans on Fuzzy Systems 24; 22(6): /TFUZZ [] Fukumoto S, Myama H Learnng Algorthms wth Regularzaton Crtera for Fuzzy Reasonng Model Journal of Innovatve Computng, Informaton and Control 26; (): [] Y J, Yubazak N, Hrota K A Proposal of SIRMs Dynamcally Connected Fuzzy Inference Model for Plural Input Fuzzy Control Fuzzy Sets and Systems 22; 25: /S65-4()35-4 [2] Shge N, Myama H, Nagamne S A Proposal of Fuzzy Inference Model Composed of Small-Number-of-Input Rule Modules Proc of 62 ISSN E-ISSN

9 Int Symp on Neural Networks: Advances n Neural Networks-Part II 29: 8-26 [3] Mke S, Myama H, Shge N, et al Fuzzy Reasonng Model wth Deleton of Rules Consstng of Small-Number-of-Input Rule Modules Journal of Japan Socety for Fuzzy Theory and Intellgent Informatcs 2: [4] Myama H, Shge N, Myama H Fuzzy Inference Systems Composed of Double-Input Rule Modules for Obstacle Avodance Problems IAENG Internatonal Journal of Computer Scence 24; 4(4): [5] Myama H, Shge N, Myama H A Proposal of SIRMs Model wth Lnear Transformaton of Input Varables Proc Jont 7th Internatonal Conference on Soft Computng and Intellgent Systems and 5th Internatonal Symposum on Advanced Intellgent Systems 24: 3-6 [6] Myama H, Shge N, Myama H SIRMs Fuzzy Inference Model wth Lnear Transformaton of Input Varables and Unversal Approxmaton Advances n Computatonal Intellgence: Proc 3th Internatonal Work Conference on Artfcal Neural Networks, Part I 25: [7] Myama H, Shge N, Myama H Some Propertes on Fuzzy Inference Systems Composed of Small Number of Input Rule Modules Advances n Fuzzy Sets and Systems 25; 2(2): [8] UCI Repostory of Machne Learnng Databases and Doman Theores Publshed by Scedu Press 63