An Empirical Study of Fuzzy Approach with Artificial Neural Network Models

Transcription

1 An Emprcal Study of Fuzzy Approach wth Artfcal Neural Networ Models Memmedaga Memmedl and Ozer Ozdemr Abstract Tme seres forecastng based on fuzzy approach by usng artfcal neural networs s a sgnfcant topc n many scentfc areas nowadays. Artfcal neural networ models are suffcent due to ther abltes to solve nonlnear problems especally fnancal researches n recent years. For these reasons, n ths paper we made a forecastng study for weely closed prces of the exchange rate of Tursh Lras (TL) to Euro between 2005 and 2009 whch has mportant effect n economcal and ndustral areas. We appled the best four networs whch are called multlayer perceptron (MLP), radal bass functon (RBF) neural networ and generalzed regresson neural networ (GRNN) to mprove forecastng fuzzy tme seres wth dfferent degrees of membershp by usng MSE performance measure. Emprcal results show that the MLP outperforms others to forecast neural networ based-fuzzy tme seres. Keywords Artfcal neural networ models, Exchange rate, Forecastng, Fuzzy approach, Tme seres. F I. INTRODUCTION UZZY set theory was ntroduced by Zadeh []. In recent years, fuzzy tme seres approach ntroduced by Song and Chssom[2],[3],[4]. Snce then, many fuzzy tme seres models have been proposed such as frst order models [5],[6],[7], hgh-order models [8],[9], hybrd models[0],[], seasonal models [2],[3], bvarate models [4],[5] and multvarate models[6],[7],[8]. These fuzzy tme seres models have been appled to varous problem domans, such as enrollment [9],[22], temperature [23,[24] and stoc ndex [25],[26]. Neural networs have become popular due to ther abltes to solve nonlnear problems n recent years. So, n ths study, a neural networ based fuzzy tme seres model has been appled to solve nonlnear problems. We made a comparson study to mprove forecastng performance of exchange rate of TL to Euro wth dfferent neural networ archtectures. Forecastng results of all neural networ models ncludng multlayer perceptron, radal bass functon and generalzed regresson neural networs are fnally compared wth each other and multlayer perceptron s the best to forecast fuzzy tme seres accordng to other artfcal neural networ models. To show Prof. Dr. M. Memmedl s wth the Statstcs Department, Faculty of Scence, Anadolu Unversty, Essehr, TURKEY. (e-mal: mmammadov@anadolu.edu.tr). Lecturer O. Ozdemr s wth the Department of Statstcs, Faculty of Scence, Anadolu Unversty, Essehr, TURKEY. (correspondng author to provde phone: ; e-mal: ozerozdemr@anadolu.edu.tr). these thngs, the remander of ths paper s organzed as follows. Secton 2 clearly revews fuzzy tme seres. Secton 3 brefly explans most commonly used artfcal neural networs. Secton 4 ncludes all emprcal analyss. Fnally, secton 5 concludes the paper. II. FUZZY TIME SERIES Song and Chssom frst proposed the defntons of fuzzy tme seres [2],[3]. The some general defntons of fuzzy tme seres are gven as follows: Let U be the unverse of dscourse, where U = { u, u2,..., un}. A fuzzy set A of U s defned by ( ) ( ) ( ) A = f u / u + f u / u f u / u, A A 2 2 A n n where f s the membershp functon of the fuzzy set A A, f : U 0,. u s an element of fuzzy set f u A [ ] s the degree of belongngness of and n. u to A and A ( ) A. f ( u ) [ 0,] Defnton : Y ( t ) ( t =...,0,,2,... ) s a subset of real numbers. Let Y ( t ) be the unverse of dscourse defned by the fuzzy set f ( t ). If F ( t ) conssts of f ( t ) ( =, 2,... ), F ( t ) s defned as a fuzzy tme seres on Y ( t ) ( t =...,0,,2,... ) [2]. Defnton 2: If there exsts a fuzzy relatonshp R( t, t), such that ( ) ( ) (, ) F t = F t R t t, where s an operator, then F ( t ) s sad to be caused by F ( t ). The relatonshp between F ( t ) and F ( t ) can be denoted by ( ) F ( t) F t. Defnton 3: Suppose F ( t ) = A and F ( t) A A =, a Issue, Volume 6, 202 4

2 fuzzy logcal relatonshp s defned as A A, where A s named as left-hand sde of the fuzzy logcal relatonshp and A the rght-hand sde. Note the repeated fuzzy logcal relatonshps are removed [3], [43]. Defnton 4: Fuzzy logcal relatonshps can be further grouped together nto fuzzy logcal relatonshp groups accordng to the same left-hand sdes of the fuzzy logcal relatonshps. For example, there are fuzzy logcal relatonshps wth the A : same left-hand sdes ( ) A A, A A 2 These fuzzy logcal relatonshps can be grouped nto a fuzzy logcal relatonshp group as follows: A A, A,... 2 Defnton 5: Suppose F ( t ) s caused by F ( t ) only, and F ( t) = F ( t ) R( t, t). For any t, f R( t, t) s ndependent of t, then F ( t ) s named a tme-nvarant fuzzy tme seres, otherwse a tme-varant fuzzy tme seres. Song and Chssom appled both tme-nvarant and tmevarant models to forecast the enrollment at the Unversty of Alabama [3],[4]. The tme-nvarant model ncludes the followng steps: () defne the unverse of dscourse and the ntervals, (2) partton the ntervals, (3) defne the fuzzy sets, (4) fuzzfy the data, (5) establsh the fuzzy relatonshps, (6) forecast, (7) defuzzfy the forecastng results. The tme-varant model ncludes the followng steps: () Defne the unverse of dscourse and the ntervals (the same as step () n the tme-nvarant model). (2) Partton the ntervals (the same as step (2) n the tmenvarant model). (3) Defne the fuzzy sets (the same as step (3) n the tmenvarant model). (4) Fuzzfy the data (the same as step (4) n the tmenvarant model). (5) Establsh the fuzzy relatonshps and forecast. (6) Defuzzfy the forecastng results. In both models, note that the establshment of fuzzy R t, t, and defuzzfcaton were the crtcal relatonshps, ( ) steps for forecastng [3],[4],[5],[8],[8], [44]. We used the followng steps n problem soluton: Step. Defnng and parttonng the unverse of dscourse. Step 2. Fuzzfcaton. Step 3. Neural Networ Tranng. Step 4. Neural Networ Forecastng. Step 5. Defuzzfcaton. Step 6. Performance Evaluaton. III. ARTIFICIAL NEURAL NETWORKS In forecastng, artfcal neural networs are mathematcal models that mtate bologcal neural networs. Artfcal neural networs consst of some elements. Determnng the elements of the artfcal neural networs ssue that affect the forecastng performance of artfcal neural networs should be consdered carefully [27],[28]. One of them s networ archtecture. However, there are not general rules for determnng the best archtecture. So, much archtecture should be tred for the correct results. There are varous types of artfcal neural networs. Let gve an overvew of the networs whch s ndcated n the best three networs for the related data of the recent study [20], [2]. Mult Layer Percepteron (MLP): MLP networs are constructed of multple layers of computatonal unts. Each neuron n one layer s drectly connected to the neurons of the subsequent hdden layer. In many applcatons, the frequently used actvaton functon s sgmod functon. Mult-layer networs use a varety of learnng technques, the most popular beng bac-propagaton. Artfcal Neural Networs usually refer to Multlayer Perceptron Neural Networs and are a popular estmator to construct nonlnear models of data. A MLP dstngushes tself by the presence of one or more hdden layers, whose computaton nodes are correspondngly called hdden neurons of hdden unts. For example, a three-layer MLP s gven n (Fg. ). The functon of hdden neurons s to ntervene between the external nput and the networ output n some useful manner [29]. MLP has been appled successfully to dffcult problems by tranng n a supervsed algorthm nown as the error bacpropagaton algorthm. Ths learnng algorthm conssts of two drectons through the dfferent layers of the networ: forward and bacward drectons. In the forward drecton, an nput data s appled to the nput nodes of the networ and ts error propagates through the networ layer by layer. Fnally, a set of outputs s produced as an actual response of the networ. Durng the forward drecton, the synaptc weghts of the networs are not changed whle, durng the bacward drecton, the synaptc weghts are altered n accordance wth an error correcton rule. The defnte response of the output layer s subtracted absolutely from an expected response to produce an error sgnal. Ths error sgnal s then propagated bacward through the networ [2]. Issue, Volume 6, 202 5

3 learn. Tranng the MLP-networ wth the bacpropagaton rule guarantees that a local mnmum of the error surface s found, though ths s not necessarly the global one. In order to speed up the tranng process, a momentum term s often ntroduced nto the update formula [42]: ( ) η δ α ( ) w t + = a + w t (5) p p Fgure. Archtecture of a three-layer MLP Durng the processng n a MLP-networ, actvatons are propagated from nput unts through hdden unts to output a w are unts. At each unt, the weghted nput actvatons summed and a bas parameter net = a w + θ θ s added. () The resultng networ nput net s then passed through a sgmod functon (the logstc functon) n order to restrct the value range of the resultng actvaton a to the nterval [0, ]. = + e (2) a net The networ learns by adaptng the weghts of the connectons between unts, untl the correct output s produced. MLP networs use a varety of learnng technques, the most popular beng bac-propagaton [2]. It performs a gradent descent search on the error surface. The weght update w,.e. the dfference between the old and the new value of weght w, s here defned as: Radal Bass Functon (RBF): That networ type s consstng of an nput layer, a hdden layer of radal unts and an output layer of lnear unts. Typcally, the radal layer has exponental actvaton functons and the output layer a lnear actvaton functons. RBF networ s an alternatve to the more wdely used MLP networ and s less computer tme consumng for networ tranng. RBF networ conssts of three layers: an nput layer, a hdden layer and an output layer. The nodes wthn each layer are fully connected to the prevous layer. The nput varables are each assgned to the nodes n the nput layer and they pass drectly to the hdden layer wthout weghts. The transfer functons of the hdden nodes are RBF [30]. RBF networs are beng used for functon approxmaton, pattern recognton and tme seres predcton problems. Such networs have the unversal approxmaton property [3], are dealt well n the theory of nterpolaton [32] and arse naturally as regularzed solutons of ll-posed problems [33]. Ther smple structure enables learnng n stages, gves a reducton n the tranng tme, and ths has led to the applcaton of such networs to many practcal problems [34]. RBF networs have tradtonally been assocated wth radal functons n a three-layer networ (see Fg. 2) consstng of an nput layer, a hdden layer of radal unts and an output layer of lnear unts [35], [36]. w = ηa δ (3) p p, where ( )( ) ( ) δ ap ap tp ap, f s an output unt δ p = ap ap pw, f s a hdden unt here, t p s the target output vector whch the networ must (4) Fgure 2. Archtecture of a RBF Networ. The radal bass functon determnes the output wth nput Issue, Volume 6, 202 6

4 varable x and dstance from center µ. As the nput varable approaches the center, the output becomes larger. As the radal bass functon, the Gaussan functon s often used. It may be wrtten as f ( x) = exp ( x µ ) 2 2 2σ (6) where x : nput, µ : center, σ : wdth of receptve feld. Output of RBF networ s expressed by a lnear combnaton of the radal bass functons. It may be wrtten as n y = w φ (7) where = functon [45]. w : connecton weght, φ : output of bass Generalzed Regresson Neural Networs (GRNN): That type of networs s a nd of Bayesan networ. GRNN has exactly four layers: nput, a layer of radal centers, a layer of regresson unts, and output. Ths layer must be traned by a clusterng algorthm. Thn of t as a normalzed RBF networ n whch there s a hdden unt centered at every tranng case. A GRNN s based on nonlnear regresson theory and often used as a popular statstcal tool for functon approxmaton. GRNN s one varant of the RBF networ, unle the standard RBF, the weghts of these networs can be calculated analytcally [37]. GRNN was devsed by Specht [38], castng a statstcal method of functon approxmaton nto a neural networ form. The GRNN, le the MLP, s able to approxmate any functonal relatonshp between nputs and outputs [39]. Structurally, the GRNN resembles the MLP. However, unle the MLP, the GRNN does not requre an estmate of the number of hdden unts to be made before tranng can tae place. Furthermore, the GRNN dffers from the classcal MLP n that every weght s replaced by a dstrbuton of weght whch mnmzes the chance of endng up n local mnma. Therefore, no test and verfcaton sets are requred [40]. GRNN has exactly four layers: nput, a layer of radal centers, a layer of regresson unts, and output. Ths layer must be traned by a clusterng algorthm. Thn of t as a normalzed RBF networ n whch there s a hdden unt centered at every tranng case. Fgure 3 shows the general structure of the GRNN [4]. Fgure 3. General structure of the GRNN. IV. EMPIRICAL ANALYSIS Ths study uses weely closed prces of the exchange rate of TL to Euro between 2005 and 2009 as the forecastng target. Emprcal analyss shows preparng a neural networ based fuzzy tme seres model to mprove forecastng performance and show forecastng performance year by year accordng to performance measure called mean square error (MSE). After obtanng all results, we compared the results for all artfcal neural networs by usng MSE from 2005 to 2009 and calculated overall results. We can explan all steps for problem soluton. Step. Defnng and parttonng the unverse of dscourse The unverse of dscourse for observatons, U = startng, endng, s defned. After the length of ntervals, [ ] l, s determned, the U can be parttoned nto equal length ntervals u, u2,..., u, b b =,... and ther correspondng mdponts m, m2,..., m respectvely. b ( ) ( ) ub = startng + b l, startng + b l, startng b l, startng b l mb = We can show dfferent length of ntervals wth startng and endng ponts accordng to exchange rate of TL to Euro for all years n Table. Table. Length of ntervals for all years Start End Interval Length For example, for the year 2005, we have U = [.58,.86] Issue, Volume 6, 202 7

5 and the length of nterval s set to Step 2. Fuzzfcaton Each lngustc observaton, A, can be fuzzfed nto a set of degrees of membershp, = b A µ / u µ / u2... µ / u b Step 3. Neural Networ Tranng A large data set s necessary for tranng a neural networ and we used weely closng prces of the exchange rate of TL to Euro for the years from 2005 to Many studes have used a convenent rato to separate n-samples from out-of samples rangng from 70%:30% to 90%:0%. Hence, we chose the data from January to October for our tranng (nsample) and November and December for forecastng (out- F t = A s taen sample). So the rato s about 83%:7%. ( ) as nput and F ( t) A = s taen as output snce fuzzy logcal relatonshp s defned as A A. Step 4. Neural Networ Forecastng By applyng the process, we use all tranng data for tranng the neural networ, so we can forecast all the degrees of membershp for out of sample data. Step 5. Defuzzfcaton We used weghted averages to defuzzfy the degrees of membershp as follows; forecast( t) = = µ m = t µ t µ t shows the forecasted degrees of membershp and m represents correspondng mdponts. Step 6. Performance Evaluaton We choose MSE for performance measure for performance evaluaton. It s calculated as follows; MSE = n ( forecast( t) observaton( t) ) 2 t= + n n s the number of observatons, ncludng n-sample and n- out-of-sample observatons. We traned and forecasted the data for all years wth MLP, RBF and GRNN. The best results of all are obtaned wth MLP and ts results by all years and overall performance can be seen n Secton 5. V. CONCLUSION Fuzzy approach and artfcal neural networs become effectve tool for researchers by forecastng fuzzy tme seres. The relaton of these has advantage to mprove forecastng performance especally n handlng nonlnear systems. Hence, n ths study we amed to handle a nonlnear problem to apply neural networ-based fuzzy tme seres model. Dfferng from prevous studes, we used varous degrees of membershp n establshng fuzzy relatonshps and we performed dfferent neural networ models to mprove forecastng performance. To demonstrate comparson between these models we used a data set of exchange rate of Tursh Lras (TL) to Euro for the years Emprcal results show that the multlayer perceptron s the best to forecast fuzzy tme seres n most commonly used artfcal neural networ models. Tme seres forecastng by usng artfcal neural networs s an mportant ssue n many scentfc researches n recent years. Artfcal neural networs are suffcent due to ther abltes to solve nonlnear problems nowadays. In ths paper we made a forecastng study for weely closed prces of the exchange rate of TL to Euro between 2005 and 2009 whch has mportant effect n economcal and ndustral areas. We appled the best four networs whch are called MLP, RBF and GRNN to mprove forecastng fuzzy tme seres wth dfferent degrees of membershp by usng MSE performance measure. Frst, by usng the exchange rate of TL to Euro for the years whch s a large data set for tranng a neural networ s used for forecastng target and separated nto nsample (estmaton) and out-of-sample (forecastng). The rato was 83%:7%. Second, we used the best four networs for tranng and forecastng. In Table 2, 3 and 4, we can see the results of MSE for varous artfcal neural networ models after forecastng for fnancal tme seres. In Table 2, all results are shown accordng to years for GRNN best models n all other GRNN models. Table 2. MSE values for neural networ models for GRNN , , , , , Overall In Table 3, all results are shown accordng to years for MLP best models n all other MLP models. Table 3. MSE values for neural networ models for MLP , , , , , Overall Issue, Volume 6, 202 8

6 Accordng to the frst two tables, n general, we can say that MLP models are better results by comparng all years and overall results. In Table 4, all results are shown accordng to years for RBF best models n all other RBF models. Table 4. MSE values for neural networ models for RBF , , , , ,00274 Overall After Table 4 and accordng to overall results n Table 5, we can fnally express that MLP models wthout fuzzy approach for a large data set of exchange rate outperform other model types by yearly comparson and overall results. Table 5. MSE overall values for all neural networ models Models MSE GRNN MLP RBF Now, by usng neural networ-based fuzzy tme seres models, we can obtan Table 6, 7 and 8. So, we can see the results of MSE for tang nto account fuzzy approach by usng varous artfcal neural networ models n proposed method to forecast ths tme seres. In Table 6, we can see the all results accordng to all years for GRNN-based fuzzy tme seres best models n all other GRNN-based fuzzy tme seres models whch are generated by STATISTICA software. Table 6. MSE values for GRNN-based fuzzy tme seres models , , , , ,00264 Overall In Table 7, all results are shown accordng to years for MLP-based fuzzy tme seres best models n all other MLPbased fuzzy tme seres models whch are generated by STATISTICA software. Accordng to the frst two tables, n general, we can say that MLP-based fuzzy tme seres models are better results by comparng all years and overall results accordng to GRNNbased fuzzy tme seres models. In Table 8, all results are shown accordng to years for RBF-based fuzzy tme seres best models n all other RBFbased fuzzy tme seres models. Table 7. MSE values for MLP-based fuzzy tme seres models , , , , ,0050 Overall Table 8. MSE values for RBF-based fuzzy tme seres models , , , , ,00852 Overall After Table 8 and accordng to overall results n Table 9, we can fnally express that MLP-based fuzzy tme seres models wth fuzzy approach for a large data set of weely closed prces of the exchange rate of TL to Euro between 2005 and 2009 outperform other model types wth usng fuzzy approach by yearly comparson and overall results. Table 9. MSE overall values for all neural networ-based fuzzy tme seres models Models MSE F-GRNN F-MLP F-RBF All emprcal results show that the MLP s the best to forecast fuzzy tme seres n most commonly used artfcal neural networ models. Also, MLP-based fuzzy tme seres models outperform artfcal neural networ models wthout fuzzy approach for Table 0. We can see the general results of all model types n Table 0. Table 0. MSE overall values for both neural networ-based fuzzy tme seres models and artfcal neural networ models Models MSE GRNN MLP RBF F-GRNN F-MLP F-RBF In terms of the yearly comparson, the MLP model Issue, Volume 6, 202 9

7 performed better than other models for demonstratng networs by usng STATISTICA 6.0 software. We can see the same results from the overall performance. Average of all years performance of neural networ based fuzzy tme seres by MLP model s better than average of all years performance of the other. We can say that MLP neural networs wthout fuzzy approach are the second and other neural networ-based fuzzy tme seres models are the last for performance evaluatons of forecastng accordng to weely closed prces of the exchange rate of TL to Euro between 2005 and 2009 by usng STATISTICA 6.0 software. Therefore, fuzzy approach wth neural networ based s better by usng ust MLP archtecture. Wthout MLP, artfcal neural networ models can have better results accordng to other neural networbased fuzzy tme seres models such as GRNN or RBF. REFERENCES [] L.A. Zadeh, Fuzzy Sets, Informaton and Control, 8, 965, pp [2] Q. Song and B. S. 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M., 20, A neuro-computatonal ntellgence analyss of the global consumer software pracy rates, Expert Systems wth Applcatons, 38, pp [4] Mostafa, M. M., 200, Forecastng stoc exchange movements usng neural networs: Emprcal evdence from Kuwat. Expert Systems wth Applcatons. [42] M. Mammadov, E. Tas, An Improved Verson of Bacpropagaton Algorthm wth Effectve Dynamc Learnng Rate and Momentum, Proceedngs of the 9 th WSEAS Internatonal Conference on Appled Mathematcs, Istanbul, Turey, May 27-29, 2006, pp [43] M. Memmedl, O. Ozdemr, A Comparson Study of Performance Measures and Length of Intervals n Fuzzy Tme Seres by Neural Networs, Proceedngs of the 8 th WSEAS Internatonal Conference on System Scence and Smulaton n Engneerng, Genova, Italy, October 7-9, 2009, pp [44] M. Memmedl, O. Ozdemr, An Applcaton of Fuzzy Tme Seres to Improve ISE Forecastng, WSEAS Transactons on Mathematcs, Issue, Volume 9, January 200, pp Issue, Volume 6,

8 [45] D. Aydın, M. Mammadov, A Comparatve Study of Hybrd, Neural Networs and Nonparametrc Regresson Models n Tme Seres Predcton, WSEAS Transactons on Mathematcs, Issue 0, Volume 8, October 2009, pp Memmedaga Memmedl was born n Azerbaan n 947. He receved a dploma at Bau State Unversty n 97. He receved a PhD dploma from Russan Academy of Scence n 977. From , he has wored as a Assocate Professor; from 2007, as a Professor Dr. at Department of Statstcs at Anadolu Unversty, Essehr, Turey. Now, he s head of Department of Statstcs, Faculty of Scence, Anadolu Unversty, Essehr, Turey. He has publshed many ournals and conference papers n hs research areas. Hs research nterests are Game Theory, Control Theory, Optmzaton, Artfcal Neural Networs, Fuzzy Modelng and Nonparametrc Regresson Analyss. Ozer Ozdemr Ozer Ozdemr was born n Turey n 982. He receved hs B.Sc., M.Sc. degrees n statstcs n the Department of Statstcs at Anadolu Unversty, Turey, respectvely n 2005 and n He s currently dong hs Ph.D. n the same place. He has wored as a Research Assstant from and as a Lecturer from 2008 n the Department of Statstcs at Anadolu Unversty, Turey. He has publshed over 20 nternatonal conference papers and ournals n hs research areas. Hs research nterests nclude Smulaton, Artfcal Neural Networs, Fuzzy Logc, Fuzzy Modelng, Tme Seres, Computer Programmng, Statstcal Software and Computer Applcatons n Statstcs. Issue, Volume 6, 202 2