An Evolutionary Method of Neural Network in System Identification

Transcription

1 Internatonal Journal of Intellgent Informaton Systems 206; 5(5): do: 0.648/.s ISSN: (Prnt); ISSN: (Onlne) An Evolutonary Method of Neural Network n System Identfcaton Shumng T. Wang, Ch-Yen Shen, Yu-Ju Chen 2, Chuo-Yean Chang 3, Rey-Chue Hwang, * Department of Electrcal Engneerng, I-Shou Unversty, Kaohsung Cty, Tawan, R.O.C. 2 Department of Informaton Management, Cheng Shu Unversty, Kaohsung Cty, Tawan, R.O.C. 3 Department of Electrcal Engneerng, Cheng Shu Unversty, Kaohsung Cty, Tawan, R.O.C. Emal address: smwang@su.edu.tw (Shumng T. Wang), cyshen@su.edu.tw (Ch-Yen Shen ), ychen@csu.edu.tw (Yu-Ju Chen), cychang@csu.edu.tw (Chuo-Yean Chang), rchwang@su.edu.tw (Rey-Chue Hwang) * Correspondng author To cte ths artcle: Shumng T. Wang, Ch-Yen Shen, Yu-Ju Chen, Chuo-Yean Chang, Rey-Chue Hwang. An Evolutonary Method of Neural Network n System Identfcaton. Internatonal Journal of Intellgent Informaton Systems. Vol. 5, No. 5, 206, pp do: 0.648/.s Receved: September 9, 206; Accepted: October 2, 206; Publshed: October 20, 206 Abstract: Ths paper presents an evolutonary method for calculatng the mportant degree (ID) of ndvdual nput varable of well-traned neural network (NN). The mportance of each nput varable of neural network could be dstngushed n accordance wth ID value obtaned. In ths research, several lnear and nonlnear systems dentfcatons were frstly studed and smulated. From the smulaton results shown, the evolutonary method proposed s qute promsng and accurate for the estmaton of system s parameters. In other worlds, the method proposed could be used for data mnng n the real applcatons. In order to verfy our nference vew, the evaporaton process of thn flm was studed ether. It s a real case of ndustral applcaton. Agan, the studed results show that the method proposed ndeed has the superorty and potental n the area of data mnng. Keywords: Evolutonary, Important Degree, Neural Network, System Identfcaton. Introducton It s well known that system dentfcaton s a method whch can dentfy the mathematcal model of an unknown system from the measurements of system s nputs and outputs. It has been employed nto many areas such as ndustral process, control system, economc forecastng, socal scence and so on. Many classes of system dentfcaton have been studed such as Volterra and Wener seres [], NARMAX models [2] and neural network (NN) [3-7]. Generally, the steps lke model hypothess, estmaton of parameters and system verfcaton are ndspensable for dentfyng an unknown system. Recently, data mnng has been wdely used n the applcatons of sgnal processng and system dentfcaton. It s the process of analyzng data from dfferent perspectves and then extracts the hdden nformaton from a large data base. It can help the researcher to grasp the useful nformaton that mght be gnored and mssed n the process of data analyss. The tasks of data mnng generally nclude four classes whch are classfcaton, clusterng, regresson, and assocaton rule learnng [8-4]. In whch, regresson analyss conssts of the graphc and analytc methods. It ams to obtan the relatonshps between response varable (output) and other predctor varables (nputs) for an unknown system. Its goal s to express the response varable as a functon of the predctor varables. The maor uses of regresson methodology nclude model specfcaton and parameter estmaton [5]. Model specfcaton s a very mportant step n the applcaton of system dentfcaton. Its obectve s the assessment of the relatve values of ndvdual predctor varables on the predcton of the response. Bascally, all relevant varables n the database must be analyzed n order to fnd whch varable has the hgher correlaton wth the response. Thus, the parameter estmaton of the model s necessary to the regresson analyss. Due to the powerful learnng and adaptve capabltes, NN model has been wdely studed n the area of data mnng. Many research artcles have been proposed [6-22]. NN has the ablty to deal wth lots of lnear or nonlnear modelng

2 Internatonal Journal of Intellgent Informaton Systems 206; 5(5): problems. Through the learnng from data provded, NN s able to generate a mappng between nput and output pars bypassng the complcated statstcal analyss steps. In general, the applcatons of NN model n data mnng can be classfed nto four maor categores: () Feed-forward NN, whch s manly used n the areas of predcton and pattern recognton; (2) Feedback NN, whch s manly used for assocatve memory and optmzaton calculaton; (3) Self-organzaton NN, whch s manly used for cluster analyss; (4) Random NN, whch has the advantages of assocatve memory and mage processng [23]. In ths research, the studes of NN versus regresson analyss were studed. The assessment of the relatve values of ndvdual predctor varables (nputs) on the predcton of the response (output) was analyzed based on NN technque. The supervsed learnng feed-forward NN was manly studed. And, the error back-propagaton (BP) learnng algorthm was used as the tranng rule of NN model. The detals of NN structure and ts learnng algorthm are descrbed n Secton 2. Secton 3 presents some relevant experments and results. At last, a concluson s gven n Secton NN Model and Its Learnng Algorthm The NN structure commonly known as mult-layered feed-forward net s used n ths study. A three-layered feed-forward NN model as shown n Fgure s the selected topology. Each layer s connected to a layer above t n a feed-forward manner n the sense that no feed-back from the same layer or a layer above. All connectons have a multplyng weght assocated wth them. Tranng s equvalent to fnd the proper weghts for all connectons such that a desred output s generated for a gven nput set. value of node. Fgure 2. The dagram of sgmod functon. Here, we use a smple three-layered NN model to descrbe the evolutonal method we proposed for calculatng the mportant degree (ID) of ndvdual nput varable of a well-traned NN [24-25]. In Fgure, t can be clearly found that the hdden nodes outputs ( H, H 2,..., Hn ) and network output (Y) can be expressed as the followng nonlnear math forms to the nputs ( x, x2..., xm ). Y = /( + exp( H n n = = /( + exp( ω H + θo )) (2) m = ω x + θ )) (3) Y = /( + exp( ω (/( + exp( ωx + θ ))) + θo)) ω f (v) = m = (4) s the strength of connecton between hdden node and nput node and ω s the strength of connecton between hdden node and output node Y. θo and θ are bas terms. The sgmod functon s an ncreasng functon whch has the output values wthn the range [0, ], thus, the followng relatonshps could be nferred. v H ω x + θ (5) Fgure. A three-layered feed-forward neural network.. In NN model, the nonlnear actvaton functon of all nodes s the sgmod functon. Its math form s expressed as the followng equaton and ts dagram s shown n Fgure 2. f ( v) v = + e Where, v ω, ω s the strength of connecton = x between node and node n the layer below and x () s the Y ω H + θ 0 (6) Y ω ( ω x + θ ) + θ (7) Accordng to the nferences of Eqs. (5), (6) and (7), the mportant degree (ID) and the percentage mportant degree (PID) of nput x to output Y are defned by NT ω k = ID = ( k) ω ( k) x ( k) (8) 0

3 77 Shumng T. Wang et al.: An Evolutonary Method of Neural Network n System Identfcaton PID ID = m = ID (9) mean zero and varance three and they have the values n the range of [-3, 3]. Table lsts the statstcs of (x, x 2, x 3, x 4, x 5 ). where, NT s the total number of nput number (category) of nput varables. 3. Experments and Results 3.. Lnear and Nonlnear Regresson Models and m s the The am of regresson s to express the response varable as a functon of the predctor varables. For lnear regresson, the response varable s supposed to have the lnear relatonshp wth the predctor varables. Denotng the response varable by Y and the m predctor varables by x, x 2,, x m, the lnear relatonshp takes the form Y = a0 + ax + a2x amxm + ε (0) In expresson, a0, a, a2,..., am are unknown parameters needed to be determned. ε s the nose term. Nonlnear regresson s a form of regresson analyss n whch the response varable s modeled by a functon whch s a nonlnear combnaton of the predctor varables. For nstance, a nonlnear functon s gven as 2 x2 Y = a0 + ax + a2 + ε () + exp( x ) In expresson, the relatonshp between Y and (x, x 2 ) are nonlnear. In general, t s very dffcult to obtan an exact closed-form expresson for the unknown model n the nonlnear regresson analyss. The data are usually ftted by a method of successve approxmatons. The present adaptve methods are very successful n lnear regresson. The applcaton of these methods to nonlnear regresson, however, requres several smplfcatons. And, only the approxmate model could be obtaned NN vs. Lnear Regresson Model A lnear regresson model s gven as [24] Y = x 0 + x (2) +.x The ratos of the coeffcents among nput varables are x :x 2 =0:, x :x 3 = 00:, and x 2 :x 3 =0:. That means the degrees of mportance of x to Y s 0 tmes x 2 to Y, and 00 tmes x 3 to Y, f x, x 2 and x 3 have the same statstc dstrbutons. In our studes, Eq. (2) was frstly consdered and 500 ponts were generated from the equaton. Then, we assume ths lnear model s an unknown system whch needs to be dentfed. In system s dentfcaton process, fve varables, x, x 2, x 3, x 4 and x 5 are collected and assumed to be the possble nfluencng factors. Fve varables are generated by unformly dstrbuted random numbers wth 3 x Table. The statstcs of (x, x 2, x 3, x 4, x 5). Varable Range Mean Varance x -3~ x 2-3~ x 3-3~ x 4-3~ x 5-3~ A NN wth sze m-4- was used to model ths lnear equaton. The number of nput nodes (m) of network s based on the varables used. In ths study, 400 ponts were used for NN s tranng and 00 ponts were used for testng. Table 2 lsts the testng mean absolute errors (MAE) of NN modelng wth dfferent nput varables. From the values of MAE, we conclude that NN has the effcent learnng to ths unknown system. Table 2. The MAEs of NN modelng wth dfferent nput varables. NN Sze Input Varables (x,x 2,x 3) (x,x 2,x 3,x 4) (x,x 2,x 3,x 4,x 5) MAE The correspondng ID value of each nput varable x n the well-traned NN models s then calculated. Table 3 s ID and PID values calculated for NN models wth three, four and fve nputs, respectvely. From the table results shown, a phenomenon could be observed. For sze 3-4- NN, the PID rato of (x, x 2, x 3 ) s (:0.0:0.00). For sze 4-4- NN, the PID rato of (x, x 2, x 3, x 4 ) s (:0.3:0.0:0.007). For sze 5-4- NN, the PID rato of (x, x 2, x 3, x 4, x 5 ) s (:0.:0.007: :0.0035). It s clearly found that the PID ratos of (x, x 2, x 3 ) n three NN models are all very close to the rato (:0.:0.0) of the coeffcents of (x, x 2, x 3 ) n the orgnal lnear equaton (2). The ID and PID values of x 4 and x 5 are much smaller than other varables. Ths phenomenon shows that the nonlnear NN model s capable of evaluatng the mportance degree of each nput varable to the output. And, x 4 and x 5 could be treated as the useless terms or noses due to ther low ID and PID values. Table 3. ID and PID values calculated by three NN models. NN Sze x ID (PID) (89.265%) (88.766%) (89.036%) x 2 ID (PID) (9.846%) (0.067%) (9.872%) x 3 ID (PID) (0.889%) (.03%) (0.952%) x 4 ID (PID) (0.53%) (0.02%) x 5 ID (PID) (0.2%)

4 Internatonal Journal of Intellgent Informaton Systems 206; 5(5): In order to evdence the valdty of the method we proposed, 500 ponts were regenerated. x, x 2, x 3, x 4 and x 5 are unformly dstrbuted random numbers wth dfferent ranges. Table 4 lsts the statstcs of (x, x 2, x 3, x 4, x 5 ). Table 4. The statstcs of (x, x 2, x 3, x 4, x 5). Varable Range Mean x -~ x 2-0~ x 3-5~ x 4 -~ x 5-5~ From the statstcs of Table 4, we conclude that the average absolute value of x 2 should be 0 tmes the average absolute value of x and the average absolute value of x 3 should be 5 tmes the average absolute value of x. Multplyng the coeffcent of each varable n Eq. (2), the mportant degrees of x, x 2 and x 3 to Y should be ::0.05. Table 5 lsts the testng mean absolute errors (MAE) of NN modelng wth dfferent nput varables. From the values of MAE, we also conclude that NN has the effcent learnng to the system. Table 5. The MAEs of NN modelng wth dfferent nput varables. NN Sze Input Varables (x,x 2,x 3) (x,x 2,x 3,x 4) (x,x 2,x 3,x 4,x 5) MAE Agan, we calculated ID and PID values from three well-traned NN models. Table 6 lsts ID and PID values calculated by NN models wth three, four and fve nputs, respectvely. From the table results shown, the rato of PID of (x, x 2, x 3 ) s (:0.982:0.049) for sze 3-4- NN model. For sze 4-4- NN model, the rato of NPID of (x, x 2, x 3, x 4 ) s (:0.990:0.0498:0.0027). For sze 5-4- NN, the rato of PID of (x, x 2, x 3, x 4, x 5 ) s (:0.985:0.0475:0.0006: ). It s clearly found that the ratos of (x, x 2, x 3 ) n three NN models are all very close to (::0.05) of (x, x 2, x 3 ) n the orgnal lnear equaton. The ID and PID values of x 4 and x 5 are much smaller than x, x 2 and x 3. Table 6. ID and PID values calculated by three NN models. NN Sze x ID (PID) (49.24%) (48.95%) (49.0%) x 2 ID (PID) (48.36%) (48.47%) (48.38%) x 3 ID (PID) (2.40%) (2.44%) (2.33%) x 4 ID (PID) (0.3%) (0.03%) x 5 ID (PID) (0.5%) Same as the prevous studes, ths phenomenon shows that the nonlnear NN model s able to evaluate the mportant degree of each nput varable to the output. And, x 4 and x 5 are nferred to be the useless terms or noses due to ther low ID and PID values NN vs. Nonlnear Regresson Model In ths sesson, the nonlnear system dentfcaton by NN model was studed contnuously. A nonlnear equaton gven by [24] Y = x + (3) x2 + xx2 x3 500 ponts were generated from the equaton. x, x 2, x 3, x 4 and x 5 are assumed to be the possble nfluencng factors. All fve varables are unformly dstrbuted random numbers wth the statstcs lsted n Table 7. Table 7. The statstcs of (x, x 2, x 3, x 4, x 5). Varable Range Mean Varance x +~ x 2 +~ x 3 +~ x 4 +~ x 5 +~ In our study, 400 ponts were used for NN s tranng and 00 ponts were used for testng. Table 8 presents the MAEs of NN modelng wth dfferent nput varables. It shows that NN has the effcent learnng to the system. Table 8. The MAEs of NN modelng wth dfferent nput varables. NN Sze Input Varables (x,x2,x3) (x,x2,x3,x4) (x,x2,x3,x4,x5) MAE Table 9 presents ID and PID values calculated by NN models wth three, four and fve nputs, respectvely. From the table results shown, the ratos of ID and PID of (x, x 2, x 3 ) n three NN modelng systems are almost equal. Same as the studes n lnear system, the ID and PID values of x 4 and x 5 are much smaller than x, x 2 and x 3. In other words, the varables (x, x 2, x 3 ) can be concluded as the real useful nputs to Y and (x 4, x 5 ) could be nferred to be the nose terms. Table 9. ID and PID values calculated by three NN models. NN Sze x ID (PID) (46.03%) (45.47%) (45.62%) x 2 ID (PID) (45.57%) (46.2%) (46.03%) x 3 ID (PID) (8.3%) (8.33%) (8.8%) x 4 ID (PID) (0.08%) (0.0%) x 5 ID (PID) (0.06%) In fact, unlke lnear modelng, no clear nformaton about the mportant degree of ndvdual nput varable to output could be obtaned n nonlnear modelng. However, from the PID results shown n Table 9, we found that the mportant degrees of x and x 2 are almost the same. Ths phenomenon could be observed from Eq. (3) when both x and x 2 have the same statstc dstrbuton.

5 79 Shumng T. Wang et al.: An Evolutonary Method of Neural Network n System Identfcaton Smlar as prevous study, the nonlnear system dentfcaton by NN model s studed contnuously. A nonlnear equaton gven by [25] y = * sn( π * x * sn( π * x 3 / 2) / 2) + 4 * sn( π * x 2 / 2) (4) 000 ponts were generated from the equaton and then the system was assumed to be unknown. 800 ponts were used for NN s tranng and 200 ponts were used for testng. Four varables, x, x 2, x 3 and x 4 are assumed to be the possble nfluencng factors to the system s output y. Four nput varables are unformly dstrbuted random numbers wth the statstcs lsted n Table 0. Table 0. The statstcs of (x, x 2, x 3, x 4). Varable Range Mean Varance x 0~ x 2 0~ x 3 0~ x 4 0~ A sze of 4-5- NN was used to perform the model s dentfcaton. Table presents the smulaton results ncludng MAEs of the tranng and test of NN and ID and PID calculatons of (x, x 2, x 3, x 4 ) for NN s modelng wth four nputs. Table. The smulaton results of NN s modelng for nonlnear equaton. Tranng MAE Test MAE Varables x x 2 x 3 x 4 ID PID 53.63% 28.53% 7.2% 0.63% In ths study, due to the ranges of (x, x 2, x 3, x 4 ) are all n [0, ], thus the angles of sne functon are wthn the range of [0, π/2]. From the nonlnear Eq. (4), t can be observed that the mportant degree of (x, x 2, x 3 ) should be (:0.5: ). From the smulaton results shown, PID rato of (x, x 2, x 3, x 4 ) s (:0.5320:0.3209:0.07). The PID rato of (x, x 2, x 3 ) s very close to the rato (:0.5: ) of the coeffcents (x, x 2, x 3 ) n the orgnal nonlnear equaton. The ID and PID values of x 4 are very small. Thus, x 4 could be treated as the useless term or nose. In above studes of NN versus nonlnear regresson models, the smulaton results show that NN model s also capable of obtanng the mportant degree of each nput varable to the system output. In order to further prove our pont of vew, the example of real ndustral system dentfcaton s contnuously studed n the followng secton NN vs. Industral System It s known that touch panel (TP) has become an ndspensable part for many electronc applances such as computer, moble phone, tcket vendor, etc. Usually, a thn plate wth coatng flm wll be stuck on the screen of TP as a decoraton and protecton flm. Bascally, the flm s coatng process s accomplshed by the evaporator under the vacuum condton. However, before the evaporaton process s taken, the relevant control parameters of evaporator are needed to be set precsely n order to ensure the whole coatng process could be accomplshed successfully. Thus, these control parameters could be treated as the nfluencng factors of evaporaton process. Transmttance s an mportant optcal property of TP flm. Its value s hghly correlated wth the nfluencng factors of evaporaton process. The relatonshp between transmttance and ts relevant nfluencng factors s very complex and nonlnear. Thus, how to use NN to fnd the real nfluencng factors of TP flm s transmttance s the am of the study n ths secton. In our research, TP flm wth two layers coatng s studed. The coatng targets are Cr and Cr O. The nformaton of data 2 3 collected for the experments ncludes the value of quartz crystal deposton montor (x ), the rotaton speed of holder (x 2 ), the substrate poston of panel placed (x 3 ), Cr thckness (x 4 ), Cr O thckness (x 5 ) and TP transmttance (y). The complex 2 3 relatonshp between y and ts possble nfluencng factors (x, x 2, x 3, x 4, x 5 ) s expected to be obtaned by NN model. In order to farly demonstrate the feasblty of NN model and the computatonal technque proposed, two data sets named -, -2, are re-organzed randomly from the orgnal data base. For each data set, 00 ponts are used for NN s tranng and 44 ponts are used for testng. Except MAE, the mean absolute percentage error (MAPE) s also used as the measurement of NN performance. Table 2. The error statstcs performed by fve-nput NN model. Inputs: x, x 2, x 3, x 4, x % 2.03% % 2.8% Avg % 2.0% Table 3. ID and PID values calculated by fve-nput NN model. Tranng Data x x 2 x 3 x 4 x 5 ID(-) ID(-2) Avg PID(-) 8.24% 2.77%.% 45.5% 42.37% PID(-2) 4.27% 8.53% 0.42% 42.56% 34.22% Avg. 6.26% 0.65% 0.76% 44.03% 38.30% Test Data x x 2 x 3 x 4 x 5 ID(-) ID(-2) Avg PID(-) 8.68% 3.93%.3% 44.67% 4.59% PID(-2) 4.34% 7.94% 0.45% 42.83% 34.44% Avg. 6.5% 0.93% 0.79% 43.75% 38.0% Table 2 lsts the MAEs and MAPEs of transmttance estmatons performed by a fve-nput NN model wth sze From MAPEs shown, we conclude that NN model has the effcent tranng. ID and PID values for all nputs are calculated and lsted n Table 3. From the values of PID

6 Internatonal Journal of Intellgent Informaton Systems 206; 5(5): shown, t s clearly found that Cr thckness (x 4 ) and Cr O 2 3 thckness (x 5 ) are two most mportant nfluencng factors to TP transmttance. The value of quartz crystal deposton montor (x ) and the rotaton speed of holder (x 2 ) also have the certan mpacts to transmttance. In these fve nput varables, the substrate poston of panel placed (x 3 ) could be gnored due to ts small PID value. In order to prove our concluson, the transmttance estmatons by NN model wth dfferent nput combnatons are redone. Table 4 lsts MAEs and MAPEs of transmttance estmatons performed by NN model wth dfferent four-nput combnatons. Table 5 lsts MAEs and MAPEs of transmttance estmatons performed by NN model wth dfferent three-nput combnatons. From the results shown n these tables, t s clearly found that the concluson we made n accordance wth the values of ID and PID s correct. In other words, the evolutonary NN technque we proposed s very promsng and potental n the real applcatons of system dentfcaton. Table 4. The error statstcs performed by NN model wth four nputs. Inputs: x, x 2, x 4, x %.76% % 2.9% Avg %.98% Inputs: x, x 3, x 4, x % 2.62% % 3.45% Avg % 3.03% Inputs: x 2, x 3, x 4, x % 2.59% % 3.24% Avg % 2.9% Table 5. The error statstcs performed by NN model wth three nputs. Inputs: x, x 4, x % 2.5% % 2.77% Avg % 2.64% Inputs: x 2, x 4, x %.74% % 2.9% Avg %.96% Inputs: x 3, x 4, x % 5.33% % 5.7% Avg % 5.25% 4. Concluson Ths research presents an evolutonary NN technque for the applcaton of system dentfcaton. The lnear and nonlnear systems and the real ndustral system were studed and smulated. The study results obvously show that the novel technque proposed ndeed has ts feasblty and superorty n the applcaton of system dentfcaton. The computatonal methods denoted by ID and PID can extract the useful and mportant nputs to the system output. In other words, ths study proposes a new system dentfcaton technque based on supervsed NN and ths technque has the potental n the area of data mnng wth large database. In ths research, the actvaton functon for all nodes of NN s the sgmod functon. The ID and PID values defned are also derved from the characterstc of sgmod functon. Thus, t s beleved that NN model wth any node s transfer functon whch has the same characterstc as sgmod functon certanly has the smlar capablty n the applcaton of data mnng. For nstance, the hyperbolc tangent functon s also a choce. It s well known that NN wth BP learnng algorthm usually has the local mnmum problem n ts learnng process. An ll-learnng NN model mght have the generalzaton and accuracy problems, f NN model really plunged nto the local mnmum. Ths condton mght cause the ncorrect ID and PID values calculated from the ll-traned NN model. Thus, we have to emphasze that NN model must have an effcent learnng for ensurng the correct ID and PID nformaton could be obtaned. Acknowledgements Ths research was supported by the Mnstry of Scence and Technology, Tawan, R.O.C. under Contracts No. MOST E References [] M. Schetzen, "The Volterra and Wener Theores of Nonlnear Systems". Wley, 980. [2] S. A. Bllngs, "Nonlnear System Identfcaton: NARMAX Methods n the Tme, Frequency, and Spato-Temporal Domans". Wley, 203. [3] O. Nelles, "Nonlnear System Identfcaton: From Classcal Approaches to Neural Networks". Sprnger Verlag, 200. [4] M. Letta, Dynamc multvarate B-splne neural network desgn usng orthogonal least squares algorthm for non-lnear system dentfcaton, 204 8th Internatonal Conference on System Theory, Control and Computng, ICSTCC 204, pp , 204. [5] K. J. Ndhl Wlfred, S. Sreera, B. Vay, V. Bagyaveereswaran, System dentfcaton usng artfcal neural network, IEEE Internatonal Conference on Crcut, Power and Computng Technologes, ICCPCT 205, July 5, 205.

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