León, Monterrey, Nuevo León, MÉXICO 4

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1 Statstcal Characterzaton and Optmzaton of Artfcal Neural Networks n Tme Seres Forecastng: The One-Perod Forecast Case Caracterzacón Estadístca y Optmzacón de Redes Neuronales Artfcales para Pronóstco de Seres de Tempo: Pronóstco de un Solo Período María Angélca Salazar Agular 1, Gullermo J. Moreno Rodríguez 2 and Maurco Cabrera-Ríos 4 1 Dvsón de Posgrado en Ingenería de Sstemas, Facultad de Ingenería Mecánca y Eléctrca, Unversdad Autónoma de Nuevo León, Monterrey, Nuevo León, MÉXICO 4 Correspondng Author, Ph: +52(81) Ingenería de Tráfco y Optmzacón de Capacdad, Avantel, Monterrey, Nuevo León, MÉXICO {angy@yalma.fme.uanl.m 1, gmoreno@avantel.com.m 2, mcabrera@mal.uanl.m 4 } Abstract Artcle receved on Augost 9, 25; accepted on September 8, 26 Tme seres forecastng s an actve area for the applcaton of Artfcal Neural Networks (ANNs). Although the selecton of an ANN has been greatly smplfed, t remans a challenge to adequately determne the ANN s parameters. In ths work a method based on statstcal analyss and optmzaton technques s proposed to select the ANN s parameters for applcaton n tme seres forecastng. The results on the successful applcaton of the method n a real demand forecastng problem for the telecommuncatons ndustry are also reported.. Keywords: Artfcal Neural Networks, Tme Seres Forecastng, Desgn and Analyss of Eperments. Resumen Los pronóstcos de seres de tempo consttuyen un área actva para la aplcacón de Redes Neuronales Artfcales (RNAs). Aunque la seleccón de una RNA para tal aplcacón se ha smplfcado grandemente, la falta de un método establecdo para asgnar los parámetros de las RNAs de una manera adecuada sgue sendo un reto. En este trabajo se propone una metodología basada en técncas estadístcas y optmzacón para la seleccón de parámetros de una RNA para el pronóstco de seres de tempo. La metodología propuesta se demuestra por medo de su aplcacón en un problema real de pronóstco de demanda en la ndustra de las telecomuncacones. Palabras Clave: Redes Neuronales Artfcales, Seres de tempo, Análss y Dseño de Epermentos, Pronóstcos. 1. Introducton Throughout hstory, forecastng the behavor of phenomena of nterest has been mportant. Ths fact s reflected n the dversty of forecastng applcatons n practcally all knowledge areas. When quanttatve nformaton regardng the behavor of a partcular phenomenon wth respect to tme s avalable, then one has a tme seres. Forecasts n tme seres are commonly generated through tradtonal statstcal technques. The body of knowledge that encompasses these technques s called Tme Seres Analyss. Forecasts are generally the bases for decson-makng at all levels rangng from the day-to-day operatonal decsons to the long-term strategcal ones n many organzatons. Gven ts mportance, t s not surprsng that forecastng had become a very actve research area (Makrdaks and Wheelwrght, 1987; Zhang, 24). Use of lnear forecastng methods such as movng averages, eponental smoothng, lnear regresson and tme seres decomposton have domnated the arena for decades. Especally sgnfcant n terms of usage s the technque called Auto-Regressve Integrated Movng Average (ARIMA), developed by Bo and Jenkns (1976). In spte of the wdespread use of these lnear methods, the estence of nonlnear relatonshps n the tme seres can lmt ther applcaton n many cases (Makrdaks et al., 1982). Nonlnear relatonshps are, ndeed, not uncommon at all n realty, thus makng t necessary to resort to technques capable to adequately reflect such behavor. Artfcal Neural Networks (ANNs) have been suggested as an alternatve forecastng technque owng to ther ablty to appromate nonlnear behavor. In fact, ANNs have also been shown to be compettve even when the case s one of lnearty (Wdrow et al., 1994; Hwarng, 21; Mederos et al., 21; Zhang, 21).

2 7 María Angélca Salazar,Gullermo J. Moreno and Maurco Cabrera The frst use of ANNs n tme seres forecastng can be traced back to Hu n 1964 (Zhang, et al., 1998). However, Hu s work could not be completed because a tranng algorthm for multlayer ANNs was not avalable at the tme. Such algorthm, known now as the backpropagaton algorthm, was developed n 1974 by Paul Werbos, nevertheless, ts potental was not mmedately recognzed by the researchers n ANNs. It was untl 1986 when the backpropagaton algorthm was fnally used by Rumelhart et al., (1986) to develop ANNs to be appled n tme seres forecastng. ANNs have been ganng ground year by year snce then (Zhang et al., 1998). A work by Werbos (1988) supported the use of ANNs through results from a seres of nstances n whch backpropagaton-traned ANNs performed better than several tradtonal statstcal technques such as lnear regresson and Bo-Jenkns method. In recent years, ANNs have become very popular as modelng and analyss ads n areas as dverse as fnances, medcne, energy generaton, engneerng, and envronmental scences among many others, and ncludng tme seres forecastng (Maer et al., 2). A large number of papers that make use ANNs for predcton can be found n the lterature. For nstance, some works use ANNs to predct the qualty of ar (Gardner and Dorlng, 1999; Kolehmanen et al., 21; Nska et al., 24); another one (Kuo et al., 1999) compares the performance of ANNs and the autoregressve mean average n forecastng the demand n a chan of Chnese supermarkets. The results n ths last one once agan favored the ANNs. Recent studes regardng the applcaton of ANNs n Operatons Research, ncludng forecastng n fnances, can be found n Zhang (24) and Smth et al., (2). The fact that there s postve evdence n favor of ANNs n forecastng, should not gve the false mpresson that ther use s wthout challenges. In fact, forecastng precson hghly depends on key decsons regardng several parameters as well as the archtecture of the ANN (Zhang, 24). Some of these decsons can be made whle the ANN s beng bult, but others must be made a pror. Due to the lack of a standard method to make these decsons, n ths work a method s proposed that allows settng the ANN s parameters to guarantee an adequate forecastng performance. The proposed method s based on statstcal analyss and optmzaton of the ANNs performance measures (PMs). These PMs are defned as functons of forecastng errors for our purposes. The method conssts on carryng out a statstcal desgn of eperments where each factor nvolved corresponds to one ANN adjustable parameter. The eperment s results are then characterzed through one regresson model ftted to each PM. Each regresson model s later used as the objectve functon n an optmzaton problem where the parameters are the decson varables. The soluton to the optmzaton problem specfes the levels at whch the ANN parameters should be set to generate relable forecasts. In ths work, the decsons regardng the applcaton of ANNs n tme seres forecastng are descrbed frst, followed by a detaled eplanaton of the proposed method. Fnally, the use of the method s demonstrated through a real case study n a local telecommuncatons company. Emphass n ths manuscrpt has been gven to descrbng the method and demonstratng ts use n cases of one-perod forecasts. Future publcatons wll approach the more complcated case of multple-perod forecastng. 2 Decsons nvolved n the applcaton of ANNs n tme seres forecastng Development of an ANN for tme seres forecastng nvolves the crtcal task of specfyng an archtecture (or topology) for the ANN. Ths archtecture must be defned n terms of the number of neurons n each of the three layers of the ANN proposed for the stated purpose.e. the nput layer, the hdden layer, and the output layer. The number of neurons n the nput layer determnes the amount of hstorcal data ponts or lags that wll be used to generate the forecast. Because only one-perod forecasts are consdered here, the output layer wll contan only the neuron correspondng to the sngle forecast. On the other hand, the number of neurons n the hdden layer determnes the capablty of the ANN to appromate the nonlnear relatonshps between the lags and the resultng forecasts (Zhang et al., 1998). Several authors have studed the choce of an adequate number of neurons n the hdden layer (Zhang, 24; Hansen et al., 23; Seton et al., 25), however, to the date there s not a defntve way to solve ths problem. Preprocessng the nput data to mprove an ANN s performance has been recommended by several authors n the lterature. Among them, Pramuthu et al. (1998) documented the mportance of preprocessng. It has been observed that tranng an ANN the process by whch the ANN learns nput-output patterns- can actually become easer through dfferent transformatons of the nput data. Preprocessng can be used to reduce the spread of the data n the search space

3 Statstcal Characterzaton and Optmzaton of Artfcal Neural Networks n Tme Seres Forecastng 71 (Hansen et al., 23) and s therefore an mportant factor to consder when tryng to mprove the performance of an ANN. On the other hand, selecton of a tranng algorthm s another mportant factor for most applcatons. Tme seres forecastng s not the ecepton n ths case. It s our premse that the decsons here descrbed can be made n a smultaneous and systematc manner through the use of statstcal desgn of eperments and optmzaton technques as lad out n the method detaled n the ensung secton. 3 Proposed method Fgure 1 schematcally shows the method proposed to set the parameters of the ANN n a smultaneous and systematc manner. The method s presented here as applcable to generc feedforward backpropagaton ANNs. The detals concernng tme seres forecastng are covered later n ths manuscrpt. The basc dea behnd the method was the representaton of the ANN as a system wth controllable varables that eert nfluence over certan ANN s outputs. In ths way, by consderng the ANN parameters as controllable varables, we can eplore ther respectve ranges and characterze ther effect on key PMs (outputs) through regresson technques. Ths characterzaton, n turn, wll allow the use of analyss of varance (ANOVA) to screen sgnfcant controllable varables as well as determne f two or more PMs depend on dsjont subsets of these varables. In ths last case, ndependent optmzaton problems can be created thereby smplfyng ths task. Because the ultmate goal s to set these parameters to obtan the best possble PM values, the problem ndeed falls n the optmzaton realm. The steps of the method outlned n Fgure 1 can be descrbed as follows: Descrpton of the ANN as a system. The objectve n ths step s to specfy the characterstcs of the ANN to be used. It s necessary to dentfy the ANN controllable parameters as well as to defne the ANN s PMs to be ncluded n the study. Desgn and Analyss of Eperments. In ths step, the focus s on plannng, eecutng and nterpretng the results of a statstcally desgned eperment that ncludes the prevously dentfed parameters and PMs. Metamodelng. A regresson model must be obtaned through proper statstcal technques (ncludng resdual analyss) to descrbe the response surface correspondng to each PM as a functon of the controllable parameters. Optmzaton Problem. An optmzaton problem s bult wth the metamodels obtaned n the prevous step as objectve functons. Soluton. Solve the optmzaton problem through proper technques.e. multple crtera optmzaton f necessary and multple startng ponts.

4 72 María Angélca Salazar,Gullermo J. Moreno and Maurco Cabrera Parameters Type of ANN 1) Descrpton of the ANN as a system. Responses 2) Desgn and Analyss of Eperments Characterzaton 3) Metamodelng 4) Optmzaton Problems Multple Startng Ponts 5) Soluton Fg.1. Proposed Method for settng an ANN s parameters The ratonale for the proposed method obeys the pont of vew of an eperment, where planned changes are ntroduced n a system wth the objectve to analyze the varaton nduced n PMs of nterest. Because the sze of the eperment to be run depends on the number of controllable parameters, the use of an adequate epermental desgn s crtcal. If there are a few parameters, then a full factoral desgn wll suffce. A factoral desgn contans as many epermental runs as the complete enumeraton of the combnatons of values to be sampled per parameter (factor). For eample, f four parameters are to be studed, each vared at three, four, three and fve values or levels, then the factoral desgn wll have =8 epermental runs. For our purposes, then, under each combnaton of controllable parameters (run) an ANN wll be bult and traned to obtan a quantfcaton of the predcton qualty through the chosen PMs as the response of the eperment. It s advsable to consder at least three dfferent values for each parameter where possble to allow the determnaton of response curvature rght from the begnnng Once the eperment s complete, t s followed by ts analyss amng to characterze the varaton nduced n the PMs. In order to acheve ths, an ANOVA s run based on an underlyng full quadratc regresson model of each PM as a functon of the controllable parameters. The regresson model proposed s shown n Equaton (1), where the resduals, ε, are assumed to be dentcally ndependently normally dstrbuted wth a mean of and an unknown, but constant varance. y = β k k k 1 k 2 + β + β + β j j = 1 = 1 = 1 j= ε (1)

5 Statstcal Characterzaton and Optmzaton of Artfcal Neural Networks n Tme Seres Forecastng 73 In Equaton (1), dependent varable y represents the value of the PM of nterest, β s the ntercept of the model, 2 β s the regresson coeffcent assocated to the varaton of, corresponds to the th parameter, β s the regresson coeffcent for, and s the regresson coeffcent for the nteracton between and ; and k s the total β j number of controllable parameters n the eperment. The regresson coeffcents are typcally computed through a squares reducton technque, avalable n most commercal statstcal software packages. Checkng the resduals assumptons to verfy the adequacy of the model s also convenently carred out through these same packages. Fnally, each of the resultng regresson models (one per PM) s ncluded as an objectve functon n an optmzaton problem, where fndng the parameter settngs to acheve the best possble value for the objectve functon s sought. At the end, the optmzaton problem wll most lkely be nonlnear and nonconve. Ths makes the optmzaton problem hard to solve, and therefore the hghest aspraton tends to be fndng an attractve local optmal soluton. The proposed method was put n practce n a local telecommuncatons company for testng purposes. The case study that demonstrates the applcaton of the method and the results s dscussed net. 4 Case study In ths case study, a real problem n a local telecommuncatons company s approached. The company s ultmate objectve, as the vast majorty of companes n the world, s to be proftable through provdng a hgh servce level to ts customers. Its major resource s a transmsson network, loosely defned as a set of nterconnected devces wth a fnte transmsson capacty. Customers demand dfferent servces at dfferent levels through such network n a stochastc fashon and at some ponts n tme challengng the nstalled network capacty. Because decsons on capacty epanson take tme and equpment purchase and nstallaton are not mmedate, t s crtcal for the company to be able to estmate the demand behavor at least for the net perod of tme. Relable forecasts should allow the decson-maker to plan the requred capacty epansons to meet the epected demand wthout havng too much unutlzed nstalled capacty. The company provded hstorc network utlzaton data organzed n monthly perods.e. a tme seres. Nonlnearty seemed evdent from the begnnng. Ths observaton was further confrmed wth the poor forecasts from a prevously tred lnear regresson. However, as none of the prevous was completely conclusve, we reled on the fact that ANNs have been shown to be compettve even n cases of lnearty (Hwarng, 21; Mederos et al., 21; Zhang, 21). Based on these peces of nformaton, t was decded that an ANN would be a sensble choce. The parameter selecton was approached through the method prevously proposed n secton 3 as descrbed n the followng secton Descrpton of the ANN as a system Fgure 2 shows the ANN selected for ts use as a forecastng model. Such choce follows from ths ANN s well-studed and demonstrated unversal appromaton capabltes (Whte, 199; Hornk, 1989); as well as ts documented good performance n forecastng (Zhang, et al., 1998; Zhang, 24; Lao et al., 25; Hansen, et al., 24). The one shown n fgure 2 s a three-layer feedforward backpropagaton ANN. Its frst layer of neurons receves hstorc data, whch s then sent to the ntermedate layer (the hdden layer) for processng. The hdden layer determnes the degree of fleblty n the forecastng model accordng to the number of neurons that t contans. Once processed by the hdden layer, the nformaton s fnally sent to the output layer, whch n ths case has a sngle output neuron (for the sngle-perod forecast). Referrng to fgure 2, ANN nputs Y = t m, t m +1,... t n the current forecastng applcaton, corresponded to the m lags prevous to perod t + 1, whch s to be forecasted. Determnng m s crucal because t ndcates the etent n hstorc data to whch a partcular data pont s potentally correlated. The number of neurons n the hdden layer allows modelng nonlnearty, hence ts mportance as a decson varable. A number of neurons larger than necessary would result n overtranng, whch mples the loss of the ANN s predcton capablty. If too few neurons are ncluded, then the model wll not be fleble enough to accommodate certan degree of nonlnearty. Fnally, j

6 74 María Angélca Salazar,Gullermo J. Moreno and Maurco Cabrera the neuron n the output layer computes the forecast for perodt + 1, represented by +1. The transfer functon for the hdden neurons was the hyperbolc tangent, and the dentty functon for the output neuron. Y t Y t+1 Output Layer b t+1 V t+1,j Hdden Layer b j W j... Input Layer Yt Yt-1 Yt-m+1 Yt-m Fg. 2. Three-layer feedforward backpropagaton ANN Addtonally n Fgure 2, W j, = 1,2,..., neurons, j = 1,..., m s the weght appled n the ncomng arc to hdden neuron j from nput neuron, V t + 1, = 1,2,..., neurons s the weght appled n the ncomng arc to neuron t +1 (output neuron) from neuron j (hdden neuron). These weghts modfy the nformaton that passes through ther respectve arc and can be understood as fttng parameters. Specal weghts b j and b t +1 are known as bases. There s a large collecton of algorthms n the lterature to tran an ANN,.e. fndng all ANN weght values. In ths case, two algorthms were tred: Levenberg-Marquardt s (lm), whch s valued for ts convergence speed, and Bayesan Regularzaton (br), valued for ts capablty to avod overtranng. Both of these algorthms have been shown to be qute compettve, although for dfferent reasons as eplaned before (Bshop, 1995; Hagan et al., 1996). Two parameters that are assocated wth the hstorc data for analyss were ncluded: transformaton and scale; transformaton ncluded two optons: ether leave the data as demand ponts on each perod or used them as the dfference between two adjacent perods; on the other hand, scale mpled the opton to ether normalze the data to fall wthn the range [-1, 1], or use ther orgnal scale., The levels on both parameters have been suggested as analyss optons n the ANN lterature. In order to measure the ANN forecastng error, the mean square error (MSE), the mean absolute error (MAE), the largest-n-magntude postve error or over-predcton (S_Pred), and the largest-n-magntude negatve error or underpredcton (B_Pred) were used. Both MSE and MAE have been wdely utlzed n the lterature, whle S_Pred and B_Pred were mportant measures n ths case to show the worst case of unsatsfed demand and the worst case of unutlzed nstalled capacty n the network respectvely..

7 Statstcal Characterzaton and Optmzaton of Artfcal Neural Networks n Tme Seres Forecastng Desgn and Analyss of Eperments The net step n the method was to carry out a statstcally desgned eperment. To ths end, number of lags was vared wthn the range [2,6]; number of hdden neurons n the range [2,7]; transformaton n {none, dfferences}; scale n {orgnal, [-1,1] }, and algorthm n {lm,br}. Three levels were consdered for lags and neurons, whle two levels for the rest of the parameters as specfed above. The PMs were MSE, MAE, and B_Pred y S_Pred. Specfc values for the ANN parameters are as follows: lags = {2,3,6}, neurons = {2,5,7}, transformaton = {none, dfferences}, scale = {orgnal, [-1,1] } and algorthm = {lm, br }. The labels 1, 2 and 3 were used to denote the levels of these parameters and correspond to the order of the lsted values. When conductng the eperment accordng to secton 4.1, t was deemed adequate to use a factoral desgn whch resulted n a total of 72 combnatons. Table 1 shows the epermental results when the tranng algorthm was Levenberg-Marquardt s. Table 1 Epermental results wth Levenberg-Marquardt s tranng algorthm Neurons Lags Scale Transformaton Algorthm MAE MSE B_Pred S_Pred

8 76 María Angélca Salazar,Gullermo J. Moreno and Maurco Cabrera 4.3. Metamodelng The resultng eperment was then used to characterze the results through the use of ANOVA. Ths analyss, along wth the factoral structure and the regresson metamodel, allows the assessment of the statstcal contrbuton of the parameters ndependently, as well as n second-order nteractons. Second order regresson models were obtaned for each of the PMs. The resultng coeffcents are shown n Table 2. Table 2 Regresson coeffcents for each ANN performance measure Regresson Term MAE MSE B_Pred S_Pred Constant Because usng regresson metamodels and ANOVA mples assumptons regardng the resduals, namely normalty, ndependence and constant varance, t s mportant that these assumptons be verfed through resdual analyss to make sure that the subsequent conclusons are statstcally vald. Fgure 3 shows an eample on how resdual analyss can be carred out graphcally as a frst appromaton. Ths fgure shows the analyss pertanng to B_Pred. The graphs on the left allow verfyng f the resduals are followng a normal dstrbuton. In the graph n the upper left corner, a straght lne pattern s sought and n the graph n the lower left corner a bell shape smlar to the normal dstrbuton should be appromately evdent. The graphs n the rght have the purpose of checkng for ndependence of the resduals vs. the model predctons (fts) and vs. the run order. The behavor of the resduals n these graphs should be random. These two graphs also help to see f there was a dramatc change n the spread of the resduals (nonconstant varance). A seres of formal statstcal tests usually follows the graphcal analyss. In ths case, after statstcal testng, the concluson was that there was no sgnfcant volaton to the

9 Statstcal Characterzaton and Optmzaton of Artfcal Neural Networks n Tme Seres Forecastng 77 resdual assumptons. Further detals on these tests can be found n Devore (1995). A smlar analyss was carred out for each PM, wth smlar conclusons Normal Probablty Plot 2 Resduals vs. Fts Percentle Resduals Resduals Fts 6 8 Resduals Hstogram Resduals vs. Run Order 24 2 Frequency Resduals Resduals Run Order 6 Fg. 3. Resdual Analyss for the regresson model correspondng to B_Pred Accordng to the proposed method, metamodelng must be followed by the creaton of optmzaton problems. Ths s descrbed n the followng secton Optmzaton problems and soluton On each of the optmzaton problems, the objectve functon z s defned by the metamodel representng the PM of nterest. In general, the resultng optmzaton problems wll have the followng format: Fnd Mnmze Subject to z = l β + k = 1 Z I β + u + = {1,2,..., k } to k = 1 β 2 I I + k 1 k = 1 j = + 1 β j j where the decson varables, represent the th ANN controllable parameter through the varaton of whch the objectve functon z s to be mnmzed. Scalar k stands for the number of controllable parameters whose values must

10 78 María Angélca Salazar,Gullermo J. Moreno and Maurco Cabrera fall wthn a preset lower bound and a preset upper bound. These bounds usually correspond to the lmts of the l regon eplored n the epermental desgn. Fnally, the decson varables must be postve ntegers as ndcated by the last constrant. In our case study k = 5, corresponds to the parameters: neurons, lags, scale, transformaton and algorthm. Solvng each optmzaton problem ndependently and usng multple startng ponts to ncrease the probablty of fndng an attractve local optmum, t was found that the nput data should be handled n dfferences and scaled to fall wthn the range [-1,1]. The number of lags (now n dfferences) was determned to be 6 perods, and the ANN should have 7 hdden neurons. The tranng algorthm was prescrbed to be Levenberg-Marquardt s. An ANN wth these resultng characterstcs provdes forecasts wth performance as shown n Fgure 4. It can be observed that the model s ndeed a good appromaton to the tme seres. In fact, ths confguraton concded wth the best epermental combnaton n Table 1. An adequate forecastng performance s hghly probable after the applcaton of the proposed method. u Demand Unts Real ANN Tme (months) Fg. 4. ANN predctons vs. the real tme seres It s necessary to pont out, however, that although we dscussed the ssue of overtranng prevously, n ths partcular case t was not eplctly consdered. The reason had to do wth keepng the objectve at hand smple: testng the methodology. A future publcaton wll nclude our complete results for multple-perod forecastng where crossvaldaton was used to avod overtranng. The fnal ANN, whose performance s depcted n Fgure 4 was the best one found by the optmzaton procedure when mnmzng each one of the PMs. Ths result s a clear ndcaton of a large degree of correlaton wthout conflct among the PMs. Had the optmzaton problems arrved to solutons dfferent to each other, selectng a combnaton would have not been a trval task. In such case t would have been necessary to use multple crtera optmzaton technques. These technques try to fnd the best compromses between all PMs, whch are formally called Paretoeffcent solutons. Because treatment of multple crtera optmzaton procedures s beyond the scope of ths work, the nterested reader s referred to Deb (24) and Hellermeer (21). Also, some recent applcatons of multple crtera optmzaton n manufacturng usng Data Envelopment Analyss can be found n Cabrera-Ríos et al. (22,24), Castro J. M. et al. (24) and n Castro C. E et al. (23).

11 Statstcal Characterzaton and Optmzaton of Artfcal Neural Networks n Tme Seres Forecastng 79 In ths case study, the computatonal tools were Matlab TM, Mntab TM and MS Ecel TM. The frst one was used to buld the ANNs, the second one was used for statstcal analyss, and the thrd one for optmzaton purposes through ts XL Solver module. The complete results of ths work are avalable upon request to the authors. 5 Conclusons and future work In ths work, a method to set the parameters for an ANN n the contet of tme seres forecastng was proposed. The method nvolves the coordnated use of desgn of eperments, analyss of varance, lnear regresson and optmzaton. A case study where the method was mplemented and tested n a local telecommuncatons company was presented to demonstrate the use and performance of the method. Among the most attractve features of the proposed method we can lst the followng: (1) t uses well-establshed and relable analytcal technques, (2) ts mplementaton does not requre hghly specalzed code, and (3) t makes the nterrelatonshps between several key ANN parameters transparent. The proposed method can be appled to ANNs wth more complcated archtectures, such as those used to forecast n future perods. A follow-up publcaton consderng ths case s under preparaton n our research group. References 1. Bshop C. M., Neural Networks for Pattern Recognton, Oford Unversty Press, Oford, UK, Bo G. E. P., and Jenkns G. M., Tme Seres Analyss: Forecastng and Control, San Francsco, CA: Holden-Day, EUA, Cabrera-Ríos M., Castro J. M., and Mount-Campbell C. A., Multple qualty crtera optmzaton n reactve nmold coatng wth a data envelopment analyss approach II: a case wth more than three performance measures, Journal of Polymer Engneerng, Vol. 24, No. 4, 24, Cabrera-Ríos M., Castro J. M., and Mount-Campbell C. A., Multple qualty crtera optmzaton n n-mold coatng (IMC) wth a data envelopment analyss approach, Journal of Polymer Engneerng, Vol. 22, No. 5, 22, Castro C. E., Cabrera-Ríos M., Llly B., Castro J. M., and Mount-Campbell C. A., Identfyng the best compromse between multple performance measures n njecton holdng (IM) usng data envelopment analyss (DEA), Journal of Integrated Desgn and Process Scence, Vol. 7, No. 1, 23, Castro J. M., Cabrera-Ríos M., and Mount-Campbell C. A., Modellng and Smulaton n reactve polymer processng, Modellng and Smulaton n Materals Scence and Engneerng, Vol. 12, No. 3, 24, S121-S Deb K., Mult-Objectve Optmzaton Usng Evolutonary Algorthms, Edtoral Wley, NY, EUA, Devore J. L., Probablty and Statstcs for Engneerng the Scences, 4ta Edton, Dubury Press, Calforna Polytechnc State Unversty, EUA, Gardner M. W., and Dorlng S. R., Artfcal neural networks (the mult-layer perceptron) - a revew of applcatons n the atmospherc scences, Atmospherc Envronment, Vol. 33, 1999, Hagan M. T., Demuth H. B., and Beale M., Neural Network Desgn, PWS Publshng Company, EUA, Hansen J. V., and Nelson R. D., Forecastng and recombnng tme-seres components by usng neural networks, Journal of the Operatons Research Socety, No. 54, 23, Hllermeer C., Nonlnear Multobjectve Optmzaton: A Generalzed Homotopy Approach, Basel: Brkhauser Verlag, Hornk K., Stnchcombe M., and Whte H., Multlayer feedforward networks are unversal appromators, Neural Networks, Vol. 2, No. 5, 1989, Hu M. J. C., Applcaton n the Adalne system to weather forecastng, Master Thess, Techncal Report , Stanford electronc Laboratores, Stanford, CA, June Hwarng H. B., Insghts nto neural-network forecastng of tme seres correspondng to ARMA (p,q) structures, Omega: The Internatonal Journal of Management Scence, Vol. 29, No. 3, 21,

12 8 María Angélca Salazar,Gullermo J. Moreno and Maurco Cabrera 16. Kolark T., and G. Rudorfer, Tme seres forecastng usng neural networks, Conference Proceedngs of the Internatonal Conference on APL APL Quote Quad, Vol. 25, No. 1, 1994, Kolehmanen M., Martkanen H., Russkanen J., Neural networks and perodc components used n ar qualty forecastng, Atmospherc Envronment, Vol. 35, 21, Lao K-P, and Fldes R., The accuracy of a procedural approach to specfyng feedforward neural networks for forecastng, Computers & Operatons Research, Vol. 32, No. 2, 25, Maer H. R., and Dandy G. C., Neural networks for the predcton and forecastng of water resources varables: a revew of modellng ssues and applcatons, Envronment Modellng & Software, Vol. 15, 2, Makrdaks S., Anderson A., Carbone R., Fldes R., Hbbon M., Lewandowsk R., Newton J., Parsen E., and Wnkley R., The accuracy of etrapolaton (tme seres) methods: Results of a forecastng competton, Journal of Forecastng, Vol. 1, 1982, Makrdaks S., and Wheelwrght S. C., The Handbook of Forecastng a Manager s Gude, 2da Edton, Edtoral Wley, NY, EUA, Mederos M. C., and Pedrera C. E., What are the effects of forecastng lnear tme seres wth neural networks? Logstc and Transportaton Revew, Vol. 31, No. 3, 21, Nska Harr, Hltunen Ter, Karppnen Ar, Russkanen J. and Kolehmanen M., Evolvng the neural network model for forecastng ar polluton tme seres, Engneerng Applcatons of Artfcal Intellgence, Vol. 17, 24, Pramuthu S., H. Ragavan, and M. Shaw, Usng feature constructon to mprove the performance of neural networks, Management Scence, Vol. 44, No. 3, 1998, R. J. Kuo and k. C. Xue, Fuzzy neural networks wth applcaton to sales forecastng, Fuzzy sets and y systems, Vol. 18, No. 2, 1999, Rumelhart D-E., Hnton G. E., and Wllans R. J., Learnng representatons by backpropagatng errors, Nature, 323 (6188), 1986, Seton R. S., McMurtrey S., Mchalopoulos J. O., and Smth A. M., Employee turnover : a neural network soluton, Computers & Operatons Research, Vol. 32, No. 1, 25, Smth K. A. and Gupta JND, Neural networks n busness: technques and applcatons for the operatons researcher, Computers and Operatons Research, Vol. 27, Num , 2, Werbos P. J., Generalzaton of backpropagaton wth applcatons to a recurrent gas market model, Neural Networks, Vol. 1, 1988, Whte H., Connectonst nonparametrc regresson: Multlayer feedforward networks can learn arbtrary mappngs, Neural Networks, Vol. 3, No. 5, 199, Wdrow B., Rumelhart D., and Lehr M. A., Neural networks: Applcatons n ndustry, busness and scence, Communcatons of the ACM, Vol. 37, No. 3, 1994, Zhang G., Patuwo E., and Hu Y. M., Forecastng wth artfcal neural networks the state of the art, Internatonal Journal of Forecastng, Vol.14, No. 1, 1998, Zhang G. P., Neural Networks n Busness Forecastng, Idea Group Publshng, Georga State Unversty, EUA, Zhang G. P., and Hu M. Y., A smulaton study of artfcal neural networks for nonlnear tme seres forecastng, Computers & Operatons Research, Vol. 28, 21, Acknowledgements The authors are grateful to the CONACYT for the scholarshp granted to Ms. Salazar for her graduate studes.

13 Statstcal Characterzaton and Optmzaton of Artfcal Neural Networks n Tme Seres Forecastng 81 María Angélca Salazar Agular obtaned her B.S. n Computer Systems Engneerng from Insttuto Tecnológco de Querétaro (ITQ) (24), and her M.S. n Systems Engneerng from the Graduate Program n Systems Engneerng at Unversdad Autónoma de Nuevo León (25). Her nterests nclude Artfcal Intellgence and Operatons Research. She currently holds an appontment as an Assstant Professor at ITQ n the Department of Systems and Computng. Gullermo J. Moreno Rodríguez obtaned hs B.S. n Electrcal and Communcatons Engneerng from ITESM- Monterrey (1994). Hs nterests nclude all aspects of telecommuncatons network management. He s currently the manager of the dvson of Traffc Engneerng and Capacty Optmzaton of Avantel, Méco. Maurco Cabrera Ríos obtaned hs B.S. n Industral and Systems Engneerng from ITESM-Monterrey (1996), and hs M.S. and Ph.D. n the same dscplne from The Oho State Unversty (1999, 22). Hs research nterests are n general related to the applcaton of optmzaton to manufacturng and servces. Currently, he holds an appontment as an Assocate Professor at the Graduate Program n Systems Engneerng, FIME, Unversdad Autónoma de Nuevo León. He s a member of the Natonal System of Researchers, Level 1. http: //yalma.fme.uanl.m/~maurco/