A Recurrent Neural Network based Forecasting System for Telecommunications Call Volume

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1 App. Math. Inf. Sc. 7, No. 5, (2013) 1643 Apped Mathematcs & Informaton Scences An Internatona Journa A Recurrent Neura Network based Forecastng System for Teecommuncatons Ca Voume Pars Mastorocostas, Constantnos Has, Dmtrs Varsams and Stergan Dova Department of Informatcs and Communcatons, Technoogca Educaton Insttute of Serres, Serres, Greece Receved: 27 Dec. 2012, Revsed: 29 Apr. 2013, Accepted: 2 May Pubshed onne: 1 Sep Abstract: A recurrent neura network based forecastng system for teecommuncatons ca voume s proposed n ths work. In partcuar, the forecaster s a Bock Dagona Recurrent Neura Network wth nterna feedback. Mode s performance s evauated by use of rea word teecommuncatons data, where an extensve comparatve anayss wth a seres of exstng forecasters s conducted, ncudng both tradtona modes as we as neura and fuzzy approaches. Keywords: teecommuncatons data voume forecastng, neura modeng, bock dagona recurrent neura network 1 Introducton Durng the ast twenty years the proferaton of mobe phones, aong wth the exposve growth n teecommuncatons servces have turned ths scentfc fed to an mportant and deveopng ndustry. Lke any other busness, carrers seek proft maxmzaton and cost reducton, by managng network traffc effectvey, by optmzng ther nfrastructure usage and by predctng future trends. In order to acheve ther objectves, teecommuncatons managers and operaton offcers are n contnuous need of accurate forecastng modes of the ca voume. In ths perspectve, ths work ams at addressng the probem of ca voume forecastng for the case of a arge organzaton: a arge Greek Unversty s nvestgated, where new teephone numbers are added day and a varyng demand for outgong trunks exsts. The Unversty s database stores the tota number, as we as the number of the natona, the nternatona and the mobe cas per empoyee and per month. It has been notced that the ca cassfcaton nto dfferent categores reveas certan and dfferent patterns between destnatons. Intay, tradtona forecastng methods - mosty statstca - were apped to ths partcuar probem. These methods can be dvded nto three major categores: The frst method empoyed s the Na ve Forecast 1 [1], whch uses the most recent observaton as a forecast for the next tme nterva. A method that takes nto account the seasona factors s the Lnear Extrapoaton wth Seasona Adjustment (LESA, [2]): After seasonaty has been removed from the orgna data, near extrapoaton s apped n order to forecast the future vaues of the seres by empoyng the trend cyce component. Fnay, the projected trend cyce component s adjusted, makng use of the dentfed seasona factors. LESA M assumes mutpcatve seasonaty, whe LESA ADD s based on addtve seasonaty. The second cassc group of tme seres anayss technques ncudes the exponenta smoothng methods, where a partcuar observaton of the tme seres s expressed as a weghted sum of the prevous observatons. The weghts for the prevous data vaues consttute a geometrc seres and become smaer as the observatons move further nto the past. In processes wthout trend Smpe Exponenta Smoothng (SES) s apped, whe near trend and seasona data are accommodated by Hot s [3] and Wnters [4] methods, respectvey. Addtonay, mutpcatve seasona modes (Wnters MS) as we as addtve seasona modes (Wnters AS) exst [5]. The ast category concerns tme seres that exhbt damped trend, whch refers to a regresson component for the trend n the updatng equaton, expressed by means of a dampenng factor. In these cases, some modfcatons of SES can be apped n order to dea wth compex types of trend. An exponenta smoothng mode wth damped trend and addtve seasonaty (DAMP AS) and ts Correspondng author e-ma: mast@teser.gr Natura Scences Pubshng Cor.

2 1644 P. Mastorocostas et a.: A Recurrent Neura Network based Forecastng System... mutpcatve seasonaty counterpart (DAMP MS) aso exst, whe a damped trend mode on tme seres wth no seasonaty (DAMP NoS) can be ftted [5]. Fnay, the Auto Regressve Integrated Movng Average method (ARIMA) and Seasona ARIMA (SARIMA), whch anayze statonary unvarate tme seres data, have been very popuar [6]. It presumes weak statonarty, equay spaced ntervas or observatons, and at east 30 to 50 observatons. The we estabshed methods mentoned above have been studed n [2]. Lnear modes are aso suggested by the ITU Recommendaton E.507 for trend forecastng n teecommuncatons data [7]. It has been proved that Damped trend modes and SARIMA are the most effectve, when apped to the probem n hand, compared to the rest of the tradtona methods. Computatona Integence methods were frst apped n [8,9,10], where fuzzy and neurofuzzy systems were empoyed. In [8] a statc fuzzy mode s presented (OLS FFM, Orthogona Least Squares based Fuzzy Forecastng Mode), where an automated mode budng process, based on the Orthogona Least Squares agorthm, performs nput seecton, determnes the number of fuzzy rues, whch are of Takag Sugeno Kang type [11], forms the consequent parts of the fuzzy rues and tunes the consequent parameters. Recurrent neurofuzzy forecasters are deveoped n [9] and [10]. In [9] the Locay Recurrent Neurofuzzy Forecastng System (LR NFFS) s presented: the consequent part of each rue consst of a three ayer recurrent neura network, wth nterna feedback at the neurons of the hdden and output ayers, and a snge nput common to the premse and consequent parts. In [10] the Recurrent Neurofuzzy Forecaster (ReNNFOR) s ntroduced, whch s a respectve recurrent system of reduced compexty, wth unt feedback at the hdden ayer s neurons. A three modes exhbted sgnfcanty ameorated predcton capabtes wth respect to the tradtona statstca methods, wth the recurrent forecasters requrng ony the vaue of the most recent observaton, snce they are capabe of dentfyng the tempora reatons va the nterna feedback connecton. A sparse recurrent neura network, where the feedback connectons are restrcted to between pars of state varabes s suggested n [12], referred to as the Bock dagona recurrent neura network (BDRNN). It s a two ayer network, a feedback ayer comprsng the bock dagona nks, and an output ayer where the state varabes are combned to formuate the network s output. The BDRNN archtecture offers a sgnfcant feature: t consttutes a sutabe choce for modeng nonnear processes, where the overa dynamcs can be decomposed nto a set of ower order dynamcs. Under these condtons, the bocks of BDRNN can be used to mode these sub dynamcs by adaptng propery the respectve bock weghts. In ths perspectve, a forecastng system, based on BDRNN, s proposed n ths work. Its performance s compared wth famar forecastng approaches, ke a seres of seasonay adjusted near extrapoaton methods, Exponenta Smoothng Methods, the SARIMA method, aong wth the aforementoned fuzzy and neura forecasters. The rest of the paper s organzed as foows: n Secton 2, a bref presentaton of the BDRNN s gven and the modeng method s descrbed. Secton 3 hosts the expermenta resuts and a comparatve anayss, whe the concudng remarks are gven n Secton 4. 2 The forecaster s structure and the modeng method 2.1 BDRNN The bock dagona recurrent neura network s a two ayer network. The hdden ayer conssts of pars of neurons (bocks); there are feedback connectons between the neurons of each par, ntroducng dynamcs to the network, whe the output ayer s statc. Therefore, the BDRNN can be consdered as a speca form of ocay recurrent gobay feedforward network [13, 14]. The mpementaton of the proposed agorthm s straghtforward, wth no precondtons requred. The confguraton of BDRNN s presented n Fg. 1, where a snge nput snge output (SISO) BDRNN s shown, snce a SISO network w be used n the seque. For the sake of smpcty, a BDRNN wth four bocks of neurons s shown. Fg. 1: Confguraton of the Bock Dagona Recurrent Neura Network. The operaton of a SISO BDRNN wth N neurons at the hdden ayer s descrbed by the foowng set of state equatons: x(k) f a (W x(k 1)+B u(k)) y(k) f b (C x(k)) (1a) (1b) Natura Scences Pubshng Cor.

3 App. Math. Inf. Sc. 7, No. 5, (2013) / where f a s a N-eement vector ncudng the neuron actvaton functons of the hdden ayer, and f b s the actvaton functon of the output ayer. The actvaton functons are both chosen to be the sgmod functon 1 e 2 z f(z) 1+e 2 z. u(k) s the nput of the network at tme k. x(k) [x (k)] s a N-eement vector, comprsng the outputs of the hdden ayer. In partcuar, x (k) s the output of the -th hdden neuron at tme k. y(k) s the output of the network at tme k. B [b ] and C [c ] are N-eement nput and output weght vectors, respectvey. W [w, j ] s the N N bock dagona feedback matrx. In partcuar, 0 f j 0 f j and j 1 and s odd w, j 0 f j and j+ 1 and s even 0 otherwse The feedback [ matrx, ] W, s bock dagona: W dag W (1),...,W (N 2 ) ; each dagona eement, correspondng to a bock of recurrent neurons, has a bock submatrx n the form [ W () w2,2 w 2,2+1 ], 1,2,..., N (2) w 2+1,2 w 2+1,2+1 2 The above equaton descrbes the genera case of BDRNN, caed the BDRNN wth free form submatrces. A speca case of BDRNN conssts of scaed orthogona submatrces takng the foowng form W () [ ] w2,2 w 2,2+1 w 2,2+1 w 2,2 [ ] w (1) w (2) w (2) w (1), 1,2,..., N 2 (3) From (2) and (3) becomes evdent that the Free Form BDRNN conssts of feedback submatrces wth four dstnct eements, thus provdng a greater degree of freedom compared to the Scaed Orthogona BDRNN, whch has two weghts at each feedback submatrx. Nevertheess, as dscussed n [12], the atter network exhbts superor modeng capabtes than the Free Form BDRNN. In vew of the above, the state equatons (1) for the case of the Scaed Orthogona BDRNN are wrtten n the foowng form: x 2 1 (k) f a ( b 2 1 u(k)+w (1) w (2) x 2 (k 1) x 2 1 (k 1)+ ) 1,2,..., N 2 ( x 2 (k) f a b 2 u(k) w (2) x 2 1 (k 1)+ w (1) x 2 (k 1) ) 1,2,..., N 2 (4a) (4b) y(k) f b ( N j1c j x j (k) 2.2 The modeng method ) (4c) In terature there exst two earnng schemes especay desgned for ths knd of neura network [12, 15]. However, they are both desgned for ensurng network stabty: n [12] a smpe gradent descent agorthm s used, wth the addton of a feedfoward neura network whch ams at keeps stabe earnng, operatng n parae. A smar task s performed more effcenty n [15] wthout the need for an addtona network, by transformng the whoe earnng process to a constraned optmzaton probem. In ths work the stabty ssue s not nvestgated, snce the earnng process has an adaptaton scheme for the magntude of the weght changes, thus confnng the weght space. The earnng agorthm s based on Resent Backpropagaton (RPROP, [16]). RPROP agorthm was orgnay deveoped for statc neura networks and consttutes one of the best performng frst order earnng methods for neura networks, snce t exhbts mproved performance characterstcs by remedyng the drawbacks nherent to gradent descent earnng [17]. A varaton of RPROP for recurrent modes s gven n [18], where a smar behavor has been reported. In the case of BDRNN, the agorthm s modfed n order to take nto consderaton the speca features of the BDRNN, requrng cacuaton of the error gradents for the feedback weghts. Let us consder a tranng data set of nput output pars. The Mean Squared Error s seected as the error measure, where y d (k) s the actua observaton: E 1 (y(k) y d (k)) 2 (5) (t) and + E(t 1) are the ordered dervatves [19] θ θ of E wth respect} to a network s weght θ (θ b,c,w (1),w (2) ) at the present, t, and the precedng, t 1, epochs, respectvey. The earnng method s descrbed as foows: (a) For a weghts θ, ntaze the step szes (1) 0 Repeat (b) For a weghts θ, compute the error gradent: + E(t) θ (c) For a weghts θ, update step szes: (c.1) If + E(t) θ + E(t 1) θ (t) > 0 then mn η + } (t 1), max (6a) Natura Scences Pubshng Cor.

4 1646 P. Mastorocostas et a.: A Recurrent Neura Network based Forecastng System... (c.2) Ese f + E(t) θ (c.3) Ese (t) + E(t 1) θ max (t) < 0 then η (t 1), mn } (t 1) (6b) (6c) (d) Update weghts θ : ( + ) E(t) θ (t+ 1)θ (t)+ θ (t)θ (t) sgn (t) θ (7) Unt convergence where the step szes are bounded by mn, max n order to avod overfow/underfow probems of foatng pont varabes. The ncrease and attenuaton factors are usuay set n + [1.1,1.6] and n [0.4,0.95], respectvey. A four parameters are determned usng the tra and error approach. The adaptaton mechansm descrbed above has the advantage of correatng the step szes not to the sze of the dervatves but to ther sgns. Hence, whenever a parameter moves aong a drecton reducng E (the dervatves at successve epochs have the same sgn), ts step sze s ncreasng ndependenty of the sze of the dervatve. In ths way, the step szes can suffcenty ncrease when needed, even at the fna stage of the earnng process when the szes of the dervatves are rather sma. Addtonay, when changes n the sgn of the dervatve occur, the step sze s dmnshng to prevent the error measure from oscatng. As mentoned n [20], due to the tempora reatons exstng n a dynamc system, the extracton of the ordered parta dervatves s not straghtforward and s accompshed va a set of recursve equatons. Cacuaton of the error gradents s based on the use of Lagrange mutpers, as shown beow: b 2 1 w (1) b 2 c 2 1 λx,2 1 (k) u(k) f a(k,2 1) } (8a) c 2 λx,2 (k) u(k) f a(k,2) } (8b) λy (k) x 2 1 (k) f b(k) } (9a) λy (k) x 2 (k) f b(k) } (9b) λx,2 1 (k) x 2 1 (k 1) f a(k,2 1)+ +λ x,2 (k) x 2 (k 1) f a(k,2) } (10a) w (2) λx,2 1 (k) x 2 (k 1) f a(k,2 1) λ x,2 (k) x 2 1 (k 1) f a(k,2) } (10b) wth the Lagrange mutpers derved as foows: λ y (k) 2 (y(k) y d (k)) (11) λ x,2 1 (k)λ y (k) c 2 1 f b(k)+ + λ x,2 1 (k+ 1) w (1) f a(k+ 1,2 1) λ x,2 (k+ 1) w (2) f a(k+ 1,2) (12a) λ x,2 (k)λ y (k) c 2 f b(k)+ + λ x,2 1 (k+ 1) w (2) f a(k+ 1,2 1)+ + λ x,2 (k+ 1) w (1) f a(k+ 1,2) (12b) where f a(k + 1,) and f b(k) are the dervatves of x (k + 1) and y(k), wth respect to ther arguments. Equatons (12) are backward dfference equatons that can be soved for k 1,...,1 usng the boundary condtons: λ x,2 1 ( )λ y ( ) c 2 1 f b( ) λ x,2 ( )λ y ( ) c 2 f b( ) (13a) (13b) It shoud be noted that the earnng method s deveoped for the SISO Scaed Orthogona BDRNN. Nevertheess, the suggested method s genera, aso appcabe to mut nput mutpe output BDRNN s of any form, provded that sght modfcatons are made, based on the structura dfferences. 3 Expermenta resuts 3.1 Data presentaton accuracy measures The ca data come from the Ca Deta Records (CDR) of the Prvate Branch Exchange (PBX) of a arge Unversty wth more than empoyees and students, and an extended teecommuncatons nfrastructure wth more than teephones. The data set covers a perod of 10 years, January 1998 to December 2007, and conssts of the monthy cas to natona and mobe destnatons. Cas to natona destnatons comprse amost haf the voume of the tota outgong cas from the campus. On the other hand, cas to mobe destnatons are subject to hgher tarffs and they demonstrate an ncreasng trend durng the ast ffteen years. Accordng to the Internatona Teecommuncaton Unon (ITU), mobe servces Natura Scences Pubshng Cor.

5 App. Math. Inf. Sc. 7, No. 5, (2013) / attaned a penetraton rate of % for Greece n 2009 [21]. The data set s dvded to two subsets: The tranng set, whch s used to perform parameter tunng, and the testng set, whch s used for the evauaton of the forecasts. The tranng set s chosen to be 9 years ong (108 observatons) and the vadaton set 1 year ong (12 observatons). Due to the varaton of days beongng n dfferent months,.e. February has 28 whe January has 31 days, a data are normazed accordng to (X k s the actua vaue): y d (k)x k /12 no ofdaysnmonthk (14) Due to ts recurrent nature, the forecaster s traned n parae mode. Snce the testng set s not used n mode fttng, the testng set s forecasts are genune forecasts and can be used to evauate the forecastng abty of each mode. The forecastng accuracy can be evauated by means of three accuracy measures ( s the sze of the data set and y(k) s the forecaster s output). The smaer vaue of each statstc ndcates the better ft of the method to the observed data. The Root Mean Squared Error (RMSE): k E 1 f (y(k) y d (k)) 2 The Mean Absoute Percentage Error (MAPE): MAPE 100 y d (k) y(k) y d (k) (15a) (15b) The The s U-statstc, whch aows a reatve comparson of forma methods wth nput output approaches and aso squares the errors nvoved, so that arge errors are gven much more weght than sma errors. It s gven by: U 1 1 ( ) y(k+1) yd (k+1) 2 y d (k) (15c) ( ) yd (k+1) y d (k) 2 y d (k) The tme seres of natona and mobe cas are hosted n Fg. 2a and 2b, respectvey. It s evdent that there exsts a dstnct seasona pattern, whch s made prevaent from the mnmum that occurs n August. Moreover, the number of cas to mobe destnatons shows an ncreasng trend whch comports wth reports on mobe servces penetraton [21]. 3.2 Comparatve anayss In order to nvestgate whether the recurrent neura system s capabe of dscoverng the tempora Number of natona cas Number of mobe cas JUL 2001 JAN 2001 JUL 2000 JAN 2000 JUL 1999 JAN 1999 JUL 1998 JAN 1998 JUL 2001 JAN 2001 JUL 2000 JAN 2000 JUL 1999 JAN 1999 JUL 1998 JAN 1998 JUL 2002 JAN 2002 (a) JUL 2002 JAN 2002 (b) DEC 2007 JUL 2007 JAN 2007 JUL 2006 JAN 2006 JUL 2005 JAN 2005 JUL 2004 JAN 2004 JUL 2003 JAN 2003 Date Tranng set DEC 2007 JUL 2007 JAN 2007 JUL 2006 JAN 2006 JUL 2005 JAN 2005 JUL 2004 JAN 2004 JUL 2003 JAN 2003 Date Tranng set Vadaton set Vadaton set Fg. 2: Monthy number of outgong cas to (a) natona and (b) mobe destnatons. dependences of both tme seres of outgong cas, a snge nput snge output structure has been seected. The nput s the number of the natona/mobe cas of the prevous month (u(k)y d (k 1)). Severa BDRNN s wth dfferent structura characterstcs are examned and the seecton of the fna mode parameter combnaton s based on the crtera of (a) effectve predcton of the ca voume and (b) forecasters of moderate compexty. The seected structura characterstcs and the earnng parameters vaues of the fna modes are gven n Tabe 1. In order to compare the proposed forecaster wth exstng estabshed forecasters that were apped to ths partcuar probem n the past, a comparatve anayss wth fourteen other modes s conducted, on the bass of the accuracy measures mentoned n Subsecton 3.1. Three of the modes come from the fed of computatona ntegence (the statc fuzzy system descrbed n [8] and the recurrent neurofuzzy systems LR NFFS and ReNFFOR), whe the other eeven are we estabshed Natura Scences Pubshng Cor.

6 1648 P. Mastorocostas et a.: A Recurrent Neura Network based Forecastng System... Tabe 1: Structura characterstcs and earnng parameters Type of cas No of bocks n+ n mn max o Epochs Natona E Mobe E Tabe 2: Comparatve anayss (testng data set) Type of cas Natona cas Mobe cas Mode RMSE MAPE Thes U RMSE MAPE Thes U Proposed Forecaster LR NFFS ReNFFOR OLS FM NF LESA M LESA ADD SES Hots Lnear Wnter s MS Wnter s AS Damped NoS Damped MS Damped AS SARIMA statstca forecastng modes. The resuts for each one of these modes are presented n Tabe 2; bod numbers ndcate best ft. The performance of the competng rvas s taken from the correspondng references. The best ft modes for each ca data category are depcted n the foowng pots. We choose to present the forecasts produced by the proposed BDRNN mode, aong wth ts cosest fuzzy/neurofuzzy and ts cosest cassc compettors. In Fg. 3 the reader may see a comparson for the best ft modes for the case of natona cas. The pot reveas that the proposed mode foows the varaton of the actua tme seres coser than ts compettors, a fact that s aso evdent from ts better performance statstcs. In the same pot the 95% upper (UCL) and ower (LCL) confdence eves are depcted. These were estmated durng the SARIMA fttng process. It shoud aso be stressed that a three forecasts ft we wthn these confdence ntervas and woud bear scrutny wth even tghter confdence. Accordng to Fdes et a., damped trend modes are consdered a benchmarorecastng method for a others to beat [22]. In ths sense the proposed BDRNN mode exhbts exceptona performance. Vsua observaton of Fg. 4 reveas the dfferences between the proposed forecaster and ts best rvas for the case of mobe cas. The BDRNN gves better forecast, n the sense that t foows the pattern of the evouton of the seres more cosey as t dentfes a mnma and maxma. The 95% confdence ntervas for the forecasts Number of natona cas JAN DEC Actua data BDRNN forecast LR-NFFS forecast SARIMA forecast LCL UCL JUN MAY APR MAR FEB Date DEC NOV OCT SEP AUG JUL Fg. 3: Comparson of the forecastng abty of the proposed BDRNN forecaster wth the best representatves of the two rva mode fames and the observed number of cas to natona destnatons. 95% confdence nterva s aso depcted. were estmated durng the fttng process of the Damped MS mode. 4 Concusons In ths paper a recurrent neura network has been proposed and has been evauated by havng been apped Natura Scences Pubshng Cor.

7 App. Math. Inf. Sc. 7, No. 5, (2013) / servces n ther organzaton, ether for proft maxmzaton or for unnecessary cost reducton. Number of mobe cas JAN DEC Actua data BDRNN forecast LR-NFFS forecast DAMPED MS forecast LCL UCL JUN MAY APR MAR FEB DEC NOV OCT SEP AUG JUL Acknowedgement The authors woud ke to thank the members of the Arstote Unversty s Teecommuncatons Center for ther contrbuton of anonymsed data. The authors wsh to acknowedge fnanca support provded by the Research Commttee of the Technoogca Educaton Insttute of Serres under grant SAT/IC/ /10. Date Fg. 4: Comparson of the forecastng abty of the proposed BDRNN forecaster wth the best representatves of the two rva mode fames and the observed number of cas to mobe destnatons. 95% confdence nterva s aso depcted. on rea word teecommuncatons data. The forecaster s a two ayer network, whose hdden ayer conssts of pars of nterconnected neurons wth nterna feedback. Its modeng quates have been nvestgated through a comparatve anayss wth a seres of we estabshed forecastng modes and recent fuzzy and neurofuzzy forecasters. Two dfferent types of cas, accordng to ther destnaton, have been examned. These are cas to natona and cas to mobe destnatons. They were chosen because the former represents more than 50% of the outgong ca voume and the ater corresponds to more than 1/3 of the teecommuncatons costs for the organzaton under study. Accordng to the resuts, the proposed forecaster s a promsng computatona ntegence s approach to the probem of teecommuncatons ca voume forecastng, snce t s capabe of capturng the tme seres dynamcs through ts nterna feedback connectons. Therefore t does not requre pror knowedge of the order of the tme seres or computatonay expensve nput seecton agorthms. Moreover, ts structure s qute smper than those of fuy recurrent networks or networks wth output feedback. From the anayss t s concuded that t can cope wth the probem better than ts rvas, ether the tradtona forecastng methods or the computatona ntegence ones. Forecastng may act as a vauabe too for teecommuncatons managers and s used for network traffc management, nfrastructure optmzaton and pannng, and the scenaro pannng process. Makng use of hstorca data, managers may predct future demand by creatng a reasonaby accurate forecast of the ca voume. Ths may be used to make educated strategc decsons on how to charge and b teecommuncatons References [1] S. Makrdaks, S. Wheewrght, V. McGee, Forecastng: Methods and Appcatons, 3rd ed., Wey, New York, (1998). [2] C. Has, S. Goudos, J. Sahaos, Seasona Decomposton and Forecastng of Teecommuncaton Data: A comparatve Case Study, Technoogca Forecastng & Soca Change, 73, (2006). [3] C. Hot, Forecastng Trends and Seasonas by Exponentay Weghted Movng Averages, ONR Memorandum, 52 Carnege Insttute of Technoogy, Pttsburg (1957). Reprnted: Internatona Journa of Forecastng, 20, 5-10 (2004). [4] P. Wnters, Forecastng Saes by Exponentay Weghted Movng Averages, Management Scence, 6, (1960). [5] E. Gardner Jr., Exponenta Smoothng: The State of the Art, Journa of Forecastng, 4, 1-28 (1985). [6] G. Box, G. Jenkns, Tme Seres Anayss: Forecastng and Contro, 2nd ed., Hoden-Day, San Francsco, (1976). [7] G. Madden, T. Joachm, Forecastng Teecommuncatons Data wth Lnear Modes, Teecommuncatons Pocy, 31, (2007). [8] P. Mastorocostas, C. Has, S. Dova, D. Varsams, Forecastng of Teecommuncatons Tme-seres va an Orthogona Least Squares-based Fuzzy Mode, Proceedngs of 21st IEEE Internatona Conference on Fuzzy Systems, (2012). [9] P. Mastorocostas, C. Has, A Computatona Integence Forecastng System for Teecommuncatons Tme Seres, Engneerng Appcatons of Artfca Integence 25, (2012). [10] P. Mastorocostas, C. Has, ReNFFor: A Recurrent Neurofuzzy Forecaster for Teecommuncatons Data, Neura Computng and Appcatons, (2012). [11] T. Takag, M. Sugeno, Fuzzy dentfcaton of systems and ts appcatons to modeng and contro, IEEE Transactons on Systems, Man, and Cybernetcs, 15, (1985). [12] S. Svakumar, W. Robertson, W. Phps, On-Lne Stabzaton of Bock-Dagona Recurrent Neura Networks, IEEE Transactons on Neura Networks, 10, (1999). [13] A. Tso, A. Back, Locay Recurrent Gobay Feedforward Networks: A Crtca Revew of Archtectures, IEEE Transactons on Neura Networks, 5, (1994). [14] P. Frascon, M. Gor, and G. Soda, Loca feedback mutayered networks, Neura Computaton, 4, (1992). Natura Scences Pubshng Cor.

8 1650 P. Mastorocostas et a.: A Recurrent Neura Network based Forecastng System... [15] P. Mastorocostas, J. Theochars, A Stabe Learnng Method for Bock-Dagona Recurrent Neura Networks: Appcaton to the Anayss of Lung Sounds, IEEE Transactons on Systems, Man, and Cybernetcs Part B, 36, (2006). [16] M. Redmer, H. Braun, A Drect Adaptve Method for Faster Backpropagaton Learnng: The RPROP Agorthm, Proceedngs of IEEE Internatona Jont Conference on Neura Networks, (1993). [17] C. Ige, M. Husken, Emprca Evauaton of the Improved RPROP Learnng Agorthms, Neurocomputng, 50, (2003). [18] P. Mastorocostas, Resent Back Propagaton Learnng Agorthm for Recurrent Fuzzy Neura Networks, IET Eectroncs Letters, 40, (2004). [19] P. Werbos, Beyond Regresson: new toos for predcton and anayss n the behavora scences, Ph.D. Thess, Harvard Unv., Cambrdge, MA (1974). [20] S. Pche, Steepest Descent Agorthms for Neura Network Controers and Fters, IEEE Transactons on Neura Networks, 5, (1994). [21] ITU-T ICTeye Reports, Avaabe onne : (2010). [22] R. Fdes, K. Nkoopouos, S. Crone, A. Syntetos, Forecastng and Operatona Research: a Revew, Journa of the Operatona Research Socety, 59, (2008). Pars Mastorocostas was born n Thessaonk, Greece, n He receved the Dpoma and Ph.D. degrees n Eectrca & Computer Engneerng from Arstote Unversty of Thessaonk, Greece. Presenty he serves as Professor at the Dept. of Informatcs & Communcatons, Technoogca Educaton Insttute of Serres, Greece. Hs research nterests ncude fuzzy and neura systems wth appcatons to dentfcaton and cassfcaton processes, data mnng and scentfc programmng. user behavor modeng and profng, cassfcaton and characterzaton. Dmtrs Varsams was born n Serres, Greece, n He receved the B.Sc. degree n Mathematcs from the Dept. of Mathematcs, Unversty of Ioannna, Greece, the M.Sc. n Theoretca Computer Scence, Contro and Systems Theory, and the Ph.D. degrees from the Dept. of Mathematcs, Arstote Unversty of Thessaonk, Greece. Presenty he serves as Lecturer at the Dept. of Informatcs & Communcatons, Technoogca Educaton Insttute of Serres, Greece. Hs research nterests ncude scentfc programmng, computatona methods, contro system theory and numerca anayss. Stergan Dova was born n Thessaonk, Greece, n She receved the B.Sc. degree from the Dept. of Informatcs & Communcatons, Technoogca Educaton Insttute of Serres, Greece. Presenty, she s competng her M.Sc. n Informaton Systems at the Unversty of Macedona, Greece. Her research nterests ncude fuzzy systems, neura networks, agorthms anayss and scentfc computng. Constantnos Has was born n Thessaonk, Greece, n He receved the B.Sc., M.Sc. and Ph.D. degrees from the Dept. of Physcs, Arstote Unversty of Thessaonk, Greece, and the M.Sc. degree n Informaton Systems from the Unversty of Macedona, Greece. Presenty he serves as Assstant Professor at the Dept. of Informatcs & Communcatons, Technoogca Educaton Insttute of Serres, Greece. Hs research nterests ncude the appcaton of data mnng technques on communcatons & networkng probems, forecastng, Natura Scences Pubshng Cor.