Research Article A Network Traffic Prediction Model Based on Quantum-Behaved Particle Swarm Optimization Algorithm and Fuzzy Wavelet Neural Network

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1 Dscrete Dynamcs n ature and Socety Volume 26, Artcle ID 43556, pages Research Artcle A etwork Traffc Predcton Model Based on Quantum-Behaved Partcle Swarm Optmzaton Algorthm and Fuzzy Wavelet eural etwork Kun Zhang, Zhao Hu, Xao-Tng Gan, and Jan-Bo Fang School of Mathematcs and Statstcs, Chuxong ormal Unversty, Chuxong, Yunnan 675, Chna Correspondence should be addressed to Kun Zhang; zhangkunpost@qqcom Receved 2 ovember 25; Revsed 4 January 26; Accepted 3 February 26 Academc Edtor: Ahmed Kattan Copyrght 26 Kun Zhang et al Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted Due to the fact that the fluctuaton of network traffc s affected by varous factors, accurate predcton of network traffc s regarded as a challengng task of the tme seres predcton process For ths purpose, a novel predcton method of network traffc based on QPSO algorthm and fuzzy wavelet neural network s proposed n ths paper Frstly, quantum-behaved partcle swarm optmzaton (QPSO) was ntroduced Then, the structure and operaton algorthms of WF are presented The parameters of fuzzy wavelet neural network were optmzed by QPSO algorthm Fnally, the QPSO-FW could be used n predcton of network traffc smulaton successfully and evaluate the performance of dfferent predcton models such as BP neural network, RBF neural network, fuzzy neural network, and FW-GA neural network Smulaton results show that QPSO-FW has a better precson and stablty n calculaton At the same tme, the QPSO-FW also has better generalzaton ablty, and t has a broad prospect on applcaton Introducton Wth the rapd development of computer network technology, network applcatons have nfltrated every corner of humansocetyandplayanmportantrolenvarousndustres and stuatons Snce the network topology structure s gradually complcated, the problem of network s emergences and congeston are more and more serous Through montorng and accuracy predcton of network traffc, t can prevent network congeston and can effectvely mprove the utlzaton rate of the network [] In general, the network traffc data s a knd of tme seres data and the problem of network traffc predcton s to forecast future network traffc rate varatons as precsely as possble based on the measured hstory The tradtonal predcton model, such as Markov model [2], ARMA (Autoregressve Movng Average) model [3], ARIMA (Autoregressve Integrated Movng Average) model [4], and FARIMA (Fractonal Autoregressve Integrated Movng Average) [5] model, has been proposed As the network traffc s affected by many factors, the network traffc tme seres show qute obvous multscale, long-range dependence, and nonlne characterstc The methods mentoned above have the weakness of lowlevel effcency [6] An artfcal neural network (A) s an analyss paradgm that s roughly modeled after the massvely parallel structure of the bran Artfcal neural networks can be thought of as black box devces that accept nputs and produceoutputsandareabletogvebetterperformancen dealng wth the nonlnear relatonshps between the output and the nput theoretcally [7] Although artfcal neural networks have been successfully used for modelng complex nonlnear systems and predctng sgnals for a wde range of engneerng applcatons, artfcal neural networks (As) have lmted ablty to characterze local features, such as dscontnutes n curvature, jumps n value or other edges [8] These local features, whch are located n tme and/or frequency, typcally embody mportant process-crtcal nformaton such as aberrant process modes or faults The fuzzy neural networks (F) are the hybrd systems whch combne both advantages of the fuzzy systems and artfcal neural networks The F possesses the merts of

2 2 Dscrete Dynamcs n ature and Socety the low-level learnng and computatonal power of neural networks, and the hgh-level human knowledge representaton and thnkng of fuzzy theory [9] A fuzzy wavelet neural network (FW) s a new network structure that combnes wavelet theory wth fuzzy logc and s The synthess of a fuzzy wavelet neural nference system ncludes the determnaton of the optmal defntons of the premse and the consequent part of fuzzy IF-THE rules [] However, many fuzzy neural network models, ncludng FW, have common problems derved from ther fundamental algorthm [] For example, the desgn process for F and FW combned tapped delays wth the backpropagaton (BP) algorthm to solve the dynamc mappng problems [2] Unfortunately, the BP tranng algorthm has some nherent defects [3, 4], such as low learnng speed, exstence of local mnma, and dffculty n choosng the proper sze of network to sut a gven problem Thus the systems whch employ basc fuzzynferencetheorymakethedegreeofeachruleextremely small and often make t underflow when the dmenson of the task s large In such a stuaton, the learnng and nference cannot be carred out correctly As a varant of PSO, quantum-behaved partcle swarm optmzaton (QPSO) s a novel optmzaton algorthm nspred by the fundamental theory of partcle swarm and features of quantum mechancs such as the use of Schrödnger equaton and potental feld dstrbuton [5] As a global optmzaton algorthm, the QPSO can seek many local mnmaandthusncreasethelkelhoodoffndngtheglobal mnmum Ths advantage of the QPSO can be appled to neural networks to optmze the topology and/or weght parameters [6] In order to predct the network traffc more accurately, a predcton model of network traffc based on QPSO algorthm and fuzzy wavelet neural network s proposed n ths paper The network traffc data s traned by QPSO and fuzzy wavelet neural network and weghts are progressvely updated untl the convergence crteron s satsfed The objectve functon to be mnmzed by the QPSO algorthm s the predcted error functon The rest of ths paper s arranged as follows Secton 2 gves a bref ntroducton to classcal PSO algorthm and quantum-behaved partcle swarm optmzaton (QPSO) algorthm In Secton 3, the fuzzy wavelet neural network s ntroduced and the fuzzy wavelet neural network based on QPSO (QPSO-FW) algorthm s presented n detal In Secton 4, smulaton results are presented Performance metrcs of the several predcton methods are analyzed and compared n Secton 5 Fnally, some conclusons are gven n Secton 6 2 Quantum-Behaved Partcle Swarm Optmzaton 2 Classcal Partcle Swarm Optmzaton Partcle swarm optmzaton (PSO) s an evolutonary computaton technque that s proposed by Kennedy and Eberhart n 995 [7] Smlarly to other genetc algorthms (GA), PSO s ntalzed wth a populaton of random solutons However, t s unlke GA, PSO does not have operators, such as crossover and mutaton In the PSO algorthm, each potental soluton, called partcles, moves around n a multdmensonal search space wth a velocty constantly updated by the partcle s own experence and the experence of the partcle s neghbors or the experence of the whole swarm [8] In the PSO, each partcle keeps track of ts coordnates n the search space whch are assocated wth the best soluton t has acheved so far and ths value s called pbest Another best value that s tracked by the global verson of the partcle swarm optmzer s the overall best value, and ts locaton, obtaned so far by any partcle n the populaton [9] Ths locaton s called gbest The process for mplementng the global verson of PSO s gven by the followng steps Step Intalze a populaton (array) of partcles wth random postons and veloctes n the D-dmensonal problem space For a D-dmensonal problem wth number of partcles, the poston vector X t and velocty vector V t are represented as where =,2,, X t =[Xt,,Xt,2,,Xt,D ], () V t =[V t,,vt,2,,vt,d ], Step 2 For each partcle, evaluate the desred optmzaton ftness functon n D varables Step 3 Compare each partcle s ftness evaluaton wth the partcle s pbest If the current value s better than pbest, then set the pbest valueequaltothecurrentvalueandthe pbest locaton equal to the current locaton n D-dmensonal space Step 4 Compare the ftness evaluaton wth the populaton s overall prevous best If the current value s better than gbest, then reset gbest to the current partcle s array ndex and value Step 5 Update the velocty and poston of the partcle accordng to (2) and (3), respectvely One has V t+ X t+ =Vt +c r (pbest t Xt ) (2) +c 2 r 2 (gbest t d Xt ), =Xt +Vt+, (3) where c and c 2 are two postve constants, known as the cogntve and socal coeffcents, whch control the relatve proporton of cognton and socal nteracton, respectvely, and the values of c and c 2 were decreased wth each teraton [2] r and r 2 are two random values n the range [, ] V t, X t,andpbestt arethevelocty,poston,andthepersonal best of th partcle n dth dmenson for the tth teraton, respectvely The gbest t d s the dth dmenson of best partcle n the swarm for the tth teraton

3 Dscrete Dynamcs n ature and Socety 3 Step 6 Loop to Step 2 untl a stop crteron s met, usually a suffcently good ftness or a maxmum number of teraton generatons 22 Quantum-Behaved Partcle Swarm Optmzaton Motvated by concepts n quantum mechancs and partcle swarm optmzaton, Sun et al proposed a new verson of PSO, quantum-behaved partcle swarm optmzaton (QPSO) [2] In the QPSO, the state of a partcle s depcted by a wave functon ψ(x, t), nstead of poston and velocty The probablty densty functon of the partcle s poston s ψ(x, t) 2 n poston X [22] Assume that, at teraton t, partcle moves n Ddmensonal space wth a δ potental well centered at pbest t,j on the dth dmenson The wave functon at teraton t+s gven by the followng equaton: ψ(x t+ )= exp ( Xt pbestt L t L t ), (4) where L t s the standard devaton of the double exponental dstrbuton, varyng wth teraton number t Hencethe probablty densty functon Q s defned as Q(X t+ )= ψ(xt+ ) 2 = L t exp ( 2 Xt+ pbestt L t and the probablty dstrbuton functon F s gven by the followng equaton: F(X t+ )= exp ( 2 Xt+ pbestt L t ) (5) ) (6) By usng Monte-Carlo method, the jth component of poston X at teraton t+can obtan by the followng equaton: X t+ =pbestt ± 2 Lt ln ( u t+ ), (7) where u t+ s a unform random number n the nterval [, ] The value of L t+ s calculated as L t =2α Ct d Xt, (8) where parameter α s known as the contracton-expanson (CE) coeffcent, whch can be tuned to control the convergence speed of the algorthms [23] C t s the mean best poston (mbest) and s defned as C t =(C t,ct 2,,Ct D ) =( M M = pbest t M,, pbest t M (9),2 M,, pbest t,d M ), = = where M stheszeofthepopulatonhencethepostonof the partcle s updated accordng to the followng equaton: X t+ =pbestt ±α Ct d Xt ln ( u t+ ) () From (4) and (), the new poston of the partcle s calculated as X t+ pbest t { +α Ct d Xt ln ( = { pbest t α Ct d Xt ln ( { u t+ u t+ ) k (5, ], ) k (, 5], () where k s a random number n the range [, ] α s lnearly decreasng factor from to 3 wth teraton as α t =α max α max α mn t, (2) t max where t max s the maxmum number teraton used n algorthm 3 Fuzzy Wavelet eural etwork Based on QPSO 3 The Wavelet Base Functon In L 2 (R),awaveletdctonary s constructed by dlatng and translatng from a wavelet base functonψ(t) of zero average [24]: + ψ (t) dt = (3) whch s dlated wth a scale parameter a and translated by b ψ a,b (t) = ψ (t b) a> (4) a a 32 Fuzzy Wavelet eural etwork The basc archtecture of fuzzy wavelet neural network could be descrbed as a set of Takag-Sugeno models Assume that there are r rules n therulebaseandthetakag-sugenofuzzyf-thenrulesare usually n the followng form: R :IF x s F,,x r s F r, THE y =α +α x + +α r x r, (5) where x,x 2,,x r are nput of T-S rule, F j s the th lngustc varable value of the jth nput, whch s a fuzzy set characterzed by wavelet functon α j s constant coeffcents whch are usually referred to as consequent parameters determned durng the tranng process Fgure shows the archtecture of the proposed FW modelng The FW s a 4-layer feedforward network and detaled descrptons and equatons for each layer are gven here

4 4 Dscrete Dynamcs n ature and Socety x x n u( ) u( ) u( ) u( ) Fgure : The archtecture of FW Layer (nput varables layer) Ths layer s the nput sgnals of the FW and each node of ths layer, respectvely, represents an nput lngustc varable The node output and thenodenputarerelatedby I () =x O () =I () =,2,,n, Σ Σ y y m (6) where I () and O () are, respectvely, the nput and output of th node n Layer Layer 2 (membershp functons layer) In ths layer, nodes represent fuzzy sets n the antecedents of fuzzy rules The outputs of ths layer are the values of the membershp functons The membershp functon s wavelet functon and s often taken as u j =φ( x b j a j = cos ( x b j 2a j ) ) exp ( 2 (x b j a j 2 ) ), (7) where u j s the jth membershp functon of x and j =,2,,p a j,bj are the dlaton and translaton parameter of wavelet functon In ths layer, the relaton between the output and nput s represented as I (2) =O (), O (2) =u (I (2) ) = cos ( O() 2a j b j ) exp ( 2 (O() where =,2,,n, j=,2,,p a j b j 2 ) ), (8) Layer 3 (rule layer) In ths layer, the number of rules s equal tothenumberofnodestheoutputcanbecalculatedas follows accordng to the AD (mn) operaton [2]: O (3) =I (3) = mn (O (2) j,o(2) 2j,,O(2) nj ) (9) Layer 4 (output layer) Ths layer conssts of output nodes The output are gven by I (4) k = m p O (3) ω(3) = j= y k =O (4) k =, (2) I (4) k m = p j= O(3) (2) To tran the parameters of FW, backpropagaton (BP) tranng algorthm s extensvely used as a powerful tranng method whch can be appled to the forward network archtecture [25] For ths purpose, mean square error (MSE) s selected as performance ndex whch s gven by J= 2 (D Y)T (D Y), (22) where D and Y are current and desred output values of network, correspondngly The all adjustable parameters of FW can be calculated by the followng formulas: ω (3) (t+) =ω (3) (t) η J ω (3) (t) +α Δω(3) (t), b (2) (t+) =b (2) J (t) η b (2) (t) +α Δb(2) (t), a (2) (t+) =a (2) (t) η J a (2) (t) +α Δa(2) (t), (23) where t represents the backward step number and η and α are the learnng and the momentum constants, dfferng n the ranges to and to 9, respectvely 33 Fuzzy Wavelet eural etwork Traned by QPSO Algorthm Computatonal ntellgence has ganed popularty n tranng of neural networks because of ther ablty to fnd a global soluton n a multdmensonal search space The QPSO algorthm s a global algorthm, whch has a strong ablty to fnd global optmstc results and QPSO algorthm has proven to have advantages than the classcal PSO due ts less control parameters [26] Therefore, by combnng the QPSO wth the fuzzy wavelet neural network, a new algorthm referred to as QPSO-FW algorthm s formulated n ths paper When QPSO algorthm s used to tran the FW model, a decson vector represents a partcular group of network

5 Dscrete Dynamcs n ature and Socety 5 parameters ncludng the connecton weght, the dlaton and translaton parameter It s further denoted as 9 8 X =(a,,,a,p,,a n,p,b,,,b,p,,b n,p,ω,,, ω,p,,ω m,p ), (24) where a and b are the dlaton and translaton parameter of wavelet functon n Layer 2 ω l,k are the connecton weght n (2) Snce a component of the poston corresponds to a network parameter, FW s structured accordng the partcle s poston vector Tranng the correspondng network by nputtng the tranng samples, we can obtan an error value computed by (22) In a word, the mean square error s adopted as the objectve functon to be mnmzed n FW based on QPSO The specfc procedure for the QPSO-FW algorthm canbesummarzedasfollows Step Defne the structure of the FW accordng to the nput and output sample Step 2 Treat the poston vector of each partcle as a group of network parameter by (24) Step 3 Intalze the populaton by randomly generatng the poston vector X of each partcle and set pbest =X Step 4 Evaluate m and α of QPSO algorthm usng (9) and (2), respectvely Step 5 Conclude the objectve functon of each partcle by (22) Step 6 (update pbest) Each partcle s current ftness value s compared wth prevous best value pbestif the current value s better than the pbest value, then set the pbest value to the current value Step 7 (update gbest) Determne the swarm best gbest as mnmum of all the partcles pbest Step 8 Judge the stoppng crtera, f the maxmal teratve tmes are met, stop the teraton, and the postons of partcles are the optmal soluton Otherwse, the procedure s repeated from Step 4 4 Smulaton Results Expermentaldatasetconsstsofhoursobservatonswhch comes from montorng the traffc between clents n our campus network and servers The mnmal tme nterval n network traffc tme seres s seconds Fgure 2 shows the normalzaton of network traffc tme seres In ths paper, the desgn of a dscrete flter predctor conssts n fndng the relaton between the future data x(n) and the past observatons x(n ),x(n 2),,x(n M), where M sthenumberofconsderednputelementsthe Fgure 2: The normalzaton of network traffc tme seres predctor relatonshp can be descrbed by the followng convoluton sum [27]: x (n) = M k= h o (k) x (n k), (25) where h o (k) (k =,2,,M)sthefltercoeffcentvector x(n k) s an k-step backward sample x(n) s the desred output In order to test the performance of the predcton model, the front s the tranng data, and the latter 2 s the predcton data The number of tme seres wndows was set as 3, whch meant that the forth of measurement data would be predcted from the past three of measurement data In the establshed predcton model base on QPSO- FW algorthm, the number of membershp functons s fve, the number of nput varables layer nodes s three, and the number of output layer nodes s one The populaton sze of QPSO algorthm s 2 partcles and D-dmensonal search space of partcle s 45 t max s 3 α max s and α mn s 5 The CE coeffcent decreases lnearly from to 3 durng the search process accordng to (2) and After 5 tmes teratons, the cost functon J of QPSO-FW neural network was 2887 The connecton weght between nput varables layer andoutputlayersgvenby a= [ [ ], [ ] b= [ [ ], [ ]

6 6 Dscrete Dynamcs n ature and Socety ω= 436, [ 965 ] [ 884 ] (26) where y sthemeanvalueandsgvenby y = = y (3) It can be seen that,f MSE =, the predcton performance s perfect, and f MSE =, the predcton s a trval predctor whch statstcally predcts the mean of the actual value If MSE >, t means that the predcton performance s worse than that of trval predctor [29] (3) MAPE (Mean Absolute Percentage Error) can calculate the predcton error as a percentage of the actual value MAPE s defned as MAPE = = y y y % (3) (4) Coeffcent correlaton s the covarance of the two varables dvded by the product of ther ndvdual standard devatons It s a normalzed measurement of how the two varables are lnearly related The coeffcent of correlaton (r) s gven as follows: where a s the dlaton parameter of the wavelet functon b s the translaton parameter of wavelet functon ω s theconnectonweghtoftheoutputlayerfgure3shows the membershp functons of nput varable x(t), x(t ), and x(t 2) n FW unts Fgure 4 shows the QPSO- FW convergence curves These predcton results show that the QPSO-FW model s an effectve, hgh-accuracy predcton model of network traffc n Fgure 5 5 Performance Metrcs In order to evaluate the predcton model more comprehensvely, the followng performance metrcs are used to estmate the predcton accuracy () MSE (mean square error) s a scale-dependent metrc whch quantfes the dfference between the predcted values andtheactualvaluesofthequanttybengpredctedby computng the average sum of squared errors [28]: MSE = = (y y ) 2, (27) where y s the actual value, y s the predcted value, and s the total number of predctons (2) MSE (ormalzed Mean Square Error) s defned as MSE = σ 2 (y y ) 2, (28) where σ 2 denotes the varance of the actual values durng the predcton nterval and s gven as follows: σ 2 = = = (y y) 2, (29) r= COV (Y, Y), (32) σ Y σ Y where σ Y and σ Y ndcate the standard devaton of the actual andthepredctedvaluesaregvenby(33) σ Y = = (y y) 2 (33) COV(X, Y) s the covarance between X and YItsobtaned as follows: COV = = (x x) (y y) (34) Values for the correlaton coeffcent range are [, ]If r=, there s a perfect postve correlaton between the actual and the predcted values, whereas r = ndcates a perfect negatve correlaton If r =, we have a complete lack of correlaton among the datasets (5) Coeffcent of effcency (CE) s defned as CE = = (y y ) 2 = (y y) 2, (35) where the doman of the effcency coeffcent s (, ] If CE =, there s a perfect ft between the observed andthepredcteddatawhenthepredctoncorrespondsto estmatng the mean of the actual values, CE = IfCE (, ], t ndcates that the average of the actual values s a better predctor than the analyzed predcton method The closer CE s to, the more accurate the predcton s In order to test the QPSO-FW method s valdty and accuracy, we carred out the experment whch s compared wth the other methods The predcton model based on BP

7 Dscrete Dynamcs n ature and Socety Input varable x(t) (a) Membershp functons of nput varable x(t) n FW unts 5 5 Input varable x(t ) (b) Membershp functons of nput varable x(t ) n FW unts 5 5 Input varable x(t 2) (c) Membershp functons of nput varable x(t 2) n FW unts Fgure 3: The membershp functons of nput varable x(t), x(t ),andx(t 2) n FW unts Mean square error Iteraton Fgure 4: The QPSO-FW convergence curves Table : Performance comparson of the sx predcton methods Method MSE MSE MAPE r CE QPSO-FW BP RBF F FW-GA ARIMA neural network s 3 layers, n whch the number of nput layer nodes s 3, the number of hdden layer nodes s 7, the number of output layer nodes s, and the number of teratons s In the predcton model based on RBF neural network, the radal based dstrbute functon spread s 5 The predcton model based on F neural network s 4 layers and the number of membershp functons s fve The number of teratons s The archtecture of predcton model based on fuzzy wavelet neural network and genetc algorthm (FW-GA) s the same wth the QPSO-FW method The populaton sze of GA s The crossover type s one-pont crossover, and crossover rate s 6 Mutaton rate s and the number of teratons s 5 [3] The predcton model based on ARIMA model s bult The estmaton of the model parameters s done usng Maxmum Lkelhood Estmaton and the best model s chosen as ARIMA (5,, 3) [3] Fgures 6, 7, 8, 9, and show that predcton results wth BP, RBF, F, and FW-GA neural network and ARIMA model, respectvely The performance comparson of the four predcton methods s shown n Table From Table, one can look further nto the predcton performance among the four predcton models By comparng the value of MSE, MSE, MAPE, coeffcent correlaton, and CE, QPSO- FW demonstrates better predcton accuracy than the other three methods Therefore, the expermental results n ths secton show that the predcton method based on QPSO- FW s much more effectve than BP, RBF, F, FW- GA, and ARIMA It can be seen that the predcton method based on QPSO-FW s a better method to predct the tme seres of network traffc In order to test the predcton stablty of each mode, the fve predcton methods were predcted tmes, respectvely Fgures, 2, and 3 show that predcton results wth QPSO- FW models are much more stable than BP, F, and FW-GA 6 Concluson Predctng the drecton of movements of network traffc s mportant as they enable us to detect potental network traffc jam spots Snce the network traffc s affected by many factors, the data of network traffc have the volatlty and self-smlarty features and the network traffc predcton becomes a challenge task In ths paper, the QPSO-FW method has been presented to predct the network traffc The QPSO-FW combnes the QPSO, whch has the mert of powerful global exploraton capablty, wth FW whch can extract the mappng relatons between the nput and output data The parameters of FW neural network are obtaned by quantum-behaved partcle swarm optmzaton (QPSO) and the tme seres of network traffc data was modeled by QPSO-FW Fnally, experments showed that QPSO-FW model has faster and better performance n

8 8 Dscrete Dynamcs n ature and Socety The network traffc Error Predcted value Actual value (a) The predcted results wth QPSO-FW method (b) The error between the actual value and ts predcted value Fgure 5: Predcton wth QPSO-FW model The network traffc Predcted value Actual value (a) The predcted results wth BP neural network Error (b) The error between the actual value and ts predcted value Fgure 6: Predcton results wth BP neural network The network traffc Predcted value Actual value (a) The predcted results wth RBF method Error (b) The error between the actual value and ts predcted value Fgure 7: Predcton results wth RBF neural network The network traffc Predcted value Actual value (a) The predcted results wth F method Error (b) The error between the actual value and ts predcted value Fgure 8: Predcton results wth F neural network

9 Dscrete Dynamcs n ature and Socety 9 The network traffc Predcted value Actual value (a) The predcted results wth FW-GA method Error (b) The error between the actual value and ts predcted value Fgure 9: Predcton results wth FW-GA method The network traffc Predcted value Actual value (a) The predcted results wth ARIMA (5,, 3) model Error (b) The error between the actual value and ts predcted value Fgure : Predcton results wth ARIMA method MAPE s MSE s QPSO-FW BP F FW-GA (a) MAPE hstogram of four predcton models QPSO-FW BP F FW-GA (b) MSE hstogram of four predcton models Fgure : MAPE and MSE hstogram of four predcton models MSE s Coeffcent of effcency (CE) s QPSO-FW BP F FW-GA (a) MSE hstogram of fve predcton models QPSO-FW BP F FW-GA (b) CE hstogram of fve predcton models Fgure 2: MSE and CE hstogram of four predcton models

10 Dscrete Dynamcs n ature and Socety Coeffcent correlaton (r) s QPSO-FW BP F FW-GA Fgure 3: Coeffcent correlaton hstogram of four predcton models predcton of nonlnear and nonstatonary tme seres than many pure neural networks Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper Acknowledgment Ths research was partally supported by the atonal atural Scence Foundaton of Chna (no 26) References [] H Mahmassan, Dynamc network traffc assgnment and smulaton methodology for advanced system management applcatons, etworks and Spatal Economcs, vol,no3,pp , 2 [2] A oguera, P Salvador, R Valadas et al, Markovan modellng of nternet traffc, n etwork Performance Engneerng, vol 5233 of Lecture otes n Computer Scence, pp 98 24, Sprnger, Berln, Germany, 2 [3] SYChangandH-CWu, ovelfastcomputatonalgorthm of the second-order statstcs for autoregressve movng-average processes, IEEE Transactons on Sgnal Processng, vol 57, no 2, pp , 29 [4] M-C Tan, S C Wong, J-M Xu, Z-R Guan, and P Zhang, An aggregaton approach to short-term traffc flow predcton, IEEE Transactons on Intellgent Transportaton Systems,vol, no, pp 6 69, 29 [5] Y Shu, Z Jn, L Zhang, L Wang, and O W W Yang, Traffc predcton usng FARIMA models, n Proceedngs of the IEEE Internatonal Conference on Communcatons (ICC 99), vol2, pp , IEEE, Vancouver, Canada, June 999 [6] R L, J-Y Chen, Y-J Lu, and Z-K Wang, WPAFIS: combne fuzzy neural network wth multresoluton for network traffc predcton, Chna Unverstes of Posts and Telecommuncatons,vol7,no4,pp88 93,2 [7] K Zhao, Global robust exponental synchronzaton of BAM recurrent Fs wth nfnte dstrbuted delays and dffuson terms on tme scales, Advances n Dfference Equatons, vol 24, artcle 37, 24 [8] EARyng,GLBlbro,andJ-CLu, Focusedlocallearnng wth wavelet neural networks, IEEE Transactons on eural etworks,vol3,no2,pp34 39,22 [9] D Ln, X Wang, F an, and Y Zhang, Dynamc fuzzy neural networks modelng and adaptve backsteppng trackng control of uncertan chaotc systems, eurocomputng,vol73,no6 8, pp , 2 [] R H Abyev and O Kaynak, Fuzzy wavelet neural networks for dentfcaton and control of dynamc plants a novel structure and a comparatve study, IEEE Transactons on Industral Electroncs,vol55,no8,pp333 34,28 [] H Iyatom and M Hagwara, Adaptve fuzzy nference neural network, Pattern Recognton, vol 37, no, pp , 24 [2] C-H Lee and C-C Teng, Identfcaton and control of dynamc systems usng recurrent fuzzy neural networks, IEEE Transactons on Fuzzy Systems,vol8,no4,pp ,2 [3] W L and Y Hor, An algorthm for extractng fuzzy rules basedonrbfneuralnetwork, IEEE Transactons on Industral Electroncs,vol53,no4,pp ,26 [4] G K Venayagamoorthy, Onlne desgn of an echo state network based wde area montor for a multmachne power system, eural etworks, vol 2, no 3, pp 44 43, 27 [5] W Fang, J Sun, and W Xu, A new mutated quantum-behaved partcle swarm optmzer for dgtal IIR flter desgn, EURASIP JournalonAdvancesnSgnalProcessng,vol29,ArtcleID , 7 pages, 29 [6] L dos Santos Coelho and V C Maran, Partcle swarm approach based on quantum mechancs and harmonc oscllator potental well for economc load dspatch wth valve-pont effects, Energy Converson and Management,vol49,no,pp , 28 [7] L D S Coelho, Gaussan quantum-behaved partcle swarm optmzaton approaches for constraned engneerng desgn problems, Expert Systems wth Applcatons, vol 37, no 2, pp , 2 [8] R Pol, Analyss of the publcatons on the applcatons of partcle swarm optmsaton, JournalofArtfcalEvolutonand Applcatons,vol28,ArtcleID68575,pages,28 [9] R C Eberhart and Y Sh, Computatonal Intellgence: Concepts to Implementatons, Elsever, Phladelpha, Pa, USA, 29 [2] S L Sabat, L dos Santos Coelho, and A Abraham, MES- FET DC model parameter extracton usng Quantum Partcle Swarm Optmzaton, Mcroelectroncs Relablty, vol 49, no 6, pp , 29 [2] J Sun, B Feng, and W Xu, Partcle swarm optmzaton wth partcles havng quantum behavor, n Proceedngs of the IEEE Congress on Evolutonary Computaton (CEC 4), pp325 33, Portland, Ore, USA, June 24 [22] M Clerc and J Kennedy, The partcle swarm-exploson, stablty, and convergence n a multdmensonal complex space, IEEE Transactons on Evolutonary Computaton, vol6,no, pp58 73,22 [23] J Sun, W Fang, V Palade, X Wu, and W Xu, Quantumbehaved partcle swarm optmzaton wth Gaussan dstrbuted

11 Dscrete Dynamcs n ature and Socety local attractor pont, Appled Mathematcs and Computaton, vol28,no7,pp ,2 [24] S Mallate, A Wavelet Tour of Sgnal Processng: The Sparse Way, Elsever, Phladelpha, Pa, USA, 29 [25] F-J Ln, C-H Ln, and P-H Shen, Self-constructng fuzzy neural network speed controller for permanent-magnet synchronous motor drve, IEEE Transactons on Fuzzy Systems, vol 9, no 5, pp , 2 [26] S M Mkk and A A Kshk, Quantum partcle swarm optmzaton for electromagnetcs, IEEE Transactons on Antennas and Propagaton, vol 54, no, pp , 26 [27] F H T Vera, G R Banch, and L L Lee, A network traffc predcton approach based on multfractal modelng, Journal of Hgh Speed etworks,vol7,no2,pp83 96,2 [28] M Barabas, G Boanea, A B Rus, V Dobrota, and J Domngo- Pascual, Evaluaton of network traffc predcton based on neural networks wth mult-task learnng and multresoluton decomposton, n Proceedngs of the IEEE 7th Internatonal Conference on Intellgent Computer Communcaton and Processng (ICCP ), pp 95 2, Cluj-apoca, Romana, August 2 [29] Y Lang, Real-tme VBR vdeo traffc predcton for dynamc bandwdth allocaton, IEEE Transactons on Systems, Man and Cybernetcs Part C: Applcatons and Revews, vol34,no,pp 32 47, 24 [3]J-TTsa,T-KLu,andJ-HChou, Hybrdtaguch-genetc algorthm for global numercal optmzaton, IEEE Transactons on Evolutonary Computaton, vol8,no4,pp , 24 [3] K Papagannak, Taft, Z-L Zhang, and C Dot, Long-term forecastng of Internet backbone traffc, IEEE Transactons on eural etworks,vol6,no5,pp 24,25

12 Advances n Operatons Research Volume 24 Advances n Decson Scences Volume 24 Appled Mathematcs Algebra Volume 24 Probablty and Statstcs Volume 24 The Scentfc World Journal Volume 24 Internatonal Dfferental Equatons Volume 24 Volume 24 Submt your manuscrpts at Internatonal Advances n Combnatorcs Mathematcal Physcs Volume 24 Complex Analyss Volume 24 Internatonal Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Mathematcs Volume 24 Volume 24 Volume 24 Volume 24 Dscrete Mathematcs Volume 24 Dscrete Dynamcs n ature and Socety Functon Spaces Abstract and Appled Analyss Volume 24 Volume 24 Volume 24 Internatonal Stochastc Analyss Optmzaton Volume 24 Volume 24