Journal of Economics and Business

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1 Journal of Economcs and Busness 64 (2012) Contents lsts avalable at ScVerse ScenceDrect Journal of Economcs and Busness A multple adaptve wavelet recurrent neural networ model to analyze crude ol prces Tang Mngmng, Zhang Jnlang Department of Resources Technology, College of Resources Scence and Technology, Bejng Normal Unversty, No. 19 Xnjeouwa Street, Bejng , PR Chna a r t c l e n f o Artcle hstory: Receved 23 September 2011 Receved n revsed form 6 March 2012 Accepted 15 March 2012 Keywords: Multple wavelet recurrent neural networ Crude ol prce forecastng Gold prce a b s t r a c t Internatonal crude ol prces are an mportant part of the economy, and trends n changng ol prces have an effect on fnancal marets. Tradtonal hybrd analyss methods for nternatonal crude ol prces, such as wavelet transform and bac propagaton neural networ (BPNN), see synergy effects by seuentally flterng data through dfferent models. However, these estmaton methods cause loss of nformaton through the ntroducton of bases n each flterng step, whch are aggregated throughout the process when model assumptons are volated, and the tradtonal BPNN model does not have forecastng ablty. In ths study, we constructed a multple wavelet recurrent neural networ (MWRNN) smulaton model, n whch trend and random components of crude ol and gold prces were consdered. The wavelet analyss was utlzed to capture multscale data characterstcs, whle a real neural networ (RNN) was utlzed to forecast crude ol prces at dfferent scales. Fnally, a standard BPNN was added to combne these ndependent forecasts from dfferent scales nto an optmal predcton of crude ol prces. The smulaton results showed that the model has hgh predcton accuracy. The desgned neural networ s able to predct ol prces wth an average error of 4.06% for testng and 3.88% for tranng data. Ths forecastng model would be able to predct the world crude ol prces wth any commercal energy source prces nstead of the gold prces Elsever Inc. All rghts reserved. Correspondng author. Tel.: E-mal addresses: tang@mal.bnu.edu.cn, tangmngmng126@126.com (T. Mngmng), jnlang@bnu.edu.cn (Z. Jnlang) /$ see front matter 2012 Elsever Inc. All rghts reserved.

2 276 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) Introducton The global economy has not mproved snce the Amercan fnancal crss n October Internatonal ol prces have been n a recesson gven the serous lac of maret confdence. Ol plays an mportant role n the economy, and the broad consensus s that hgher ol prces affect gold prces and fnancal marets (Brown & Yucel, 2002; Jones, Leby, & Pa, 2004). The man methods for predctng ol prces are uanttatve, such as an econometrc model or a stochastc model. Most methods assume that a crude ol prce seres mples a lnear relatonshp. In fact, crude ol prces are a very complex nonlnear tme seres and show very complex characterstcs. Crude ol prces are nfluenced by objectve economc laws as well as poltcs and prcng systems. Therefore, establshng an effectve predcton model based on general tme seres analyss s dffcult. Generally, economc and maret factors are consdered to be the man causes of surgng and fallng global ol prces. By consderng dfferent economc factors, many researchers have attempted to descrbe and even predct changes n ol prces. Barone-Ades, Bourgon, and Gannopoulos (1998) used sem-emprcal euatons to forecast ol prces. Noel (2000) nvestgated the mpact of ol prces on the producton of crude ol and gas, whle Yasnder (1983) studed the relatonshp between ol prces and economc stablty. Gulen (1998) used the contnuty prncple to predct prces of West Texas Intermedate (WTI) crude ol. The GARCH model was used by Morana (2001) to forecast shortterm ol prces. Mrmran and L (2004) forecasted ol prces n the Amercan maret usng neural networs and genetc algorthms. Lanza, Manera, and Govannn (2005) used error correcton models to predct ol prces. Abramson and Fnzza (1991) attempted to predct ol prces usng neural networ models. Xe, Yu, Xu, and Wang (2006) used a support vector machne method to forecast ol prces. Azadeh, Arab, and Behfard (2010) modeled long-term ol prces wth an adaptve ntellgent algorthm. Gor, Ludovs, and Cerrtell (2007) forecasted ol prces and consumpton usng three dfferent scenaros. Taba and Fetosa (2009, 2010) devoted proactve nowledge to developng new predctve models, ncludng for nonlnear and chaotc behavor of a tme seres. In recent years, practtoners have emphaszed decomposton methods to capture drfts or spes relatve to major economc aggregatons. He, Xe, Chen, and La (2009) employed three wavelet varants to estmate the rs value n the ol maret. Slva, Legey, and Slva (2010) used a wavelet decomposton to forecast ol prce trends. Tsung, Hsao, and Yeh (2011) used wavelet transformaton to decompose orgnal data and a smple RNN wth a three-layer archtecture to forecast stoc maret movements by sendng all wavelet coeffcents to the nput layer of only one RNN, whch cannot analyze trends and forecast crude ol prces on multple scales. Jammaz and Alou (2011) developed a crude ol prce forecastng model, and a smoothed sgnal after wavelet decomposton was used as tranng data to construct the bac propagaton neural networ (BPNN). To determne synergy effects, seuental flterng data were assumed n the tradtonal hybrd models. However, ths estmaton method led to nformaton loss because bases were ntroduced n each flterng step when model assumptons were volated. Therefore, the tradtonal BPNN model does not have forecastng ablty. All of these researchers gnored the real dynamcal relatonshp between crude ol and gold prces. In ths paper, a new, real dynamcal model to predct crude ol prces was establshed by consderng gold prces. The dynamc propertes of recurrent neural networ (RNN) were combned wth wavelet decomposton to analyze and forecast a seres of nternatonal crude ol prces, and fnally, a standard BPNN was added to combne these ndependent forecasts from dfferent scales nto an optmal predcton of crude ol prces. Wavelet analyss was used to capture multscale data characterstcs, and RNN was used to smulate crude ol prces. Ths model wll be useful n determnng polces on nternatonal crude ol prce estmatons. The structure of ths paper s as follows. A bref overvew on wavelet transform s presented n Secton 2. Secton 3 descrbes the MWRNN. The data used and the exploratory analyss carred out on the data are explaned n Secton 4. Fnally, Secton 5 descrbes the model estmaton and provdes a comparson of densty forecastng accuracy.

3 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) Contnuous wavelet transform (CWT) and dscrete wavelet transform (DWT) Wavelet transforms are well-nown and useful tools for varous sgnal-processng applcatons (Burrus, Gopnath, & Guo, 1998; Morlet, 1981; Thullard, 2001). A CWT s used to decompose a sgnal nto wavelets and s an excellent tool for mappng the changng propertes of nonstatonary sgnals. CWT s the best method for the analyss of nternatonal crude ol and gold prce seres. CWT s a convoluton between a data seuence and a mother wavelet. The mother wavelet s a,b(t) = a 1/2 ( (t b) a ), a, b R, a /= 0 (1) where a s the scalng factor and b s the translatng factor. The CWT of ol prce seres p(t) s defned as follows: ( P(a, b) = a 1/2 p(t) (t ) b) dt = p(t), a a,b(t) (2) where P(a,b) are wavelet coeffcents. Generally, mother wavelet functons nclude Mexcan hat wavelets, Morlet wavelets and others. In ths paper, the Daubeches wavelet (Malenejad & Derl, 2005) was used to analyze the characterstcs of ol prces. The Daubeches wavelet satsfes the admssble condton: ˆ (ω) 2 C = dω < ω (3) In E. (3), ˆ (ω) s the Fourer transform spectrum of (t). The nversed transform of E. (2) s as follows: P(a, p(t) = C 1 b) a,b(t) a 2 da db (4) The negatve freuency of ˆ (ω) has no meanng; therefore, E. (4) can be turned nto the followng: p(t) = 2C 1 P(a, b) a,b(t)/a 2 dadb (5) 0 0 CWT ncreases the data redundancy of the crude ol prce seres and reduces the speed of operaton. To accelerate the computng speed and to reduce data redundancy, the scale and dsplacement parameters of DWT can be scaled up. The dscrete mother wavelet functons j,(t) are defned as follows: j,(t) = a j/2 (a j 0 0 t b 0 ), j, Z (6) In a J level decomposton, j = J. The dlated and translated bass functons at dfferent resoluton levels are descrbed by the scalng functon, whch s called the father wavelet, j, (t), and s gven by j, (t) = a j/2 (a j 0 0 t b 0 ), j, Z (7) By usng these two basc wavelet functons, the DWT of an ol prce seres can be approxmately expanded to the followng: P(t) j, j, (t) s J, J, (t) + d J, J,(t) d 1, 1,(t) s J, J, (t) j + d j, J,(t) j where s J, and d j, are the smooth and detal component coeffcents representng the trend and random components of crude ol prces, respectvely, and s J, = J, (t)p(t) dt, d j, = j,(t)p(t) dt. (8)

4 278 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) Theory and model of multple wavelet recurrent neural networs (MWRNNs) To develop the forecastng hybrd model, the wavelet method was coupled wth recurrent neural archtecture, whch s called a multple wavelet recurrent neural networ (MWRNN) Basc theory of MWRNNs The basc dea of a MWRNN s that wavelet analyss s used to capture multscale data characterstcs of crude ol prces, whle RNN forecasts crude ol prces at each scale. When each estmaton provdes a unue perspectve about the underlyng rs evolutons, the partal nformaton sets can be pooled together usng an addtonal BPNN. By addng a fnal artfcal networ, the BPNN can provde an estmaton wth a hgher degree of accuracy and relablty, fnd complex nonlnear patterns among ensemble members and then mae a forecast. At each tme step j, the nput unt of the RNN s gven by {x, = j 1, j 2,..., j m, m = tapped delayed lne length ; = level of DWT}. The RNN computes the hdden unt, {y j,, = hdden nodes number, = level of DWT}, and the output unt z j s as follows: { y j = RNN f (x j 1,, x j 2,,...x j m, ) (9) z j = RNN g(y j ) In E. (9), RNN f and RNN g are sgmod and lnear functons. By usng a tranng nput seuence collecton and correspondng desred outputs, RNN can be traned to approxmate a hgh complex functon. At the end, a standard three-layer BPNN was added to combne the ndependent forecasts nto an optmal predcton of crude ol prces The MWRNN model Artfcal neural networs (ANNs) are very attractve for an economst n determnng the relatonshp between gold and crude ol prces. ANNs do not provde any type of descrpton, but they are a beneft to the modeler n systematcally developng and tranng the networ. In general, decsons relatng to the tranng factors depend on the modeler s experence and judgment. Unfortunately, the major shortcomng n applyng the standard feed forward ANNs to ol prce analyss s that the storage elements are or can be gnored n a networ. In the feed forward archtecture, the output at a partcular tme depends on the nput at that moment, whch s not the best result for ol prce analyss and predcton. To overcome ths lmtaton, temporal dependence should be ntroduced usng a feed forward networ wth a tapped delay lne (TDL) technue. A recurrent networ or a partally recurrent networ can be used. The recurrent networ prmarly has a forward connecton and ncludes a reduced set of feedbac connectons. The smplest and most popular way to address a temporal seres s to convert t nto an addtonal pattern on the nput layer of a networ, nown as the TDL procedure (Graves & Pedrycz, 2009). The sldng wndow method for capturng delayed nput data has been wdely used n the lterature (Azadeh, Arab, & Behfard, 2010). A recent emprcal dscovery suggested that tme-scale nformaton s nontrval. To further understand the complcated behavor of the data, freuency scale nformaton and tme-scale nformaton need to be ncorporated. Based on the standard archtecture of Elman s RNN, we constructed a MWRNN smulaton model. Fg. 1 shows the MWRNN structure. Crude ol and gold prces were all consdered n ths model. Wavelet analyss was used to address the multscale nonlnear nature of the data. The nonlnear ensemble RNN method ncorporated partal nformaton sets to produce more accurate forecastng results by combnng the dfferent ndependent forecasts n RNN nto an optmal one. RNN s a three-layer RNN for the th trend components of crude ol and gold prces. In Fg. 1, K and M represent the number of trend components of crude ol and gold prces n DWT; P j and G j represent the crude ol and gold prces at tme j; P,j and G,j represent the th trend component coeffcents of P j and G j ; P s represents j the output values of crude ol prces at tme j. o and g are the target prces of crude ol and gold at the th output layer unt n RNN; U o j and U g j represent the weghts between the th hdden and

5 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) Fg. 1. Structure of MWRNN (wavelet analyss was used to capture the multscale data characterstcs of crude ol prces, whle RNN was used to forecast crude ol prces at each scale. P,j and G,j are the th trend components coeffcents of P j and G j at tme j. RNN s a three-layer RNN for the th trend components of crude ol and gold prces. At the end, a standard three-layer BPNN was added to combne the ndependent forecasts nto an optmal predcton of crude ol prces). the jth nput layer unts of crude ol and gold prces n RNN; W o and W g represent the weghts between the th hdden and the th output layer unts of crude ol and gold prces n RNN; W j represents the set of all weghts used n RNN; T j represents the weght between the hdden layer and the output layer n RNN. The major shortcomng n applyng the standard feed forward ANN to ol prce analyss s that the networ does not consder storage elements. Td o and Td g are the th TDL sectons of crude ol and gold prces n RNN, Td o = {P,j j = 1, 2,..., m, m} and Td g = {G,j j = 1, 2,..., n, n}. Td o m and Td g m represent the frst TDL sectons of crude ol and gold prces n RNN. f s a hdden layer actvaton functon, l s the number of layers employed n the networ, m represents the order of memory and n represents the number of crude ol and gold prce seres. In ths study, we collected data for 34 months, and n = 34.

6 280 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) A three-layer RNN for each trend component of crude ol and gold prces was constructed. The nput layer comprses two TDL (Td o and Td g ) sectons n RNN. The th hdden unt of RNN receves a net nput: h = 1 U o j P,j + 1 U g j G,j (10) Here, P,j and G,j represent the th trend component s coeffcents (s J, n E. (8)) of crude ol and gold prces at tme j. The th hdden layer unt wll produce the followng output: 1 1 v = f (h ) = f U o j P,j + U g j G,j (11) In E. (11), v s the output of the th hdden layer unt, f s the hdden layer actvaton functon, m s the orders of memory. The th gold prce output unt wll receve an nput unt: 1 1 h g = W g v j = W g f U o j P,j + U g j G,j (12) In E. (12), h g s the nput value of the th gold prce output unt, W g are the weghts between the th hdden layer unt and the th output layer unt of gold prces n RNN. The th crude ol prce output unts wll receve an nput value unt: 1 1 h o = W o v = W o f U o j P,j + U g j G,j (13) In E. (13), h o s nput value of the th crude ol prce output unt, and W o are weghts between the th hdden layer unt and the th output layer unt of crude ol prces n RNN. The output gold and crude ol prces are as follows: ( ) G s, =g(hg ) = g W g v = g 1 1 W g f U o j P,j + U g j G,j (14) ( ) P s, =g(ho, ) = g W o v = g 1 1 W o j f U o j P,j + U g j G,j (15) In Es. (14) and (15), P s, and G s, are the output values of the th trend component s coeffcents of crude ol and gold prces at tme j, and h g and h o are the nput values of the th gold prce and crude ol output unt. The error functon of RNN depends only on the weghts W j, and s gven by the followng: [ ] E [W j ] = 1 ( o P s 2, )2 + ( g G s, )2 (16) where o and g are eual to the target prces of crude ol and gold at the th output layer unt n RNN and the neural networ tranng procedure. Accordng to the gradent type update rule, the weghts are updated usng the followng euaton: W new j = E W j + W old j (17)

7 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) Fg. 2. Internatonal crude ol prces and nternatonal gold prces. where and are the Learnng Rate and Momentum parameters. After tranng, Td o m and Td g n represent the ntal nputs redoundng to predct and compare the predcton results wth the orgnal P,j and G,j. Ths approach helps test the valdaton of RNN at dfferent levels of the trend components. A standard three-layer BPNN was added as an analyzer to combne P, ( = 1,..., K) and gve the fnal output of P. If the th hdden unt of ths BNPP receves a net nput K h j = U P s (18),j =1 then t wll produce the followng output: ( K v j = f (h j ) = f =1 U P,j ) The fnal crude ol prce output unt wll receve ( ) h j = W v j = W f (h j ) = W f U o P,j and the gold and crude ol prce output wll be ( ) ( ( )) P s = g(h j ) = g W v j = g W f U o P,j The error functon of the BPNN only depends on the weghts W. The weghts are updated accordng to gradent type update rule for E. (16). 4. Emprcal studes 4.1. The crude ol prce data Two ol prce seres were used n ths paper. Brent and West Texas Intermedate (WTI) represent the European and US ol marets, respectvely. Ol prce data were easly obtaned from 1946 to From 1946 to 1970, ol prces ncreased from 1.63$ per barrel to 3.39$/bbl. A more eventful tme n hstory from 1970 to 2010 s summarzed n Fg. 2. Varous Mddle Eastern crses led to ol prces exceedng 10$/bbl n After reachng a pea of 37.50$/bbl n March 1981, prces fell nto 10 20$/bbl range, (19) (20) (21)

8 282 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) Fg. 3. DWT scale coeffcents of nternatonal crude ol prces. where they remaned untl Stevens (2005) attrbuted the sharp prce declne n 1986 to excess upstream capacty. Durng the perod , prces showed an upward movement, and surplus producton capacty decreased from 7 mllon bbl per day (January 2002) to less than one mllon bbl per day (October 2004). In md-2008, the prces of Brent and WTI both exceeded US$140, ndcatng the occurrence of the ol prce bubble. Ths bubble was explaned by the followng causes: an economc boom n the world s largest developng countres, partcularly Chna and Inda; restrctons n supply; dmnshng reserves; an excess of speculatve actvty, among other factors. Gold prce data were obtaned from the World Gold Councl. Prevous studes showed that gold prce fluctuatons had dfferent effects on gold producton and gold mnng stocs from country to country and mne to mne. Fg. 2 shows that gold has traded n the maret snce 1967, and ts prce has ncreased wth rapd fluctuatons from that tme (Mlls & Mlls, 2004). In our model, 60% data were used to tran, 10% to valdate, and 30% to test the MWRNN. A part of the tranng data was also used to valdate the MWRNN. To trac valdaton errors, random samples from the tranng data set formed the valdaton set for each applcaton, and the sze of the valdaton set was eual to 10% of the total data set. Valdaton errors decreased at the begnnng of the tranng but began to ncrease at some pont, although tranng errors stll decreased MWRNN parameter settng The performance of dfferent recurrent networ archtectures wth dfferent orders of memory was evaluated. To compare the effects of dfferent orders of memory n MWRNN whle holdng as many other factors as possble constant, the memory orders, m, from 1 to 30 n ncrements of 1 were vared. For each value of m, 20 tme smulatons were run. In each smulaton, predcton values were generated for every neural networ except for the ntal nput TDL. The networ was traned usng the mean suared error cost functon, and a smple algorthm wth a learnng rate of 0.1 for a maxmum of 200 epochs was used. Updates occurred at the end of each TDL, and the error was propagated bac along the full length of the TDL. If the smulaton exceeded 200 epochs or the error was larger than , the smulaton was determned a falure. The best parameters of the networ are those wth a mnmum valdaton error, and learnng was stopped at that pont to avod over-fttng the data. After the tranng and evaluaton of MWRNN, the model s ablty to predct ol prce ratos was also checed.

9 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) Fg. 4. DWT scale coeffcents of nternatonal gold prces. Fg. 5. DWT detaled coeffcents of nternatonal crude ol prces Expermental results and dscusson A lac of useful nformaton can severely mpar the learnng process because any pror mathematcal descrpton that encodes structural nformaton cannot be used n RNNs; the encodng structural nformaton has to be extracted drectly from the nput data. For the same reason, trval and unnecessary nputs may also mpede learnng. Although RNNs are blac-box models, any avalable physcal nsght or exstng system nowledge should be used when selectng nputs durng the data preprocessng phase. The wavelet decompostons of annual crude ol and gold prces are shown n Fgs The horzontal axes represent the tme horzon. The vertcal axes n Fgs. 3 and 4 represent the ampltude of the trend components coeffcents, and represent the random components coeffcents n Fgs. 5 and 6. The unts of the orgnal sgnal and the correspondng extracted porton (trend and random components) are all measured n $/barrel. Because annual data are used, the frst level of detal represents varatons wthn 1 or 2 years, whle the next level of detal represents varatons wthn 2 j years. The

10 284 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) Fg. 6. DWT detaled coeffcents of nternatonal gold prces. Fg. 7. Gold prce predctons usng MWRNN. horzon corresponds to 4 8, 8 16, and year dynamcs, respectvely. All of the coeffcents are lsted from the hghest to the lowest freuency. Predctons from the proposed archtecture were compared wth both the testng and the tranng data (see Fgs. 7 and 8). In Fgs. 7 and 8, the exact predctons would le along the dagonal lnes n the mddle, and the ponts show the tranng or testng data. The error for each predcton s proportonal to the dstance from each pont to the dagonal lne, and the cluster of ponts near the dagonal lne llustrates the overall accuracy of the present model. The mean percent errors for the testng and tranng data were calculated to be 4.06% and 3.88%, respectvely. Values of MRSE (mean relatve suare error), MRAE (mean relatve absolute error), MRSD (mean relatve suare devaton) and MRAD (mean relatve absolute devaton) for dfferent orders of memory n MWRNN are lsted n Table 1. Gven the value of the crude ol prce seres, the model has the ablty to produce a good predcton about the followng year. Because the MWRNN nput varables are selected smply and are avalable for any commercal ol, couplng the present model wth any other methods for predctng average or benchmar prces wll not be very dffcult. The gold prce factor was smply added to the MWRNN,

11 T. Mngmng, Z. Jnlang / Journal of Economcs and Busness 64 (2012) Fg. 8. Gold prce predctons usng MWRNN. Table 1 Values of MRSE, MRAE, MRSD, and MRAD for dfferent orders. Orders of memory MRSE (%) MRAD (%) MRSD (%) MRAD (%) enablng the model to predct crude ol prces by consderng other commercal materal, such as coal prces and nature gas prces. The shortcomng of ths model s that t s not able to predct long-term prces. 5. Concluson A MWRNN-based neural networ ensemble learnng model to predct world crude ol prces was proposed. Ths model conssts of the desgn of an approprate RNN to predct world crude ol prces. In ths model, wavelet analyss was used to capture the mult-scale data characterstcs of crude ol prces, whle RNN was assumed to forecast crude ol prces at each scale. Because each estmaton provded unue perspectves nto the underlyng evolutons of rs, these partal nformaton sets can be pooled together to produce estmatons wth hgher accuracy and relablty. By addng a fnal artfcal networ, complex nonlnear patterns between ensemble members and forecasts were establshed. Ths forecastng model would be able to predct world crude ol prces usng any commercal energy source prces nstead of gold prces. Acnowledgments The authors would le to than Lu Xaobng, Sun Jng and Alng Yang, who provded comments for mprovng the ualty of the paper. Ths wor was supported by the Chnese Natonal Program for Hgh Technology Research and Development (grant No. 2006AA09Z336), the Natonal Natural Scence Foundaton of Chna (grant No ), the Nature Scence Foundaton of Shandong Provnce (No. 2009ZRB02103), and the Hgher Specalzed Research Fund for the Doctoral Program (No ).

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