Financial Time Series Forecasting Using Hybrid Wavelet-Neural Model

Transcription

1 50 The Iteratioal Arab Joural of Iformatio Techology, Vol. 5, No., Jauary 208 Fiacial Time Series Forecastig Usig Hybrid Wavelet-Neural Model Jovaa Božić ad Djordje Babić School of Computig, Uiversity Uio, Serbia Abstract: I this paper, we examie ad discuss results of fiacial time series predictio by usig a combiatio of wavelet trasform, eural etworks ad statistical time series aalytical techiques. The aalyzed hybrid model combies the capabilities of wavelet packet trasform ad eural etworks that ca capture hidde but crucial structure attributes embedded i the time series. The iput data is decomposed ito a wavelet represetatio usig two differet resolutio levels. For each of the ew time series, a eural etwork is created, traied ad used for predictio. I order to create a aggregate forecast, the idividual predictios are combied with statistical features extracted from the origial iput. Additioal to the coclusio that the icrease i resolutio level does ot improve the predictio accuracy, the aalysis of obtaied results idicates that the suggested model presets satisfactory predictor. The results also serve as a idicatio that deoisig process geerates more accurate results whe applied. Keywords: Time-series forecastig, wavelet packet trasform, eural etworks. Received November 23, 204; accepted Jauary 20, 206. Itroductio The fiacial time series are iheretly a ostatioary, oisy ad chaotic [23]. They are a combiatio of log ad short memory processes imbedded i oe complex sigal, explaiig why their predictio ca preset a true challege [23], especially kowig that due to its cogeital complexity, traditioal statistical methods perform poorly i this field. All of this actually suggests that there is o complete iformatio base o which we ca perform successful forecast. Moreover, the geeral assumptio made i these cases is that the historical data of oe time series itegrates all importat features ecessary for successful predictio. Despite this complicated sceario, our goal i this paper is to ivestigate the use of oe specific wavelet trasform ad Artificial Neural Networks (ANNs) for the predictio of fiacial time series. Over the last few decades, it has become obvious that liear models do ot adequately represet oliear series, while wavelet aalysis theory has emerged as a powerful tool i the mathematical aalysis field [0]. Simply said, the wavelet trasform produces a fuctioal local decompositio of a sigal i both the time ad frequecy domais ad is ot restraied by the assumptio of statioarity [0]. Numerous publicatios describe the applicatio of wavelets i the field of fiace [7, 9]. The trasform we use i this paper is Wavelet Packet Trasform (WPT). Each wavelet trasform offers the capability of capturig key features of a uderlyig process with a limited umber of coefficiets, but this choice is drive by the fact that with the WPT the most complex ad detailed sigal aalysis is obtaied. The fudametal ad ovel cotributio of this paper is to use oe particular processig techique to decompose fiacial time series ito a set of approximatio ad detail series which are fed ito the eural etworks i the model s ext phase. The traditioal approaches to time series predictio, such as Box-Jekis or ARIMA method, assume that the time series used for the predictio process are liear ad statioary [, 4]. Therefore, these methods are obviously ot a good tool for fiacial time series predictio. O the other had, durig the past few decades the ANNs have show great applicability i time series predictio [2, 22]. Studies have compared the performace of eural etworks to Autoregressive Itegrated Movig Average ARIMA [4], with all research agreeig that ANNs perform better tha ARIMA models. Several uique features of ANNs make them a attractive forecastig tool: they are multivariate, oparametric statistical methods that ca map ay oliear fuctio without a priori assumptio about the data, yet maitai desired accuracy [22]. Numerous articles illustrate the practical cosideratios of ANNs applicability [, 8, 8]. By combiig wavelet trasform with ANNs we get a ew kid of modellig method with great predictio ability for high frequecy fiacial data. With this syergy, we gai advatages from both of the methods-the multiscale aalysis supplied by wavelet theory ad powerful learig ad traiig capability of the eural etwork. Amogst may studies that ivestigate the cocept of mixig wavelets ad eural etworks, we referece several i our research [3, 5, 5, 6].

2 Fiacial Time Series Forecastig Usig Hybrid Wavelet-Neural Model 5 I this paper, we use a time series predictio usig hybrid wavelet-eural model. I order to expose the complex uderlyig structures for deeper evaluatio, the time-series is first subjected to a wavelet-based decompositio process usig decompositio levels of two ad three. The decomposed sigal compoets are the used as iput elemets to a ew group of eural etworks where valuable iformatio is captured durig the process of the traiig phase. I the third stage of the model, the ewly obtaied time series alog with the statistical features extracted from the origial iput are directed ito the fial eural etwork where the predictio is made. Compariso of the predictio results of the two models are based o four evaluative parameters: Mea Absolute Error (MAE), Mea Absolute Percetage Error (MAPE), Mea Squared Error (MSE) ad Real Mea Squared Error (RMSE). The results show that the hybrid model did ot perfectly fulfil the predictio tasks, meaig there is a certai mismatch betwee origial ad predicted values. However, havig i mid the great umber of parameters affectig fiacial series treds ad the large umber of scearios takig place o a daily basis i global stock markets, this mismatch is quite reasoable. Strategy applied shows that fiacial time series follow a tred that is ot completely radom, ad therefore the preseted topology is capable i graspig complex iteractios ad chaotic compoets i fiacial time series. Also, the icrease i resolutio level, perhaps uexpectedly, brigs o improved results. This paper is orgaized as follows. The ext sectio presets a review of theory behid wavelets ad eural etworks. The third sectio covers all aspects of the itroduced hybrid model, icludig a detailed descriptio of the procedure for its desig. I the fourth sectio, the experimetal results are preseted to demostrate the effectiveess of the preseted hybrid strategy. The fifth sectio presets aalysis of the results. The fial sectio proposes coclusios ad recommedatios for future research. 2. The use of Wavelets ad Neural Networks This Sectio explais the value of wavelets i timeseries predictio together with eural etworks. 2.. The Value of Wavelet Trasform Oe of the drawbacks of the Fourier aalysis is that although it is possible to determie the frequecies preset i the sigal with this aalysis, it is ot possible to establish whe they actually occur [7]. This premise is surpassed with the itroductio of wavelet trasform, represetatio of the sigal whose root is comprised of wavelets (or mother wavelets or aalyzig wavelets). Wavelet trasform, through the mechaism of mother wavelets traslatio, offers precise iformatio about time ad frequecy resolutio [0]. Discrete Wavelet Trasform (DWT) is oe of the most coveiet tools for sigal aalysis [0]. With its help, we ca preset the sigal with a limited umber of coefficiets that capture iformatio at differet frequecies at distict time momets. Oe of the variatios of the DWT is WPT, where both approximatio ad detail coefficiets are decomposed (2 sets of coefficiets are produced, ulike + i the case of a DWT. With this kid of sigal presetatio, the most complex ad detail sigal presetatio is gaied. Figure shows level 2 wavelet packet decompositio. Figure. Wavelet packet decompositio of level two. As for the wavelet used i previous studies, most sigal processig researchers adopted Daubechies wavelets [20] ad oe of the most commoly used i time series aalysis is Db40.With previously specified sigal processig techique, the iput time series ca be aalyzed at multiple time resolutios; the sigal ca be smootheed util the log-term tred is idetified ad the fluctuatios aroud the tred ca be ivestigated at multiple time scales. After the decompositio, the idividual time series ca give a detailed ad more easily aalyzed view of the ier uderlyig processes Artificial Neural Networks Artificial Neural Networks (ANNs) are a class of oliear models that ca extract crucial parameters from complex high-dimesioal time series ad approximate ay oliear fuctio with a high level of accuracy as a result [, 9]. They are capable of discoverig the uderlyig patter or auto-correlatioal structure i the time series eve whe the uderlyig law is ukow or hard to determie, makig them a powerful forecastig tool i may differet fields. Despite the fact that eural etworks have bee successfully implemeted i predictio process o umerous occasios, desigig a predictor for specific fiacial time series with eural etwork is a challegig ad otrivial task. I compariso with Box Jekis, ARIMA models ad other regressive models, a larger umber of factors play a role i the eural etworks desig.

3 52 The Iteratioal Arab Joural of Iformatio Techology, Vol. 5, No., Jauary 208 Oe of the most popular ad most successfully implemeted eural etwork model is the feed forward multilayer etwork or Multi-Layer Perceptro (MLP) [24]. This type of etwork cosists of several layers that cotai odes (artificial euros). The first layer is the iput level ad receives exteral iformatio, while the last layer is the output level ad produces model solutios. Hidde layers lie i betwee these two. All of the odes i oe layer are coected to the odes i the adjacet layer by a itercoectio stregth called weights. These weights are set through a traiig algorithm, where the goal is to miimize the differece betwee the etwork`s target ad actual output. Figure 2 shows MLP etwork with oe hidde layer. Figure 2. Neural etwork with oe hidde layer. The desig of eural etwork is a difficult task cocerig the umber of factors that ifluece its performace. Although some studies [2] propose certai methods for eural etwork desig, o study has bee reported to aalytically determie geeral architecture rules for successful eural etwork desig. Oe of the most sesitive parameters are umber of layers, umber of euros i each layer, activatio fuctio (fuctio that produces a output based o iput values eterig the ode) ad learig algorithm (the way the weights are set). The umber of iput ad output layers depeds o the problem s ature (i most papers, the suggested value is oe). As for the hidde layers (iteral iformatio processig layers), it has bee poited out that oe hidde layer etwork is able to approximate most of the oliear fuctios []. The dimesio of each layer, rather the umber of euros i each layer, is oe of the most essetial parameters for the etwork s proficiecy ad successful performace [2]. Icreased traiig time ad reduced geeralizatio ability of the NN ca be a result of too may hidde odes. O the other had, if it is too few, the etwork s ability to lear will be reduced. I most cases, because of the lack of a systematic approach to eural etwork desig ad established guidelies for it, trial ad error approaches are suggested for most previously stated architecture parameters. To summarize, eural etworks are difficult to desig, require high processig ad traiig time, ad give ustable results i may situatios. However, if their architecture is correctly plaed (which demads a sigificat level of ivested time ad resources), they preset a powerful tool that ca perform may demadig tasks that liear programs caot ad ca therefore be successfully implemeted i may applicatios Statistical Feature Extractio Besides basic wavelet features, we implemeted a additioal seve statistical features, all of them commoly used i fiace ad ecoomics, ito our model i order to improve the predictio process. They are give i the Table, where x presets the time series ad value of curret observatio, presets the total umber of observatios, σ the stadard deviatio, t represets the momet of time betwee ad, ad p to p are probabilities of the sigal. Table. Statistical features ad their defiitios. Statistical feature Mea Mea Absolute Deviatio Defiitio x x i i MAD ( x... x) i xi x Variace 2 Skewess VAR SKEW ( x... x ) i i ( x... x) i x x 3 x i x 4 x i x Kurtosis KURT ( xi x) 3 i x x x x Turig Poits 0 Shao etropy t t t t H( x) p i log 2 p i 3. Hybrid Modellig Strategy The mai cocept behid the predictio method preseted i this paper is to decompose the fiacial time series, usig oe particular wavelet trasform ito a rage of frequecy scales ad to pull these idividual compoets through separate eural etworks, makig a aggregatio forecast i the fial eural etwork. The etire process of this algorithm ivolves a series of steps, icludig: statistical feature extractio, preprocessig step, wavelet aalysis, eural etworks traiig ad modellig, ad fial forecastig. We approach this issue by dividig the model ito three separate stages: A exchage rate is processed ad subjected to a wavelet-based decompositio process i order to detect uderlyig processes (features) for further evaluatio. All idividual decomposed compoets are fed ito a set of eural etworks i order to capture valuable iformatio. The outputs from the eural etwork, rather the predicted values of each compoet, alog with a set of statistical features (calculated o the origial i

4 Fiacial Time Series Forecastig Usig Hybrid Wavelet-Neural Model 53 time series) are fed ito the fial eural etwork after which the predictio is to be made. The data preparatio phase icludes the statistical parameters calculatio ad ormalizatio of the fiacial time series. Each statistical feature for a specific sample is calculated based o 0 previous samples. The ormalizatio process has to be doe i order to avoid the effect of outsized values o the model ad to faste the calculatio. It is carried out i such maer that both the iputs ad targets fall i the rage [-, ]. The iput data are decomposed ito a certai umber of sub-time series compoets with the help of wavelet decompositio. Our mai cosideratios regardig which wavelet trasform ad mother wavelet we use are: The iput time series is discretely sampled so there is a dyadic relatioship betwee resolutio scales, leadig to the use of a DWT, rather its variatio of a WPT. Wavelet fuctio which respects the complex ature of the fiacial time series leads to the use of Daubechies wavelet fuctio. The choice of optimal decompositio level, geerally depedig o the researcher s experiece ad time series ature, is oe of the most importat factors of the model s performace i the first stage. I this study, we use two ad three decompositio levels (poor results are obtaied whe usig decompositio levels greater tha three). Figure 3 shows a represetatio of the model. For simplicity, the model has bee cosidered oly for a decompositio level of two. INPUT STATISTICAL FEATURES EXTRACTION SIGNAL PREPROCESSING WAVELET PACKET TRANSFORM NETWORK NETWORK 2 NETWORK 3 NETWORK 4 NETWORK 5 OUTPUT Figure 3. Hybrid model with Wavelet packet trasform; the 2 d level of resolutio. The cocept of usig wavelet trasform i time series aalysis offers the advatage of separatig the smooth part (approximatio series) ad the irregular ad oisy (detail series) part of the sigal, which are both more stable to hadle ad easier to predict due to the filterig effect of the trasform. As a result of this stage, the objective is to exploit these series as iput sigals to a set of eural etworks i the followig stage. The secod stage of the model cosists of a set of feed forward eural etworks that uses lagged detail ad smooth coefficiets gaied from decompositio i the previous step as iput. Oe of the papers where a similar idea is preseted is [7]. Separate eural etworks are built for each decompositio level, meaig that a 2-level wavelet packet decompositio results i four, while a 3-level wavelet packet decompositio results i eight eural etworks. What ca be oticed here is that we have a small amout of cotrol over the complexity of the architecture i this stage. Oe possible way of hadlig this complexity is to test etworks with differet desigs ad compare them i order to choose the optimal oe. Havig this i mid, the problem ca be tackled by varyig a large umber of desig factors that ifluece the predictio result. This is the reaso why we vary the umber of iput odes ad why we desig the hidde layer as simply as possible. Not oly does our research cofirm that this is the best approach, but our fidigs are also supported by literature which states that the simplest model is the least likely to over fit/uder fit ad the most likely to geeralize well o the usee data [3]. All series are split ito two sets: traiig ad testig. There are o specific rules for data divisio betwee these sets, ad most researchers use a trial ad error approach. For the traiig phase, we use a method kow as the slidig widow techique where the - tuple iput goes through the etire traiig set while a sigle output is used as the target value. Oce traied, the etworks are used for predictio. As for the trasfer fuctios, we use a liear oe for the ode i the output layer ad a ta sigmoid fuctio for the odes i the hidde layer. This fuctio is the most commoly used fuctio i forecastig problems ad patter detectio because it outperforms the alteratives whe deviatios are calculated from the average behaviour [2]. All of the etworks are traied usig the Scaled Cojugate Gradiet algorithm, a supervised learig algorithm that shows liear covergece o the most of the problems [6] ad provides faster learig. I the fial stage, the idividual predictios alog with statistical features that are calculated o the origial time series are combied to geerate a aggregate forecast. All of those values serve as iputs ito the last eural etwork, where the fial output is the oe-step-ahead predicted sample. To desig the last eural etwork, we use the same parameters as we did for the etworks from the secodary stage of the model, with the oly differece beig that the umber of iput samples is fixed. Table 2 summarizes the architecture of the fial etwork i the system, depedig o the level of resolutio used i the secod stage.

5 54 The Iteratioal Arab Joural of Iformatio Techology, Vol. 5, No., Jauary 208 Table 2. The architecture details of the eural etwork from the last stage. Method Wavelet Packet Wavelet Packet Resolutio Level Number of Iputs Number of Outputs NN architecture 2 :6: 3 5 5:8: To coclude, based o a previously itroduced model, the goal is to demostrate that the oe-stepahead predictios for specific fiacial time series ca be estimated with reasoable accuracy. The predictive power of these forecasts is compared by usig a set of statistical parameters. I the ext sectio, we show that the model s ability to capture dyamical behaviour differs with the wavelet resolutio level, but ot i a way that we expect. 4. Results I this Sectio we test the proposed forecastig model by usig a specific fiacial time series. The fiacial time series used here is the official exchage rate of Euro agaist the domestic exchage rate (republic serbia diar) betwee July 2003 ad September 2007 (total legth of 024 samples). We use this particular fiacial time series because we believe that this period of exchage rate is the most suitable for testig because it cotais various dyamic chages i the exchage rate. The first 80% of it is used as sample data for the traiig phase; while the remaiig 20% is used for evaluatio of each eural etwork (this kid of divisio gives best performace results). The graph of the EUR/RSD exchage rate is illustrated i Figure 4. For all tests ad simulatios, a special MATLAB code has bee geerated. Figure 4. EUR/RSD exchage rate from July 2003 till September As for the predictio performace, the hybrid wavelet eural model is evaluated by usig four statistical parameters: MAE, MAPE, MSE ad RMSE. These parameters are defied i the followig maer: MAE y t y t t yt yˆ t MAPE *00% t y t () (2) MSE ( y y y t ) t RMSE 2 ( y y y t ) t Where y t is the real value ad ŷ is the predicted value. t Although, most of the above expressios are selfexplaatory, it is useful to poit out the followig: MAE measures the deviatios betwee the actual ad predicted values, MAPE is the average absolute percetage error, MSE is the average of the squared errors betwee the predicted ad real value, ad RMSE presets how good a variace of the estimate is. Obviously, the closer that these values are to zero, the more accurate is the predictio performace. Followig the preseted modelig strategy, the EUR/RSD exchage data is preprocessed ad decomposed ito two differet resolutio levels by WPT. The goal is to make uderlyig temporal processes of the origial exchage rate more traceable ad easier for further aalysis. Afterwards, the ew set of sub-time series is fed ito a set of eural etworks. I this phase, we also tackle the deoisig effect o the predictio performace. The goal is to check if the process of oise removal will improve the quality of the overall fial forecast. We apply deoisig i the first stage with the WPT used. Because the process of oise removal is quite complex, we otice that the model performace is particularly sesitive to the threshold parameter ad that it is very importat to determie the correct threshold value ad apply deoisig to the detail coefficiets [, 2]. The process of soft threshold has bee tested for various values from 0.0 to 0.06 with step of 0.005, ad the best results are obtaied for the threshold value of After filtratio, wavelet packet recostructio is performed to obtai a de-oised sigal that will serve as a iput sigal i the ext phase. It ca be oticed that the oise from the origial time series is removed without the ifluece of sudde glitches which meas that most of the origial sigal is preserved. This feature is oe of the biggest advatages of the wavelet packet method. I the ext stage, we ivestigate the approach of forecastig each idividual sub-time series i the set of eural etworks. All etworks are traied separately (by usig correspodig wavelet coefficiets), ad the objective is to perform a oe-day-ahead predictio for each time series. Depedig o the resolutio level used i the first stage, we trai 4 or 8 eural etworks sow i Figure 3. As for the desig of each eural etwork, we apply the same architecture for each of them. I order to determie the optimal umber of iput odes, we tested the ANNs with iput layer that cosists of, 2, 3, ad 4 odes. The stadard for determiig the optimal umber of iput odes is RMSE. As for the iteral architecture, we desig all NNs with a sigle-hidde layer. This is doe because 2 (3) (4)

6 Fiacial Time Series Forecastig Usig Hybrid Wavelet-Neural Model 55 the NN with a sigle hidde layer ca approximate ay fuctio with arbitrary precisio ad because iput layer has fewer odes. For the umber of odes i the hidde layer, we apply a priciple most ofte used i literature-okam s razor pricipl-where the umber of hidde odes equals half of the iput ad output odes (we otice that icreasig the umber of hidde odes or eve addig more layers does ot improve etworks performace). Also havig i mid that we wat to predict a sigle value, we use oe euro i output layer. The predictio results of each eural etwork are idividually combied with statistical parameters, calculated o the origial EUR/RSD time series, thus establishig the iputs for the fial eural etwork. Figure 5 compares the real time series ad the output of the last eural etwork (the simulated time series). There is a large degree of overlap i iput ad output values for both resolutio levels, with o major discrepacies. a) Wavelet packet trasform used, 2 resolutio levels. b) Wavelet packet trasform used, 3 resolutio levels. Figure 5. Visual compariso of real ad simulated sigal. The forecastig aalysis is performed based upo the results from the oe-step-ahead predictio of the preseted hybrid model. We measure the performace metrics to ivestigate how well the model captures the uderlyig tred of the movemet of the EUR/RSD exchage rate. Table 3 shows the performace metrics achieved by our model. We illustrate this performace i Figure 6 where the predictio of 00 samples for both resolutio levels is show. Table 3. Performace metrics for EUR/RSD exchage rate depedig o parameters used i model s stages. Wavelet Trasform Level of Number of NNs i Wavelet Resolutio the Secod Stage Number of Iputs i the NN i the 3 rd Stage Number of Samples Predicted MAE MAPE MSE RMSE WPT 2 Db WPT 3 Db a) Wavelet packet trasform used, 2 resolutio levels. b) Wavelet packet trasform used, 3 resolutio levels. Figure 6. Comparative view of 00 real ad predicted samples. Based o the results, the learig algorithm applied o coefficiets hadles the uderlyig structures i a satisfyig maer so it ca be cocluded that the WPT decomposes the sigal i a overall accurate ad precise way. Whe it comes to deoisig applied, the results idicate that both models are ruig well, with slightly more accurate ad stable results i the case where oise is removed. Additioally, we observe that the ability of the model to capture dyamical behavior is chagig with the resolutio level used. Although oe would thik that for lower resolutio levels, resultig i oise ad irregular sub-time series, the model would show less accurate results, we actually get less accurate results with the higher resolutio time series (i other words, whe we use the more smooth series). This pheomeo ca be summarized i the followig waythe performace of the model deteriorates with a icrease i the resolutio level, with level 2 beig the optimal decompositio. A icrease of the resolutio level greater tha 3 yields poor results ad a meaigless predictio as a result. Evidet from the precedig is that we here aalyze a exchage rate with frequet, high jumps ad peaks, which ca corrupt the predictio process to a large degree. The model tests the EUR/RSD exchage rate over a very particular time period, durig which the Republic of Serbia experieced a difficult fiacial crisis i correlatio with a geerally volatile global ecoomy. Due to this volatility, we believe that the trade market itself is ot valid eough ad that the historic data caot depict all of the iformatio required. This explais why we have boosted the model s accuracy with the itroductio of statistic features i the fial stage. Havig all of the precedig i mid, the model largely maages to predict the oe day value of the EUR/RSD exchage rate. 5. Coclusios ad Further Research I this paper, we have aalyzed predictio strategy combiig wavelet trasform, eural etworks ad statistical features for fiacial time series predictio. Accordig to our fidigs, the model presets promisig cadidates for such predictio. Moreover, it seems that a WPT has a large capacity to capture global behaviour i a fiacial time series ad thus offers rich iformatio that is used i the secod stage for the purpose of traiig, modellig ad forecastig.

7 56 The Iteratioal Arab Joural of Iformatio Techology, Vol. 5, No., Jauary 208 Aother coclusio we have draw is opposite to our expectatios-icreasig the resolutio does ot improve the system performace, idicatig that a predictio may ot ecessarily be more accurate if the sigal decompositio level is higher. I the past years, umerous hybrid models have bee ivestigated, ad they all show sigificat promise. Future research could possibly study the predictive power for log term forecasts of the same model or the utilizatio of other outside imported ecoomic idicators. Also havig i mid that the predictive power of wavelet eural hybrid model is highly sesitive to a large umber of parameters, we had to do tedious experimets ad trial-ad-error procedures i order to obtai valid results. These major weakesses ca perhaps be avoided by chagig the selectio of appropriate umber of hidde odes, traiig times ad lags, ad determiig ad settig systematic rules for these tricky tasks. Ackowledgemets This paper is partly a result of the techology developmet project fuded by the Miistry of Educatio ad Sciece of the Republic of Serbia etitled "Performace optimizatio of eergy-efficiet computig ad commuicatio systems (TR 32023)". Refereces [] Al-waza H., Ibraheem K., ad Salim A., A Hybrid Algorithm to Forecast Erolmet Based o Geetic Algorithms ad Fuzzy Time Series, The Iteratioal Arab Joural of Iformatio Techology, vol., o. 3, pp , 204. [2] Bozic J. ad Babic D., Predictig the EUR/RSD Exchage Rate Usig Wavelets ad Neural Network, i Proceedigs of Iteratioal Coferece o Applied Iteret ad Iformatio Techologies, Zrejai, pp. 08-2, 203. [3] Bozic J., Vukotic S., ad Babic D., Predictio of the RSD Exchage Rate by Usig Wavelets ad Neural Networks, i Proceedigs of the 9 th Telecommuicatios Forum, Belgrade, pp , 20. [4] Box G., Jekis G., ad Reisel G., Time Series Aalysis: Forecastig ad Cotrol, Joh Wiley ad Sos, [5] Fa S., Ji T., Gordo W., ad Rickard B., Forecastig Baltic Dirty Taker Idex by Applyig Wavelet Neural Networks, Joural of Trasportatio Techologies, vol. 3, o., pp , 203. [6] Orozco J. ad Garcia C., Detectig Pathologies From Ifat Cry Applyig Scaled Cojugate Gradiet Neural Networks, i Proceedigs of the Europea Symposium o Artificial Neural Networks, Bruges, pp , [7] Gecay R., Selcuk F., ad Whitcher B., A Itroductio to Wavelets ad Other Filterig Methods i Fiace ad Ecoomics, Academic Press, [8] Gil P., Cortes J., Heradez S., ad Aquio V., A Neural Network Scheme for Log-Term Forecastig of Chaotic Time Series, Neural Processig Letters, vol. 33, o. 3, pp , 20. [9] Hippert H., Pedreira C., ad Souza R., Neural Networks for Short-Term Load Forecastig: A Review ad Evaluatio, IEEE Trasactios o Power Systems, vol. 6, o., pp , 200. [0] Holscheider M., Wavelets: A Aalysis Tool, Oxford Uiversity Press, 999. [] Icze A., Ileaa I., ad Rotar C., The Optimizatio of Feed Forward Neural Networks Structure Usig Geetic Algorithms, i Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics, Thessaloiki, pp , [2] Kaastr I. ad Boyd M., Desigig a Neural Network for Forecastig Fiacial ad Ecoomic Time Series, Neurocomputig, vol. 0, o. 3, pp , 996. [3] Kamruzzama J., Begg R., ad Sarker R., Artificial Neural Networks i Fiace ad Maufacturig, Idea Group Ic, [4] Kamruzzama J. ad Sarker R., Comparig ANN Based Models with ARIMA for Predictio of Forex Rates, Asor Bulleti, vol. 22, o.2, pp. 2-, [5] Miu K., Lieesh M., ad Jessy C., Wavelet Neural Networks for Noliear Time Series Aalysis, Applied Mathematical Scieces, vol. 4, o. 50, pp , 200. [6] Murtagh F., Starck J., ad Reaud O., O Neuro-wavelet Modelig, Decisio Support Systems, vol. 37, o. 4, pp , [7] Ortega L. ad Khashaah K., A Neuro-wavelet Model for the Short-Term Forecastig of High- Frequecy Time Series of Stock Returs, Joural of Forecastig, vol. 33, o. 2, pp , 203. [8] Pacelli V., Bevilacqua V., ad Azzollii M., A Artificial Neural Network Model to Forecast Exchage Rates, Joural of Itelliget Learig Systems ad Applicatios, vol. 3, o. 2, pp , [9] Reaud O., Stark J., ad Murtagh F., Predictio Based o a Multiscale Decompositio, Iteratioal Joural of Wavelets, Multiresolutio ad Iformatio Processig, vol., o. 2, pp , [20] Voesch C., Blu T., ad User M., Geeralized Daubechies Wavelet Families, IEEE Trasactios o Sigal Processig, vol. 55, o. 9, pp , 2007.

8 Fiacial Time Series Forecastig Usig Hybrid Wavelet-Neural Model 57 [2] Yao J., Li Y., ad Ta C., Optio Price Forecastig usig Neural Networks, Omega, vol. 28, o. 4, pp , [22] Yao J. ad Ta, C., A Case Study o Usig Neural Networks to Perform Techical Forecastig of Forex, Neurocomputig, vol. 34, o. -4, pp , [23] Yu L., Wag S., Huag W., ad Lai K., Are Foreig Exchage Rates Predictable? Scietific Iquiry, A Joural of Iteratioal Istitute for Geeral Systems Studies, vol. 8, o. 2, pp , [24] Zhag G. ad Qi M., Neural Network Forecastig for Seasoal ad Tred Time Series, Europea Joural of Operatioal Research, vol. 60, o. 2, pp , Jovaa Bozic obtaied her Diploma Egieer degree i electrical egieerig i 2007 from the School of Electrical Egieerig, Uiversity of Belgrade, Serbia. She is curretly a PhD cadidate at the School of Computig, Departmet of Sigal processig i telecommuicatios, of the Uio Uiversity. Her mai research iterests are i the areas of time series predictio, artificial eural etworks ad wavelet-based sigal processig. She has published several papers i atioal ad iteratioal cofereces ad jourals. Djordje Babic obtaied his Diploma Egieer degree i 999 at the School of Electrical Egieerig, Uiversity of Belgrade. He defeded his doctoral thesis i the field of sigal processig i telecommuicatios at Tampere Uiversity of Techology, Filad, i From 999 to 2004, he was employed at the Istitute of Telecommuicatios, Tampere Uiversity of Techology. Sice 2008 he has bee employed at the School of Computig, Belgrade, as a associate professor i the field of etworked computer systems. He coducts research i the field of sigal processig i telecommuicatios, multirate sigal processig. He has published over 30 articles i differet iteratioal jourals ad at iteratioal cofereces.