Improved Particle Swarm Optimization For Fuzzy Based Stock Market .

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IMPROVED PARTICLE SWARM OPTIMIZATION FOR FUZZY BASEDSTOCK MARKET TURNING POINTS PREDICTIONCHAWALSAK PHETCHANCHAIA thesis submitted in fulfilment of therequirements for the award of the degree ofDoctor of Philosophy (Computer Science)Faculty of ComputingUniversiti Teknologi MalaysiaFEBRUARY 2013

iiiTo my beloved family

ivACKNOWLEDGEMENTSI would like to express my gratitude to those who have helped me in mypursuit for knowledge. I would especially like to express my deep and sinceregratitude to my supervisor Prof. Dr. Ali Bin Salamat, for his attention, continuousguidance, and support throughout the length of this study. He has greatly helped mein a lot of ways I needed to go through this study.I am grateful to him for giving him wide knowledge, time and guidance tohelp me overcome the challenges in my study.I am also immensely grateful to other faculty members for their kindcooperation, as well as to all staff of our faculty who extended their best cooperationduring my study and stay here.I would like to thank Suan Dusit Rajabhat University and Office of the CivilService Commission of Thailand for their generosity in funding the scholarshipsduring my study. Receiving these scholarships motivates me to complete my degree.I thank you for their confidence to help me achieve my goals.My deepest thanks go to my family. Their influence made me realize theimportance of education and for bearing towards them during the course of thisstudy.

vABSTRACTStock prices usually appear as a series of zigzag patterns that move in upwardand downward trends. These zigzag patterns are learned as a tool for predicting thestock market turning points. Identification of these zigzag patterns is a challengebecause they occur in multi-resolutions and are hidden in the stock prices.Furthermore, learning from these zigzag patterns for prediction of stock marketturning points involves vagueness or imprecision. To address these problems, thisresearch proposed the swarm-based stock market turning points prediction modelwhich is a combination of a zigzag patterns extraction method, and a mutationcapable particle swarm optimization method. This model also includes the stepwiseregression analysis, adaptive neuro-fuzzy classifier, and subtractive clusteringmethod. This study explores the benefits of the zigzag-based multi-ways search treedata structure to manage the zigzag patterns for extracting interesting zigzag patterns.Furthermore, the mutation capable particle swarm optimization method is used tooptimize the parameters of subtractive clustering method for finding the optimalnumber of fuzzy rules of adaptive neuro-fuzzy classifier. Stepwise regressionanalysis is used to select the important features from the curse of input dimensions.Finally, adaptive neuro-fuzzy classifier is used for learning the historical turningpoints from the selected input features and the extracted zigzag patterns to predictstock market turning points. The proposed turning points prediction model is testedusing stock market datasets which are the historical data of stocks listed ascomponents of S&P500 index of New York Stock Exchange. These data are stockprices that are either moving upward, downward, or sideways. From the findings, theproposed turning points prediction model has the potential to improve the predictiveaccuracy, and the performance of stock market trading simulation.

viABSTRAKPasaran saham selalunya muncul sebagai siri dalam corak zigzag yangbergerak sama ada dalam bentuk indeks meningkat atau indeks menurun. Corakzigzag ini dikenalpasti sebagai salah satu alat untuk untuk meramal titik perubahanpasaran saham. Untuk mengenalpasti corak zigzag adalah merupakan satu cabarankerana kerana ianya berada dalam pelbagai resolusi dan tersembunyi di dalam nilaipasaran saham. Tambahan pula, pola pembelajaran di dalam meramal titik perubahanpasaran saham melibatkan kesamaran dan ketidaketepatan terhadap corak, dan kajianini mencadangkan teknik titik perubahan pasaran saham secara kelompok melaluikombinasi di antara kaedah pengekstrakan corak zigzag dan pengoptimumankerumunan partikel boleh mutasi. Model ini juga merangkumi analisis regrasiberperingkat, pengkelas neuro kabur, dan juga pengklusteran penolakan. Kajian inimengkaji kelebihan struktur data zigzag berdasarkan pelbagai kaedah carian yangmempunyai ciri-ciri yang menampung corak zigzag yang mengekstrak corak zigzagyang menarik. Kaedah pengoptimuman kerumunan partikel boleh mutasi digunakanuntuk mengoptimum nilai parameter daripada kaedah pengklusteran penolakan untukmencari nilai optimum bagi pengkelas neuro kabur. Analisis regrasi berperingkatdigunakan untuk memilih ciri-ciri yang penting daripada dimensi input. Bagipengkelas neuro kabur pula, kefahaman mengenai statistik titik perubahan pasaransaham yang di ekstrak dari corak zigzag dan ciri-ciri input yang terpilih digunakanbagi meramal titik perubahan di masa akan datang. Ramalan titik perubahan pasaransaham yang telah diuji dengan set data pasaran saham yang terdahulu yang tersenaraisebagai komponen indeks S&P500 yang terdapat dalam Bursa Saham New York dimana data pasaran saham yang diuji adalah merangkumi statistik pasaran sahamyang meningkat, menurun dan pergerak sisi. Melalui kajian ini, model titikperubahan saham yang telah diusulkan mempunyai potensi bagi meningkatkanketepatan ramalan dan juga prestasi simulasi perdagangan pasaran saham.

viiTABLE OF iiACKNOWLEDGEMENTivABSTRACTvABSTRAKviTABLE OF CONTENTSviiLIST OF TABLESxivLIST OF FIGURESxviLIST OF ABBREVIATIONxxivLIST OF d of Problem41.3Problem Statement71.4Objectives of Research141.5Scopes of Research151.6Contributions of Research171.7Thesis organization18LITERATURE REVIEWS202.1Introduction202.2Stock market prediction212.2.1 Types of Stock Market Prediction222.2.2 Stock Market Prediction Frameworks26

viii2.2.3 Fuzzy Based Stock Market Prediction27Frameworks2.32.2.4 Stock Market Prediction Techniques292.2.5 Datasets31Turning Points Prediction342.3.1 Representation of Turning Points342.3.2 Stock Turning Points Prediction35Techniques2.4Stock Patterns Extraction382.4.1 Time Series Indexing Techniques392.4.2 Patterns Retrievals From Indexed Time43Series2.52.62.7Feature Selection442.5.1 Definition of Feature Selection442.5.2 Sequential Search Algorithms46Fuzzy Classification502.6.1 Fuzzy Sets Concepts502.6.2 Fuzzy If-Then Rules52Neuro-Fuzzy Classifiers2.7.1 Structures of Adaptive Neuro-Fuzzy5354Classifier2.7.2 Learning in Adaptive Neuro-Fuzzy57Classifier2.7.3 The Cost Function in Scaled Conjugate57Gradient2.8Techniques for Fuzzy Rules Generation592.8.1 Simple fuzzy Grid Partition602.8.2 Subtractive Clustering Method622.8.3 Converting Clusters to Initial Rules642.9 Optimization Methods662.9.1 Genetic Algorithms662.9.2 Particle Swarm Optimization712.9.3 Hybrid of Particle Swarm Optimization75

ixand Genetic Algorithms2.9.4 Evaluation of Optimization Methods3772.10 Stock Market Trading Strategies802.11 Discussion822.12 Summary84RESEARCH METHODOLOGY853.1Introduction853.2Research Operational Framework863.2.1 Phase 1: Initial Study and Data Collection863.2.2 Phase 2: Development of Zigzag Patterns92Extraction Method3.2.3 Phase 3: Development of Hybrid118Optimization Method3.2.4 Phase 4: Development and Implementation124of Stock Turning Points Prediction Model3.34SummaryFINANCIAL TIME SERIES REPRESENTATION144145FOR ZIGZAG PATTERNS EXTRACTION4.1Introduction1454.2The Proposed Model1474.2.1 Zigzag Perceptually Important Points147Identification Method4.2.2 Index ZIPs Using Zigzag Based152Multi-way Search Tree4.3ZM-Tree Traversal1544.4Experimental Results and Discussion1584.4.1 Evaluation of Zigzag Perceptually158Important Points IdentificationMethod4.4.2 Evaluation of Tree Retrievals1664.4.3 Evaluation of Zigzag Shape Failure175

xRate (ZFR)54.5Discussion1764.5Summary179A HYBRID PARTICLE SWARM OPTIMIZATION180AND GENETIC ALGORITHMS METHOD FORGLOBAL OPTIMIZATION PROBLEMS5.1Introduction1805.2Solving Global Optimization Problems1825.3The Basic Concepts of Particle Swarm183Optimization and Genetic Algorithms65.3.1 Particle Swarm Optimization Method1835.3.2 Genetic Algorithms1845.4The Proposed Hybrid Approach1855.5Experimental Results1855.5.1 Benchmark Functions1855.5.2 Performance Measurement1865.5.3 The Results1875.6Discussion1965.7Summary199SWARM BASED FUZZY TURNING POINTS200PREDICDTION MODEL6.1Introduction2006.2Stock Turning Points Representation and202Prediction6.3The Proposed Turning Points Prediction204Model6.4Experimental Results2046.4.1 Datasets2056.4.2 Parameters Setup for The209Prediction Model6.4.3 Comparisons of Prediction Results211

xi6.4.4 Results of Feature Selection2126.4.5 Results of Prediction Accuracy2176.4.6 Results of Rate of Returns2206.4.7 Results of Rate of Successfully227Trading S2367.1Introduction2367.3Contribution of the Study2377.5Future Work2397.6Summary240REFERENCESAppendix A241261-263

xiiLIST OF TABLESTABLE NO.1.1TABLEPAGEIssues in turning points prediction with solved and12unsolved issues2.1Types of stock prediction target with problems23remain unsolved2.2The used datasets in stock market prediction312.3The summary of PSO and GA hybridization by76incorporating the mutation operation into the PSOparticles3.1Summary of the problem formulation893.2Selected stocks from S&P500 index components913.3Features to be selected as inputs to adaptive neuro-128fuzzy classifier3.4Zigzag patterns specification for each dataset1303.5Example of prediction results from turning points140prediction model3.6An example of the generated turning points from141the predicted trend classes4.1ZIPs collected by the ZIP identification function151from the 17 points synthetic time series4.2The vertical distance threshold (vdthres) values for166evaluating the tree pruning approach4.3The zigzag shape failure rate (ZFR) of stocks for176different retrieval methods5.1Parameters setting for PSO, GA, APSO, and MPSO1865.2Mean fitness values of Griewank function1875.3Mean fitness values of Rastrigin function189

xiii5.4Mean fitness values of Rosenbrock function1915.5The 2-Way ANOVA results for Griewank function1935.6Analysis results for the benchmark functions for194dimension of 10 in term of standard deviation(s.t.d.)6.1Selected stocks from S&P500 index components2066.2The sets of selected targets oscillation size and210trading time frame6.3The parameters setting for mutation capable particle210swarm optimization method (MPSO)6.4The parameters setting for subtractive clustering211method6.5The selected features and number of generated213fuzzy rules for stocks whichtheir testing periods arein the upward trend6.6The selected features and number of generated214fuzzy rules for stocks which their testing periods arein the sideways trend6.7The selected features and number of generated215fuzzy rules for stocks which their testing periods arein the downward trend6.8Overall comparisons of prediction accuracy of217upward trend stocks6.9Overall comparisons of prediction accuracy of218sideways trend stocks6.10Overall comparisons of prediction accuracy of219downward trend stocks6.11The rate of return with the best target set of upward221trend stocks6.12The rate of return with the best target set of223sideways trend stocks6.13The rate of return with the best target set ofdownward trend stocks225

xiv6.14The 2-Way ANOVA results for rate of return2276.15The rate of successfully trading operation for228SFTPP model

xvLIST OF FIGURESFIGURE NO.1.1TITLEPAGEPlot of stock prices with trends and points of peaks6and troughs which represent the turning points forstock of Akamai Technologies Inc. (AKAM)1.2Plot of stock prices and trends representing ofbuying/selling pointsforstockof7AkamaiTechnologies Inc. (AKAM)2.1Plot of stock prices with trends and points of peaks35and troughs which represent the turning points forstock of United Technologies Corp. (UTX)2.2Representation of piecewise linear rporation (IBM)2.3Piecewise aggregate approximation (PAA) and its40original time series of S&P500 index2.4The measurements of point importance for PIP41identifications2.5Pseudo code of the PIP identification process422.6Some Perceptually important points (PIPs) and the42original time series of S&P500 index2.7Feature selection method diagrams (a) Filter46method, (b) Wrapper method2.8Sequential forward selection algorithm472.9Sequential backward selection algorithm482.10Triangular membership function512.11Membership functions of grade point average52(GPA)

xvi2.12Architecture of adaptive neuro-fuzzy classifier552.13Fuzzy subspaces of two input variables X1 and X2,60each variable is divided into K subspaces of A1, ,AK2.14Chromosome representation in genetic presentation2.15A cycle of genetic algorithms672.16Two points crossover operation in genetic69algorithms2.17Particle swarm optimization flowchart723.1Research operational framework873.2The stock market time series zigzag patterns93extraction (STZE) method3.3The vertical distance (VD) measurement of the94point pc where the black lines denote the stocktime series segment, the red dash line denotes theVD, and the blue line denotes a straight lineconnecting between the first point and the lastpoint of the time series segment3.4measurements of the96Pseudo code of the algorithm for ZIP identification98The illustration ofandpoint pc3.5process3.6Pseudo code of the algorithm for GetZIPS function99for using in ZIP identification process3.7Condition of determining zigzag turning signals100(ZTS)3.8A node structure of the ZM-Tree1023.9Pseudo code of algorithm inserting a new key into104the ZM-Tree3.10Pseudo code of seeking a position for inserting anew key to the ZM-Tree105

xvii3.11Pseudo code of traversing the ZM-Tree algorithm1073.12The illustration of mapping 7 important points112(x1.x7) into 6 time series segments(T1.T6)3.13Zigzag shape of important points; (a) and (b) are115the demonstration of zigzag shape success pointsand (c) and (d) are the demonstration of zigzagshape failure points3.14The algorithm of calculating the zigzag shape116failure rate (ZFR)3.15Plot of example important points in the list Z1173.16Flowchart of mutation capable particle swarm121optimization (MPSO) algorithm3.17Swarm based fuzzy turning points prediction126(SFTPP) model3.18An example of the collected zigzag perceptually131important points3.19Pseudo code for algorithm of transforming the132zigzag patterns to zigzag trends3.20The plot of the series of trend classes1333.21Initial membership functions of three generated136fuzzy rules on the x1 feature3.22Example of membership functions after learning1374.1Synthetic time series of 17 data points1484.2Steps 1 to 8 of collecting ZIPs from the 17 points149synthetic time series based on ZIP identificationmethod4.3Step 9 to 14 of collecting ZIPs from the 17 points150synthetic time series based on ZIP identificationmethod4.4The step by step of creating ZM-Tree from thecollected ZIPs153

xviii4.5Steps of ZM-Tree traversal, dark grey nodes are156visited nodes and light grey nodes are nodes do notyet son of mean square error (MSE) of160Communications (FTR) stock4.7different number of collected points of differentthree methods; ZIP, PLR, and PIP for FTR stock4.8The plot of close prices for Intel Corporation161(INTC) stock4.9Comparison of mean square error (MSE) of162different number of collected points of differentthree methods; ZIP, PLR, and PIP for INTC stock4.10The plot of close prices for International Game163Technology (IGT)4.11Comparison of mean square error (MSE) of164different number of collected points of differentthree methods; ZIP, PLR, and PIP for IGT stock4.12The time series shape of the first ten important165points from different methods and the originaltime series of INTC4.13Mean square error (MSE) of FTR based on tree167pruning approach with different values of verticaldistance thresholds4.14Mean square error (MSE) of INTC based on tree167pruning approach with different values of verticaldistance thresholds4.15Mean square error (MSE) IGT based on tree168pruning approach with different values of verticaldistance thresholds4.16Mean square error (MSE) of the reconstructedtime series of FTR, INTC, and IGT169

xix4.17The retrieved time series with the 1% oscillation170size threshold for stock Intel Corporation (INTC)4.18The retrieved time series with the 3% oscillation170size threshold for stock Intel Corporation (INTC)4.19The retrieved important points with the 5%171oscillation size threshold for Intel Corporation(INTC) stock4.20Mean square error (MSE) of the reconstructed172time series from the retrieved important points andthe original time series of the three stocks; FTR,INTC, and IGT4.21The retrieved time series with the 1% oscillation173size threshold and 1 days trading time frame sizethreshold INTC4.22The retrieved time series with the 3% oscillation174size threshold and 3 days trading time frame sizethreshold for INTC4.23The retrieved time series with the 5% oscillation174size threshold and 5 days trading time frame sizethreshold for INTC5.1Mean fitness values plots of Griewank function188experimental results with population size of 20,40, and 80 for problem dimension of 10, 20 and 30for each case of population size5.2Mean fitness values plots of Rastrigin function190experimental results with population size of 20,40, and 80 for problem dimension of 10, 20 and 30for each case of population size5.3Mean fitness values plots of Rosenbrock functionexperimental results with population size of 20,40, and 80 for problem dimension of 10, 20 and 30for each case of population size192

xx5.4Standard deviations of fitness values for Griewank195function with different population size anddifferent algorithms5.5Standard deviations of fitness values for Rastrigin195function with different population sizes anddifferent 96Buying points and selling points according to the203Rosenbrock function in different algorithms6.1changing of the price trends of Exelon Corporation(EXC)6.2Price and trend line in upward trend during the207testing period for International Business MachinesCorporation (IBM) stock6.3Price and trend line in sideways trend during the208testing period for Exelon Corporation (EXC) stock6.4Price and trend line in downward trend during the208testing period for Akamai Technologies, Inc.(AKAM) stock6.5Fuzzy rules generated from stock AKAM with 4216input variables and 4 fuzzy rules6.6The comparison of the rate of return (ROR) of222overall methods for stocks in upward trend6.7The comparison of the rate of return of overall224methods for stocks in sideways trend6.8The comparison of the rate of return of overall226methods for stocks in downward trend6.9Plot of the rate of success trade for the best229performance sets generated by SFTPP model6.10A plot of generated buy/sell positions by SFTPP230model for an upward trend stock of IBM6.11A plot of generated buy/sell positions by SFTPPmodel for a sideways trend stock of EXC231

xxi6.12A plot of generated buy/sell positions by SFTPPmodel for a downward trend stock of AKAMLIST OF ABBREVIATIONSANFC-Adaptive Neuro-Fuzzy ClassifierANFIS-Adaptive Neuro-Fuzzy Inference SystemANN-Artificial Neural NetworksAPSO-Adaptive Particle Swarm OptimizationB&H-Buy&Hold trading modelGA-Genetic AlgorithmsKNN-K-Nearest NeighboursMPSO-Mutation Capable Particle Swarm OptimizationPIP-Perceptually Important PointPLR-Piecewise Linear RepresentationPSO-Particle Swarm Optimizations.t.d-Standard DeviationSCG-Scaled Conjugate GradientSFTPP-Swarm Based Fuzzy Turning Prediction ModelVD-Vertical DistanceZIP-Zigzag Perceptually Important PointZM-Tree-Zigzag Based Multi-Way Search Tree231

xxiiLIST OF APPENDICESAPPENDIXATITLETechnical indicator formulaPAGE261

CHAPTER 1INTRODUCTION1.1OverviewA large fraction of attention from the data mining community has focused ontime series data. This is plausible and highly anticipated since time series data is aby-product in virtually every human endeavor, including biology (Titsias et al.,2012), finance (Liu et al., 2012), geology (Morton et al., 2011), space exploration(Lafleur and Saleh, 2010), and human motion analysis (Akiduki et al., 2011). Thestudy of time series dates back to the 1960s, where the analysts focused mainly onfinancial data such as stock market movements. Common tasks on classic time seriesanalysis include prediction, finding trends, seasonality, etc.Financial or stock market prediction can be considered as an attractive tasksince it is able to gain amount of money which people who trade in financial or stockmarkets usually focus their determination to the market timing for taking action tobuy, hold, or sell (Chang et al., 2011). Unfortunately, stock market prediction is notan easy task, due to the fact that stock market is essentially dynamic, nonlinear,complicated, nonparametric, imprecise, and chaotic in nature (Jung et al., 2011; Liuet al., 2011; Ozer and Ertokatli, 2011; Peters, 1994).Financial time series has high volatility, where the time series change as thestock markets move in and out of different periods, or in other words, stock marketshows the variation of stock prices as upward and downward direction overtime

2(Golosnoy et al., 2011). In addition, stock market's movements are affected by manymacro-economic factors such as political events, firms’ policies, general economicconditions, investors' expectations, institutional investors' choices, movement ofother stock markets, and psychology of investors (Chang et al., 2009).Thosefactors drive stock prices moving in upward, downward, or sideways trends. Stockprices are determined solely by interaction of demand and supply. Furthermore, stockprices tend to move in trends (Edwards et al., 2007b) . Shifts in demand and supplycauses reversals in trends and can be detected in charts (Bauer and Dahlquist, 1999).Finally, chart patterns tend to repeat themselves (Brown, 2012; Canelas et al., 2012;Edwards et al., 2007a). Hence the shifts of demand and supply influence the stockand will affect the stock price. However, technical analysts believe that the market isalways correct , all factors are already factored into the demand and supply curves,and, thus, the price of the company’s stock (Kirkpatrick, 2007; Schwager, 2012).As mentioned above, the stock prices often move up and down. Obviously,considering price movement behaviors after an uptrend movement, the stock oftenoppositely changes the trend to the down trend movement. Conversely, after thedown trend ends, the stock trend often changes the direction to the uptrend again.The trends frequently change the directions to upward and downward trends subsequentially. The changing points of upward trends to the downward trends areknown as peaks and the changing points of the downward trends to the upwardtrends are known as troughs. In other words, a peak will appear when the stock priceswhich is in an upward trend is interrupted and the stock prices start to move in thedownward trend, and conversely, a trough will appear when the stock prices which isin a downward trend is interrupted and the stock prices start to move in the upwardtrend. The term “zigzag pattern” has been used to describe the peaks and troughs thatinvestors can lay down on a chart that they are viewing (Edwards, et al., 2007b),however, the significant zigzag patterns are unobvious, contaminated with noise, orhidden in the data and, hence, are difficult to be discovered and interpreted.Zigzag patterns is one of stock price patterns that experts use along with someother patterns such as reversal patterns (Bouchentouf et al., 2011), or Elliott waves(Brown, 2012; Richard, 2003) to predict the future price movement. Unfortunately,

3experts predict the stock market based on vague, imperfect and uncertain knowledgerepresentation because they usually use the raw data which usually consist of highdimensionality, is imprecise, and uncertain, in their stock market time series. Alongwith the development of artificial intelligence; for example, machine learning anddata mining, a number of researchers attempted to build automatic decision supportsystems to predict stock market (Chan and Franklin, 2011; Wen et al., 2010). Anumber of artificial intelligent methods have been applied for stock prediction suchas neural networks (Chaigusin et al., 2008; Chang et al., 2012; Hajizadeh et al.,2012; Pino et al., 2008), evolutionary methods (Hsu, 2011; Wang et al., 2012),support vector machine (Wen, et al., 2010; Zhao et al., 2012), etc. However, asstock market prediction relates to imprecise concepts and imprecise reasoningdecision (Zadeh, 1975), therefore fuzzy logic is seen as a choice for knowledgerepresentation and is applied in stock market prediction (Atsalakis and Valavanis,2009a; Boyacioglu and Avci, 2010a; Liu et al., 2012; Wei, 2011).Fuzzy logic, introduced by Zadeh (1965, 1975), is a form for reasoningmethod with vague knowledge. A fuzzy based model is known as a preferableapproach among a number of available models for making prediction. It is essentialfor the prediction model that closely corresponds to the way experts work likeinteractive problem solving and explanation facilities to justify the decision making.However, among above approaches, using a single method for stock marketprediction may produce the poor result with low accuracy or high error comparing tothe actual values.Obviously, by nature, the stock market prediction problemrequires the combination of a number of techniques together instead of exclusivesingle technique to increase the prediction performance (Atsalakis and Valavanis,2009b; Wang, et al., 2012). Recently, researchers combined fuzzy logic techniquewith neural networks (Agrawal et al., 2010; Boyacioglu and Avci, 2010b), particleswarm optimization (Liu, et al., 2012) , genetic algorithms (Chang, et al., 2012) order to improve the prediction performance. The results reported that, obviously,the hybridizations of fuzzy logic with other methods produce better predictionperformance than their basic single methods. However, in the fuzzy based methods,the appropriate number of generated fuzzy rules is important because it affects the

4prediction performance, thus the optimal number of fuzzy rules is still an issue andrequired to be improved.Searching for the appropriate number of fuzzy rules has been widely studied.However, a number of researchers used the subtractive clustering method (Chiu,1994)to solve the problems (Esfahanipour and Mardani, 2011; Torun andTohumoglu, 2011; Zanaganeh et al., 2009) because it is able to find an appropriatenumber of clusters which correspond to a number of fuzzy rules. However, thesubtractive clustering method requires some predetermine parameters to search anumber of clusters. Some optimization methods; e.g. particle swarm optimization,and genetic algorithm were used to find the optimal values of these parameters (Chenet al., 2008; Shahram, 2011; Zanaganeh, et al., 2009).1.2Background of ProblemPrediction of stocks is generally believed to be a very difficult task. There areseveral attempts to predict stock market in order to help investors to make decisionof buying a stock at the bottom and selling it at the top in the range. The points wherestock prices change their trend directions are called turning points (Bao and Yang,2008). The turning point of changing the trend direction from an upward trend to thedownward trend is called the peak, and the turning point of changing the trenddirection from a downward trend to an upward trend is called the trough point(Siegel, 2000). Predicting price behaviors on the financial market such as trends andturning points have been considered as important tasks and have been widelydiscussed (Bao and Yang, 2008; Chang, et al., 2012; Li and Deng, 2008; Ni et al.,2011; Poddig and Huber, 1999).In general, the markets do not exclusively move in one direction, but theymove in upward and downward directions sub-sequentially by a series of zigzagdirections (Edwards, et al., 2007b).These zigzag directions form a series ofconsecutive zigzag waves which represent the obvious peaks and troughs. The

5direction of each pair of a peak and a trough constitutes a market trend of upward,downward, and sideways trends. An upward trend is a series of consecutively higherpeaks and troughs; a downward trend is a series of consecutively lower peaks andtroughs; finally, a sideways trend is a series of horizontal peaks and troughs(Edwards, et al., 2007b; Siegel, 2000).Naturally, the prediction of financial time series trends relies on the discoveryof strong empirical turning points in observations of the system (Li, 2009; Liu andKwong, 2007). Turning points, obviously, position nearby or at the peaks andtroughs of the time series (Bao and Yang, 2008). Nevertheless, since these turningpoints are often masked by noise, and hidden in the price movement, thus, theaccurate prediction of trends and turning points is very difficult. Many researchershave attempted to predict stock market based on learning from turning points, whichthe experimental results showed that learning from the historical turning pointsaffected the stock market prediction performance (Bao, 2007; Bao and Yang, 2008;Chang, et al., 2012; Li, 2009). In order to predict the stock market in the accurateway, discovery and learning from the zigzag patterns are very important since thezigzag patterns represent the zigzag moving trends of prices consisting of

stock market turning points. The proposed turning points prediction model is tested using stock market datasets which are the historical data of stocks listed as components of S&P500 index of New York Stock Exchange. These data are stock prices that are either moving upward, downward, or sideways. From the findings, the

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