Stock Market Forecasting Using Hidden Markov Model: A New Approach

Transcription

1 Stock Market Forecastng Usng Hdden Markov Model: A New Approach Md. Raful Hassan and Bakunth Nath Computer Scence and Software Engneerng The Unversty of Melbourne, Carlton 3010, Australa. {mrhassan, bnath}@cs.mu.oz.au Abstract Ths paper presents Hdden Markov Models (HMM) approach for forecastng stock prce for nterrelated markets. We apply HMM to forecast some of the arlnes stock. HMMs have been extensvely used for pattern recognton and classfcaton problems because of ts proven sutablty for modellng dynamc systems. However, usng HMM for predctng future events s not straghtforward. Here we use only one HMM that s traned on the past dataset of the chosen arlnes. The traned HMM s used to search for the varable of nterest behavoural data pattern from the past dataset. By nterpolatng the neghbourng values of these datasets forecasts are prepared. The results obtaned usng HMM are encouragng and HMM offers a new paradgm for stock market forecastng, an area that has been of much research nterest lately. Key Words: HMM, stock market forecastng, fnancal tme seres, feature selecton 1. Introducton Forecastng stock prce or fnancal markets has been one of the bggest challenges to the AI communty. The objectve of forecastng research has been largely beyond the capablty of tradtonal AI research whch has manly focused on developng ntellgent systems that are supposed to emulate human ntellgence. By ts nature the stock market s mostly complex (non-lnear) and volatle. The rate of prce fluctuatons n such seres depends on many factors, namely equty, nterest rate, securtes, optons, warrants, merger and ownershp of large fnancal corporatons or companes etc. Human traders can not consstently wn n such markets. Therefore, developng AI systems for ths knd of forecastng requres an teratve process of knowledge dscovery and system mprovement through data mnng, knowledge engneerng, theoretcal and data-drven modellng, as well as tral and error expermentaton. The stock markets n the recent past have become an ntegral part of the global economy. Any fluctuaton n ths market nfluences our personal and corporate fnancal lves, and the economc health of a country. The stock market has always been one of the most popular nvestments due to ts hgh returns [1]. However, there s always some rsk to nvestment n the Stock market due to ts unpredctable behavour. So, an ntellgent predcton model for stock market forecastng would be hghly desrable and would of wder nterest. A large amount of research has been publshed n recent tmes and s contnung to fnd an optmal (or nearly optmal) predcton model for the stock market. Most of the forecastng research has employed the statstcal tme seres analyss technques lke autoregresson movng average (ARMA) [2] as well as the multple regresson models. In recent years, numerous stock predcton systems based on AI technques, ncludng artfcal neural networks (ANN) [3, 4, 5], fuzzy logc [6], hybrdzaton of ANN and fuzzy system [7, 8, 9], support vector machnes [10] have been proposed. However, most of them have ther own constrants. For nstance, ANN s very much problem orented because of ts chosen archtecture. Some researchers have used fuzzy systems to develop a model to forecast stock market behavour. To buld a fuzzy system one requres some background expert knowledge. In ths paper, we make use of the well establshed Hdden Markov Model (HMM) technque to forecast stock prce for some of the arlnes. The HMMs have been extensvely used n the area lke speech recognton, DNA

2 sequencng, electrcal sgnal predcton and mage processng, etc. In here, HMM s used n a new way to develop forecasts. Frst we locate pattern(s) from the past datasets that match wth today s stock prce behavour, then nterpolate these two datasets wth approprate neghbourng prce elements and forecast tomorrow s stock prce of the varable of nterest. Detals of the proposed method are provded n Secton 3. The remander of the paper s organsed as follows: Secton 2 provdes a bref overvew on HMM; Secton 4 lsts expermental results obtaned usng HMM and compares wth results obtaned usng ANN; and fnally Secton 5 concludes the paper. 2. HMM as a Predctor A Hdden Markov Model (HMM) s a fnte state machne whch has some fxed number of states. It provdes a probablstc framework for modellng a tme seres of multvarate observatons. Hdden Markov models were ntroduced n the begnnng of the 1970 s as a tool n speech recognton. Ths model based on statstcal methods has become ncreasngly popular n the last several years due to ts strong mathematcal structure and theoretcal bass for use n a wde range of applcatons. In recent years researchers proposed HMM as a classfer or predctor for speech sgnal recognton [11, 12, ], DNA sequence analyss [], handwrtten characters recognton [], natural language domans etc. It s clear that HMM s a very powerful tool for varous applcatons. The advantage of HMM can be summarzed as: HMM has strong statstcal foundaton It s able to handle new data robustly Computatonally effcent to develop and evaluate (due to the exstence of establshed tranng algorthms). It s able to predct smlar patterns effcently [] Rabner [] tutoral explans the bascs of HMM and how t can be used for sgnal predcton. The next sesson descrbes the HMM n bref The Hdden Markov Model Hdden Markov Model s characterzed by the followng 1) number of states n the model 2) number of observaton symbols 3) state transton probabltes 4) observaton emsson probablty dstrbuton that characterzes each state 5) ntal state dstrbuton For the rest of ths paper the followng notatons wll be used regardng HMM N = number of states n the model M = number of dstnct observaton symbols per state (observaton symbols correspond to the physcal output of the system beng modelled) T = length of observaton sequence O = observaton sequence,.e., O 1,O 2,O 3, O T Q = state sequence q 1,q 2,., q T n the Markov model A = {a j } transton matrx, where a j represents the transton probablty from state to state j B={b j (O t )} observaton emsson matrx, where b j (O t ) represent the probablty of observng O t at state j π = {π } the pror probablty, where π represent the probablty of beng n state at the begnnng of the experment,.e., at tme t = 1 λ = (A,B,π) the overall HMM model. As mentoned above the HMM s characterzed by N, M, A, B and π. Thea j,b (O t ), and π have the propertes j =, b ( Ot ) = 1, = j a 1 π 1 and a, b ( O ), π 0 for all, j, t. j t t To work wth HMM, the followng three fundamental questons should be resolved 1. Gven the model λ= (A,B,π) howdowecompute P(O λ), the probablty of occurrence of the observaton sequence O = O 1,O 2,..,O T. 2. Gven the observaton sequence O and a model λ, howdowechooseastatesequenceq 1,q 2,..,q T that best explans the observatons. 3. Gven the observaton sequence O and a space of models found by varyng the model parameters A, B and π, how do we fnd the model that best explans the observed data. There are establshed algorthms to solve the above questons. In our task we have used the forward-backward algorthm to compute the P(O λ), Vterb algorthm to resolve problem 2, and Baum-Welch algorthm to tran the HMM. The detals of these algorthms are gven n the tutoral by Rabner []. 3. Usng HMM for Stock market forecastng In ths secton we develop an HMM based tool for tme seres forecastng, for nstance for the stock market

3 forecastng. Whle mplementng the HMM, the choce of the model, choce of the number of states and observaton symbol (contnuous or dscrete or mult-mxture) become a tedous task. For nstance we have used left-rght HMM wth 4 states. In our problem, for smplcty, we consder 4 nput features for a stock that s the openng prce, closng prce, hghest prce, and the lowest prce. The next day s closng prce s taken as the target prce assocated wth the four nput features. Our observatons here beng contnuous rather than dscrete, we choose emprcally as many as 3 mxtures for each state for the model densty. For the pror probabltyπ, a random number was chosen and normalzed so that. The dataset = 1 beng contnuos, the probablty of emttng symbols from a state can not be calculated. For ths reason a threedmensonal Gaussan dstrbuton was ntally chosen as the observaton probablty densty functon. Thus, we have j c ℵ[ O, U ] = N π = 1 b ( O) µ, 1 j N where O = vector of observatons beng modelled c = mxture coeff. for the m-th mxture n state j, M c = 1 where m= 1 µ = mean vector for the m-th mxture component n state j U = Covarance matrx for the m-th mxture component n state j ℵ = Gaussan densty. In the experment, our objectve was to predct the next day s closng prce for a specfc stock market share usng aforementoned HMM model. For tranng the model, past one and a half years (approxmately) daly data were used and recent last three month s data were used to test the effcency of the model. The nput and output data features were as follows: Input: openng, hgh, low, and closng prce Output: next day s closng prce The dea behnd our new approach n usng HMM s that of usng the tranng dataset for estmatng the parameter set (A, B,π ) of the HMM. For a specfc stock at the market close we know day s prce values of the four varables (open, hgh, low, close), and usng ths nformaton our objectve s to predct next day s closng prce. A forecast of any of the four varables for the next day ndeed wll be of tremendous value to the traders and nvestors. Usng the traned HMM, lkelhood value for current day s dataset s calculated. For example, say the lkelhood value for the day s, then from the past dataset usng the HMM we locate those nstances that wouldproducethesame or nearest to the lkelhood value. That s we locate the past day(s) where the stock behavour s smlar to that of the current day. Assumng that the next day s stock prce should follow about the same past data pattern, from the located past day(s) we smply calculate the dfference of that day s closng prce and next to that day s closng prce. Thus the next day s stock closng prce forecast s establshed by addng the above dfference to the current day s closng prce. 4. Expermentaton: Tranng and Testng In order to tran the HMM, we dvded the dataset nto two sets, one tranng set and one test (recall) set. For example, we traned an HMM usng the daly stock data of Southwest for the perod 18 December 2002 to 29 September 2004 to predct the closng prce on 30 September The traned HMM produced lkelhood value of for the stock prce on 29 September Usng ths traned HMM and the past data, we located a (closer) lkelhood value on 01 July Fgure 1 shows the smlartes between these two datasets (stock prces on 30 September 2004 and 01 July 2003). It seems qute logcal that 29 September 2004 stock behavour should follow the behavour that of 01 July Prce 18 MatchedData(Past) Open Hgh Low Close Varables Known Data (Today) Fgure 1. The current day s stock prce behavour matched wth past day s prce data Prce Day Actual Prce Predcted Prce Fgure 2. Actual Vs. Predcted stock prce

4 Table 1. Dataset along wth the matched past dataset Today s data 29 Sep 2004 Matched data pattern usng HMM 01 Jul 2003 Next day s data 02 Jul 2003 Openng prce Hgh prce Low prce Closng prce $.63 $.73 $.49 $.62 $.1 $.2 $.83 $. $.36 Predcted closng prce (30 Sep 2004) Actual closng prce (30 Sep 2004) $.85 $.85 Predcted Prce Actual Prce Table 2. Informaton on tranng and test datasets Stock Tranng Data Test Data Name From To From To Brtsh /09/ /09/ /09/ /01/2005 Delta 27/12/ /08/ /09/2004 /11/2004 Southwest 18/12/ /07/ /07/2004 /11/2004 Ryanar Holdngs Ltd. 06/05/ /12/ /12/2004 /03/2005 Fgure 3. The correlaton between predcted and actual closng stock prce We, therefore, calculated the dfference between the closng prces on 01 July 2003 and the next day 02 July That s $.36-$. =$0.23.Then ths dfference s added to the closng prce on 29 September 2004 to forecast closng prce for 30 September Table 1 shows the predcted and the actual prces of stock on 30 September Fgure 2 shows the predcton accuracy of the model and Fgure 3 shows the correlaton among the predcted values and the actual values of the stock. For the aforementoned experment the mean absolute percentage error (MAPE) = 2.01, and R 2 = Stock prce forecasts for some arlnes We have traned four HMM for four dfferent stock prce. Usng the same tranng dataset we traned four dfferent (same archtecture) ANN. Then we predcted the next few day s closng prce of these three stocks usng the HMMs n aforementoned method and we predcted the same day s closng prce usng ANN. The Table 2 shows the nformaton of the tranng dataset and the test dataset whle the Table 3 shows the predcton accuracy of these two models. Table 3. Predcton accuracy of ANN and the proposed method ANN (MAPE) Proposed method (MAPE) Brtsh Delta Southwest Ryanar Holdngs Ltd Concluson ANN s well researched and establshed method that has been successfully used to predct tme seres behavour from past datasets. In ths paper, we proposed the use of HMM, a new approach, to predct unknown value n a tme seres (stock market). It s clear from Table 3 that the mean absolute percentage errors (MAPE) values of the two methods are qute smlar. Whlst, the prmary weakness wth ANNs s the nablty to properly explan the models. Accordng to Repley the desgn and learnng for feed-forward networks are

5 hard. Judd [18] and Blum and Rvest [19] showed ths problem to be NP- complete. The proposed method usng HMM to forecast stock prce s explanable and has sold statstcal foundaton. The results show potental of usng HMM for tme seres predcton. In our future work we plan to develop hybrd systems usng AI paradgms wth HMM to further mprove accuracy and effcency of our forecasts. 6. References [1] Kuo R J, Lee L C and Lee C F (1996), Integraton of Artfcal NN and Fuzzy Delph for Stock market forecastng, IEEE Internatonal Conference on Systems, Man, and Cybernetcs, Vol. 2, pp [2] Kmoto T, Asakawa K, Yoda M and Takeoka M (1990), Stock market predcton system wth modular neural networks, Proc. Internatonal Jont Conference on Neural Networks, San Dego, Vol. 1, pp [3] Whte H (1998), Economc Predcton Usng Neural Networks: The Case of IBM Daly Stock Returns, Proceedngs of the Second Annual IEEE Conference on Neural Networks, Vol. 2, pp [4] Chang W C, Urban T L and Baldrdge G W (1996), A Neural Network Approach to Mutual Fund Net Asset Value Forecastng. Omega, Vol. 24 (2), pp [5] Km S H and Chun S H (1998), Graded forecastng usng an array of bpolar predctons: applcaton of probablstc neural networks to a stock market ndex. Internatonal Journal of Forecastng, Vol., pp [6] Romah Y and Shen Q (2000), Dynamc Fnancal Forecastng wth Automatcally Induced Fuzzy Assocatons, Proceedngs of the 9 th nternatonal conference on Fuzzy systems, pp [7] Thammano A (1999), Neuro-fuzzy Model for Stock Market Predcton, Proceedngs of the Artfcal Neural Networks n Engneerng Conference, ASME Press, New York, pp [8] Abraham A, Nath B and Mahant P K (2001), Hybrd Intellgent Systems for Stock Market Analyss, Proceedngs of the Internatonal Conference on Computatonal Scence. Sprnger, pp [9] Raposo R De C T and Cruz A J De O (2004), Stock Market predcton based on fundamentalst analyss wth Fuzzy-Neural Networks. [10] Cao L and Tay F E H (2001), Fnancal Forecastng Usng Support Vector Machnes, Neural Computaton and Applcaton, Vol. 10, pp [11] Huang X, Ark Y, Jack M (1990), Hdden Markov Models for speech recognton. Ednburgh Unversty Press. [12] Jelnek F, Kaufmann M, Mateo C S (1990), Selforganzed language modellng for speech recognton, n Readngs n Speech Recognton (Eds. Alex Wabel and Ka-Fu Lee), Morgan Kaufmann, San Mateo, Calforna, pp [] Xe H, Anreae P, Zhang M, Warren P (2004), Learnng Models for Englsh Speech Recognton, Proceedngs of the 27 th Conference on Australasan Computer Scence, pp [] Lebert M A (2004), Use of runs statstcs for pattern recognton n genomc DNA sequences. Journal of Computatonal Bology, Vol. 11, pp [] Vncarell A and Luettn J (2000), Off-lne cursve scrpt recognton based on contnuous densty HMM, Proceedngs of the 7 th Internatonal Workshop on Fronters n Handwrtng Recognton, Amsterdam, pp [] L Z, Wu Z, He Y and Fule C (2005), Hdden Markov model-based fault dagnostcs method n speed-up and speed-down process for rotatng machnery. Mechancal Systems and Sgnal Processng, Vol. 19(2), pp [] Rabner R L (1989), A Tutoral on Hdden Markov Models and Selected Applcatons n Speech Recognton, Proceedngs of the IEEE, Vol. 77(2), pp [18] Judd J S, (1990), Neural Network desgn and Complexty of Learnng. MIT Press, USA. [19] Blum A L and Rvest R L, (1992), Tranng a 3-node Neural Networks s NP- complete. Neural Networks, Vol. 5, pp