lssvm-abc algorithm for stock price prediction

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International Journal of Computer Trends and Technology (IJCTT) volume 7 number 2Jan 2014

ISSN: 2231-2803 www.internationaljournalssrg.org Page 81

LSSVM-ABC Algorithm for Stock Price predictionOsman Hegazy

1, Omar S. Soliman

2and Mustafa Abdul Salam

3

1, 2(Faculty of Computers and Informatics, Cairo University, Egypt)

3(Higher Technological Institute (H.T.I), 10th of Ramadan City, Egypt)

ABSTRACT : In this paper, Artificial BeeColony (ABC) algorithm which inspired from the

behavior of honey bees swarm is presented. ABC isa stochastic population-based evolutionaryalgorithm for problem solving. ABC algorithm,which is considered one of the most recently swarmintelligent techniques, is proposed to optimize leastsquare support vector machine (LSSVM) to predict

the daily stock prices. The proposed model is basedon the study of stocks historical data, technicalindicators and optimizing LSSVM with ABCalgorithm. ABC selects best free parameters

combination for LSSVM to avoid over-fitting andlocal minima problems and improve predictionaccuracy. LSSVM optimized by Particle swarm

optimization (PSO) algorithm, LSSVM, and ANNtechniques are used for comparison with proposedmodel. Proposed model tested with twenty datasets

representing different sectors in S&P 500 stockmarket. Results presented in this paper show thatthe proposed model has fast convergence speed,and it also achieves better accuracy than compared

techniques in most cases.

Keywords -Least square support vector machine,

Artificial Bee Colony, technical indicators, and

stock price prediction.I. INTRODUCTIONThe stock market field is characterized by

unstructured nature, data intensity, noise, andhidden relationships. Also the behavior of a stocktime series is close to random. For these reasonsthe stock market prediction is not a simple task. Infield of stock market, there are several predictiontools that have been used since years ago.

Fundamental and technical analyses were the firsttwo methods used to forecast stock prices. ARIMA(Autoregressive Integrated Moving Average) is oneof the most commonly used time series predictionmethods [1]. ARIMA method deals with linear, butit is difficult to deal with nonlinear feature in timeseries [2]. GARCH (Generalized AutoregressiveConditional Heteroskedasticity) linear time seriesprediction method is also used [3]. Artificial NeuralNetworks (ANNs) tool which is a branch of

Computational Intelligence (CI) is become one ofthe most commonly machine learning techniquesused in stock market prediction. ANNs predictionmethod is used to overcome the limitations ofabove methods. In most cases ANNs suffer fromover-fitting problem due to the large number ofparameters to fix, and the little prior userknowledge about the relevance of the inputs in theanalyzed problem [4]. Support vector machines(SVMs) have been developed as an alternative thatavoids the above prediction models limitations.

Their practical successes can be attributed to solidtheoretical foundations based on VC-theory [5].SVM computes globally optimal solutions, unlikethose obtained with ANN, which tend to fall intolocal minima [6]. Least squares support vectormachine (LSSVM) method which is presented in[7], is a reformulation of the traditional SVMalgorithm. LSSVM uses a regularized least squaresfunction with equality constraints, leading to alinear system which meets the Karush-Kuhn-Tucker (KKT) conditions for obtaining an optimal

solution. Although LSSVM simplifies the SVMprocedure, the regularization parameter and thekernel parameters play an important role in theregression system. Therefore, it is necessary toestablish a methodology for properly selecting theLSSVM free parameters, in such a way that theregression obtained by LSSVM must be robustagainst noisy conditions, and it does not need prioriuser knowledge about the influence of the freeparameters values in the problem studied [8].The perceived advantages of evolutionary

strategies as optimization methods motivated theauthors to consider such stochastic methods in thecontext of optimizing SVM. A survey andoverview of evolutionary algorithms (EAs) isfound in [9]. Since 2005, D. Karaboga and hisresearch group have been studying the ArtificialBee Colony (ABC) algorithm and its applicationsto real world problems. Karaboga and Basturk haveinvestigated the performance of the ABC algorithmon unconstrained numerical optimization problems
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which is found in [10], [11], [12] and its extendedversion for the constrained optimization problemsin [13]. Neural network trained with ABCalgorithm is presented in [14], [15]. ABCAlgorithm based approach for structural

optimization is presented in [16]. Applying ABCalgorithm for optimal multi-level thresholding isintroduced in [17]. ABC was used for MR brainimage classification in [18], cluster analysis in [19],face pose estimation in [20], and 2D protein foldingin [21]. A new hybrid ABC algorithm for robustoptimal design and manufacturing is introduced in[22]. Hybrid artificial bee colony-based approachfor optimization of multi-pass turning operations isused in [23]. Intrusion detection in AODV-basedMANETs using ABC and negative selection

algorithms is presented in [24]. A study on ParticleSwarm Optimization (PSO) and ABC algorithmsfor multilevel thresholding is introduced in [25].Heuristic approach for inverse kinematics of robotarms is found in [26]. A combinatorial ABCAlgorithm applied for traveling salesman problemis introduced in [27]. Identifying nuclear powerplant transients using the discrete binary ABCAlgorithm is presented in [28]. An ABC-AIShybrid approach to dynamic anomaly detection inAODV-based MANETs is found in [29]. Modified

ABC algorithm based on fuzzy multi-objectivetechnique for optimal power flow problem ispresented in[30]. ABC optimization was used formulti-area economic dispatch in [31]. MMSEdesign of nonlinear volterra equalizers using ABCalgorithm is introduced in [32]. Hybrid seekeroptimization algorithm for global optimization isfound in [33].

This paper proposes a hybrid LSSVM-ABC model.The performance of LSSVM is based on the

selection of hyper parameters C (cost penalty), (insensitive-loss function) and (kernel parameter).ABC will be used to find the best parametercombination for LSSVM.

The paper is organized as follows: Section IIpresents the Least square support vector machinealgorithm; Section III presents the Artificial BeeColony algorithm; Section IV is devoted for the

proposed model and implementation of ABCalgorithms in stock prediction; In Section V theresults are discussed. The main conclusions of thework are presented in Section VI.

II. LEAST SQUARE SUPPORT VECTORMACHINE

Least squares support vector machines(LSSVM) are least squares versions of supportvector machines (SVM), which are a set ofrelated supervised learning methods that analyzedata and recognize patterns, and which are usedfor classification and regression analysis. In thisversion one finds the solution by solving a setof linear equations instead of a convex quadraticprogramming (QP) problem for classical SVMs.LSSVM classifiers, were proposed by Suykens and

Vandewalle [34].

Let X is pn input data matrix and y is 1n

output vector. Given then

iii yx 1},{ training data

set, wherepRxi and Ryi , the LSSVM goal

is to construct the function yxf )( , which

represents the dependence of the output iy on the

inputi

x . This function is formulated as

bxWxf T )()(

Where Wand )(x :np RR are

1n column vectors, and Rb . LSSVMalgorithm [8] computes the function (1) from asimilar minimization problem found in the SVMmethod [6]. However the main difference is thatLSSVM involves equality constraints instead ofinequalities, and it is based on a least square costfunction. Furthermore, the LSSVM method solves

a linear problem while conventional SVM solves aquadratic one. The optimization problem and theequality constraints of LSSVM are defined asfollows:

(1)

TT

beweCwwbewj

2

1

2

1),,(min

,, (2)


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ii

T

i ebxwy )(

Where e is the 1n error vector, 1 is a 1n

vector with all entries 1, and

RCis the

tradeoff parameter between the solution size andtraining errors. From (2) a Lagranian is formed,

and differentiating with respect to aebw ,,,

( a is Largrangian multipliers), we obtain

yae

b

W

IZ

ICI

ZIT

T

0

0

0

01

00

1000

00

Where I represents the identity matrix and

T

nxxxZ )](),...,2(),1([

From rows one and three in (3) aZw T andaCe

Then, by defining the kernel matrix TZZK ,

and the parameter 1C , the conditions for

optimality lead to the following overall solution

ya

b

IK

T 0

1

10

Kernel function K types are as follows:

Linear kernel xxxxK Tii ),( Polynomial kernel of degree d:

dT

ii cxxxxK )/1(),(

Radial basis function RBF kernel :)/exp(),( 2

2ii xxxxK

MLP kernel :

)tanh(),( xkxxxK Tii

III. ARTIFICIAL BEE COLONYALGORITHM

Artificial Bee Colony (ABC) algorithmwas proposed by Karaboga in 2005 for realparameter optimization [35]. ABC is consideredrecent swarm intelligent technique. It is inspired bythe intelligent behavior of honey bees. The colonyof artificial bees consists of three groups of bees:employed, onlooker and scout bees. Half of thecolony composed of employed bees and the restconsist of the onlooker bees. The number of foodsources/nectar sources is equal with the employedbees, which means one nectar source is responsiblefor one employed bee. The aim of the whole colonyis to maximize the nectar amount. The duty ofemployed bees is to search for food sources(solutions). Later, the nectars amount (solutions

qualities/fitness value) is calculated. Then, theinformation obtained is shared with the onlookerbees which are waiting in the hive (dance area).The onlooker bees decide to exploit a nectar sourcedepending on the information shared by theemployed bees. The onlooker bees also determinethe source to be abandoned and allocate itsemployed bee as scout bees. For the scout bees,

their task is to find the new valuable food sources.They search the space near the hive randomly [36].In ABC algorithm, suppose the solution space ofthe problem is D-dimensional, where D is thenumber of parameters to be optimized.

The fitness value of the randomly chosen site isformulated as follows:

The size of employed bees and onlooker bees areboth SN, which is equal to the number of foodsources. There is only one employed bee for eachfood source whose first position is randomlygenerated.. In each iteration of ABC algorithm,each employed bee determines a new neighboringfood source of its currently associated food source

(4)

(5)

(6)

(7)

(8)

(9)

(3)

(11)

(10)


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and computes the nectar amount of this new foodsource by

( )

Where;

[]

If the new food source is better than that ofprevious one, then this employed bee moves to newfood source, otherwise it continues with the oldone.

After all employed bees complete the searchprocess; they share the information about theirfood sources with onlooker bees. An onlookerbee evaluates the nectar information taken fromall employed bees and chooses a food source witha probability related to its nectar amount byEquation:

Where is the fitness value of the solution iwhich is proportional to the nectar amount ofthe food source in the position i and SN is thenumber of food sources which is equal to thenumber of employed bees.

Later, the onlooker bee searches a new solution inthe selected food source site, the same way asexploited by employed bees. After all the employedbees exploit a new solution and the onlooker beesare allocated a food source, if a source is found that

the fitness hasnt been improved for apredetermined number of cycles (limit parameter),it is abandoned, and the employed bee associatedwith that source becomes a scout bee. In thatposition, scout generates randomly a new solutionby Equation:

(

) (14)

Where;

is random number in range [0 , 1].

,

are the lower and upper borders in the

jth dimension of the problem space.

The ABC algorithm is shown in Fig. 1.

Fig.1 the ABC algorithm.

IV. THE PROPOSED MODELThe proposed model is based on the study

of stock historical data (High, Low, Open, Close,and Volume). Then technical indicators arecalculated from these historical data to be used asinputs to proposed model. After that LSSVM isoptimized by ABC algorithm to be used in theprediction of daily stock prices. LSSVM optimized

by PSO algorithm, standard LSSVM, and ANNtrained with Scaled Conjugate gradient (SCG)algorithm, which is one of the best back-propagation derivatives, are used as benchmarksfor comparison with proposed model. The proposedmodel architecture contains six inputs vectorsrepresent the historical data and five derivedtechnical indicators from raw datasets, and oneoutput represents next price.

(13)

(12)


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Proposed model phases are summarized in Fig 2.

Fig. 2 the proposed model phases.

The technical indicators, which are calculated from

the raw datasets, are as follows:

Relative Strength Index (RSI): A technicalmomentum indicator that compares themagnitude of recent gains to recent losses in anattempt to determine overbought and oversoldconditions of an asset. The formula forcomputing the Relative Strength Index is asfollows.

RSI = 100- [100 / (1+RS)]

Where RS = Avg. of x days up closesdividedby average of x daysdown closes.

Money Flow Index (MFI): This onemeasures the strength of money in and out of asecurity. The formula for MFI is as follows.

Money Flow (MF) = TP *V

Where, TP is typical price, and V is moneyvolume

Money Ratio (MR) is calculated as:

MR = (Positive MF / Negative MF)

MFI = 100(100/ (1+MR)).

Exponential Moving Average (EMA): Thisindicator returns the exponential movingaverage of a field over a given period of time.EMA formula is as follows.

EMA = [ *T Close] + [1-* Y EMA].

Where T is Todays close and Y is Yesterdays

close.

Stochastic Oscillator (SO): The stochasticoscillator defined as a measure of thedifference between the current closing price ofa security and its lowest low price, relative toits highest high price for a given period of

time. The formula for this computation is asfollows.

%K = [(CPLP) / (HPLP)] * 100

Where, CP is Close price, LP is Lowest price,HP is Highest Price, and LP is Lowest Price.

Moving Average Convergence/Divergence(MACD): This function calculates differencebetween a short and a long term movingaverage for a field. The formulas forcalculating MACD and its signal as follows.

MACD= [0.075*E] - [0.15*E]

Where, E is EMA (CP)Signal Line = 0.2*EMA of MACD

V. RESULTS AND DISCUSSIONSThe proposed and compared models were

trained and tested with datasets for twentycompanies cover all sectors in S&P 500 stockmarket. Datasets period are from Jan 2009 to Jan2012. All datasets are available in [37]. Datasetsare divided into training part (70%) and testing part(30%).LSSVM-ABC algorithm parameters are shown in

table (1).Table (1) LSSVM-ABC algorithm parameters.

No. of bees epochs LSSVM kernel20 bees 50 RBF kernel

LSSVM-PSO parameters are shown in table (2).Table (2) LSSVM-PSO algorithm parameters.

No. of Particles epochs LSSVM kernel20 particles 50 RBF kernel

ANN parameters are found in table (3).Table (3) ANN model parameters.

Trainingalgorithm

Inputlayer

Hiddenlayer

epochs Outputlayer

SCG 6nodes

11nodes

1000 1node

Fig. 3 outlines the ANN structure used in thispaper.

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

Computing MSE for( LSSVM-ABC, LSSVM-PSO,LSSVM and ANN)

Testing LSSVM-ABC model with new data

Optimizing and training LSSVM with ABC algorithm

Feature Extraction and selection (Techincal Indicators)

Data Acquisition and Preprocessing (Stocks Histoical data)


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Fig. 3 ANN model structureFig. 4 to Fig. 23 outline the application ofProposed LSSVM-ABC model compared withLSSVM-PSO, LSSVM and ANN-BP algorithms atdifferent datasets representing different stockmarket sectors.In Fig. 4, Fig. 7, Fig. 8 and Fig. 9 which representresults of four companies in informationtechnology sector (Adobe, Oracle, CISCO, and HP)results show that proposed LSSVM-ABC, andLSSVM-PSO models are achieving lowest errorvalue with little advance to proposed model.

In Fig. 5, and Fig. 6 which represent results of twoother companies in information technology sector(Amazon, and Apple), results show that ANN isfallen in overfitting problem, since the datasetshave fluctuations. LSSVM-ABC algorithm is thebest one with lowest error value and could easilyovercome local minima and overfitting problems.Fig. 10, and Fig. 11 which represent results offinancial sector companies (American Express, andBank of New york), we can remark that thepredicted curve using the proposed LSSVM-ABCand LSSVM-PSO are most close to the real curveswhich achieve best accuracy, followed by LSSVM,

while ANN-BP is the worst one.Fig. 12 represents result of Honeywell companywhich represents industrials stock sector. Proposedmodel could easily overcome local minimumproblem which ANN suffers from.Fig. 13 and Fig.14 outline the application of theproposed algorithm to hospera and lifetechnologies companies in health stock sector.From figures one can remark the enhancement inthe error rate achieved by the proposed model.Fig. 12, Fig.16, Fig.17, Fig.18, and Fig.19 outlinethe results of testing proposed model for (Coca-

Cola, Exxon-mobile, AT&T, FMC, and dukeenergy) companies which represent different stocksectors. LSSVM-ABC and LSSVM-PSO canconverge to global minima with lowest error valuecompared to LSSVM and ANN (especially influctuation cases).Fig. 20, Fig.21, Fig.22, and Fig.23 represent resultsfor (Ford, FEDX, MMM, and PPL) companies indifferent stock sectors. The results of proposed

model has little advance compared with LSSVMand ANN since the data sets is normal and have nofluctuations, but LSSVM-ABC has fastconvergence speed compared to other models.

Table 4 outlines Mean Square Error (MSE)performance function for proposed and comparedmodels. It can be remarked that the LSSVM-ABC,and LSSVM-PSO techniques always gives anadvance over LSSVM and ANN trained with SCGalgorithms in all performance functions and in alltrends and sectors.Fig. 24 outlines comparison between LSSVM-ABC, LSSVM-PSO, LSSVM and ANN algorithmsaccording to MSE function.

Fig. 4 Results for Adobe Company.

Fig. 5 Results for Amazon Company.


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Fig. 6 Results for Apple Company.

Fig. 7 Results for Oracle Co.

Fig. 8 Results for CISCO Co.

Fig. 9 Results for HP company.

Fig. 10 Results for American Express Company

Fig. 11 Results Bank of New York company.


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Fig. 12 Results for Coca-Cola company.

Fig. 13 Results for Honey Well company.

Fig. 14 Results for Hospera Co.

.Fig. 15 Results for Life Tech. Company

Fig. 16 Results for Exxon-Mobile. Company.

Fig. 17 Results for AT&T. Company


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Fig. 18 Results for FMC. Company.

Fig. 19 Results for Duke. Company.

Fig. 20 Results for Ford. Company

Fig. 21 Results for FEDX. Company.

Fig. 22 Results for MMM. Company.

Fig. 23 Results for PPL. Company


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Table (4) Mean Square Error (MSE) for proposed

algorithm.

ANNLSSVMLSSVM-PSO

LSSVM-ABC

AlgorithmCompany

0.69790.55290.53170.5310Adobe22.0156.19814.53154.5286Amazon143.019.48635.59245.5915Apple0.75980.68070.63140.6320Oracle0.39290.36230.31110.3115CISCO1.31460.97520.77250.7727HP1.51160.87610.79050.7904Am.Expres0.96460.93940.48390.4841NY Bank2.05600.80960.68230.6809Coca-Cola2.18531.33710.95740.9563H. Well1.51620.89360.86940.8695Hospera2.41621.01950.77130.7718Life Tech.1.30801.30161.10001.0997Exx.Mob.0.42410.36840.29110.2916AT & T3.08431.75291.58811.5881FMC0.58970.37090.17350.1710Duke0.26280.26600.24720.2473FORD1.94541.52851.49481.4948FEDX2.41621.01950.77130.7718MMM0.30730.29070.29190.2920PPL

Fig. 24 MSE of proposed algorithm.

Convergence speed of the LSSVM-ABC, LSSVM-PSO, and ANN models are shown in Fig. 25, Fig.26 and Fig. 27 respectively. From figures one cannotice that the proposed model has fastconvergence to global minimum after less thantwenty iteration with lowest error value followedby LSSVM-PSO model, while ANN has worst

convergence speed and has highest error value evenafter training for long period (one thousanditerations).

Fig. 25 Convergence curve of LSSVM-ABC.

Fig. 26 Convergence curve of LSSVM-PSO.

Fig. 27 Convergence curve of ANN model


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VI. CONCLUSIONArtificial Bee Colony algorithm (ABC), a

population-based iterative global optimizationalgorithm is used to optimize LSSVM for stockprice prediction. ABC algorithm is used in

selection of LSSVM free parameters C (costpenalty), (insensitive-loss function) and (kernelparameter). The proposed LSSVM-ABC modelConvergence to a global minimum can beexpected. Also proposed model overcome the over-fitting problem which found in ANN, especially incase of fluctuations in stock sector. LSSVM-ABCalgorithm parameters can be tuned easily. Optimumfound by the proposed model is better thanLSSVM-PSO and LSSVM. Proposed modelconverges to global minimum faster than LSSVM-

PSO model. LSSVM-ABC and LSSVM-PSOachieves the lowest error value followed bystandard LSSVM, while ANN-BP algorithm is theworst one.

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lssvm-abc algorithm for stock price prediction

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