Research ArticleStock Market Forecasting Using Restricted GeneExpression Programming

Bin Yang Wei Zhang and Haifeng Wang

School of Information Science and Engineering Zaozhuang University Zaozhuang China

Correspondence should be addressed to Bin Yang batsi126com

Received 29 June 2018 Revised 17 December 2018 Accepted 6 January 2019 Published 5 February 2019

Academic Editor Carmen De Maio

Copyright copy 2019 Bin Yang et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Stock index prediction is considered as a difficult task in the past decade In order to predict stock index accurately this paperproposes a novel prediction method based on S-system model Restricted gene expression programming (RGEP) is proposed toencode and optimize the structure of the S-system A hybrid intelligent algorithm based on brain storm optimization (BSO) andparticle swarm optimization (PSO) is proposed to optimize the parameters of the S-system model Five real stock market pricessuch as Dow Jones Index Hang Seng Index NASDAQ Index Shanghai Stock Exchange Composite Index and SZSE ComponentIndex are collected to validate the performance of our proposedmethod Experiment results reveal that our method could performbetter than deep recurrent neural network (DRNN) flexible neural tree (FNT) radial basis function (RBF) backpropagation (BP)neural network and ARIMA for 1-week-ahead and 1-month-ahead stock prediction problems And our proposed hybrid in-telligent algorithm has faster convergence than PSO and BSO

1 Introduction

Stock market plays a leading and crucial role in the marketmechanism which connects the savers and investors [1 2])e operating mechanism of the stock market reflects thesituation of national economy and is recognized as the signalsystem of the national economy [3 4] Because of someuncontrollable factors such as economic growth economiccycle interest rate fiscal revenue and expenditure moneysupply and price the prediction of the stock market index isconsidered to be a difficult job [5ndash7]

Many machine learning (ML) methods containing statis-tical models artificial neural networks and hybrid predictionmodels have been proposed to model and predict the stockindex As a classical statistical model the ARIMA model hasproposed to predict theNewYork Stock Exchange (NYSE) andNigeria Stock Exchange (NSE) and the results revealed that theARIMA model performed better for short-term prediction[8ndash10] Compared with the ARIMAmodel the artificial neuralnetwork (ANN) model has more strong prediction andmodeling ability Adebiyi et al made the comparison ofARIMA andANNmodels for stock price prediction and found

that the stock forecasting model based on ANN approach hadsuperior performance over ARIMA models [11]

In the past decades many ANN models have beenemployed for solving real problems especially stock marketprices forecasting [12 13] Dong et al presented back-propagation (BP) neural networks for stock prediction [14]Feedforward ANN was proposed to predict price movementof the stock market [15] Akita et al proposed a novel deeplearning method based on paragraph vector and long short-term memory (LSTM) to predict the Tokyo Stock Exchange[16] Rout et al used the radial basis function (RBF) neuralnetwork to forecast DJIA and SampP 500 stock indices [17]Wang et al proposed a novel method based on complex-valued neural network (CVNN) and Cuckoo search (CS)algorithm to forecast stock price [18] Chen et al presentedthe flexible neural tree (FNT) ensemble technique to analyze7-year Nasdaq-100 main index values and 4-year NIFTYindex values [19]

However the existing methods mainly trained the blackbox with the training sample )e model could change itsinternal structure and parameters to make it approximate tothe training sample )e gained model could not display the

HindawiComputational Intelligence and NeuroscienceVolume 2019 Article ID 7198962 14 pageshttpsdoiorg10115520197198962

distinct input-output relationship and deeply understandthe internal mechanisms of real-world problems And inmost of these methods all variables are input into themodels which easily lead to overfitting problem Recentlythe methods based on mathematical formulations have beenproposed to predict time series which could clearly indicatethe mathematical relationship between the input data andoutput data Zuo et al proposed that gene expressionprogramming (GEP) was utilized to identify differentialequation for time series prediction [20] Graff et al proposedgenetic programming (GP) to forecast time series [21]Grigioni et al proposed a modified power-law mathematicalmodel to predict the blood damage sustained by red cellswith the load history [22] Mina et al proposed a beta-function formula to forecast the maxillary arch form [23]Chen et al identified ordinary differential equations (ODEs)to forecast the small time scale traffic measurements dataand proved that the ODE model was more feasible andefficient than ANN models [24]

As a classical nonlinear differential equation theS-systemmodel has been proposed to predict time series andidentify genetic networks Zhang and Yang proposed arestricted additive tree (RAT) to represent the S-systemmodel for stock market index forecasting [25] However theRAT method has nonlinear structure and is implementedinconveniently In this paper a novel stock index predictionmethod based on S-system model is proposed Restrictedgene expression programming (RGEP) is proposed to en-code and optimize the structure of S-system In order tooptimize the parameters of the S-system model accurately anew hybrid intelligent algorithm based on the brain stormoptimization (BSO) algorithm and particle swarm optimi-zation (PSO) algorithm is proposed

Dow Jones Index Hang Seng Index and NASDAQIndex are old and famous stock indexes in the world whichare usually utilized to reflect the development of the globaleconomy Shanghai Stock Exchange Composite Index andSZSE Component Index represent the general trend ofChinarsquos stock market and economic development )ese fivestock indexes have been considered as the standard datasetsto evaluate the performance of stock prediction models[26ndash30] )us Dow Jones Index Hang Seng Index NAS-DAQ Index Shanghai Stock Exchange Composite Indexand SZSE Component Index are collected to validate theperformance of our proposed method

2 Background Concepts andRelated Technologies

21 Data Description Let stock time series data to be[X1 X2 XT] (T is the number of time points) Gen-erally the data from the past time points are used to predictthe data at the current time point Figure 1 shows an exampleof data partition with m input variables )e data in the boxare utilized as the input vector and the data on the right sideof the box is the prediction value Two forecasting strategies1-week-ahead (m 7) and 1-month-ahead (m 30) areutilized in this paper

22 S-SystemModel )e S-systemmodel has a complex andpowerful structure which captures the dynamic nature ofthe real system and achieves a good performance in theterms of precision and flexibility [31 32] )e ith nonlineardifferential equation in S-system is described as follows

dXi

dt αi1113945

N

j1X

gij

j minus βi1113945

N

j1X

hij

j (1)

where N is the number of equations Xi is the ith variable αi

and βi are the rate constants of production function andconsumption function and gij and hij are the kinetic orders

23 Brain Storm Optimization Algorithm Brain storm op-timization (BSO) algorithm is a new swarm intelligenceoptimization algorithm which was proposed by Shi in theyear 2011 [33] In BSO the cluster algorithm is proposed tosearch the local optimal solution and the global optimalsolution is obtained through the comparison of all localoptimal solutions Mutation strategy is utilized to enhance thediversity of the algorithm and avoid obtaining local optimalsolution [34] )e BSO process is described as follows

(1) Initialize the population and generate N potentialsolutions (x1 x2 xN)

(2) )e k-means clustering algorithm is utilized to dividethe N individuals into k classes )e fitness value ofeach individual is calculated )e best individual ineach category is selected as the central individual

(3) Select randomly the central individual of a class andmutate it with a random disturbance

(4) Update the individual with the following fourmethods

(a) Select randomly a class (the probability is pro-portional to the number of individuals in eachclass) A new individual (xsprime) is generated byadding the random perturbation to the centralindividual (xs) which is defined as follows

xsprime xs + ζ times N(μ σ) (2)

where N(μ σ) is the Gaussian random function and ζ is thefactor that balances the random number which is defined asfollows

X1 X2 X3 X4 Xm

X2 X3 X4 X5 Xm+1

Xm+1

Xm+2

X3 X4 X5 Xm+1 Xm+2 Xm+3

X4 X5 Xm+1 Xm+2 Xm+3 Xm+4

Figure 1 Data structure

2 Computational Intelligence and Neuroscience

ζ log sig(05lowastmax_iterationminus current_iteration)

k1113888 1113889

lowast rand()

(3)where log sig is a logarithmic S-transform functionmax_interation is the maximum number of iterations in thealgorithm current_interation is the number of current it-erations is the gradient which is utilized to control thelogarithmic S-transformation function and rand() is therandom number in the interval [0 1]

(b) Randomly select a class and an individual in theselected class A new individual is created with theselected individual and Gaussian value by equations(2) and (3)

(c) Select randomly two classes and two central in-dividuals from the two classes are utilized as thecandidate individuals xs1 and xs2 which are fusedwith the following formula

xs λ times xs1 +(1minus λ) times xs2 (4)

where λ is a random number in the interval [0 1]

After merging the candidate individuals the individual isupdated according to the formula (2)

(d) Two candidate individuals xs1 and xs2 are selectedrandomly from the two selected classes )e fusionand updating operators are implemented withequations (2) and (4)

After the new individual is generated its fitness value iscalculated Compared with the fitness values of the candidateindividuals the individuals with the better fitness values areselected to the next generation When N new individuals aregenerated enter the next iteration process

(5) When the maximum iteration number is reachedalgorithm stops otherwise go to step (2)

24 Particle Swarm Optimization Algorithm )e particleswarm optimization (PSO) algorithm is a classical swarmintelligent method [35] In PSO each potential solution ispresented by a particle A swarm of particles [x1 x2 xN]

moves in order to search the food source with the movingvelocity vector [v1 v2 vN] At each step each particlesearches the optimal position separately in the space whichis recorded in a vector Pbesti )e global optimal position issearched among all the particles which is kept as Gbest [36]

At each step a new velocity for the particle i is updatedby the following equation

vi(t + 1) wlowast vi(t) + c1r1 Pbesti minus xi(t)1113872 1113873

+ c2r2 Gbest(t)minusxi(t)( 1113857(5)

where w is the inertia weight and impacts on the convergencerate of PSO which is calculated adaptively as w

(max_iterationminus current_iteration(2lowastmax_iteration)) + 04(max_interation is the maximum number of iterations in the

algorithm and current_interation is the number of currentiterations) c1 and c2 are the positive constants and r1 and r2are uniformly distributed random numbers in [0 1]

With the updated velocities each particle changes itsposition according to the following equation

xi(t + 1) xi(t) + vi(t + 1) (6)

3 Methods

31 Restricted Gene Expression Programming )e restrictedgene expression programming (RGEP) as the improvedversion of GEP was proposed to identify the S-system modelfor gene regulatory network (GRN) inference [37] )eflowchart of RGEP is described as follows

(1) Initialize the population One example of chromosomein population is depicted in Figure 2 Each chromo-some contains two genes and each gene contains headpart and tail part which are created randomly usingthe function set (F) and variable set (T)

F lowast1lowast2lowast3 lowastn

T x1 x2 xm R1113864 11138651113896 (7)

where lowastn is an operation of n variables multiplying xi is thevariable m is the number of input variables and R is theconstant

In order to make the chromosome similar to theS-system each gene is allocated the corresponding pa-rameters For gene 1 αi is given as its coefficient and eachvariable is given exponent gij For gene 2 βi is given as itscoefficient and each variable is given exponent hij Two genesare connected by the subtraction operation (minus) Figure 3shows the expression tree (ET) of Figure 2 and its corre-sponding S-system model is expressed as follows

dxi

dt αix

gi13 x

gi21 x

gi32 minus βix

hi12 x

hi24 x

hi31 x

hi43 (8)

(2) According to the given fitness function evaluate thepopulation with the training samples In this processthe S-system model is solved by the fourth-orderRungendashKutta method [38] For the differentialequation (dydt) f(x y) the solution is as follows

k1 f(x(t) y(t))

k2 f x(t) +h

2 y(t) + hlowast

k1

21113888 1113889

k3 f x(t) +h

2 y(t) + hlowast

k2

21113888 1113889

k4 f x(t) + h y(t) + hlowast k3( 1113857

y(t + 1) y(t) + hlowastk1 + 2k2 + 2k3 + k4

6

(9)

Computational Intelligence and Neuroscience 3

where h is the step size

(3) If the optimal solution appears RGEP is terminatedotherwise turn to (4)

(4) Selection recombination and mutation are used forreproduction of each chromosome which are in-troduced in Reference [37]

In the initial stage of structural optimization the sym-bols of the chromosome in RGEP are randomly selectedincluding function symbols and variable symbols Withtraining data reproduction operators are used to optimizeand change the chromosomal symbols in the optimizationprocess )e optimized S-system structure does not containall the input variables According to the training data RGEPcould automatically select the appropriate input variables InFigure 2 we can find that the coefficients αi and βi and theexponents gi1 gi2 gi3 hi1 hi2 hi3 and hi4 are needed to beoptimized In this paper the parameters in each chromo-some are optimized by a hybrid intelligent algorithm basedon BSO algorithm and PSO algorithm

32 Hybrid Optimization Algorithm )e BSO algorithm issuitable for solving the problem of multipeak and high-dimensional function )e PSO algorithm has theadvantages of easy realization high accuracy and fast

convergence But these two methods are easy to convergeprematurely and fall into local optimum In order to im-prove the diversity of population a novel hybrid intelligentalgorithm based on BSO and PSO (BSO-PSO) is proposedIn the BSO-PSO algorithm the half of individuals areselected randomly and optimized by BSO And the otherindividuals are optimized by PSO )e flowchart is de-scribed in Figure 4

33 Time Series Data Forecasting Using S-System )eflowchart of time series forecasting using the S-systemmodelis described in Figure 5 During the training phase theS-system model is optimized according to the genetic op-erators of RGEP hybrid intelligent algorithm and trainingdataset During the test phase the optimal S-system is usedto make the prediction of the stock index )e detailedprocess is described as follows

331 Training Phase

(1) Initialize the S-system population with the structureand parameters Each S-system is encoded as theRGEP chromosome which is described in Figure 2

(2) With the training samples the S-system is solved byequation (4) and the fitness value of each S-system iscalculated Search the best S-system according to thefitness values If the optimal model is found thealgorithm stops

(3) Selection recombination and mutation are used tosearch the optimal structure of the S-system Go tostep (2)

(4) At some iterations in RGEP BSO-PSO algorithm isused to optimize the parameters of RGEP chro-mosomes In this process the structure of theS-system model is fixed According to the structureof the model the number of parameters (αi βigij and hij) is counted With the hybrid intelligentalgorithm search and update the optimal parametersof each S-system

332 Testing Phase With the data at the previous time pointthe optimal S-system model obtained in the training phase issolved and the data at the current time point are predictedRepeat this procedure until that the data at all testing timepoints have been predicted According to the predicted dataand target data the predicted error is calculated

4 Results and Discussion

41 Data and Evaluation Standard Five stock indexes suchas Dow Jones Index (DJI) Hang Seng Index (HSI) NAS-DAQ Index (NASI) SSE (Shanghai Stock Exchange)Composite Index (SSEI) and SZSE Component Index(SZSEI) are proposed to test the performance of our methodSeventy percent of the data are used for training and 30 ofthe data are used for testing )e descriptions of five stockindexes are listed in Table 1

lowast2 lowast2 x3 x1 x2 x1 x2

lowast2 lowast3 x2 x4 x1 x3 x1

Head Tail

Gene 1

Gene 2

x3

x5 x2

x5 x1

x1

αi

βi

gi1 gi2 gi3 gi4 gi5 gi6 gi7 gi8

hi1 hi2 hi3 hi4 hi5 hi6 hi7 hi8

Figure 2 )e phenotype of chromosome in RGEP withparameters

ndash

lowast2

lowast2 x3

x1 x2 x3x1

dxidt

αi βi

gi3gi2

gi1

hi3hi2 hi4

lowast2

x2

hi1

x4

lowast3

Figure 3 )e expression tree of chromosome in RGEP withparameters

4 Computational Intelligence and Neuroscience

RMSE (root mean square error) MAP (mean absolutepercentage) and MAPE (mean absolute percentage error)R2 (coefficient of multiple determinations for multiple

regressions) ARV (average relative variance) and VAF(variance accounted for) are proposed to evaluate the per-formance of our method [30 39]

RMSE

1N

1113944

N

i1f

itarget minusf

iforecast1113872 1113873

2

11139741113972

MAP maxfitarget minusfi

forecast

11138681113868111386811138681113868

11138681113868111386811138681113868

fiforecast

times 100⎛⎝ ⎞⎠

MAPE 1N

1113944

N

i1

fitarget minusfi

forecast

11138681113868111386811138681113868

11138681113868111386811138681113868

fiforecast

⎛⎝ ⎞⎠ times 100

R2

1minus1113936

Ni1 fi

target minusfiforecast1113872 1113873

2

1113936Ni1 fi

target minusf1113872 11138732

ARV 1113936

Ni1 fi

target minusfiforecast1113872 1113873

2

1113936Ni1 fi

forecast minusf1113872 11138732

VAF 1minus1113936

Ni1 fi

target minusyiforecast1113872 1113873

2

1113936Ni1 fi

target1113872 11138732

⎛⎜⎝ ⎞⎟⎠ times 100

(10)

where N is the number of stock sample points fitarget is the

real stock value at the ith time point fiforecast is the predicting

stock value at the ith time point and f is the mean of stockindexes

42 Prediction Results In order to test the performance ofour method clearly five states of the art methods (DeepRecurrent Neural Network (DRNN) [40] FNT [19] RBFNN[17] BPNN [14] and ARIMA [8]) are also used to predictfive stock indexes

For 1-week-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x71113864 1113865

in the RGEP method By optimizing S-systemmodels by ourmethod we could obtain the optimal phenotypes and ex-pression trees (ETs) with five stock indexes which are de-scribed in Figure 6 Five optimal S-systemmodels gained arelisted in Table 2 for five stock datasets)e forecasting resultsof five stock indexes by our method are depicted in Figure 7From Figure 7 it can be clearly seen that the predictingcurves are very near to the target ones and the errors arenearly zero

Comparison results of different prediction modelsrsquoperformance on five stock indexes are listed in Table 3 FromTable 3 among the past five states of the art methods theDRNN model performs best for five stock indexes pre-diction But in terms of six indicators (RMSE MAP ARVMAPE R2 and VAF) our proposed method has betterperformance than the DRNN model In terms of RMSE ourmethod is 348 lower than DRNN for DJI dataset 464lower than DRNN for HSI dataset 404 lower than DRNNfor NASI dataset 198 lower than DRNN for SSEI dataset

Initialize N individuals

Select randomly

N2 individuals N2 individuals

Optimize with BSO Optimize with PSO

Calculate fitness values

Obtain the new generation

Is the satisfiedsolution found

Algorithm stops

YesNo

Figure 4 )e flowchart of BSO-PSO algorithm

Training

Optimization ofS-system

Gain the optimal S-system model

Test phasebegin

let Xlowast X be m lowast n matrices as the targeted and forecasting test data respectively Assign the last row of the training data as the initial condition to Y

for i = 1 to m dobegin

integrate the system of the best S-system for a step with the numerical integration method

assign the solution to the ith row of Xtake the ith row of Xlowast as the Y

endThe error can be calculated using X and Xlowast

end

Structure optimization using genetic operators of

RGEP

Parameter optimization using BSO-PSO algorithm

Figure 5 )e flowchart of time series data forecasting usingS-system

Computational Intelligence and Neuroscience 5

Table 1 Parameters of five stock indexes

Parameters DJI HSI NASI SSEI SZSEI

Time interval 121990ndash12292017

121991ndash12292017

121990ndash12292017

111996ndash12292017

122008ndash12302016

Train data for week-aheadprediction 4936 4666 4936 3866 1528

Test data for week-ahead prediction 2115 2000 2115 1657 655Train data for month-aheadprediction 4918 4649 4918 3849 1511

Test data for month-aheadprediction 2108 1992 2108 1649 647

Gene 1

Gene 2

ndash1365846

1912372

Head Tail

lowast2 x2 x5

lowast2 x1 x3

ndash04793 ndash00765

ndash19772 ndash24786

x1 x5 x4 x6 x3 x7

x7 x3 x5 x3 x2 x6

ndash

lowast2 lowast2

ndash1365846 1912372

x2 x5 x1 x3

ndash04793 ndash00765 ndash19772 ndash24786

(a)

Gene 1

Gene 2

ndash138218

156987

ndash

lowast2 lowast2

x1 x1 x3

Head Tail

lowast2 lowast2 x1

lowast2 x1 x3

ndash1012ndash15661

ndash20551 ndash149348

x3 x5 x7 x3 x5 x7

x3 x5 x2 x7 x6 x6

ndash36556

ndash149348ndash20551

156987ndash138218

ndash15661

lowast2

x3 x5

ndash1012 ndash36556

(b)

Gene 1

Gene 2

ndash00011

01105

ndash

lowast3 lowast2

x2x5 x2 x1

Head Tail

lowast3 x4 x3

lowast2 x2 x1

ndash12356 ndash41503

ndash76373 ndash38423

x5 x2 x4 x7 x6 x2

x7 x3 x5 x3 x2 x6

ndash07905

ndash00011 01105

ndash76373 ndash38423

x4

ndash12356 ndash41503 ndash07905

(c)

Gene 1

Gene 2

ndash17056

052302

Head Tail

lowast2 x1 x2

lowast2 x1 x5

12053 ndash02955

45115 ndash11571

x2 x3 x4 x1 x3 x7

x6 x2 x5 x3 x3 x1

ndash

lowast2 lowast2

x1 x2 x1 x5

ndash17056 052302

12053 ndash02955 45115 ndash11571

(d)

Figure 6 Continued

6 Computational Intelligence and Neuroscience

Gene 1

Gene 2

ndash101789

148636

Head Tail

lowast2 x3 x2

lowast2 x1 x6

ndash07625 ndash31649

ndash84503 ndash56786

x3 x3 x5 x5 x2 x7

x1 x2 x6 x2 x4 x1

ndash

lowast2 lowast2

x3 x2 x1 x6

ndash101789 148636

ndash07625 ndash31649 ndash84503 ndash56786

(e)

Figure 6 )e optimal phenotypes and expression trees for a-week-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

Table 2 Optimal S-system models of five stock datasets for a-week-ahead prediction

Type of datasets Optimal S-system modelDJI _f minus1365846xminus04793

2 xminus007655 minus 1912372xminus19772

1 xminus247863

HSI _f minus138218xminus156611 xminus1012

3 xminus365565 minus 156987xminus20551

1 xminus1493483

NASI _f minus00011xminus123564 xminus41503

3 xminus079055 minus 01105xminus76373

2 xminus384231

SSEI _f minus17056x120531 xminus02955

2 minus 052302x451151 xminus11571

5SZS _f minus101789xminus07625

3 xminus316492 minus 148636xminus84503

1 xminus567866

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 7 Continued

Computational Intelligence and Neuroscience 7

and 74 lower than DRNN for SZSEI dataset In terms ofARV our method is 587 lower than DRNN for DJIdataset 671 lower than DRNN for HSI dataset 688lower than DRNN for NASI dataset 369 lower thanDRNN for SSEI dataset and 165 lower than DRNN forSZSEI dataset In terms ofMAPE ourmethod is 375 lowerthan DRNN for DJI dataset 48 lower than DRNN for HSIdataset 429 lower than DRNN for NASI dataset 352lower than DRNN for SSEI dataset and 18 lower thanDRNN for SZSEI dataset In terms of VAF our method iscloser to 100 than DRNN for five stock indexes It could be

seen clearly that our proposed method could improve theprediction accuracy sharply

For 1-month-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x301113864 1113865

in the RGEP method With five stock indexes we obtain fiveoptimal phenotypes and expression trees (ETs) which aredescribed in Figure 8 According to five ETs the S-systemmodels gained are listed in Table 4 )e forecasting results offive stock indexes by our method are depicted in Figure 9From Figure 9 we could see clearly that the predicting andtarget curves are very close

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 7)e prediction and actual results for a-week-ahead predictionwith five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) and SZSEI (e)

Table 3 Comparison results of six methods for a-week-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005411 69911 000113 073146 099887 99992DRNN 00083 84909 0002735 11696 099726 99981FNT 0015427 14907 000613 18023 098939 99933

RBFNN 0016188 24473 001057 22631 098943 99927BPNN 0049026 22721 0060301 55923 09397 99328ARIMA 0052472 20924 0071326 59773 092867 99230

HSI

Our method 0008725 68824 001116 097359 098883 99984DRNN 0016272 13384 0033924 18718 096608 99944FNT 0020128 19261 0065331 22759 093467 99915

RBFNN 0023406 2532 0072545 27067 092756 99885BPNN 0035987 44738 012867 41357 087133 99729ARIMA 0013361 13817 0026688 15367 097331 99963

NASI

Our method 0008324 79168 0001707 10035 099829 99979DRNN 0013969 11069 0005465 1757 099453 99941FNT 0016468 32352 0006859 25336 099314 99918

RBFNN 003669 37371 0027513 45327 097249 99591BPNN 0046 175 0042533 5973 095747 99353ARIMA 0049849 1846 0093189 531 090681 99245

SSEI

Our method 0008105 9957 000535 11271 099465 99962DRNN 0010107 12959 0008481 17396 099152 99941FNT 0014559 18931 0018903 22848 09811 99878

RBFNN 0014681 2006 0018024 21804 098198 99876BPNN 0035922 32768 0091613 69046 090839 99256ARIMA 0020766 20533 0029814 39766 097019 99752

SZSEI

Our method 0016959 16079 0009762 24933 099024 99851DRNN 0018315 19783 0011685 30419 098831 99826FNT 0018571 2167 0012189 30233 098781 99821

RBFNN 0023881 31222 0018031 38187 098197 99704BPNN 0027297 41441 0027768 40751 097223 99614ARIMA 0029022 26983 002844 48583 097156 99563

8 Computational Intelligence and Neuroscience

Gene 1

Gene 2

26495

50785

Head Tail

lowast2 x10 x18

lowast2 x14 x4

04093 12433

ndash18385 ndash14525

x10 x25 x22 x18 x28 x12

x11 x3 x7 x23 x28 x10

ndash

lowast2 lowast2

26495 50785

x10 x18 x14 x4

04093 12433 ndash18385 ndash14525

(a)

Gene 1

Gene 2

24693

54397

ndash

lowast2 x9

x9 x26

ndash3638122267

5439724693

ndash102498

Head Tail

lowast2 x26

x11 x23

ndash3638122267

ndash102498

x12 x18 x2 x26 x30 x7

x1 x9 x20 x22 x26 x19

x9

x9

(b)

Gene 1

Gene 2

38864

22829

Head Tail

lowast2 x3 x29

lowast2 x17 x5

ndash18748 ndash26364

ndash12427 ndash37315

x25 x12 x1 x5 x3 x2

x23 x21 x2 x2 x7 x16

ndash

lowast2 lowast2

x3x29 x17 x5

38864 22829

ndash12427 ndash37315ndash18748 ndash26364

(c)

Gene 1

Gene 2

ndash41868

1030168

Head Tail

lowast2 x21 x30

lowast2 x19 x28

ndash016 ndash0288

ndash0398 0734

x12 x12 x1 x20 x22 x17

x1 x30 x6 x1 x3 x2

ndash

lowast2 lowast2

x21 x30 x19 x28

ndash41868 1030168

ndash016 ndash0288 ndash0398 0734

(d)

Figure 8 Continued

Computational Intelligence and Neuroscience 9

Table 4 Optimal S-system models of five stock datasets for a-month-ahead prediction

Type of datasets Optimal S-system modelDJI _f 26495x04093

10 x1243318 minus 50785xminus18385

14 xminus145254

HSI _f 24693x222679 xminus36381

26 minus 54397xminus1024989

NASI _f 38864x187483 xminus26364

29 minus 22829xminus1242717 xminus37315

5SSEI _f minus41868xminus016

21 x028830 minus 1030168xminus0398

19 x073428

SZSEI _f 10205x0621423 x36797

27 minus 88024xminus2357914 xminus4797

16

Gene 1

Gene 2

10205

88024

Head Tail

lowast2 x23 x27

lowast2 x14 x16

06214 36797

ndash23579 ndash4797

x1 x8 x15 x23 x19 x1

x4 x12 x23 x20 x4 x4

ndash

lowast2 lowast2

x23 x27 x14 x16

10205 88024

06214 36797 ndash23579 ndash4797

(e)

Figure 8 )e optimal phenotypes and expression trees for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 9 Continued

10 Computational Intelligence and Neuroscience

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

ζ log sig(05lowastmax_iterationminus current_iteration)

k1113888 1113889

lowast rand()

(3)where log sig is a logarithmic S-transform functionmax_interation is the maximum number of iterations in thealgorithm current_interation is the number of current it-erations is the gradient which is utilized to control thelogarithmic S-transformation function and rand() is therandom number in the interval [0 1]

(b) Randomly select a class and an individual in theselected class A new individual is created with theselected individual and Gaussian value by equations(2) and (3)

(c) Select randomly two classes and two central in-dividuals from the two classes are utilized as thecandidate individuals xs1 and xs2 which are fusedwith the following formula

xs λ times xs1 +(1minus λ) times xs2 (4)

where λ is a random number in the interval [0 1]

After merging the candidate individuals the individual isupdated according to the formula (2)

(d) Two candidate individuals xs1 and xs2 are selectedrandomly from the two selected classes )e fusionand updating operators are implemented withequations (2) and (4)

After the new individual is generated its fitness value iscalculated Compared with the fitness values of the candidateindividuals the individuals with the better fitness values areselected to the next generation When N new individuals aregenerated enter the next iteration process

(5) When the maximum iteration number is reachedalgorithm stops otherwise go to step (2)

24 Particle Swarm Optimization Algorithm )e particleswarm optimization (PSO) algorithm is a classical swarmintelligent method [35] In PSO each potential solution ispresented by a particle A swarm of particles [x1 x2 xN]

moves in order to search the food source with the movingvelocity vector [v1 v2 vN] At each step each particlesearches the optimal position separately in the space whichis recorded in a vector Pbesti )e global optimal position issearched among all the particles which is kept as Gbest [36]

At each step a new velocity for the particle i is updatedby the following equation

vi(t + 1) wlowast vi(t) + c1r1 Pbesti minus xi(t)1113872 1113873

+ c2r2 Gbest(t)minusxi(t)( 1113857(5)

where w is the inertia weight and impacts on the convergencerate of PSO which is calculated adaptively as w

(max_iterationminus current_iteration(2lowastmax_iteration)) + 04(max_interation is the maximum number of iterations in the

algorithm and current_interation is the number of currentiterations) c1 and c2 are the positive constants and r1 and r2are uniformly distributed random numbers in [0 1]

With the updated velocities each particle changes itsposition according to the following equation

xi(t + 1) xi(t) + vi(t + 1) (6)

3 Methods

31 Restricted Gene Expression Programming )e restrictedgene expression programming (RGEP) as the improvedversion of GEP was proposed to identify the S-system modelfor gene regulatory network (GRN) inference [37] )eflowchart of RGEP is described as follows

(1) Initialize the population One example of chromosomein population is depicted in Figure 2 Each chromo-some contains two genes and each gene contains headpart and tail part which are created randomly usingthe function set (F) and variable set (T)

F lowast1lowast2lowast3 lowastn

T x1 x2 xm R1113864 11138651113896 (7)

where lowastn is an operation of n variables multiplying xi is thevariable m is the number of input variables and R is theconstant

In order to make the chromosome similar to theS-system each gene is allocated the corresponding pa-rameters For gene 1 αi is given as its coefficient and eachvariable is given exponent gij For gene 2 βi is given as itscoefficient and each variable is given exponent hij Two genesare connected by the subtraction operation (minus) Figure 3shows the expression tree (ET) of Figure 2 and its corre-sponding S-system model is expressed as follows

dxi

dt αix

gi13 x

gi21 x

gi32 minus βix

hi12 x

hi24 x

hi31 x

hi43 (8)

(2) According to the given fitness function evaluate thepopulation with the training samples In this processthe S-system model is solved by the fourth-orderRungendashKutta method [38] For the differentialequation (dydt) f(x y) the solution is as follows

k1 f(x(t) y(t))

k2 f x(t) +h

2 y(t) + hlowast

k1

21113888 1113889

k3 f x(t) +h

2 y(t) + hlowast

k2

21113888 1113889

k4 f x(t) + h y(t) + hlowast k3( 1113857

y(t + 1) y(t) + hlowastk1 + 2k2 + 2k3 + k4

6

(9)

Computational Intelligence and Neuroscience 3

where h is the step size

(3) If the optimal solution appears RGEP is terminatedotherwise turn to (4)

(4) Selection recombination and mutation are used forreproduction of each chromosome which are in-troduced in Reference [37]

In the initial stage of structural optimization the sym-bols of the chromosome in RGEP are randomly selectedincluding function symbols and variable symbols Withtraining data reproduction operators are used to optimizeand change the chromosomal symbols in the optimizationprocess )e optimized S-system structure does not containall the input variables According to the training data RGEPcould automatically select the appropriate input variables InFigure 2 we can find that the coefficients αi and βi and theexponents gi1 gi2 gi3 hi1 hi2 hi3 and hi4 are needed to beoptimized In this paper the parameters in each chromo-some are optimized by a hybrid intelligent algorithm basedon BSO algorithm and PSO algorithm

32 Hybrid Optimization Algorithm )e BSO algorithm issuitable for solving the problem of multipeak and high-dimensional function )e PSO algorithm has theadvantages of easy realization high accuracy and fast

convergence But these two methods are easy to convergeprematurely and fall into local optimum In order to im-prove the diversity of population a novel hybrid intelligentalgorithm based on BSO and PSO (BSO-PSO) is proposedIn the BSO-PSO algorithm the half of individuals areselected randomly and optimized by BSO And the otherindividuals are optimized by PSO )e flowchart is de-scribed in Figure 4

33 Time Series Data Forecasting Using S-System )eflowchart of time series forecasting using the S-systemmodelis described in Figure 5 During the training phase theS-system model is optimized according to the genetic op-erators of RGEP hybrid intelligent algorithm and trainingdataset During the test phase the optimal S-system is usedto make the prediction of the stock index )e detailedprocess is described as follows

331 Training Phase

(1) Initialize the S-system population with the structureand parameters Each S-system is encoded as theRGEP chromosome which is described in Figure 2

(2) With the training samples the S-system is solved byequation (4) and the fitness value of each S-system iscalculated Search the best S-system according to thefitness values If the optimal model is found thealgorithm stops

(3) Selection recombination and mutation are used tosearch the optimal structure of the S-system Go tostep (2)

(4) At some iterations in RGEP BSO-PSO algorithm isused to optimize the parameters of RGEP chro-mosomes In this process the structure of theS-system model is fixed According to the structureof the model the number of parameters (αi βigij and hij) is counted With the hybrid intelligentalgorithm search and update the optimal parametersof each S-system

332 Testing Phase With the data at the previous time pointthe optimal S-system model obtained in the training phase issolved and the data at the current time point are predictedRepeat this procedure until that the data at all testing timepoints have been predicted According to the predicted dataand target data the predicted error is calculated

4 Results and Discussion

41 Data and Evaluation Standard Five stock indexes suchas Dow Jones Index (DJI) Hang Seng Index (HSI) NAS-DAQ Index (NASI) SSE (Shanghai Stock Exchange)Composite Index (SSEI) and SZSE Component Index(SZSEI) are proposed to test the performance of our methodSeventy percent of the data are used for training and 30 ofthe data are used for testing )e descriptions of five stockindexes are listed in Table 1

lowast2 lowast2 x3 x1 x2 x1 x2

lowast2 lowast3 x2 x4 x1 x3 x1

Head Tail

Gene 1

Gene 2

x3

x5 x2

x5 x1

x1

αi

βi

gi1 gi2 gi3 gi4 gi5 gi6 gi7 gi8

hi1 hi2 hi3 hi4 hi5 hi6 hi7 hi8

Figure 2 )e phenotype of chromosome in RGEP withparameters

ndash

lowast2

lowast2 x3

x1 x2 x3x1

dxidt

αi βi

gi3gi2

gi1

hi3hi2 hi4

lowast2

x2

hi1

x4

lowast3

Figure 3 )e expression tree of chromosome in RGEP withparameters

4 Computational Intelligence and Neuroscience

RMSE (root mean square error) MAP (mean absolutepercentage) and MAPE (mean absolute percentage error)R2 (coefficient of multiple determinations for multiple

regressions) ARV (average relative variance) and VAF(variance accounted for) are proposed to evaluate the per-formance of our method [30 39]

RMSE

1N

1113944

N

i1f

itarget minusf

iforecast1113872 1113873

2

11139741113972

MAP maxfitarget minusfi

forecast

11138681113868111386811138681113868

11138681113868111386811138681113868

fiforecast

times 100⎛⎝ ⎞⎠

MAPE 1N

1113944

N

i1

fitarget minusfi

forecast

11138681113868111386811138681113868

11138681113868111386811138681113868

fiforecast

⎛⎝ ⎞⎠ times 100

R2

1minus1113936

Ni1 fi

target minusfiforecast1113872 1113873

2

1113936Ni1 fi

target minusf1113872 11138732

ARV 1113936

Ni1 fi

target minusfiforecast1113872 1113873

2

1113936Ni1 fi

forecast minusf1113872 11138732

VAF 1minus1113936

Ni1 fi

target minusyiforecast1113872 1113873

2

1113936Ni1 fi

target1113872 11138732

⎛⎜⎝ ⎞⎟⎠ times 100

(10)

where N is the number of stock sample points fitarget is the

real stock value at the ith time point fiforecast is the predicting

stock value at the ith time point and f is the mean of stockindexes

42 Prediction Results In order to test the performance ofour method clearly five states of the art methods (DeepRecurrent Neural Network (DRNN) [40] FNT [19] RBFNN[17] BPNN [14] and ARIMA [8]) are also used to predictfive stock indexes

For 1-week-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x71113864 1113865

in the RGEP method By optimizing S-systemmodels by ourmethod we could obtain the optimal phenotypes and ex-pression trees (ETs) with five stock indexes which are de-scribed in Figure 6 Five optimal S-systemmodels gained arelisted in Table 2 for five stock datasets)e forecasting resultsof five stock indexes by our method are depicted in Figure 7From Figure 7 it can be clearly seen that the predictingcurves are very near to the target ones and the errors arenearly zero

Comparison results of different prediction modelsrsquoperformance on five stock indexes are listed in Table 3 FromTable 3 among the past five states of the art methods theDRNN model performs best for five stock indexes pre-diction But in terms of six indicators (RMSE MAP ARVMAPE R2 and VAF) our proposed method has betterperformance than the DRNN model In terms of RMSE ourmethod is 348 lower than DRNN for DJI dataset 464lower than DRNN for HSI dataset 404 lower than DRNNfor NASI dataset 198 lower than DRNN for SSEI dataset

Initialize N individuals

Select randomly

N2 individuals N2 individuals

Optimize with BSO Optimize with PSO

Calculate fitness values

Obtain the new generation

Is the satisfiedsolution found

Algorithm stops

YesNo

Figure 4 )e flowchart of BSO-PSO algorithm

Training

Optimization ofS-system

Gain the optimal S-system model

Test phasebegin

let Xlowast X be m lowast n matrices as the targeted and forecasting test data respectively Assign the last row of the training data as the initial condition to Y

for i = 1 to m dobegin

integrate the system of the best S-system for a step with the numerical integration method

assign the solution to the ith row of Xtake the ith row of Xlowast as the Y

endThe error can be calculated using X and Xlowast

end

Structure optimization using genetic operators of

RGEP

Parameter optimization using BSO-PSO algorithm

Figure 5 )e flowchart of time series data forecasting usingS-system

Computational Intelligence and Neuroscience 5

Table 1 Parameters of five stock indexes

Parameters DJI HSI NASI SSEI SZSEI

Time interval 121990ndash12292017

121991ndash12292017

121990ndash12292017

111996ndash12292017

122008ndash12302016

Train data for week-aheadprediction 4936 4666 4936 3866 1528

Test data for week-ahead prediction 2115 2000 2115 1657 655Train data for month-aheadprediction 4918 4649 4918 3849 1511

Test data for month-aheadprediction 2108 1992 2108 1649 647

Gene 1

Gene 2

ndash1365846

1912372

Head Tail

lowast2 x2 x5

lowast2 x1 x3

ndash04793 ndash00765

ndash19772 ndash24786

x1 x5 x4 x6 x3 x7

x7 x3 x5 x3 x2 x6

ndash

lowast2 lowast2

ndash1365846 1912372

x2 x5 x1 x3

ndash04793 ndash00765 ndash19772 ndash24786

(a)

Gene 1

Gene 2

ndash138218

156987

ndash

lowast2 lowast2

x1 x1 x3

Head Tail

lowast2 lowast2 x1

lowast2 x1 x3

ndash1012ndash15661

ndash20551 ndash149348

x3 x5 x7 x3 x5 x7

x3 x5 x2 x7 x6 x6

ndash36556

ndash149348ndash20551

156987ndash138218

ndash15661

lowast2

x3 x5

ndash1012 ndash36556

(b)

Gene 1

Gene 2

ndash00011

01105

ndash

lowast3 lowast2

x2x5 x2 x1

Head Tail

lowast3 x4 x3

lowast2 x2 x1

ndash12356 ndash41503

ndash76373 ndash38423

x5 x2 x4 x7 x6 x2

x7 x3 x5 x3 x2 x6

ndash07905

ndash00011 01105

ndash76373 ndash38423

x4

ndash12356 ndash41503 ndash07905

(c)

Gene 1

Gene 2

ndash17056

052302

Head Tail

lowast2 x1 x2

lowast2 x1 x5

12053 ndash02955

45115 ndash11571

x2 x3 x4 x1 x3 x7

x6 x2 x5 x3 x3 x1

ndash

lowast2 lowast2

x1 x2 x1 x5

ndash17056 052302

12053 ndash02955 45115 ndash11571

(d)

Figure 6 Continued

6 Computational Intelligence and Neuroscience

Gene 1

Gene 2

ndash101789

148636

Head Tail

lowast2 x3 x2

lowast2 x1 x6

ndash07625 ndash31649

ndash84503 ndash56786

x3 x3 x5 x5 x2 x7

x1 x2 x6 x2 x4 x1

ndash

lowast2 lowast2

x3 x2 x1 x6

ndash101789 148636

ndash07625 ndash31649 ndash84503 ndash56786

(e)

Figure 6 )e optimal phenotypes and expression trees for a-week-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

Table 2 Optimal S-system models of five stock datasets for a-week-ahead prediction

Type of datasets Optimal S-system modelDJI _f minus1365846xminus04793

2 xminus007655 minus 1912372xminus19772

1 xminus247863

HSI _f minus138218xminus156611 xminus1012

3 xminus365565 minus 156987xminus20551

1 xminus1493483

NASI _f minus00011xminus123564 xminus41503

3 xminus079055 minus 01105xminus76373

2 xminus384231

SSEI _f minus17056x120531 xminus02955

2 minus 052302x451151 xminus11571

5SZS _f minus101789xminus07625

3 xminus316492 minus 148636xminus84503

1 xminus567866

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 7 Continued

Computational Intelligence and Neuroscience 7

and 74 lower than DRNN for SZSEI dataset In terms ofARV our method is 587 lower than DRNN for DJIdataset 671 lower than DRNN for HSI dataset 688lower than DRNN for NASI dataset 369 lower thanDRNN for SSEI dataset and 165 lower than DRNN forSZSEI dataset In terms ofMAPE ourmethod is 375 lowerthan DRNN for DJI dataset 48 lower than DRNN for HSIdataset 429 lower than DRNN for NASI dataset 352lower than DRNN for SSEI dataset and 18 lower thanDRNN for SZSEI dataset In terms of VAF our method iscloser to 100 than DRNN for five stock indexes It could be

seen clearly that our proposed method could improve theprediction accuracy sharply

For 1-month-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x301113864 1113865

in the RGEP method With five stock indexes we obtain fiveoptimal phenotypes and expression trees (ETs) which aredescribed in Figure 8 According to five ETs the S-systemmodels gained are listed in Table 4 )e forecasting results offive stock indexes by our method are depicted in Figure 9From Figure 9 we could see clearly that the predicting andtarget curves are very close

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 7)e prediction and actual results for a-week-ahead predictionwith five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) and SZSEI (e)

Table 3 Comparison results of six methods for a-week-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005411 69911 000113 073146 099887 99992DRNN 00083 84909 0002735 11696 099726 99981FNT 0015427 14907 000613 18023 098939 99933

RBFNN 0016188 24473 001057 22631 098943 99927BPNN 0049026 22721 0060301 55923 09397 99328ARIMA 0052472 20924 0071326 59773 092867 99230

HSI

Our method 0008725 68824 001116 097359 098883 99984DRNN 0016272 13384 0033924 18718 096608 99944FNT 0020128 19261 0065331 22759 093467 99915

RBFNN 0023406 2532 0072545 27067 092756 99885BPNN 0035987 44738 012867 41357 087133 99729ARIMA 0013361 13817 0026688 15367 097331 99963

NASI

Our method 0008324 79168 0001707 10035 099829 99979DRNN 0013969 11069 0005465 1757 099453 99941FNT 0016468 32352 0006859 25336 099314 99918

RBFNN 003669 37371 0027513 45327 097249 99591BPNN 0046 175 0042533 5973 095747 99353ARIMA 0049849 1846 0093189 531 090681 99245

SSEI

Our method 0008105 9957 000535 11271 099465 99962DRNN 0010107 12959 0008481 17396 099152 99941FNT 0014559 18931 0018903 22848 09811 99878

RBFNN 0014681 2006 0018024 21804 098198 99876BPNN 0035922 32768 0091613 69046 090839 99256ARIMA 0020766 20533 0029814 39766 097019 99752

SZSEI

Our method 0016959 16079 0009762 24933 099024 99851DRNN 0018315 19783 0011685 30419 098831 99826FNT 0018571 2167 0012189 30233 098781 99821

RBFNN 0023881 31222 0018031 38187 098197 99704BPNN 0027297 41441 0027768 40751 097223 99614ARIMA 0029022 26983 002844 48583 097156 99563

8 Computational Intelligence and Neuroscience

Gene 1

Gene 2

26495

50785

Head Tail

lowast2 x10 x18

lowast2 x14 x4

04093 12433

ndash18385 ndash14525

x10 x25 x22 x18 x28 x12

x11 x3 x7 x23 x28 x10

ndash

lowast2 lowast2

26495 50785

x10 x18 x14 x4

04093 12433 ndash18385 ndash14525

(a)

Gene 1

Gene 2

24693

54397

ndash

lowast2 x9

x9 x26

ndash3638122267

5439724693

ndash102498

Head Tail

lowast2 x26

x11 x23

ndash3638122267

ndash102498

x12 x18 x2 x26 x30 x7

x1 x9 x20 x22 x26 x19

x9

x9

(b)

Gene 1

Gene 2

38864

22829

Head Tail

lowast2 x3 x29

lowast2 x17 x5

ndash18748 ndash26364

ndash12427 ndash37315

x25 x12 x1 x5 x3 x2

x23 x21 x2 x2 x7 x16

ndash

lowast2 lowast2

x3x29 x17 x5

38864 22829

ndash12427 ndash37315ndash18748 ndash26364

(c)

Gene 1

Gene 2

ndash41868

1030168

Head Tail

lowast2 x21 x30

lowast2 x19 x28

ndash016 ndash0288

ndash0398 0734

x12 x12 x1 x20 x22 x17

x1 x30 x6 x1 x3 x2

ndash

lowast2 lowast2

x21 x30 x19 x28

ndash41868 1030168

ndash016 ndash0288 ndash0398 0734

(d)

Figure 8 Continued

Computational Intelligence and Neuroscience 9

Table 4 Optimal S-system models of five stock datasets for a-month-ahead prediction

Type of datasets Optimal S-system modelDJI _f 26495x04093

10 x1243318 minus 50785xminus18385

14 xminus145254

HSI _f 24693x222679 xminus36381

26 minus 54397xminus1024989

NASI _f 38864x187483 xminus26364

29 minus 22829xminus1242717 xminus37315

5SSEI _f minus41868xminus016

21 x028830 minus 1030168xminus0398

19 x073428

SZSEI _f 10205x0621423 x36797

27 minus 88024xminus2357914 xminus4797

16

Gene 1

Gene 2

10205

88024

Head Tail

lowast2 x23 x27

lowast2 x14 x16

06214 36797

ndash23579 ndash4797

x1 x8 x15 x23 x19 x1

x4 x12 x23 x20 x4 x4

ndash

lowast2 lowast2

x23 x27 x14 x16

10205 88024

06214 36797 ndash23579 ndash4797

(e)

Figure 8 )e optimal phenotypes and expression trees for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 9 Continued

10 Computational Intelligence and Neuroscience

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

where h is the step size

(3) If the optimal solution appears RGEP is terminatedotherwise turn to (4)

(4) Selection recombination and mutation are used forreproduction of each chromosome which are in-troduced in Reference [37]

In the initial stage of structural optimization the sym-bols of the chromosome in RGEP are randomly selectedincluding function symbols and variable symbols Withtraining data reproduction operators are used to optimizeand change the chromosomal symbols in the optimizationprocess )e optimized S-system structure does not containall the input variables According to the training data RGEPcould automatically select the appropriate input variables InFigure 2 we can find that the coefficients αi and βi and theexponents gi1 gi2 gi3 hi1 hi2 hi3 and hi4 are needed to beoptimized In this paper the parameters in each chromo-some are optimized by a hybrid intelligent algorithm basedon BSO algorithm and PSO algorithm

32 Hybrid Optimization Algorithm )e BSO algorithm issuitable for solving the problem of multipeak and high-dimensional function )e PSO algorithm has theadvantages of easy realization high accuracy and fast

convergence But these two methods are easy to convergeprematurely and fall into local optimum In order to im-prove the diversity of population a novel hybrid intelligentalgorithm based on BSO and PSO (BSO-PSO) is proposedIn the BSO-PSO algorithm the half of individuals areselected randomly and optimized by BSO And the otherindividuals are optimized by PSO )e flowchart is de-scribed in Figure 4

33 Time Series Data Forecasting Using S-System )eflowchart of time series forecasting using the S-systemmodelis described in Figure 5 During the training phase theS-system model is optimized according to the genetic op-erators of RGEP hybrid intelligent algorithm and trainingdataset During the test phase the optimal S-system is usedto make the prediction of the stock index )e detailedprocess is described as follows

331 Training Phase

(1) Initialize the S-system population with the structureand parameters Each S-system is encoded as theRGEP chromosome which is described in Figure 2

(2) With the training samples the S-system is solved byequation (4) and the fitness value of each S-system iscalculated Search the best S-system according to thefitness values If the optimal model is found thealgorithm stops

(3) Selection recombination and mutation are used tosearch the optimal structure of the S-system Go tostep (2)

(4) At some iterations in RGEP BSO-PSO algorithm isused to optimize the parameters of RGEP chro-mosomes In this process the structure of theS-system model is fixed According to the structureof the model the number of parameters (αi βigij and hij) is counted With the hybrid intelligentalgorithm search and update the optimal parametersof each S-system

332 Testing Phase With the data at the previous time pointthe optimal S-system model obtained in the training phase issolved and the data at the current time point are predictedRepeat this procedure until that the data at all testing timepoints have been predicted According to the predicted dataand target data the predicted error is calculated

4 Results and Discussion

41 Data and Evaluation Standard Five stock indexes suchas Dow Jones Index (DJI) Hang Seng Index (HSI) NAS-DAQ Index (NASI) SSE (Shanghai Stock Exchange)Composite Index (SSEI) and SZSE Component Index(SZSEI) are proposed to test the performance of our methodSeventy percent of the data are used for training and 30 ofthe data are used for testing )e descriptions of five stockindexes are listed in Table 1

lowast2 lowast2 x3 x1 x2 x1 x2

lowast2 lowast3 x2 x4 x1 x3 x1

Head Tail

Gene 1

Gene 2

x3

x5 x2

x5 x1

x1

αi

βi

gi1 gi2 gi3 gi4 gi5 gi6 gi7 gi8

hi1 hi2 hi3 hi4 hi5 hi6 hi7 hi8

Figure 2 )e phenotype of chromosome in RGEP withparameters

ndash

lowast2

lowast2 x3

x1 x2 x3x1

dxidt

αi βi

gi3gi2

gi1

hi3hi2 hi4

lowast2

x2

hi1

x4

lowast3

Figure 3 )e expression tree of chromosome in RGEP withparameters

4 Computational Intelligence and Neuroscience

RMSE (root mean square error) MAP (mean absolutepercentage) and MAPE (mean absolute percentage error)R2 (coefficient of multiple determinations for multiple

regressions) ARV (average relative variance) and VAF(variance accounted for) are proposed to evaluate the per-formance of our method [30 39]

RMSE

1N

1113944

N

i1f

itarget minusf

iforecast1113872 1113873

2

11139741113972

MAP maxfitarget minusfi

forecast

11138681113868111386811138681113868

11138681113868111386811138681113868

fiforecast

times 100⎛⎝ ⎞⎠

MAPE 1N

1113944

N

i1

fitarget minusfi

forecast

11138681113868111386811138681113868

11138681113868111386811138681113868

fiforecast

⎛⎝ ⎞⎠ times 100

R2

1minus1113936

Ni1 fi

target minusfiforecast1113872 1113873

2

1113936Ni1 fi

target minusf1113872 11138732

ARV 1113936

Ni1 fi

target minusfiforecast1113872 1113873

2

1113936Ni1 fi

forecast minusf1113872 11138732

VAF 1minus1113936

Ni1 fi

target minusyiforecast1113872 1113873

2

1113936Ni1 fi

target1113872 11138732

⎛⎜⎝ ⎞⎟⎠ times 100

(10)

where N is the number of stock sample points fitarget is the

real stock value at the ith time point fiforecast is the predicting

stock value at the ith time point and f is the mean of stockindexes

42 Prediction Results In order to test the performance ofour method clearly five states of the art methods (DeepRecurrent Neural Network (DRNN) [40] FNT [19] RBFNN[17] BPNN [14] and ARIMA [8]) are also used to predictfive stock indexes

For 1-week-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x71113864 1113865

in the RGEP method By optimizing S-systemmodels by ourmethod we could obtain the optimal phenotypes and ex-pression trees (ETs) with five stock indexes which are de-scribed in Figure 6 Five optimal S-systemmodels gained arelisted in Table 2 for five stock datasets)e forecasting resultsof five stock indexes by our method are depicted in Figure 7From Figure 7 it can be clearly seen that the predictingcurves are very near to the target ones and the errors arenearly zero

Comparison results of different prediction modelsrsquoperformance on five stock indexes are listed in Table 3 FromTable 3 among the past five states of the art methods theDRNN model performs best for five stock indexes pre-diction But in terms of six indicators (RMSE MAP ARVMAPE R2 and VAF) our proposed method has betterperformance than the DRNN model In terms of RMSE ourmethod is 348 lower than DRNN for DJI dataset 464lower than DRNN for HSI dataset 404 lower than DRNNfor NASI dataset 198 lower than DRNN for SSEI dataset

Initialize N individuals

Select randomly

N2 individuals N2 individuals

Optimize with BSO Optimize with PSO

Calculate fitness values

Obtain the new generation

Is the satisfiedsolution found

Algorithm stops

YesNo

Figure 4 )e flowchart of BSO-PSO algorithm

Training

Optimization ofS-system

Gain the optimal S-system model

Test phasebegin

let Xlowast X be m lowast n matrices as the targeted and forecasting test data respectively Assign the last row of the training data as the initial condition to Y

for i = 1 to m dobegin

integrate the system of the best S-system for a step with the numerical integration method

assign the solution to the ith row of Xtake the ith row of Xlowast as the Y

endThe error can be calculated using X and Xlowast

end

Structure optimization using genetic operators of

RGEP

Parameter optimization using BSO-PSO algorithm

Figure 5 )e flowchart of time series data forecasting usingS-system

Computational Intelligence and Neuroscience 5

Table 1 Parameters of five stock indexes

Parameters DJI HSI NASI SSEI SZSEI

Time interval 121990ndash12292017

121991ndash12292017

121990ndash12292017

111996ndash12292017

122008ndash12302016

Train data for week-aheadprediction 4936 4666 4936 3866 1528

Test data for week-ahead prediction 2115 2000 2115 1657 655Train data for month-aheadprediction 4918 4649 4918 3849 1511

Test data for month-aheadprediction 2108 1992 2108 1649 647

Gene 1

Gene 2

ndash1365846

1912372

Head Tail

lowast2 x2 x5

lowast2 x1 x3

ndash04793 ndash00765

ndash19772 ndash24786

x1 x5 x4 x6 x3 x7

x7 x3 x5 x3 x2 x6

ndash

lowast2 lowast2

ndash1365846 1912372

x2 x5 x1 x3

ndash04793 ndash00765 ndash19772 ndash24786

(a)

Gene 1

Gene 2

ndash138218

156987

ndash

lowast2 lowast2

x1 x1 x3

Head Tail

lowast2 lowast2 x1

lowast2 x1 x3

ndash1012ndash15661

ndash20551 ndash149348

x3 x5 x7 x3 x5 x7

x3 x5 x2 x7 x6 x6

ndash36556

ndash149348ndash20551

156987ndash138218

ndash15661

lowast2

x3 x5

ndash1012 ndash36556

(b)

Gene 1

Gene 2

ndash00011

01105

ndash

lowast3 lowast2

x2x5 x2 x1

Head Tail

lowast3 x4 x3

lowast2 x2 x1

ndash12356 ndash41503

ndash76373 ndash38423

x5 x2 x4 x7 x6 x2

x7 x3 x5 x3 x2 x6

ndash07905

ndash00011 01105

ndash76373 ndash38423

x4

ndash12356 ndash41503 ndash07905

(c)

Gene 1

Gene 2

ndash17056

052302

Head Tail

lowast2 x1 x2

lowast2 x1 x5

12053 ndash02955

45115 ndash11571

x2 x3 x4 x1 x3 x7

x6 x2 x5 x3 x3 x1

ndash

lowast2 lowast2

x1 x2 x1 x5

ndash17056 052302

12053 ndash02955 45115 ndash11571

(d)

Figure 6 Continued

6 Computational Intelligence and Neuroscience

Gene 1

Gene 2

ndash101789

148636

Head Tail

lowast2 x3 x2

lowast2 x1 x6

ndash07625 ndash31649

ndash84503 ndash56786

x3 x3 x5 x5 x2 x7

x1 x2 x6 x2 x4 x1

ndash

lowast2 lowast2

x3 x2 x1 x6

ndash101789 148636

ndash07625 ndash31649 ndash84503 ndash56786

(e)

Figure 6 )e optimal phenotypes and expression trees for a-week-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

Table 2 Optimal S-system models of five stock datasets for a-week-ahead prediction

Type of datasets Optimal S-system modelDJI _f minus1365846xminus04793

2 xminus007655 minus 1912372xminus19772

1 xminus247863

HSI _f minus138218xminus156611 xminus1012

3 xminus365565 minus 156987xminus20551

1 xminus1493483

NASI _f minus00011xminus123564 xminus41503

3 xminus079055 minus 01105xminus76373

2 xminus384231

SSEI _f minus17056x120531 xminus02955

2 minus 052302x451151 xminus11571

5SZS _f minus101789xminus07625

3 xminus316492 minus 148636xminus84503

1 xminus567866

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 7 Continued

Computational Intelligence and Neuroscience 7

and 74 lower than DRNN for SZSEI dataset In terms ofARV our method is 587 lower than DRNN for DJIdataset 671 lower than DRNN for HSI dataset 688lower than DRNN for NASI dataset 369 lower thanDRNN for SSEI dataset and 165 lower than DRNN forSZSEI dataset In terms ofMAPE ourmethod is 375 lowerthan DRNN for DJI dataset 48 lower than DRNN for HSIdataset 429 lower than DRNN for NASI dataset 352lower than DRNN for SSEI dataset and 18 lower thanDRNN for SZSEI dataset In terms of VAF our method iscloser to 100 than DRNN for five stock indexes It could be

seen clearly that our proposed method could improve theprediction accuracy sharply

For 1-month-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x301113864 1113865

in the RGEP method With five stock indexes we obtain fiveoptimal phenotypes and expression trees (ETs) which aredescribed in Figure 8 According to five ETs the S-systemmodels gained are listed in Table 4 )e forecasting results offive stock indexes by our method are depicted in Figure 9From Figure 9 we could see clearly that the predicting andtarget curves are very close

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 7)e prediction and actual results for a-week-ahead predictionwith five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) and SZSEI (e)

Table 3 Comparison results of six methods for a-week-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005411 69911 000113 073146 099887 99992DRNN 00083 84909 0002735 11696 099726 99981FNT 0015427 14907 000613 18023 098939 99933

RBFNN 0016188 24473 001057 22631 098943 99927BPNN 0049026 22721 0060301 55923 09397 99328ARIMA 0052472 20924 0071326 59773 092867 99230

HSI

Our method 0008725 68824 001116 097359 098883 99984DRNN 0016272 13384 0033924 18718 096608 99944FNT 0020128 19261 0065331 22759 093467 99915

RBFNN 0023406 2532 0072545 27067 092756 99885BPNN 0035987 44738 012867 41357 087133 99729ARIMA 0013361 13817 0026688 15367 097331 99963

NASI

Our method 0008324 79168 0001707 10035 099829 99979DRNN 0013969 11069 0005465 1757 099453 99941FNT 0016468 32352 0006859 25336 099314 99918

RBFNN 003669 37371 0027513 45327 097249 99591BPNN 0046 175 0042533 5973 095747 99353ARIMA 0049849 1846 0093189 531 090681 99245

SSEI

Our method 0008105 9957 000535 11271 099465 99962DRNN 0010107 12959 0008481 17396 099152 99941FNT 0014559 18931 0018903 22848 09811 99878

RBFNN 0014681 2006 0018024 21804 098198 99876BPNN 0035922 32768 0091613 69046 090839 99256ARIMA 0020766 20533 0029814 39766 097019 99752

SZSEI

Our method 0016959 16079 0009762 24933 099024 99851DRNN 0018315 19783 0011685 30419 098831 99826FNT 0018571 2167 0012189 30233 098781 99821

RBFNN 0023881 31222 0018031 38187 098197 99704BPNN 0027297 41441 0027768 40751 097223 99614ARIMA 0029022 26983 002844 48583 097156 99563

8 Computational Intelligence and Neuroscience

Gene 1

Gene 2

26495

50785

Head Tail

lowast2 x10 x18

lowast2 x14 x4

04093 12433

ndash18385 ndash14525

x10 x25 x22 x18 x28 x12

x11 x3 x7 x23 x28 x10

ndash

lowast2 lowast2

26495 50785

x10 x18 x14 x4

04093 12433 ndash18385 ndash14525

(a)

Gene 1

Gene 2

24693

54397

ndash

lowast2 x9

x9 x26

ndash3638122267

5439724693

ndash102498

Head Tail

lowast2 x26

x11 x23

ndash3638122267

ndash102498

x12 x18 x2 x26 x30 x7

x1 x9 x20 x22 x26 x19

x9

x9

(b)

Gene 1

Gene 2

38864

22829

Head Tail

lowast2 x3 x29

lowast2 x17 x5

ndash18748 ndash26364

ndash12427 ndash37315

x25 x12 x1 x5 x3 x2

x23 x21 x2 x2 x7 x16

ndash

lowast2 lowast2

x3x29 x17 x5

38864 22829

ndash12427 ndash37315ndash18748 ndash26364

(c)

Gene 1

Gene 2

ndash41868

1030168

Head Tail

lowast2 x21 x30

lowast2 x19 x28

ndash016 ndash0288

ndash0398 0734

x12 x12 x1 x20 x22 x17

x1 x30 x6 x1 x3 x2

ndash

lowast2 lowast2

x21 x30 x19 x28

ndash41868 1030168

ndash016 ndash0288 ndash0398 0734

(d)

Figure 8 Continued

Computational Intelligence and Neuroscience 9

Table 4 Optimal S-system models of five stock datasets for a-month-ahead prediction

Type of datasets Optimal S-system modelDJI _f 26495x04093

10 x1243318 minus 50785xminus18385

14 xminus145254

HSI _f 24693x222679 xminus36381

26 minus 54397xminus1024989

NASI _f 38864x187483 xminus26364

29 minus 22829xminus1242717 xminus37315

5SSEI _f minus41868xminus016

21 x028830 minus 1030168xminus0398

19 x073428

SZSEI _f 10205x0621423 x36797

27 minus 88024xminus2357914 xminus4797

16

Gene 1

Gene 2

10205

88024

Head Tail

lowast2 x23 x27

lowast2 x14 x16

06214 36797

ndash23579 ndash4797

x1 x8 x15 x23 x19 x1

x4 x12 x23 x20 x4 x4

ndash

lowast2 lowast2

x23 x27 x14 x16

10205 88024

06214 36797 ndash23579 ndash4797

(e)

Figure 8 )e optimal phenotypes and expression trees for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 9 Continued

10 Computational Intelligence and Neuroscience

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

RMSE (root mean square error) MAP (mean absolutepercentage) and MAPE (mean absolute percentage error)R2 (coefficient of multiple determinations for multiple

regressions) ARV (average relative variance) and VAF(variance accounted for) are proposed to evaluate the per-formance of our method [30 39]

RMSE

1N

1113944

N

i1f

itarget minusf

iforecast1113872 1113873

2

11139741113972

MAP maxfitarget minusfi

forecast

11138681113868111386811138681113868

11138681113868111386811138681113868

fiforecast

times 100⎛⎝ ⎞⎠

MAPE 1N

1113944

N

i1

fitarget minusfi

forecast

11138681113868111386811138681113868

11138681113868111386811138681113868

fiforecast

⎛⎝ ⎞⎠ times 100

R2

1minus1113936

Ni1 fi

target minusfiforecast1113872 1113873

2

1113936Ni1 fi

target minusf1113872 11138732

ARV 1113936

Ni1 fi

target minusfiforecast1113872 1113873

2

1113936Ni1 fi

forecast minusf1113872 11138732

VAF 1minus1113936

Ni1 fi

target minusyiforecast1113872 1113873

2

1113936Ni1 fi

target1113872 11138732

⎛⎜⎝ ⎞⎟⎠ times 100

(10)

where N is the number of stock sample points fitarget is the

real stock value at the ith time point fiforecast is the predicting

stock value at the ith time point and f is the mean of stockindexes

42 Prediction Results In order to test the performance ofour method clearly five states of the art methods (DeepRecurrent Neural Network (DRNN) [40] FNT [19] RBFNN[17] BPNN [14] and ARIMA [8]) are also used to predictfive stock indexes

For 1-week-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x71113864 1113865

in the RGEP method By optimizing S-systemmodels by ourmethod we could obtain the optimal phenotypes and ex-pression trees (ETs) with five stock indexes which are de-scribed in Figure 6 Five optimal S-systemmodels gained arelisted in Table 2 for five stock datasets)e forecasting resultsof five stock indexes by our method are depicted in Figure 7From Figure 7 it can be clearly seen that the predictingcurves are very near to the target ones and the errors arenearly zero

Comparison results of different prediction modelsrsquoperformance on five stock indexes are listed in Table 3 FromTable 3 among the past five states of the art methods theDRNN model performs best for five stock indexes pre-diction But in terms of six indicators (RMSE MAP ARVMAPE R2 and VAF) our proposed method has betterperformance than the DRNN model In terms of RMSE ourmethod is 348 lower than DRNN for DJI dataset 464lower than DRNN for HSI dataset 404 lower than DRNNfor NASI dataset 198 lower than DRNN for SSEI dataset

Initialize N individuals

Select randomly

N2 individuals N2 individuals

Optimize with BSO Optimize with PSO

Calculate fitness values

Obtain the new generation

Is the satisfiedsolution found

Algorithm stops

YesNo

Figure 4 )e flowchart of BSO-PSO algorithm

Training

Optimization ofS-system

Gain the optimal S-system model

Test phasebegin

let Xlowast X be m lowast n matrices as the targeted and forecasting test data respectively Assign the last row of the training data as the initial condition to Y

for i = 1 to m dobegin

integrate the system of the best S-system for a step with the numerical integration method

assign the solution to the ith row of Xtake the ith row of Xlowast as the Y

endThe error can be calculated using X and Xlowast

end

Structure optimization using genetic operators of

RGEP

Parameter optimization using BSO-PSO algorithm

Figure 5 )e flowchart of time series data forecasting usingS-system

Computational Intelligence and Neuroscience 5

Table 1 Parameters of five stock indexes

Parameters DJI HSI NASI SSEI SZSEI

Time interval 121990ndash12292017

121991ndash12292017

121990ndash12292017

111996ndash12292017

122008ndash12302016

Train data for week-aheadprediction 4936 4666 4936 3866 1528

Test data for week-ahead prediction 2115 2000 2115 1657 655Train data for month-aheadprediction 4918 4649 4918 3849 1511

Test data for month-aheadprediction 2108 1992 2108 1649 647

Gene 1

Gene 2

ndash1365846

1912372

Head Tail

lowast2 x2 x5

lowast2 x1 x3

ndash04793 ndash00765

ndash19772 ndash24786

x1 x5 x4 x6 x3 x7

x7 x3 x5 x3 x2 x6

ndash

lowast2 lowast2

ndash1365846 1912372

x2 x5 x1 x3

ndash04793 ndash00765 ndash19772 ndash24786

(a)

Gene 1

Gene 2

ndash138218

156987

ndash

lowast2 lowast2

x1 x1 x3

Head Tail

lowast2 lowast2 x1

lowast2 x1 x3

ndash1012ndash15661

ndash20551 ndash149348

x3 x5 x7 x3 x5 x7

x3 x5 x2 x7 x6 x6

ndash36556

ndash149348ndash20551

156987ndash138218

ndash15661

lowast2

x3 x5

ndash1012 ndash36556

(b)

Gene 1

Gene 2

ndash00011

01105

ndash

lowast3 lowast2

x2x5 x2 x1

Head Tail

lowast3 x4 x3

lowast2 x2 x1

ndash12356 ndash41503

ndash76373 ndash38423

x5 x2 x4 x7 x6 x2

x7 x3 x5 x3 x2 x6

ndash07905

ndash00011 01105

ndash76373 ndash38423

x4

ndash12356 ndash41503 ndash07905

(c)

Gene 1

Gene 2

ndash17056

052302

Head Tail

lowast2 x1 x2

lowast2 x1 x5

12053 ndash02955

45115 ndash11571

x2 x3 x4 x1 x3 x7

x6 x2 x5 x3 x3 x1

ndash

lowast2 lowast2

x1 x2 x1 x5

ndash17056 052302

12053 ndash02955 45115 ndash11571

(d)

Figure 6 Continued

6 Computational Intelligence and Neuroscience

Gene 1

Gene 2

ndash101789

148636

Head Tail

lowast2 x3 x2

lowast2 x1 x6

ndash07625 ndash31649

ndash84503 ndash56786

x3 x3 x5 x5 x2 x7

x1 x2 x6 x2 x4 x1

ndash

lowast2 lowast2

x3 x2 x1 x6

ndash101789 148636

ndash07625 ndash31649 ndash84503 ndash56786

(e)

Figure 6 )e optimal phenotypes and expression trees for a-week-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

Table 2 Optimal S-system models of five stock datasets for a-week-ahead prediction

Type of datasets Optimal S-system modelDJI _f minus1365846xminus04793

2 xminus007655 minus 1912372xminus19772

1 xminus247863

HSI _f minus138218xminus156611 xminus1012

3 xminus365565 minus 156987xminus20551

1 xminus1493483

NASI _f minus00011xminus123564 xminus41503

3 xminus079055 minus 01105xminus76373

2 xminus384231

SSEI _f minus17056x120531 xminus02955

2 minus 052302x451151 xminus11571

5SZS _f minus101789xminus07625

3 xminus316492 minus 148636xminus84503

1 xminus567866

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 7 Continued

Computational Intelligence and Neuroscience 7

and 74 lower than DRNN for SZSEI dataset In terms ofARV our method is 587 lower than DRNN for DJIdataset 671 lower than DRNN for HSI dataset 688lower than DRNN for NASI dataset 369 lower thanDRNN for SSEI dataset and 165 lower than DRNN forSZSEI dataset In terms ofMAPE ourmethod is 375 lowerthan DRNN for DJI dataset 48 lower than DRNN for HSIdataset 429 lower than DRNN for NASI dataset 352lower than DRNN for SSEI dataset and 18 lower thanDRNN for SZSEI dataset In terms of VAF our method iscloser to 100 than DRNN for five stock indexes It could be

seen clearly that our proposed method could improve theprediction accuracy sharply

For 1-month-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x301113864 1113865

in the RGEP method With five stock indexes we obtain fiveoptimal phenotypes and expression trees (ETs) which aredescribed in Figure 8 According to five ETs the S-systemmodels gained are listed in Table 4 )e forecasting results offive stock indexes by our method are depicted in Figure 9From Figure 9 we could see clearly that the predicting andtarget curves are very close

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 7)e prediction and actual results for a-week-ahead predictionwith five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) and SZSEI (e)

Table 3 Comparison results of six methods for a-week-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005411 69911 000113 073146 099887 99992DRNN 00083 84909 0002735 11696 099726 99981FNT 0015427 14907 000613 18023 098939 99933

RBFNN 0016188 24473 001057 22631 098943 99927BPNN 0049026 22721 0060301 55923 09397 99328ARIMA 0052472 20924 0071326 59773 092867 99230

HSI

Our method 0008725 68824 001116 097359 098883 99984DRNN 0016272 13384 0033924 18718 096608 99944FNT 0020128 19261 0065331 22759 093467 99915

RBFNN 0023406 2532 0072545 27067 092756 99885BPNN 0035987 44738 012867 41357 087133 99729ARIMA 0013361 13817 0026688 15367 097331 99963

NASI

Our method 0008324 79168 0001707 10035 099829 99979DRNN 0013969 11069 0005465 1757 099453 99941FNT 0016468 32352 0006859 25336 099314 99918

RBFNN 003669 37371 0027513 45327 097249 99591BPNN 0046 175 0042533 5973 095747 99353ARIMA 0049849 1846 0093189 531 090681 99245

SSEI

Our method 0008105 9957 000535 11271 099465 99962DRNN 0010107 12959 0008481 17396 099152 99941FNT 0014559 18931 0018903 22848 09811 99878

RBFNN 0014681 2006 0018024 21804 098198 99876BPNN 0035922 32768 0091613 69046 090839 99256ARIMA 0020766 20533 0029814 39766 097019 99752

SZSEI

Our method 0016959 16079 0009762 24933 099024 99851DRNN 0018315 19783 0011685 30419 098831 99826FNT 0018571 2167 0012189 30233 098781 99821

RBFNN 0023881 31222 0018031 38187 098197 99704BPNN 0027297 41441 0027768 40751 097223 99614ARIMA 0029022 26983 002844 48583 097156 99563

8 Computational Intelligence and Neuroscience

Gene 1

Gene 2

26495

50785

Head Tail

lowast2 x10 x18

lowast2 x14 x4

04093 12433

ndash18385 ndash14525

x10 x25 x22 x18 x28 x12

x11 x3 x7 x23 x28 x10

ndash

lowast2 lowast2

26495 50785

x10 x18 x14 x4

04093 12433 ndash18385 ndash14525

(a)

Gene 1

Gene 2

24693

54397

ndash

lowast2 x9

x9 x26

ndash3638122267

5439724693

ndash102498

Head Tail

lowast2 x26

x11 x23

ndash3638122267

ndash102498

x12 x18 x2 x26 x30 x7

x1 x9 x20 x22 x26 x19

x9

x9

(b)

Gene 1

Gene 2

38864

22829

Head Tail

lowast2 x3 x29

lowast2 x17 x5

ndash18748 ndash26364

ndash12427 ndash37315

x25 x12 x1 x5 x3 x2

x23 x21 x2 x2 x7 x16

ndash

lowast2 lowast2

x3x29 x17 x5

38864 22829

ndash12427 ndash37315ndash18748 ndash26364

(c)

Gene 1

Gene 2

ndash41868

1030168

Head Tail

lowast2 x21 x30

lowast2 x19 x28

ndash016 ndash0288

ndash0398 0734

x12 x12 x1 x20 x22 x17

x1 x30 x6 x1 x3 x2

ndash

lowast2 lowast2

x21 x30 x19 x28

ndash41868 1030168

ndash016 ndash0288 ndash0398 0734

(d)

Figure 8 Continued

Computational Intelligence and Neuroscience 9

Table 4 Optimal S-system models of five stock datasets for a-month-ahead prediction

Type of datasets Optimal S-system modelDJI _f 26495x04093

10 x1243318 minus 50785xminus18385

14 xminus145254

HSI _f 24693x222679 xminus36381

26 minus 54397xminus1024989

NASI _f 38864x187483 xminus26364

29 minus 22829xminus1242717 xminus37315

5SSEI _f minus41868xminus016

21 x028830 minus 1030168xminus0398

19 x073428

SZSEI _f 10205x0621423 x36797

27 minus 88024xminus2357914 xminus4797

16

Gene 1

Gene 2

10205

88024

Head Tail

lowast2 x23 x27

lowast2 x14 x16

06214 36797

ndash23579 ndash4797

x1 x8 x15 x23 x19 x1

x4 x12 x23 x20 x4 x4

ndash

lowast2 lowast2

x23 x27 x14 x16

10205 88024

06214 36797 ndash23579 ndash4797

(e)

Figure 8 )e optimal phenotypes and expression trees for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 9 Continued

10 Computational Intelligence and Neuroscience

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

Table 1 Parameters of five stock indexes

Parameters DJI HSI NASI SSEI SZSEI

Time interval 121990ndash12292017

121991ndash12292017

121990ndash12292017

111996ndash12292017

122008ndash12302016

Train data for week-aheadprediction 4936 4666 4936 3866 1528

Test data for week-ahead prediction 2115 2000 2115 1657 655Train data for month-aheadprediction 4918 4649 4918 3849 1511

Test data for month-aheadprediction 2108 1992 2108 1649 647

Gene 1

Gene 2

ndash1365846

1912372

Head Tail

lowast2 x2 x5

lowast2 x1 x3

ndash04793 ndash00765

ndash19772 ndash24786

x1 x5 x4 x6 x3 x7

x7 x3 x5 x3 x2 x6

ndash

lowast2 lowast2

ndash1365846 1912372

x2 x5 x1 x3

ndash04793 ndash00765 ndash19772 ndash24786

(a)

Gene 1

Gene 2

ndash138218

156987

ndash

lowast2 lowast2

x1 x1 x3

Head Tail

lowast2 lowast2 x1

lowast2 x1 x3

ndash1012ndash15661

ndash20551 ndash149348

x3 x5 x7 x3 x5 x7

x3 x5 x2 x7 x6 x6

ndash36556

ndash149348ndash20551

156987ndash138218

ndash15661

lowast2

x3 x5

ndash1012 ndash36556

(b)

Gene 1

Gene 2

ndash00011

01105

ndash

lowast3 lowast2

x2x5 x2 x1

Head Tail

lowast3 x4 x3

lowast2 x2 x1

ndash12356 ndash41503

ndash76373 ndash38423

x5 x2 x4 x7 x6 x2

x7 x3 x5 x3 x2 x6

ndash07905

ndash00011 01105

ndash76373 ndash38423

x4

ndash12356 ndash41503 ndash07905

(c)

Gene 1

Gene 2

ndash17056

052302

Head Tail

lowast2 x1 x2

lowast2 x1 x5

12053 ndash02955

45115 ndash11571

x2 x3 x4 x1 x3 x7

x6 x2 x5 x3 x3 x1

ndash

lowast2 lowast2

x1 x2 x1 x5

ndash17056 052302

12053 ndash02955 45115 ndash11571

(d)

Figure 6 Continued

6 Computational Intelligence and Neuroscience

Gene 1

Gene 2

ndash101789

148636

Head Tail

lowast2 x3 x2

lowast2 x1 x6

ndash07625 ndash31649

ndash84503 ndash56786

x3 x3 x5 x5 x2 x7

x1 x2 x6 x2 x4 x1

ndash

lowast2 lowast2

x3 x2 x1 x6

ndash101789 148636

ndash07625 ndash31649 ndash84503 ndash56786

(e)

Figure 6 )e optimal phenotypes and expression trees for a-week-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

Table 2 Optimal S-system models of five stock datasets for a-week-ahead prediction

Type of datasets Optimal S-system modelDJI _f minus1365846xminus04793

2 xminus007655 minus 1912372xminus19772

1 xminus247863

HSI _f minus138218xminus156611 xminus1012

3 xminus365565 minus 156987xminus20551

1 xminus1493483

NASI _f minus00011xminus123564 xminus41503

3 xminus079055 minus 01105xminus76373

2 xminus384231

SSEI _f minus17056x120531 xminus02955

2 minus 052302x451151 xminus11571

5SZS _f minus101789xminus07625

3 xminus316492 minus 148636xminus84503

1 xminus567866

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 7 Continued

Computational Intelligence and Neuroscience 7

and 74 lower than DRNN for SZSEI dataset In terms ofARV our method is 587 lower than DRNN for DJIdataset 671 lower than DRNN for HSI dataset 688lower than DRNN for NASI dataset 369 lower thanDRNN for SSEI dataset and 165 lower than DRNN forSZSEI dataset In terms ofMAPE ourmethod is 375 lowerthan DRNN for DJI dataset 48 lower than DRNN for HSIdataset 429 lower than DRNN for NASI dataset 352lower than DRNN for SSEI dataset and 18 lower thanDRNN for SZSEI dataset In terms of VAF our method iscloser to 100 than DRNN for five stock indexes It could be

seen clearly that our proposed method could improve theprediction accuracy sharply

For 1-month-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x301113864 1113865

in the RGEP method With five stock indexes we obtain fiveoptimal phenotypes and expression trees (ETs) which aredescribed in Figure 8 According to five ETs the S-systemmodels gained are listed in Table 4 )e forecasting results offive stock indexes by our method are depicted in Figure 9From Figure 9 we could see clearly that the predicting andtarget curves are very close

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 7)e prediction and actual results for a-week-ahead predictionwith five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) and SZSEI (e)

Table 3 Comparison results of six methods for a-week-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005411 69911 000113 073146 099887 99992DRNN 00083 84909 0002735 11696 099726 99981FNT 0015427 14907 000613 18023 098939 99933

RBFNN 0016188 24473 001057 22631 098943 99927BPNN 0049026 22721 0060301 55923 09397 99328ARIMA 0052472 20924 0071326 59773 092867 99230

HSI

Our method 0008725 68824 001116 097359 098883 99984DRNN 0016272 13384 0033924 18718 096608 99944FNT 0020128 19261 0065331 22759 093467 99915

RBFNN 0023406 2532 0072545 27067 092756 99885BPNN 0035987 44738 012867 41357 087133 99729ARIMA 0013361 13817 0026688 15367 097331 99963

NASI

Our method 0008324 79168 0001707 10035 099829 99979DRNN 0013969 11069 0005465 1757 099453 99941FNT 0016468 32352 0006859 25336 099314 99918

RBFNN 003669 37371 0027513 45327 097249 99591BPNN 0046 175 0042533 5973 095747 99353ARIMA 0049849 1846 0093189 531 090681 99245

SSEI

Our method 0008105 9957 000535 11271 099465 99962DRNN 0010107 12959 0008481 17396 099152 99941FNT 0014559 18931 0018903 22848 09811 99878

RBFNN 0014681 2006 0018024 21804 098198 99876BPNN 0035922 32768 0091613 69046 090839 99256ARIMA 0020766 20533 0029814 39766 097019 99752

SZSEI

Our method 0016959 16079 0009762 24933 099024 99851DRNN 0018315 19783 0011685 30419 098831 99826FNT 0018571 2167 0012189 30233 098781 99821

RBFNN 0023881 31222 0018031 38187 098197 99704BPNN 0027297 41441 0027768 40751 097223 99614ARIMA 0029022 26983 002844 48583 097156 99563

8 Computational Intelligence and Neuroscience

Gene 1

Gene 2

26495

50785

Head Tail

lowast2 x10 x18

lowast2 x14 x4

04093 12433

ndash18385 ndash14525

x10 x25 x22 x18 x28 x12

x11 x3 x7 x23 x28 x10

ndash

lowast2 lowast2

26495 50785

x10 x18 x14 x4

04093 12433 ndash18385 ndash14525

(a)

Gene 1

Gene 2

24693

54397

ndash

lowast2 x9

x9 x26

ndash3638122267

5439724693

ndash102498

Head Tail

lowast2 x26

x11 x23

ndash3638122267

ndash102498

x12 x18 x2 x26 x30 x7

x1 x9 x20 x22 x26 x19

x9

x9

(b)

Gene 1

Gene 2

38864

22829

Head Tail

lowast2 x3 x29

lowast2 x17 x5

ndash18748 ndash26364

ndash12427 ndash37315

x25 x12 x1 x5 x3 x2

x23 x21 x2 x2 x7 x16

ndash

lowast2 lowast2

x3x29 x17 x5

38864 22829

ndash12427 ndash37315ndash18748 ndash26364

(c)

Gene 1

Gene 2

ndash41868

1030168

Head Tail

lowast2 x21 x30

lowast2 x19 x28

ndash016 ndash0288

ndash0398 0734

x12 x12 x1 x20 x22 x17

x1 x30 x6 x1 x3 x2

ndash

lowast2 lowast2

x21 x30 x19 x28

ndash41868 1030168

ndash016 ndash0288 ndash0398 0734

(d)

Figure 8 Continued

Computational Intelligence and Neuroscience 9

Table 4 Optimal S-system models of five stock datasets for a-month-ahead prediction

Type of datasets Optimal S-system modelDJI _f 26495x04093

10 x1243318 minus 50785xminus18385

14 xminus145254

HSI _f 24693x222679 xminus36381

26 minus 54397xminus1024989

NASI _f 38864x187483 xminus26364

29 minus 22829xminus1242717 xminus37315

5SSEI _f minus41868xminus016

21 x028830 minus 1030168xminus0398

19 x073428

SZSEI _f 10205x0621423 x36797

27 minus 88024xminus2357914 xminus4797

16

Gene 1

Gene 2

10205

88024

Head Tail

lowast2 x23 x27

lowast2 x14 x16

06214 36797

ndash23579 ndash4797

x1 x8 x15 x23 x19 x1

x4 x12 x23 x20 x4 x4

ndash

lowast2 lowast2

x23 x27 x14 x16

10205 88024

06214 36797 ndash23579 ndash4797

(e)

Figure 8 )e optimal phenotypes and expression trees for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 9 Continued

10 Computational Intelligence and Neuroscience

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

Gene 1

Gene 2

ndash101789

148636

Head Tail

lowast2 x3 x2

lowast2 x1 x6

ndash07625 ndash31649

ndash84503 ndash56786

x3 x3 x5 x5 x2 x7

x1 x2 x6 x2 x4 x1

ndash

lowast2 lowast2

x3 x2 x1 x6

ndash101789 148636

ndash07625 ndash31649 ndash84503 ndash56786

(e)

Figure 6 )e optimal phenotypes and expression trees for a-week-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

Table 2 Optimal S-system models of five stock datasets for a-week-ahead prediction

Type of datasets Optimal S-system modelDJI _f minus1365846xminus04793

2 xminus007655 minus 1912372xminus19772

1 xminus247863

HSI _f minus138218xminus156611 xminus1012

3 xminus365565 minus 156987xminus20551

1 xminus1493483

NASI _f minus00011xminus123564 xminus41503

3 xminus079055 minus 01105xminus76373

2 xminus384231

SSEI _f minus17056x120531 xminus02955

2 minus 052302x451151 xminus11571

5SZS _f minus101789xminus07625

3 xminus316492 minus 148636xminus84503

1 xminus567866

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 7 Continued

Computational Intelligence and Neuroscience 7

and 74 lower than DRNN for SZSEI dataset In terms ofARV our method is 587 lower than DRNN for DJIdataset 671 lower than DRNN for HSI dataset 688lower than DRNN for NASI dataset 369 lower thanDRNN for SSEI dataset and 165 lower than DRNN forSZSEI dataset In terms ofMAPE ourmethod is 375 lowerthan DRNN for DJI dataset 48 lower than DRNN for HSIdataset 429 lower than DRNN for NASI dataset 352lower than DRNN for SSEI dataset and 18 lower thanDRNN for SZSEI dataset In terms of VAF our method iscloser to 100 than DRNN for five stock indexes It could be

seen clearly that our proposed method could improve theprediction accuracy sharply

For 1-month-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x301113864 1113865

in the RGEP method With five stock indexes we obtain fiveoptimal phenotypes and expression trees (ETs) which aredescribed in Figure 8 According to five ETs the S-systemmodels gained are listed in Table 4 )e forecasting results offive stock indexes by our method are depicted in Figure 9From Figure 9 we could see clearly that the predicting andtarget curves are very close

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 7)e prediction and actual results for a-week-ahead predictionwith five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) and SZSEI (e)

Table 3 Comparison results of six methods for a-week-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005411 69911 000113 073146 099887 99992DRNN 00083 84909 0002735 11696 099726 99981FNT 0015427 14907 000613 18023 098939 99933

RBFNN 0016188 24473 001057 22631 098943 99927BPNN 0049026 22721 0060301 55923 09397 99328ARIMA 0052472 20924 0071326 59773 092867 99230

HSI

Our method 0008725 68824 001116 097359 098883 99984DRNN 0016272 13384 0033924 18718 096608 99944FNT 0020128 19261 0065331 22759 093467 99915

RBFNN 0023406 2532 0072545 27067 092756 99885BPNN 0035987 44738 012867 41357 087133 99729ARIMA 0013361 13817 0026688 15367 097331 99963

NASI

Our method 0008324 79168 0001707 10035 099829 99979DRNN 0013969 11069 0005465 1757 099453 99941FNT 0016468 32352 0006859 25336 099314 99918

RBFNN 003669 37371 0027513 45327 097249 99591BPNN 0046 175 0042533 5973 095747 99353ARIMA 0049849 1846 0093189 531 090681 99245

SSEI

Our method 0008105 9957 000535 11271 099465 99962DRNN 0010107 12959 0008481 17396 099152 99941FNT 0014559 18931 0018903 22848 09811 99878

RBFNN 0014681 2006 0018024 21804 098198 99876BPNN 0035922 32768 0091613 69046 090839 99256ARIMA 0020766 20533 0029814 39766 097019 99752

SZSEI

Our method 0016959 16079 0009762 24933 099024 99851DRNN 0018315 19783 0011685 30419 098831 99826FNT 0018571 2167 0012189 30233 098781 99821

RBFNN 0023881 31222 0018031 38187 098197 99704BPNN 0027297 41441 0027768 40751 097223 99614ARIMA 0029022 26983 002844 48583 097156 99563

8 Computational Intelligence and Neuroscience

Gene 1

Gene 2

26495

50785

Head Tail

lowast2 x10 x18

lowast2 x14 x4

04093 12433

ndash18385 ndash14525

x10 x25 x22 x18 x28 x12

x11 x3 x7 x23 x28 x10

ndash

lowast2 lowast2

26495 50785

x10 x18 x14 x4

04093 12433 ndash18385 ndash14525

(a)

Gene 1

Gene 2

24693

54397

ndash

lowast2 x9

x9 x26

ndash3638122267

5439724693

ndash102498

Head Tail

lowast2 x26

x11 x23

ndash3638122267

ndash102498

x12 x18 x2 x26 x30 x7

x1 x9 x20 x22 x26 x19

x9

x9

(b)

Gene 1

Gene 2

38864

22829

Head Tail

lowast2 x3 x29

lowast2 x17 x5

ndash18748 ndash26364

ndash12427 ndash37315

x25 x12 x1 x5 x3 x2

x23 x21 x2 x2 x7 x16

ndash

lowast2 lowast2

x3x29 x17 x5

38864 22829

ndash12427 ndash37315ndash18748 ndash26364

(c)

Gene 1

Gene 2

ndash41868

1030168

Head Tail

lowast2 x21 x30

lowast2 x19 x28

ndash016 ndash0288

ndash0398 0734

x12 x12 x1 x20 x22 x17

x1 x30 x6 x1 x3 x2

ndash

lowast2 lowast2

x21 x30 x19 x28

ndash41868 1030168

ndash016 ndash0288 ndash0398 0734

(d)

Figure 8 Continued

Computational Intelligence and Neuroscience 9

Table 4 Optimal S-system models of five stock datasets for a-month-ahead prediction

Type of datasets Optimal S-system modelDJI _f 26495x04093

10 x1243318 minus 50785xminus18385

14 xminus145254

HSI _f 24693x222679 xminus36381

26 minus 54397xminus1024989

NASI _f 38864x187483 xminus26364

29 minus 22829xminus1242717 xminus37315

5SSEI _f minus41868xminus016

21 x028830 minus 1030168xminus0398

19 x073428

SZSEI _f 10205x0621423 x36797

27 minus 88024xminus2357914 xminus4797

16

Gene 1

Gene 2

10205

88024

Head Tail

lowast2 x23 x27

lowast2 x14 x16

06214 36797

ndash23579 ndash4797

x1 x8 x15 x23 x19 x1

x4 x12 x23 x20 x4 x4

ndash

lowast2 lowast2

x23 x27 x14 x16

10205 88024

06214 36797 ndash23579 ndash4797

(e)

Figure 8 )e optimal phenotypes and expression trees for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 9 Continued

10 Computational Intelligence and Neuroscience

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

and 74 lower than DRNN for SZSEI dataset In terms ofARV our method is 587 lower than DRNN for DJIdataset 671 lower than DRNN for HSI dataset 688lower than DRNN for NASI dataset 369 lower thanDRNN for SSEI dataset and 165 lower than DRNN forSZSEI dataset In terms ofMAPE ourmethod is 375 lowerthan DRNN for DJI dataset 48 lower than DRNN for HSIdataset 429 lower than DRNN for NASI dataset 352lower than DRNN for SSEI dataset and 18 lower thanDRNN for SZSEI dataset In terms of VAF our method iscloser to 100 than DRNN for five stock indexes It could be

seen clearly that our proposed method could improve theprediction accuracy sharply

For 1-month-ahead prediction problem function set isF lowast1lowast2lowast3lowast4 and variable set is T x1 x2 x301113864 1113865

in the RGEP method With five stock indexes we obtain fiveoptimal phenotypes and expression trees (ETs) which aredescribed in Figure 8 According to five ETs the S-systemmodels gained are listed in Table 4 )e forecasting results offive stock indexes by our method are depicted in Figure 9From Figure 9 we could see clearly that the predicting andtarget curves are very close

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 7)e prediction and actual results for a-week-ahead predictionwith five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) and SZSEI (e)

Table 3 Comparison results of six methods for a-week-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005411 69911 000113 073146 099887 99992DRNN 00083 84909 0002735 11696 099726 99981FNT 0015427 14907 000613 18023 098939 99933

RBFNN 0016188 24473 001057 22631 098943 99927BPNN 0049026 22721 0060301 55923 09397 99328ARIMA 0052472 20924 0071326 59773 092867 99230

HSI

Our method 0008725 68824 001116 097359 098883 99984DRNN 0016272 13384 0033924 18718 096608 99944FNT 0020128 19261 0065331 22759 093467 99915

RBFNN 0023406 2532 0072545 27067 092756 99885BPNN 0035987 44738 012867 41357 087133 99729ARIMA 0013361 13817 0026688 15367 097331 99963

NASI

Our method 0008324 79168 0001707 10035 099829 99979DRNN 0013969 11069 0005465 1757 099453 99941FNT 0016468 32352 0006859 25336 099314 99918

RBFNN 003669 37371 0027513 45327 097249 99591BPNN 0046 175 0042533 5973 095747 99353ARIMA 0049849 1846 0093189 531 090681 99245

SSEI

Our method 0008105 9957 000535 11271 099465 99962DRNN 0010107 12959 0008481 17396 099152 99941FNT 0014559 18931 0018903 22848 09811 99878

RBFNN 0014681 2006 0018024 21804 098198 99876BPNN 0035922 32768 0091613 69046 090839 99256ARIMA 0020766 20533 0029814 39766 097019 99752

SZSEI

Our method 0016959 16079 0009762 24933 099024 99851DRNN 0018315 19783 0011685 30419 098831 99826FNT 0018571 2167 0012189 30233 098781 99821

RBFNN 0023881 31222 0018031 38187 098197 99704BPNN 0027297 41441 0027768 40751 097223 99614ARIMA 0029022 26983 002844 48583 097156 99563

8 Computational Intelligence and Neuroscience

Gene 1

Gene 2

26495

50785

Head Tail

lowast2 x10 x18

lowast2 x14 x4

04093 12433

ndash18385 ndash14525

x10 x25 x22 x18 x28 x12

x11 x3 x7 x23 x28 x10

ndash

lowast2 lowast2

26495 50785

x10 x18 x14 x4

04093 12433 ndash18385 ndash14525

(a)

Gene 1

Gene 2

24693

54397

ndash

lowast2 x9

x9 x26

ndash3638122267

5439724693

ndash102498

Head Tail

lowast2 x26

x11 x23

ndash3638122267

ndash102498

x12 x18 x2 x26 x30 x7

x1 x9 x20 x22 x26 x19

x9

x9

(b)

Gene 1

Gene 2

38864

22829

Head Tail

lowast2 x3 x29

lowast2 x17 x5

ndash18748 ndash26364

ndash12427 ndash37315

x25 x12 x1 x5 x3 x2

x23 x21 x2 x2 x7 x16

ndash

lowast2 lowast2

x3x29 x17 x5

38864 22829

ndash12427 ndash37315ndash18748 ndash26364

(c)

Gene 1

Gene 2

ndash41868

1030168

Head Tail

lowast2 x21 x30

lowast2 x19 x28

ndash016 ndash0288

ndash0398 0734

x12 x12 x1 x20 x22 x17

x1 x30 x6 x1 x3 x2

ndash

lowast2 lowast2

x21 x30 x19 x28

ndash41868 1030168

ndash016 ndash0288 ndash0398 0734

(d)

Figure 8 Continued

Computational Intelligence and Neuroscience 9

Table 4 Optimal S-system models of five stock datasets for a-month-ahead prediction

Type of datasets Optimal S-system modelDJI _f 26495x04093

10 x1243318 minus 50785xminus18385

14 xminus145254

HSI _f 24693x222679 xminus36381

26 minus 54397xminus1024989

NASI _f 38864x187483 xminus26364

29 minus 22829xminus1242717 xminus37315

5SSEI _f minus41868xminus016

21 x028830 minus 1030168xminus0398

19 x073428

SZSEI _f 10205x0621423 x36797

27 minus 88024xminus2357914 xminus4797

16

Gene 1

Gene 2

10205

88024

Head Tail

lowast2 x23 x27

lowast2 x14 x16

06214 36797

ndash23579 ndash4797

x1 x8 x15 x23 x19 x1

x4 x12 x23 x20 x4 x4

ndash

lowast2 lowast2

x23 x27 x14 x16

10205 88024

06214 36797 ndash23579 ndash4797

(e)

Figure 8 )e optimal phenotypes and expression trees for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 9 Continued

10 Computational Intelligence and Neuroscience

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

Gene 1

Gene 2

26495

50785

Head Tail

lowast2 x10 x18

lowast2 x14 x4

04093 12433

ndash18385 ndash14525

x10 x25 x22 x18 x28 x12

x11 x3 x7 x23 x28 x10

ndash

lowast2 lowast2

26495 50785

x10 x18 x14 x4

04093 12433 ndash18385 ndash14525

(a)

Gene 1

Gene 2

24693

54397

ndash

lowast2 x9

x9 x26

ndash3638122267

5439724693

ndash102498

Head Tail

lowast2 x26

x11 x23

ndash3638122267

ndash102498

x12 x18 x2 x26 x30 x7

x1 x9 x20 x22 x26 x19

x9

x9

(b)

Gene 1

Gene 2

38864

22829

Head Tail

lowast2 x3 x29

lowast2 x17 x5

ndash18748 ndash26364

ndash12427 ndash37315

x25 x12 x1 x5 x3 x2

x23 x21 x2 x2 x7 x16

ndash

lowast2 lowast2

x3x29 x17 x5

38864 22829

ndash12427 ndash37315ndash18748 ndash26364

(c)

Gene 1

Gene 2

ndash41868

1030168

Head Tail

lowast2 x21 x30

lowast2 x19 x28

ndash016 ndash0288

ndash0398 0734

x12 x12 x1 x20 x22 x17

x1 x30 x6 x1 x3 x2

ndash

lowast2 lowast2

x21 x30 x19 x28

ndash41868 1030168

ndash016 ndash0288 ndash0398 0734

(d)

Figure 8 Continued

Computational Intelligence and Neuroscience 9

Table 4 Optimal S-system models of five stock datasets for a-month-ahead prediction

Type of datasets Optimal S-system modelDJI _f 26495x04093

10 x1243318 minus 50785xminus18385

14 xminus145254

HSI _f 24693x222679 xminus36381

26 minus 54397xminus1024989

NASI _f 38864x187483 xminus26364

29 minus 22829xminus1242717 xminus37315

5SSEI _f minus41868xminus016

21 x028830 minus 1030168xminus0398

19 x073428

SZSEI _f 10205x0621423 x36797

27 minus 88024xminus2357914 xminus4797

16

Gene 1

Gene 2

10205

88024

Head Tail

lowast2 x23 x27

lowast2 x14 x16

06214 36797

ndash23579 ndash4797

x1 x8 x15 x23 x19 x1

x4 x12 x23 x20 x4 x4

ndash

lowast2 lowast2

x23 x27 x14 x16

10205 88024

06214 36797 ndash23579 ndash4797

(e)

Figure 8 )e optimal phenotypes and expression trees for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 9 Continued

10 Computational Intelligence and Neuroscience

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

Table 4 Optimal S-system models of five stock datasets for a-month-ahead prediction

Type of datasets Optimal S-system modelDJI _f 26495x04093

10 x1243318 minus 50785xminus18385

14 xminus145254

HSI _f 24693x222679 xminus36381

26 minus 54397xminus1024989

NASI _f 38864x187483 xminus26364

29 minus 22829xminus1242717 xminus37315

5SSEI _f minus41868xminus016

21 x028830 minus 1030168xminus0398

19 x073428

SZSEI _f 10205x0621423 x36797

27 minus 88024xminus2357914 xminus4797

16

Gene 1

Gene 2

10205

88024

Head Tail

lowast2 x23 x27

lowast2 x14 x16

06214 36797

ndash23579 ndash4797

x1 x8 x15 x23 x19 x1

x4 x12 x23 x20 x4 x4

ndash

lowast2 lowast2

x23 x27 x14 x16

10205 88024

06214 36797 ndash23579 ndash4797

(e)

Figure 8 )e optimal phenotypes and expression trees for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c)SSEI (d) and SZSEI (e)

0 500 1000 1500 200002

04

06

08

1

Days

Nor

mal

ized

stoc

k pr

ice

Actual valuePredicted value

(a)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200004

06

08

1

Days

Actual valuePredicted value

(b)

Nor

mal

ized

stoc

k pr

ice

0 500 1000 1500 200002

04

06

08

1

12

Days

Actual valuePredicted value

(c)

Nor

mal

ized

stoc

k pr

ice

0 200 400 600 800 1000 1200 1400 160002

04

06

08

1

Days

Actual valuePredicted value

(d)

Figure 9 Continued

10 Computational Intelligence and Neuroscience

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

Six prediction models are used to forecast five stockindexes and the prediction results are listed in Table 5 FromTable 5 it can be seen that the five indicators (RMSE ARVMAPE R2 and VAF) of our method are all the best of thesesix methods with the three datasets (DJI HIS and NASI))e DRNN model has the highest MAP which are 2136829568 and 63901 respectively For SSEI and SZSEI data-sets our proposedmethod has the best performance in termsof RMSE MAP ARV MAPE R2 and VAF In terms of

ARV our method is closer to 0 than other five methods Interms of R2 our method is closer to 1 In terms of VAF ourmethod is closer to 100 )us our proposed forecastingmodel tends to be more accurate

43 Hybrid Intelligent Algorithm Analysis In order to testthe performance of our proposed hybrid intelligent algo-rithm we use BSO and PSO to optimize the parameters of

Nor

mal

ized

stoc

k pr

ice

0 100 200 300 400 500 6000

02

04

06

08

1

Days

Actual valuePredicted value

(e)

Figure 9 )e prediction and actual results for a-month-ahead prediction with five stock indexes DJI (a) HIS (b) NASI (c) SSEI (d) andSZSEI (e)

Table 5 Comparison results of six methods for a-month-ahead prediction

Stock index Method RMSE MAP ARV MAPE R2 VAF ()

DJI

Our method 0005413 69911 0001139 073002 099886 99992DRNN 0007741 21368 0002616 14501 099738 99983FNT 0012504 24418 0007481 19062 099252 99956

RBFNN 0013379 39361 0007573 25368 099243 99950BPNN 0048029 101 013547 73864 086453 99356ARIMA 0052385 93126 011662 73731 088338 99234

HSI

Our method 0008645 695 0010944 096689 098906 99984DRNN 0011502 29568 0027542 1489 097246 99972FNT 0014388 43212 0046465 17348 095353 99957

RBFNN 0044134 35289 041237 51249 058763 99592BPNN 0045971 55247 022748 53369 077252 99557ARIMA 0061245 53144 054399 70813 045601 99214

NASI

Our method 00057 79168 837E-04 083204 099916 99990DRNN 0031166 63901 0031964 53044 096804 99706FNT 0031894 93407 0037088 37881 096291 99692

RBFNN 0035863 11232 004911 39605 095089 99610BPNN 0047487 99709 0083254 75185 091675 99317ARIMA 0098081 9102 03589 12177 06411 97086

SSEI

Our method 0003073 1669 814E-04 061418 099919 99995DRNN 0008104 9957 0005335 11292 099467 99962FNT 0033005 34807 0070553 55303 092945 99372

RBFNN 004973 76098 012737 84546 087263 98574BPNN 0053661 11148 014219 94878 085781 98340ARIMA 0071626 87753 02008 13551 07992 97043

SZSEI

Our method 0017003 16007 0010063 24916 098994 99852DRNN 0045067 19483 013092 73945 088439 98959FNT 006323 25729 02267 11628 07733 97950

RBFNN 0071342 35295 032104 11772 067896 97390BPNN 0082818 3758 051956 16103 048044 96483ARIMA 0084487 1171 032894 12521 067106 96340

Computational Intelligence and Neuroscience 11

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

S-system models in the comparison experiments )rough20 runs with DJI dataset the a-week-ahead predictionresults by three evolutionary methods are listed in Table 6which contains the best value worse value mean value andstandard error (SD) of the mean of 20-run RMSEs FromTable 6 we can see that through 20 runs the best RMSEvalues by three evolutionary methods are very close but theother three indicators seem to have a big difference Ourhybrid intelligent algorithm could obtain smaller worseRMSE mean RMSE and SD than PSO and BSO whichindicates that our hybrid intelligent algorithm is morerobust and not easier to fall into local optimum than PSOand BSO

Figure 10 depicts the comparison of the RMSE con-vergence rate obtained from the application of our hybridintelligent algorithm BSO and PSO with DJI dataset fora-week-ahead prediction Figure 10 reveals that our pro-posed intelligent algorithm has faster convergence than PSOand BSO in the early stage of the optimization processWhen the number of iterations reaches 200 the RMSEconvergence rate is dropping to 10minus3 that indicates thesignificant minimization of error

44 Restricted Gene Expression Programming Analysis Inorder to test the performance of restricted gene expressionprogramming for S-system optimization the restrictedadditive tree is used to optimize the structure of theS-system model in the comparison experiments )rough20 runs with five stock indexes the a-week-ahead

prediction results by RGEP and RAT are depicted inFigure 11 which contains the best values worse valuesand mean values of 20-run RMSEs From Figure 11 itcould be clearly seen that RGEP could obtain smaller bestworse and mean RMSE values than RAT which reveal thatRGEP could search the optimal S-system model moreeasily than RAT

5 Conclusions

In this paper a novel stock prediction method based on theS-system model is proposed to forecast the stock marketAn improved gene expression programming (RGEP) isproposed to represent and optimize the structure of theS-system model A hybrid intelligent algorithm based onBSO and PSO is used to optimize the parameters of theS-system model Our proposed method is tested by pre-dicting five real stock price datasets such as DJI HISNASI SSEI and SZSEI )e results of predicting the stockprice a-week-ahead and a-month-ahead reveal that ourmethod could predict the stock index accurately andperforms better than DRNN FNT RBFNN BPNN andARIMA

)e convincing performance of our method is mainlydue to three aspects )e first is that the nonlinear ordinarydifferential equation model S-system has strong nonlinearmodeling and forecasting ability Table 6 and Figure 10 showthat our hybrid intelligent algorithm is more robust and noteasier to fall into local optimum than PSO and BSO From

005

115

225

335

445

1 50 99 148 197 246 295 344 393 442 491

RMSE

Number of iterations

PSOBSOOur hybrid intelligent algorithm

Figure 10 Comparison of error convergence characteristics of our hybrid intelligent algorithm BSO and PSO for a-week-ahead predictionusing DJI dataset

Table 6 )e averaged RMSE results of three evolutionary methods for a-week-ahead prediction

Method Best Worse Mean SDOur hybrid intelligent algorithm 0005411 00071 00061 000065BSO 0005608 00085 00072 000074PSO 0005475 00098 00079 000081

12 Computational Intelligence and Neuroscience

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

Tables 2 and 4 we can see that the optimal S-system modelscontain a portion of input variables )is is because ourmethod can automatically select the proper input variablesaccording to different stock data which also preventsoverfitting problem

Data Availability

)e five stock indexes could be downloaded freely at httpshkfinanceyahoocom

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)is work was supported by the Natural Science Foundationof China (no 61702445) Shandong Provincial NaturalScience Foundation China (no ZR2015PF007) the PhDresearch startup foundation of Zaozhuang University (no2014BS13) and Zaozhuang University Foundation (no2015YY02)

References

[1] S S Roy D Mittal A Basu and A Abraham ldquoStock marketforecasting using LASSO linear regression modelrdquo inAdvancesin Intelligent Systems and Computing vol 334pp 371-381 Springer Berlin Germany 2015

[2] N-F Chen R Roll and S A Ross ldquoEconomic forces and thestock marketrdquo Journal of Business vol 59 no 3 pp 383ndash4031986

[3] J Engelberg and C A Parsons ldquoWorrying about the stockmarket evidence from hospital admissionsrdquo Journal of Fi-nance vol 71 no 3 pp 1227ndash1250 2016

[4] B Majhi and C M Anish ldquoMultiobjective optimization basedadaptive models with fuzzy decision making for stock marketforecastingrdquo Neurocomputing vol 167 pp 502ndash511 2015

[5] S Bekiros R Gupta and C Kyei ldquoOn economic uncertaintystock market predictability and nonlinear spillover effectsrdquoNorth American Journal of Economics and Finance vol 36pp 184ndash191 2016

[6] D Onkal and G Muradoglu ldquoEffects of feedback on prob-abilistic forecasts of stock pricesrdquo International Journal ofForecasting vol 11 no 2 pp 307ndash319 1995

[7] T A Marsh and R C Merton ldquoDividend variability andvariance bounds tests for the rationality of stock marketpricesrdquo American Economic Review vol 76 no 3 pp 483ndash498 1986

[8] A A Ariyo A O Adewumi and C K Ayo ldquoStock priceprediction using the ARIMA modelrdquo in Proceedings of 16thInternational Conference on Computer Modelling and Simu-lation pp 106ndash112 East Lansing MI USA September 2015

[9] O O Mathew A F Sola B H Oladiran and A A AmosldquoPrediction of stock price using autoregressive integratedmoving average filter ((ARIMA (pdq)))rdquo Global Journal ofScience Frontier Research vol 13 no 8 pp 79ndash88 2013

[10] B Uma Devi D Sundar and P Alli ldquoAn effective time seriesanalysis for stock trend prediction using ARIMA model fornifty midcap-50rdquo International Journal of Data Mining ampKnowledge Management Process vol 3 no 1 pp 65ndash78 2013

[11] A A Adebiyi A O Adewumi and C K Ayo ldquoComparisonof ARIMA and artificial neural networks models for stockprice predictionrdquo Journal of Applied Mathematics vol 2014Article ID 614342 7 pages 2014

[12] D J Armaghani E T Mohamad M S NarayanasamyN Narita and S Yagiz ldquoDevelopment of hybrid intelligentmodels for predicting TBM penetration rate in hard rockconditionrdquo Tunnelling and Underground Space Technologyvol 63 pp 29ndash43 2017

[13] B Y Bejarbaneh E Y Bejarbaneh M F M AminA Fahimifar D Jahed Armaghani and M Z A MajidldquoIntelligent modelling of sandstone deformation behaviour

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002

RMSE

DJI HSI NASI SSEI SZSEI

RATRGEP

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Best

Wor

se

Mea

n

Figure 11 Prediction comparison of two optimization algorithms for a-week-ahead prediction with five stock indexes

Computational Intelligence and Neuroscience 13

using fuzzy logic and neural network systemsrdquo Bulletin ofEngineering Geology and the Environment vol 77 no 1pp 345ndash361 2016

[14] G Dong K Fataliyev and L Wang ldquoOne-step and multi-stepahead stock prediction using back propagation neural net-worksrdquo in Proceedings of 9th International Conference onInformation Communications amp Signal Processing pp 1ndash5Bangkok )ailand October 2014

[15] S Jabin ldquoStockmarket prediction using feed-forward artificialneural networkrdquo International Journal of Computer Appli-cations vol 99 no 9 pp 4ndash8 2014

[16] R Akita A Yoshihara T Matsubara and K Uehara ldquoDeeplearning for stock prediction using numerical and textualinformationrdquo in Proceedings of 15th International Conferenceon Computer and Information Science (ICIS) pp 1ndash6Okayama Japan June 2016

[17] M Rout B Majhi U M Mohapatra and R MahapatraldquoStock indices prediction using radial basis function neuralnetworkrdquo in Proceedings of Swarm Evolutionary andMemetic Computing vol 7677 pp 285ndash293 BhubaneswarIndia December 2012

[18] H Wang B Yang and J Lv ldquoComplex-valued neural net-work model and its application to stock predictionrdquo inProceedings of 16th International Conference on Hybrid In-telligent Systems pp 21ndash28 Marrakech Morocco November2016

[19] Y Chen B Yang and A Abraham ldquoFlexible neural treesensemble for stock index modelingrdquoNeurocomputing vol 70no 4ndash6 pp 697ndash703 2007

[20] J Zuo C-j Tang C Li C-a Yuan and A-l Chen ldquoTimeseries prediction based on gene expression programmingrdquo inProceedings of Advances in Web-Age Information Manage-ment vol 3129 pp 55ndash64 Dalian China July 2004

[21] M Graff H J Escalante F Ornelas-Tellez and E S TellezldquoTime series forecasting with genetic programmingrdquo NaturalComputing vol 16 no 1 pp 165ndash174 2016

[22] M Grigioni U Morbiducci G DrsquoAvenio G D Benedettoand C D Gaudio ldquoA novel formulation for blood traumaprediction by a modified power-law mathematical modelrdquoBiomechanics and Modeling in Mechanobiology vol 4 no 4pp 249ndash260 2005

[23] MMina A Borzabadi-Farahani A Tehranchi M Nouri andF Younessian ldquoMathematical beta function formulation formaxillary arch form prediction in normal occlusion pop-ulationrdquo Odontology vol 105 no 2 pp 229ndash236 2016

[24] Y Chen B Yang Q Meng Y Zhao and A Abraham ldquoTime-series forecasting using a system of ordinary differentialequationsrdquo Information Sciences vol 181 no 1 pp 106ndash1142011

[25] W Zhang and B Yang ldquoStock market forecasting usingS-system modelrdquo in Proceedings of International Conferenceon Intelligent amp Interactive Systems amp Applications (IISA)pp 397ndash403 Larnaca Cyprus August 2017

[26] J-C Hua F Noorian D Moss P H W Leong andG H Gunaratne ldquoHigh-dimensional time series predictionusing kernel-based Koopman mode regressionrdquo NonlinearDynamics vol 90 no 3 pp 1785ndash1806 2017

[27] Y Chen and Y Hao ldquoA feature weighted support vectormachine and K-nearest neighbor algorithm for stock marketindices predictionrdquo Expert Systems with Applications vol 80pp 340ndash355 2017

[28] M Junhai and L Lixia ldquoMultivariate nonlinear analysis andprediction of Shanghai stock marketrdquo Discrete Dynamics in

Nature and Society vol 2008 Article ID 526734 9 pages2008

[29] J Z Wang J J Wang Z G Zhang and S P Guo ldquoFore-casting stock indices with back propagation neural networkrdquoExpert Systems with Applications vol 38 no 11 pp 14346ndash14355 2011

[30] S C Nayak B B Misra and H S Behera ldquoACFLN artificialchemical functional link network for prediction of stockmarket indexrdquo Evolving Systems vol 4 pp 1ndash26 2018

[31] N Noman and H Iba ldquoInference of genetic networks usingS-system information criteria for model selectionrdquo in Pro-ceedings of 8th Annual Conference on Genetic and EvolutionaryComputation vol 1 pp 263ndash270 Seattle WA USA July 2006

[32] M Iwata K Sriyudthsak M Y Hirai and F ShiraishildquoEstimation of kinetic parameters in an S-system equationmodel for a metabolic reaction system using the Newton-Raphson methodrdquo Mathematical Biosciences vol 248 no 3pp 11ndash21 2014

[33] Y Shi ldquoBrain storm optimization algorithmrdquo in Proceedingsof Second International Conference Advances in Swarm In-telligence vol 6728 no 3 pp 1ndash14 Chongqing China June2011

[34] S Cheng Q Qin J Chen and Y Shi ldquoBrain storm opti-mization algorithm a reviewrdquo Artificial Intelligence Reviewvol 46 no 4 pp 445ndash458 2016

[35] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of IEEE International Conference on NeuralNetworks vol 4 no 8 pp 1942ndash1948 Perth AustraliaNovember 1995

[36] M Hajihassani D Jahed Armaghani and R KalatehjarildquoApplications of particle swarm optimization in geotechnicalengineering a comprehensive reviewrdquo Geotechnical andGeological Engineering vol 36 no 2 pp 705ndash722 2017

[37] B Yang S Liu and W Zhang ldquoReverse engineering of generegulatory network using restricted gene expression pro-grammingrdquo Journal of Bioinformatics and ComputationalBiology vol 14 no 5 article 1650021 2016

[38] J C Butcher and K R Jackson ldquo)e numerical analysis ofordinary differential equations RungendashKutta and generallinear methodsrdquo Mathematics of Computation vol 51no 183 p 693 1987

[39] D J Armaghani M F M Amin S Yagiz R S Faradonbehand R A Abdullah ldquoPrediction of the uniaxial compressivestrength of sandstone using various modeling techniquesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 85 pp 174ndash186 2016

[40] M Hermans and B Schrauwen ldquoTraining and analysing deeprecurrent neural networksrdquo in Proceedings of Advances inNeural Information Processing Systems pp 190ndash198 LakeTahoe NV USA December 2013

14 Computational Intelligence and Neuroscience

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom