research article the interval slope method for long-term forecasting...
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Research ArticleThe Interval Slope Method for Long-Term Forecasting of StockPrice Trends
Chun-xue Nie and Xue-bo Jin
School of Computer and Information Engineering Beijing Technology and Business University Beijing 100048 China
Correspondence should be addressed to Xue-bo Jin jinxuebobtbueducn
Received 3 November 2015 Accepted 3 February 2016
Academic Editor Doojin Ryu
Copyright copy 2016 C-x Nie and X-b Jin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A stock price is a typical but complex type of time series data We used the effective prediction of long-term time series data toschedule an investment strategy and obtain higher profit Due to economic environmental and other factors it is very difficult toobtain a precise long-term stock price predictionThe exponentially segmented pattern (ESP) is introduced here and used to predictthe fluctuation of different stock data over five future prediction intervals The new feature of stock pricing during the subintervalnamed the interval slope can characterize fluctuations in stock price over specific periods The cumulative distribution function(CDF) of MSE was compared to those of MMSE-BC and SVR We concluded that the interval slope developed here can capturemore complex dynamics of stock price trends The mean stock price can then be predicted over specific time intervals relativelyaccurately in which multiple mean values over time intervals are used to express the time series in the long term In this way theprediction of long-term stock price can be more precise and prevent the development of cumulative errors
1 Introduction
Stock price is a typical and complex type of time seriesdata The prediction of stock prices has been an activearea of research in econometrics signal processing patternrecognition and machine learning for some time Stocktraders and investors are extremely interested in stockmarketprediction because of the considerable profits that can bereaped by trading stocks Traditionally the basic method-ology for financial time series has been statistical methodssuch as autoregressive and moving average model (ARMA)autoregressive integrated moving average model (ARIMA)and generalized autoregressive conditional heteroskedasticity(GARCH) which require the linear variation in the stockprices to remain stationary In general the statistical modelscannot adapt to changes in the process Accordingly tradi-tional statistical methods cannot predict stock performancevery well when tracking the complexity of the stock markets[1]
Recently manymachine learning systems have been usedto predict stock prices These include artificial neural net(ANN) [2 3] Bayes networks [4] genetic programming [5]
support vector regression (SVR) [6] user analysis [7] sen-timent analysis [8] and hybrid networks [9ndash15] Accord-ingly machine learning methods can be used to track thecomplexity and nonstationary nature of the stock marketsin short-term prediction These methods predict long-termtrends only with great difficulty Existing methods of long-term stock predictionmainly include the following use of therecursive iteration prediction to obtain the long term predic-tion trend [12] however this method involves accumulativeerror and the cumulative error increases with the number ofsteps in the prediction process By using a moving windowalgorithm to delete older data and take in new data theprediction model can be updated in sequence [6] The lengthof the moving window also has considerable influence on theaccuracy of the modeling processThis system can only showthe mean stock price during the prediction interval and can-not show the details of changes in the stock trend during thisinterval In addition the present methods use the mean valuedirectly as a feature of the stock trend prediction Regardingthe fluctuations in larger time series the mean values of theinterval weaken the fluctuation characteristics of the timeseries and reduce the long-term accuracy of the forecast
Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 8045656 7 pageshttpdxdoiorg10115520168045656
2 Advances in Mathematical Physics
For these reasons the prediction of stock price is still aworthwhile issue
Long-term time series forecasts have other applicationssuch as the host load prediction Load prediction is crucialto efficient resource utilization in dynamic cloud computingenvironments Di et al used the exponentially segmentedpattern (ESP) [16] to predict the host load in the cloud Theyproposed the use of 9 different features to characterize therecent load fluctuation in the evidence subintervalTheywereable to predict the mean load over consecutive time intervals
In this paper the exponentially segmented pattern (ESP)is used to predict the fluctuation of stock price over con-secutive future time intervals While we give a new featureof stock price in the subinterval namely interval slope tocharacterize the stock price fluctuation over a set periodTheinterval slope can be used to determine the mean of stock inthe subinterval The support vector regression and the Bayesclassifier were used to predict the stock price trend and verifythe effectiveness of the interval slope of the stock price in thesubinterval
In this paper the following contributions are made
(i) The exponentially segmented pattern (ESP) is hereused to predict fluctuations in different stock dataover a long period and can accurately predict not onlymean stock price over a future time interval but alsothe mean stock price over consecutive future timeintervals In this way the prediction of long-termstock price can be more precise and the generation ofcumulative errors can be prevented
(ii) The use of new features of stock pricing in thesubinterval namely interval slope is here proposedto better characterize the stock price fluctuation oversome time period
The rest of the paper is organized as follows In Section 2the long-term stock price prediction model is introduced InSection 3 experiments and comparisons of different modelsare made Conclusions are given in Section 4
2 Model-Based Prediction of Long-TermTrends in Stock Price
The predictive objective is to predict the fluctuation ofopening price over a long period Multiple precise meanvalues over time interval are used to express the time serieslong-term trend
The proposed stock price trend predictionmodel involvesthe following three steps first using the ESP principle theestimated data segment is split into a set of consecutivesegments whose lengths increase exponentially The intervalslope is used to describe the features of each interval Thenmachine learning methods SVR and MMSE-BC were usedto produce the transformation model of the data which isused to predict the mean stock price for the next intervalMultiple precise mean values over time interval are usedto express the long-term trends in the time series In thisway the prediction of stock price in the long term can beperformed precisely without generating cumulative errors
History
Predict exponentiallysegmented pattern
PredictCurrent moment
2 2 4 8 16
Stoc
k pr
ice
Time axis
l1l2
l3
l4
l5
1205781
1205782
12057831205784
1205785
t0 t1 t2 t3 t4 t5
li = 2120578i minus 120578iminus1
Figure 1 Illustration of ESP and the relation of l and 120578
21 Exponentially Segmented Pattern (ESP) and Transforma-tion of Segments The objective of the current work is topredict trends in the patterns of fluctuation of stock price overthe consecutive future time intervals The most importantstep of the proposed stock price trend pattern prediction isthat the estimated data segment is split into a set of con-secutive segments by ESP principle whose lengths increaseexponentially An example of ESP is shown in Figure 1 Ata current time point 119905
0 the estimated data segment is split
into a set of consecutive segments whose lengths increasedexponentially The length of each following segment was 2119894where 119894 = 1 2 3 4 For each segment over the consecutivefuture time intervals the mean values were denoted by 119897
119894
where 119894 = 1 2 3 4 However the mean stock price over the consecutive time
intervals 119897119894is hard to predict and the mean stock price over
a single future time interval is easy to predict The length ofeach following segment is 119905
0+ 2119894 where 119894 = 1 2 3 4 The
mean predicted stock price of each time segment is given hereas 120578119894 where 119894 = 1 2 3 4 A set of the mean stock prices for a single time interval is
then availableThe aim of the current work was to predict themean stock price over the consecutive future time intervals(119897) In fact the vector 119897 can be converted from the vector 120578through the following induction according to a previouswork[16]
Suppose that the current moment is 1199050 and the user has
already predicted two mean stock prices (shown by 120578119894minus1
and120578119894 the solid red line segment) over two different intervals
([1199050 119905119894minus1] and [119905
0 119905119894]) Then for the two shaded areas which
are of equal size the mean stock price in [119905119894minus1 119905119894] can be
derived The transformation is given in
119897119894= 120578119894minus119905119894minus1
minus 1199050
119905119894minus 119905119894minus1
(120578119894minus 120578119894minus1) (1)
Here 119897119894is the predicted mean stock price in the new segment
[119905119894minus1 119905119894] corresponding to the black line segment in Figure 1
Taking into account 119905119894= 2119905119894minus1
and 1199050= 0 (1) can be
simplified further producing
119897119894= 2120578119894minus 120578119894minus1 (2)
Advances in Mathematical Physics 3St
ock
pric
e
Time axis
History Predict
Mean
Last price
Mean
Mean
Mean
1000 data samples every window has 4 data samples
tan120572 = Ki =meani minus lastpriceiminus1
d
120572
d = 4
Figure 2 Illustration of features for stock price fluctuation (119889119895= 4
days)
22 Features of Fluctuations in Stock Price The aim of thecurrent work is to predict the mean stock price over a futuretime interval (120578) starting at the current time 119905
0 Here every
future time interval is called a subinterval Based on thefeatures of stock price trend a new feature of the stock pricein the subinterval was proposed here This feature is herecalled the interval slope and it can be used to characterizethe fluctuations in stock price fluctuation over a specific timeperiod The time series is denoted in the subinterval as e =(1198901 1198902 119890
119894)119879 where 119894 = 1 2 3 4 and 119890
119894is the sample
stock price in the subinterval For example the subinterval is4 days (ie 119894 = 4) whose features for stock price fluctuationare shown in Figure 2
IntervalMean Pricemeanprice119895is themean stock price when
the conditions of the subinterval equal 119889119895
meanprice119895=1
119889119895
119889119895
sum
119894=1
119890119894 (3)
Interval Last PriceThe last price is themost recent price valuein the subinterval
Interval Slope (119870) 119870119895interval slope is the slope of linear
equation in the condition of the subinterval equals 119889119895
First the last price and the mean price of the subintervalwere computedThen the linear equation119910 = 119896119909+119887was usedto fit the samples in the subinterval that is
meanprice119895= 119870119895119889119895+ lastprice
119895minus1 (4)
This produces the following
119870119895=
meanprice119895minus lastprice
119895minus1
119889119895
(5)
Interval slope can be transformed into mean by (5)
For example the value of 119889119895can be set as follows 119889
119895=
2 4 8 16 32 The feature of stock price fluctuation based on119889119895= 4 history data and predicted data are shown in Figure 2Next future data features were predicted (the interval
mean and the interval slope) through learning history datafeatures The prediction methods are presented in the nextsection
23 Long-Term Forecasting Based on Interval Slope To verifythe effectiveness of the developed feature and assess theinterval slope of stock price in the subinterval the supportvector regression and the Bayes classifier were used for long-term forecasting Two machine learning methods SVR andMMSE-BC are used to produce the transform model of thedata and the mean stock price is used to predict the nextinterval slope
(1) Support Vector Regression (SVR) The aim of SVR algo-rithm was to minimize 120576-sensitive errors on the subset ofdata here called the support vectors SVR algorithm usesnonlinear kernel functions in order to project initial data toa higher dimensional space and project linear classifiers fromthe higher space to the original spaceThe formulation of SVRis represented as follows
min 121199082+ 119862
1
119897
119897
sum
119894=1
119871120576
119871120576=
1003816100381610038161003816119910119894minus 119908 times 120595 (119909
119894minus 119887)
1003816100381610038161003816minus 120576
1003816100381610038161003816119910119894minus 119908 times 120595 (119909
119894minus 119887)
1003816100381610038161003816ge 120576
0 otherwise
(6)
Here 119908 is a weight vector which is used to determine themaximum margin hyper plane the term 119908 is called aregularized term and it should be as flat as possible Thesecond term is the empirical error as measured by Vapnikrsquos120576-insensitive loss function 119862 is the regularization constant
The following commonly used kernel functions areincluded
Linear 119896(1199091 1199092) = ⟨119909
1 1199092⟩
Polynomial 119896(1199091 1199092) = (⟨119909
1 1199092⟩ + 119877)
119889Sigmoid 119896(119909
1 1199092) = (tanh(⟨119909
1 1199092⟩ + 119903))
Radial basis function 119896(1199091 1199092) = exp(minus119909
1minus
1199092221205902)
(2) Bayes Classifier The MMSE-BC has been considered thebest strategy that uses Bayes method with the single featuremean load based on the evaluation type A in a previouswork [16] The MMSE-BC used here was the minimizedMSE (MMSE) based Bayes classifier It is a classic supervisedlearning classifier used in data mining The formulation ofMMSE-BC is represented in
119875 (120596119894| 120594119895) =
119875 (120594119895| 120596119894) 119875 (120596
119894)
sum119898
119896=1119875 (120594119895| 120596119896) 119875 (120596
119896)
(7)
119894= 119864 (120596
119894| 120594119895) =
119898
sum
119894=1
120596119894119875 (120594119895| 120596119894) (8)
4 Advances in Mathematical Physics
Input stock dataset interval duration of predictionOutput CDF of the prediction MSE on different dataset and different methods(1) Split dataset into training dataset and testing dataset(2) for (newdataset = dataset[ 119899])lowast 119899 is data number increasing by 40 lowastdo(3) for (interval = 2 4 8 16 32) do(4) Determine the feature of the mean and interval slope in every interval(5) Predict the mean price 120578
119894 using SVR or MMSE-BC method in training dataset
lowast Use the mean and interval slope as feature of the stock price trend lowast(6) end for(7) Segment transformation based on (2) 120578rarr 119897
(8) Calculate the MSE of this dataset(9) end for(10) Statistic 80 MSE of different dataset and plot the cumulative distribution function (CDF) of MSE
Algorithm 1 Stock price trend prediction model
It is important for the Bayes classifier to compute the priorprobability distribution 119875(120596
119894) for the target states based on
the samples and compute the joint probability distribution119875(120594119895| 120596119894) for each state 120596
119894 Then the posterior probability
119875(120596119894| 120594119895) was computed according to Formula (7)
24 Trend Prediction Model The following trend predictionmodel is proposed here as a way of preventing the gener-ation of cumulative errors The proposed stock price trendpredictionmodel has the following three steps first using theESP principle the estimated data segment is split into a set ofconsecutive segments whose lengths increase exponentiallyThe interval slope is used to describe the features of eachinterval Then the machine learning methods SVR andMMSE-BC were used to produce the transformmodel of thedata and by which the mean stock price was predicted in theprediction of the next interval
First the stock opening price data were selected Secondthe time series (stock opening price data) was split into a setof a future time interval segments (120578) whose lengths increaseexponentiallyThe length of following subinterval was 119905
0+2119894
where 119894 = 1 2 3 4 Third the mean and interval slopewere computed for every subinterval and the feature datasetwas split into training dataset and prediction dataset NextMMSE-BC and SVR were trained in order to produce themodel parameters For example it can compute the priorprobability 119875(120596
119894) and the conditional probability (119875(120594
119895| 120596119894)
in (8)) and produce a boundary that leads to the largestmargin from both sets of points in SVR and predict the meanstock price and the interval slope in prediction interval overa single interval The interval slope must transform into themean of the interval based on (5) because the mean valuesover consecutive future time intervals are used to express thelong-term trends in the time series Then the mean valuesover consecutive future time intervals l can be converted fromthe vector 120578 based on (2) At last the mean squared error ofthis dataset can be calculated
In order to evaluate the performance of MMSE-BC andSVR the entire dataset prediction mean squared error wascomputed For example the price over the first 1000 tradingdays was selected for training and the price over the next 32
1000 32Train Predict
Time axis
32Predict
32Predict
1040Train
1080Train
1120Train
4200Train
Time axis
Time axis
Time axis
32Predict
32Predict
Time axis
Figure 3 Setting the time window
days was selected for prediction The entire process followsthe procedure of the trend prediction model mentionedThen the first 1040 trading daysrsquo price can be learned and thenext 32 daysrsquo price can be predicted Next each process is tofind the mean squared error of the prediction process Theprocess with higher prices prediction performance continuesto predict the future stock price The method of setting thetime window is shown in Figure 3
Algorithm 1 gives the pseudocode of the stock price trendprediction model
3 Experiments and Comparison
This section presents experiments of the trend predictionmodel on the stock open price forecasting The trend predic-tionmodel was here shown to be able to capture the dynamicsof highly nonlinear nonstationary time series
31 Evaluation Indicator To evaluate the accuracy of thesepredictions the overall mean squared error (MSE) between
Advances in Mathematical Physics 5
Table 1 Optimized parameters for the method
Method Key parameters Valuescompute method
SVRKernel
Penalty parameter 119862 of the error termSlack variables 119871
120576
Sigmoidradial basis function100001
CDF
CDF
CDF
CDF
IBM Coca Cola
Microsoft Amazon
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
01
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
020 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 4 CDF of MSE for prediction of trends in stock price
the predicted stock price values and the true values in theprediction interval can be calculated as follows
mse (119904) = 1
119904
119899
sum
119894=1
119904119894(119897119894minus 119871119894)2 (9)
Here 119904119894= 2119894minus1 119894 ge 1 119904 = sum
119899
119894=1119904119894 119897119894is the predicted mean of
testing dataset 119871119894is the true mean of testing dataset and 119899 is
the total number of the segments in the prediction interval
32 Method of Training and Evaluation Eight openingstock price data samples were selected at random for theseexperiments IBM Coca Cola Microsoft Amazon Sony
Kimberly-Clark Bank of America andWalgreens in 199911ndash20141030
SVR and MMSE-BC were here used to predict the trendsin opening stock price and some key parameters are listed inTable 1
33 Experimental Results The results of MMSE-BC and SVRwere compared to the classic mean and the interval slopeEight stock opening price data samples IBM Coca ColaMicrosoft Amazon Sony Kimberly-Clark Bank of AmericaandWalgreens in 199911ndash20141030 were compared toMSEFigures 4 and 5 show the cumulative distribution function(CDF) of MSE of different prediction methods in whichSVRrsquos kernel is sigmoid
6 Advances in Mathematical Physics
Sony Kimberly-Clark
Bank of America Walgreen
CDF
CDF
CDF
CDF
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
01
0 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 5 CDF of MSE for prediction of trends in stock price
As shown in Figures 4 and 5 the interval slope curve isabove the interval mean curveThis indicates that the intervalslopersquos cumulative probability is greater than that of theinterval mean when the value of MSE is below a certainthreshold For example the IBM interval slopersquos cumulativeprobability was larger than the interval mean curve when theMSE value was less than 100Thatmeans that 88 of theMSEvalues using interval slope were below 100 and only 52 ofthe MSE values using interval mean were below 100
It is clear that interval slopersquos performancewas better thanthat of the mean as indicated by the MMSE-BC and SVRmethods In this way the interval slope can indicate morecomplex dynamics such as change trends In contrast themean can smooth out the dynamic fluctuations in stock price
As an example of prediction results Figure 6 shows IBMstock price trend prediction that is 119905 = 3440ndash3972 by SVRbased on interval slope in which SVRrsquos kernel is a radial basisfunction
Both the mean stock price over a future time interval andthe mean stock price over consecutive future time intervals
can be predicted accurately This shows that the predictionof long-term stock price can be performed precisely withoutgenerating cumulative errors The mean stock price overconsecutive future time intervals can express future trendssuch as sharp falls slight falls concussions slight increasessharp increases falls followed by increases and increases fol-lowed by falls According to the prediction of the fluctuationof opening price over a long-term period fund allocationmodels and trading strategies can be developed in advance
4 Conclusion and Future Work
In this paper ESP which does not generate cumulativeerrors was introduced and used to predict fluctuations in theopening prices of different stocks over a long period The useof a new feature of stock price in the evidence subintervalinterval slope was proposed to characterize the stock pricefluctuation over some time period It can be concluded thatthe interval slope can capture complex dynamics such astrends in the changes in stock price
Advances in Mathematical Physics 7
3500 3600 3700 3800
210
200
190
180
170
160
Stoc
k pr
ice
Time axis
True dataTrue mean stepPredict k SVR step
Figure 6 IBM stock price trend prediction
The premise of this method of trend prediction is thatfuture markets will change gradually rather than abruptlyThe complexities of changes in stock price can greatlyincrease the difficulty of prediction Future work shouldevaluate different learning methods and even combine dif-ferent learning methods Some new methods of evaluationshould be used to evaluate the interval slope the classic meanmethod and the rate of return
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by NSFC under Grantno 61273002 the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions noCITampTCD201304025 and the Key Science and TechnologyProject of BeijingMunicipal EducationCommission ofChinano KZ201510011012
References
[1] R S Tsay Analysis of Financial Time Series JohnWiley amp SonsHoboken NJ USA 2005
[2] G Sermpinis C Dunis J Laws and C Stasinakis ldquoForecastingand trading the EURUSD exchange rate with stochastic NeuralNetwork combination and time-varying leveragerdquo DecisionSupport Systems vol 54 no 1 pp 316ndash329 2012
[3] P-C Chang D-D Wang and C-L Zhou ldquoA novel modelby evolving partially connected neural network for stock pricetrend forecastingrdquo Expert Systems with Applications vol 39 no1 pp 611ndash620 2012
[4] E Kita M Harada and T Mizuno ldquoApplication of BayesianNetwork to stock price predictionrdquo Artificial IntelligenceResearch vol 1 no 2 2012
[5] Z Er-bo M Huan and H Zhan-Gang ldquoApplying geneticprogramming to analyze moving average and long amp mid-term
trends of stock pricesrdquo Application Research of Computers vol27 no 6 2010
[6] P Meesad and R I Rasel ldquoPredicting stock market priceusing support vector regressionrdquo in Proceedings of the 2ndInternational Conference on Informatics Electronics and Vision(ICIEV rsquo13) pp 1ndash6 IEEE Dhaka Bangladesh May 2013
[7] PDondio ldquoStockmarket predictionwithout sentiment analysisusing a web-traffic based classifier and user-level analysisrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 3137ndash3146 IEEE WaileaHawaii USA January 2013
[8] M Hagenau M Hauser M Liebmann and D NeumannldquoReading all the news at the same time predicting mid-term stock price developments based on news momentumrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 1279ndash1288 Wailea HawaiUSA January 2013
[9] D-Y Xu S-L Yang and R-P Liu ldquoA mixture of HMM GAand Elman network for load prediction in cloud-oriented datacentersrdquo Journal of Zhejiang University Science C vol 14 no 11pp 845ndash858 2013
[10] R Bisoi and P K Dash ldquoA hybrid evolutionary dynamic neuralnetwork for stock market trend analysis and prediction usingunscented Kalman filterrdquo Applied Soft Computing Journal vol19 pp 41ndash56 2014
[11] B Bican and Y Yaslan ldquoA hybrid method for time seriesprediction using EMD and SVRrdquo in Proceedings of the 6th Inter-national Symposium on Communications Control and SignalProcessing (ISCCSP rsquo14) pp 566ndash569 Athens GreeceMay 2014
[12] Z Huang and M-L Shyu ldquok-NN based LS-SVM frameworkfor long-term time series predictionrdquo in Proceedings of the IEEEInternational Conference on Information Reuse and Integration(IRI rsquo10) pp 69ndash74 IEEE Las Vegas Nev USA August 2010
[13] J Fan and Y Tang ldquoAn EMD-SVR method for non-stationarytime series predictionrdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance and SafetyEngineering (QR2MSE rsquo13) pp 1765ndash1770 IEEE ChengduChina July 2013
[14] T Fletcher and J Shawe-Taylor ldquoMultiple kernel learningwith fisher kernels for high frequency currency predictionrdquoComputational Economics vol 42 no 2 pp 217ndash240 2013
[15] Q Yang C Peng Y Yu et al ldquoHost load prediction basedon PSR and EA-GMDH for cloud computing systemrdquo inProceedings of the 3rd IEEE International Conference on Cloudand Green Computing (CGC rsquo13) pp 9ndash15 IEEE KarlsruheGermany October 2013
[16] S Di D Kondo and W Cirne ldquoGoogle hostload predictionbased on Bayesian model with optimized feature combinationrdquoJournal of Parallel and Distributed Computing vol 74 no 1 pp1820ndash1832 2014
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2 Advances in Mathematical Physics
For these reasons the prediction of stock price is still aworthwhile issue
Long-term time series forecasts have other applicationssuch as the host load prediction Load prediction is crucialto efficient resource utilization in dynamic cloud computingenvironments Di et al used the exponentially segmentedpattern (ESP) [16] to predict the host load in the cloud Theyproposed the use of 9 different features to characterize therecent load fluctuation in the evidence subintervalTheywereable to predict the mean load over consecutive time intervals
In this paper the exponentially segmented pattern (ESP)is used to predict the fluctuation of stock price over con-secutive future time intervals While we give a new featureof stock price in the subinterval namely interval slope tocharacterize the stock price fluctuation over a set periodTheinterval slope can be used to determine the mean of stock inthe subinterval The support vector regression and the Bayesclassifier were used to predict the stock price trend and verifythe effectiveness of the interval slope of the stock price in thesubinterval
In this paper the following contributions are made
(i) The exponentially segmented pattern (ESP) is hereused to predict fluctuations in different stock dataover a long period and can accurately predict not onlymean stock price over a future time interval but alsothe mean stock price over consecutive future timeintervals In this way the prediction of long-termstock price can be more precise and the generation ofcumulative errors can be prevented
(ii) The use of new features of stock pricing in thesubinterval namely interval slope is here proposedto better characterize the stock price fluctuation oversome time period
The rest of the paper is organized as follows In Section 2the long-term stock price prediction model is introduced InSection 3 experiments and comparisons of different modelsare made Conclusions are given in Section 4
2 Model-Based Prediction of Long-TermTrends in Stock Price
The predictive objective is to predict the fluctuation ofopening price over a long period Multiple precise meanvalues over time interval are used to express the time serieslong-term trend
The proposed stock price trend predictionmodel involvesthe following three steps first using the ESP principle theestimated data segment is split into a set of consecutivesegments whose lengths increase exponentially The intervalslope is used to describe the features of each interval Thenmachine learning methods SVR and MMSE-BC were usedto produce the transformation model of the data which isused to predict the mean stock price for the next intervalMultiple precise mean values over time interval are usedto express the long-term trends in the time series In thisway the prediction of stock price in the long term can beperformed precisely without generating cumulative errors
History
Predict exponentiallysegmented pattern
PredictCurrent moment
2 2 4 8 16
Stoc
k pr
ice
Time axis
l1l2
l3
l4
l5
1205781
1205782
12057831205784
1205785
t0 t1 t2 t3 t4 t5
li = 2120578i minus 120578iminus1
Figure 1 Illustration of ESP and the relation of l and 120578
21 Exponentially Segmented Pattern (ESP) and Transforma-tion of Segments The objective of the current work is topredict trends in the patterns of fluctuation of stock price overthe consecutive future time intervals The most importantstep of the proposed stock price trend pattern prediction isthat the estimated data segment is split into a set of con-secutive segments by ESP principle whose lengths increaseexponentially An example of ESP is shown in Figure 1 Ata current time point 119905
0 the estimated data segment is split
into a set of consecutive segments whose lengths increasedexponentially The length of each following segment was 2119894where 119894 = 1 2 3 4 For each segment over the consecutivefuture time intervals the mean values were denoted by 119897
119894
where 119894 = 1 2 3 4 However the mean stock price over the consecutive time
intervals 119897119894is hard to predict and the mean stock price over
a single future time interval is easy to predict The length ofeach following segment is 119905
0+ 2119894 where 119894 = 1 2 3 4 The
mean predicted stock price of each time segment is given hereas 120578119894 where 119894 = 1 2 3 4 A set of the mean stock prices for a single time interval is
then availableThe aim of the current work was to predict themean stock price over the consecutive future time intervals(119897) In fact the vector 119897 can be converted from the vector 120578through the following induction according to a previouswork[16]
Suppose that the current moment is 1199050 and the user has
already predicted two mean stock prices (shown by 120578119894minus1
and120578119894 the solid red line segment) over two different intervals
([1199050 119905119894minus1] and [119905
0 119905119894]) Then for the two shaded areas which
are of equal size the mean stock price in [119905119894minus1 119905119894] can be
derived The transformation is given in
119897119894= 120578119894minus119905119894minus1
minus 1199050
119905119894minus 119905119894minus1
(120578119894minus 120578119894minus1) (1)
Here 119897119894is the predicted mean stock price in the new segment
[119905119894minus1 119905119894] corresponding to the black line segment in Figure 1
Taking into account 119905119894= 2119905119894minus1
and 1199050= 0 (1) can be
simplified further producing
119897119894= 2120578119894minus 120578119894minus1 (2)
Advances in Mathematical Physics 3St
ock
pric
e
Time axis
History Predict
Mean
Last price
Mean
Mean
Mean
1000 data samples every window has 4 data samples
tan120572 = Ki =meani minus lastpriceiminus1
d
120572
d = 4
Figure 2 Illustration of features for stock price fluctuation (119889119895= 4
days)
22 Features of Fluctuations in Stock Price The aim of thecurrent work is to predict the mean stock price over a futuretime interval (120578) starting at the current time 119905
0 Here every
future time interval is called a subinterval Based on thefeatures of stock price trend a new feature of the stock pricein the subinterval was proposed here This feature is herecalled the interval slope and it can be used to characterizethe fluctuations in stock price fluctuation over a specific timeperiod The time series is denoted in the subinterval as e =(1198901 1198902 119890
119894)119879 where 119894 = 1 2 3 4 and 119890
119894is the sample
stock price in the subinterval For example the subinterval is4 days (ie 119894 = 4) whose features for stock price fluctuationare shown in Figure 2
IntervalMean Pricemeanprice119895is themean stock price when
the conditions of the subinterval equal 119889119895
meanprice119895=1
119889119895
119889119895
sum
119894=1
119890119894 (3)
Interval Last PriceThe last price is themost recent price valuein the subinterval
Interval Slope (119870) 119870119895interval slope is the slope of linear
equation in the condition of the subinterval equals 119889119895
First the last price and the mean price of the subintervalwere computedThen the linear equation119910 = 119896119909+119887was usedto fit the samples in the subinterval that is
meanprice119895= 119870119895119889119895+ lastprice
119895minus1 (4)
This produces the following
119870119895=
meanprice119895minus lastprice
119895minus1
119889119895
(5)
Interval slope can be transformed into mean by (5)
For example the value of 119889119895can be set as follows 119889
119895=
2 4 8 16 32 The feature of stock price fluctuation based on119889119895= 4 history data and predicted data are shown in Figure 2Next future data features were predicted (the interval
mean and the interval slope) through learning history datafeatures The prediction methods are presented in the nextsection
23 Long-Term Forecasting Based on Interval Slope To verifythe effectiveness of the developed feature and assess theinterval slope of stock price in the subinterval the supportvector regression and the Bayes classifier were used for long-term forecasting Two machine learning methods SVR andMMSE-BC are used to produce the transform model of thedata and the mean stock price is used to predict the nextinterval slope
(1) Support Vector Regression (SVR) The aim of SVR algo-rithm was to minimize 120576-sensitive errors on the subset ofdata here called the support vectors SVR algorithm usesnonlinear kernel functions in order to project initial data toa higher dimensional space and project linear classifiers fromthe higher space to the original spaceThe formulation of SVRis represented as follows
min 121199082+ 119862
1
119897
119897
sum
119894=1
119871120576
119871120576=
1003816100381610038161003816119910119894minus 119908 times 120595 (119909
119894minus 119887)
1003816100381610038161003816minus 120576
1003816100381610038161003816119910119894minus 119908 times 120595 (119909
119894minus 119887)
1003816100381610038161003816ge 120576
0 otherwise
(6)
Here 119908 is a weight vector which is used to determine themaximum margin hyper plane the term 119908 is called aregularized term and it should be as flat as possible Thesecond term is the empirical error as measured by Vapnikrsquos120576-insensitive loss function 119862 is the regularization constant
The following commonly used kernel functions areincluded
Linear 119896(1199091 1199092) = ⟨119909
1 1199092⟩
Polynomial 119896(1199091 1199092) = (⟨119909
1 1199092⟩ + 119877)
119889Sigmoid 119896(119909
1 1199092) = (tanh(⟨119909
1 1199092⟩ + 119903))
Radial basis function 119896(1199091 1199092) = exp(minus119909
1minus
1199092221205902)
(2) Bayes Classifier The MMSE-BC has been considered thebest strategy that uses Bayes method with the single featuremean load based on the evaluation type A in a previouswork [16] The MMSE-BC used here was the minimizedMSE (MMSE) based Bayes classifier It is a classic supervisedlearning classifier used in data mining The formulation ofMMSE-BC is represented in
119875 (120596119894| 120594119895) =
119875 (120594119895| 120596119894) 119875 (120596
119894)
sum119898
119896=1119875 (120594119895| 120596119896) 119875 (120596
119896)
(7)
119894= 119864 (120596
119894| 120594119895) =
119898
sum
119894=1
120596119894119875 (120594119895| 120596119894) (8)
4 Advances in Mathematical Physics
Input stock dataset interval duration of predictionOutput CDF of the prediction MSE on different dataset and different methods(1) Split dataset into training dataset and testing dataset(2) for (newdataset = dataset[ 119899])lowast 119899 is data number increasing by 40 lowastdo(3) for (interval = 2 4 8 16 32) do(4) Determine the feature of the mean and interval slope in every interval(5) Predict the mean price 120578
119894 using SVR or MMSE-BC method in training dataset
lowast Use the mean and interval slope as feature of the stock price trend lowast(6) end for(7) Segment transformation based on (2) 120578rarr 119897
(8) Calculate the MSE of this dataset(9) end for(10) Statistic 80 MSE of different dataset and plot the cumulative distribution function (CDF) of MSE
Algorithm 1 Stock price trend prediction model
It is important for the Bayes classifier to compute the priorprobability distribution 119875(120596
119894) for the target states based on
the samples and compute the joint probability distribution119875(120594119895| 120596119894) for each state 120596
119894 Then the posterior probability
119875(120596119894| 120594119895) was computed according to Formula (7)
24 Trend Prediction Model The following trend predictionmodel is proposed here as a way of preventing the gener-ation of cumulative errors The proposed stock price trendpredictionmodel has the following three steps first using theESP principle the estimated data segment is split into a set ofconsecutive segments whose lengths increase exponentiallyThe interval slope is used to describe the features of eachinterval Then the machine learning methods SVR andMMSE-BC were used to produce the transformmodel of thedata and by which the mean stock price was predicted in theprediction of the next interval
First the stock opening price data were selected Secondthe time series (stock opening price data) was split into a setof a future time interval segments (120578) whose lengths increaseexponentiallyThe length of following subinterval was 119905
0+2119894
where 119894 = 1 2 3 4 Third the mean and interval slopewere computed for every subinterval and the feature datasetwas split into training dataset and prediction dataset NextMMSE-BC and SVR were trained in order to produce themodel parameters For example it can compute the priorprobability 119875(120596
119894) and the conditional probability (119875(120594
119895| 120596119894)
in (8)) and produce a boundary that leads to the largestmargin from both sets of points in SVR and predict the meanstock price and the interval slope in prediction interval overa single interval The interval slope must transform into themean of the interval based on (5) because the mean valuesover consecutive future time intervals are used to express thelong-term trends in the time series Then the mean valuesover consecutive future time intervals l can be converted fromthe vector 120578 based on (2) At last the mean squared error ofthis dataset can be calculated
In order to evaluate the performance of MMSE-BC andSVR the entire dataset prediction mean squared error wascomputed For example the price over the first 1000 tradingdays was selected for training and the price over the next 32
1000 32Train Predict
Time axis
32Predict
32Predict
1040Train
1080Train
1120Train
4200Train
Time axis
Time axis
Time axis
32Predict
32Predict
Time axis
Figure 3 Setting the time window
days was selected for prediction The entire process followsthe procedure of the trend prediction model mentionedThen the first 1040 trading daysrsquo price can be learned and thenext 32 daysrsquo price can be predicted Next each process is tofind the mean squared error of the prediction process Theprocess with higher prices prediction performance continuesto predict the future stock price The method of setting thetime window is shown in Figure 3
Algorithm 1 gives the pseudocode of the stock price trendprediction model
3 Experiments and Comparison
This section presents experiments of the trend predictionmodel on the stock open price forecasting The trend predic-tionmodel was here shown to be able to capture the dynamicsof highly nonlinear nonstationary time series
31 Evaluation Indicator To evaluate the accuracy of thesepredictions the overall mean squared error (MSE) between
Advances in Mathematical Physics 5
Table 1 Optimized parameters for the method
Method Key parameters Valuescompute method
SVRKernel
Penalty parameter 119862 of the error termSlack variables 119871
120576
Sigmoidradial basis function100001
CDF
CDF
CDF
CDF
IBM Coca Cola
Microsoft Amazon
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
01
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
020 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 4 CDF of MSE for prediction of trends in stock price
the predicted stock price values and the true values in theprediction interval can be calculated as follows
mse (119904) = 1
119904
119899
sum
119894=1
119904119894(119897119894minus 119871119894)2 (9)
Here 119904119894= 2119894minus1 119894 ge 1 119904 = sum
119899
119894=1119904119894 119897119894is the predicted mean of
testing dataset 119871119894is the true mean of testing dataset and 119899 is
the total number of the segments in the prediction interval
32 Method of Training and Evaluation Eight openingstock price data samples were selected at random for theseexperiments IBM Coca Cola Microsoft Amazon Sony
Kimberly-Clark Bank of America andWalgreens in 199911ndash20141030
SVR and MMSE-BC were here used to predict the trendsin opening stock price and some key parameters are listed inTable 1
33 Experimental Results The results of MMSE-BC and SVRwere compared to the classic mean and the interval slopeEight stock opening price data samples IBM Coca ColaMicrosoft Amazon Sony Kimberly-Clark Bank of AmericaandWalgreens in 199911ndash20141030 were compared toMSEFigures 4 and 5 show the cumulative distribution function(CDF) of MSE of different prediction methods in whichSVRrsquos kernel is sigmoid
6 Advances in Mathematical Physics
Sony Kimberly-Clark
Bank of America Walgreen
CDF
CDF
CDF
CDF
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
01
0 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 5 CDF of MSE for prediction of trends in stock price
As shown in Figures 4 and 5 the interval slope curve isabove the interval mean curveThis indicates that the intervalslopersquos cumulative probability is greater than that of theinterval mean when the value of MSE is below a certainthreshold For example the IBM interval slopersquos cumulativeprobability was larger than the interval mean curve when theMSE value was less than 100Thatmeans that 88 of theMSEvalues using interval slope were below 100 and only 52 ofthe MSE values using interval mean were below 100
It is clear that interval slopersquos performancewas better thanthat of the mean as indicated by the MMSE-BC and SVRmethods In this way the interval slope can indicate morecomplex dynamics such as change trends In contrast themean can smooth out the dynamic fluctuations in stock price
As an example of prediction results Figure 6 shows IBMstock price trend prediction that is 119905 = 3440ndash3972 by SVRbased on interval slope in which SVRrsquos kernel is a radial basisfunction
Both the mean stock price over a future time interval andthe mean stock price over consecutive future time intervals
can be predicted accurately This shows that the predictionof long-term stock price can be performed precisely withoutgenerating cumulative errors The mean stock price overconsecutive future time intervals can express future trendssuch as sharp falls slight falls concussions slight increasessharp increases falls followed by increases and increases fol-lowed by falls According to the prediction of the fluctuationof opening price over a long-term period fund allocationmodels and trading strategies can be developed in advance
4 Conclusion and Future Work
In this paper ESP which does not generate cumulativeerrors was introduced and used to predict fluctuations in theopening prices of different stocks over a long period The useof a new feature of stock price in the evidence subintervalinterval slope was proposed to characterize the stock pricefluctuation over some time period It can be concluded thatthe interval slope can capture complex dynamics such astrends in the changes in stock price
Advances in Mathematical Physics 7
3500 3600 3700 3800
210
200
190
180
170
160
Stoc
k pr
ice
Time axis
True dataTrue mean stepPredict k SVR step
Figure 6 IBM stock price trend prediction
The premise of this method of trend prediction is thatfuture markets will change gradually rather than abruptlyThe complexities of changes in stock price can greatlyincrease the difficulty of prediction Future work shouldevaluate different learning methods and even combine dif-ferent learning methods Some new methods of evaluationshould be used to evaluate the interval slope the classic meanmethod and the rate of return
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by NSFC under Grantno 61273002 the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions noCITampTCD201304025 and the Key Science and TechnologyProject of BeijingMunicipal EducationCommission ofChinano KZ201510011012
References
[1] R S Tsay Analysis of Financial Time Series JohnWiley amp SonsHoboken NJ USA 2005
[2] G Sermpinis C Dunis J Laws and C Stasinakis ldquoForecastingand trading the EURUSD exchange rate with stochastic NeuralNetwork combination and time-varying leveragerdquo DecisionSupport Systems vol 54 no 1 pp 316ndash329 2012
[3] P-C Chang D-D Wang and C-L Zhou ldquoA novel modelby evolving partially connected neural network for stock pricetrend forecastingrdquo Expert Systems with Applications vol 39 no1 pp 611ndash620 2012
[4] E Kita M Harada and T Mizuno ldquoApplication of BayesianNetwork to stock price predictionrdquo Artificial IntelligenceResearch vol 1 no 2 2012
[5] Z Er-bo M Huan and H Zhan-Gang ldquoApplying geneticprogramming to analyze moving average and long amp mid-term
trends of stock pricesrdquo Application Research of Computers vol27 no 6 2010
[6] P Meesad and R I Rasel ldquoPredicting stock market priceusing support vector regressionrdquo in Proceedings of the 2ndInternational Conference on Informatics Electronics and Vision(ICIEV rsquo13) pp 1ndash6 IEEE Dhaka Bangladesh May 2013
[7] PDondio ldquoStockmarket predictionwithout sentiment analysisusing a web-traffic based classifier and user-level analysisrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 3137ndash3146 IEEE WaileaHawaii USA January 2013
[8] M Hagenau M Hauser M Liebmann and D NeumannldquoReading all the news at the same time predicting mid-term stock price developments based on news momentumrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 1279ndash1288 Wailea HawaiUSA January 2013
[9] D-Y Xu S-L Yang and R-P Liu ldquoA mixture of HMM GAand Elman network for load prediction in cloud-oriented datacentersrdquo Journal of Zhejiang University Science C vol 14 no 11pp 845ndash858 2013
[10] R Bisoi and P K Dash ldquoA hybrid evolutionary dynamic neuralnetwork for stock market trend analysis and prediction usingunscented Kalman filterrdquo Applied Soft Computing Journal vol19 pp 41ndash56 2014
[11] B Bican and Y Yaslan ldquoA hybrid method for time seriesprediction using EMD and SVRrdquo in Proceedings of the 6th Inter-national Symposium on Communications Control and SignalProcessing (ISCCSP rsquo14) pp 566ndash569 Athens GreeceMay 2014
[12] Z Huang and M-L Shyu ldquok-NN based LS-SVM frameworkfor long-term time series predictionrdquo in Proceedings of the IEEEInternational Conference on Information Reuse and Integration(IRI rsquo10) pp 69ndash74 IEEE Las Vegas Nev USA August 2010
[13] J Fan and Y Tang ldquoAn EMD-SVR method for non-stationarytime series predictionrdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance and SafetyEngineering (QR2MSE rsquo13) pp 1765ndash1770 IEEE ChengduChina July 2013
[14] T Fletcher and J Shawe-Taylor ldquoMultiple kernel learningwith fisher kernels for high frequency currency predictionrdquoComputational Economics vol 42 no 2 pp 217ndash240 2013
[15] Q Yang C Peng Y Yu et al ldquoHost load prediction basedon PSR and EA-GMDH for cloud computing systemrdquo inProceedings of the 3rd IEEE International Conference on Cloudand Green Computing (CGC rsquo13) pp 9ndash15 IEEE KarlsruheGermany October 2013
[16] S Di D Kondo and W Cirne ldquoGoogle hostload predictionbased on Bayesian model with optimized feature combinationrdquoJournal of Parallel and Distributed Computing vol 74 no 1 pp1820ndash1832 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 3St
ock
pric
e
Time axis
History Predict
Mean
Last price
Mean
Mean
Mean
1000 data samples every window has 4 data samples
tan120572 = Ki =meani minus lastpriceiminus1
d
120572
d = 4
Figure 2 Illustration of features for stock price fluctuation (119889119895= 4
days)
22 Features of Fluctuations in Stock Price The aim of thecurrent work is to predict the mean stock price over a futuretime interval (120578) starting at the current time 119905
0 Here every
future time interval is called a subinterval Based on thefeatures of stock price trend a new feature of the stock pricein the subinterval was proposed here This feature is herecalled the interval slope and it can be used to characterizethe fluctuations in stock price fluctuation over a specific timeperiod The time series is denoted in the subinterval as e =(1198901 1198902 119890
119894)119879 where 119894 = 1 2 3 4 and 119890
119894is the sample
stock price in the subinterval For example the subinterval is4 days (ie 119894 = 4) whose features for stock price fluctuationare shown in Figure 2
IntervalMean Pricemeanprice119895is themean stock price when
the conditions of the subinterval equal 119889119895
meanprice119895=1
119889119895
119889119895
sum
119894=1
119890119894 (3)
Interval Last PriceThe last price is themost recent price valuein the subinterval
Interval Slope (119870) 119870119895interval slope is the slope of linear
equation in the condition of the subinterval equals 119889119895
First the last price and the mean price of the subintervalwere computedThen the linear equation119910 = 119896119909+119887was usedto fit the samples in the subinterval that is
meanprice119895= 119870119895119889119895+ lastprice
119895minus1 (4)
This produces the following
119870119895=
meanprice119895minus lastprice
119895minus1
119889119895
(5)
Interval slope can be transformed into mean by (5)
For example the value of 119889119895can be set as follows 119889
119895=
2 4 8 16 32 The feature of stock price fluctuation based on119889119895= 4 history data and predicted data are shown in Figure 2Next future data features were predicted (the interval
mean and the interval slope) through learning history datafeatures The prediction methods are presented in the nextsection
23 Long-Term Forecasting Based on Interval Slope To verifythe effectiveness of the developed feature and assess theinterval slope of stock price in the subinterval the supportvector regression and the Bayes classifier were used for long-term forecasting Two machine learning methods SVR andMMSE-BC are used to produce the transform model of thedata and the mean stock price is used to predict the nextinterval slope
(1) Support Vector Regression (SVR) The aim of SVR algo-rithm was to minimize 120576-sensitive errors on the subset ofdata here called the support vectors SVR algorithm usesnonlinear kernel functions in order to project initial data toa higher dimensional space and project linear classifiers fromthe higher space to the original spaceThe formulation of SVRis represented as follows
min 121199082+ 119862
1
119897
119897
sum
119894=1
119871120576
119871120576=
1003816100381610038161003816119910119894minus 119908 times 120595 (119909
119894minus 119887)
1003816100381610038161003816minus 120576
1003816100381610038161003816119910119894minus 119908 times 120595 (119909
119894minus 119887)
1003816100381610038161003816ge 120576
0 otherwise
(6)
Here 119908 is a weight vector which is used to determine themaximum margin hyper plane the term 119908 is called aregularized term and it should be as flat as possible Thesecond term is the empirical error as measured by Vapnikrsquos120576-insensitive loss function 119862 is the regularization constant
The following commonly used kernel functions areincluded
Linear 119896(1199091 1199092) = ⟨119909
1 1199092⟩
Polynomial 119896(1199091 1199092) = (⟨119909
1 1199092⟩ + 119877)
119889Sigmoid 119896(119909
1 1199092) = (tanh(⟨119909
1 1199092⟩ + 119903))
Radial basis function 119896(1199091 1199092) = exp(minus119909
1minus
1199092221205902)
(2) Bayes Classifier The MMSE-BC has been considered thebest strategy that uses Bayes method with the single featuremean load based on the evaluation type A in a previouswork [16] The MMSE-BC used here was the minimizedMSE (MMSE) based Bayes classifier It is a classic supervisedlearning classifier used in data mining The formulation ofMMSE-BC is represented in
119875 (120596119894| 120594119895) =
119875 (120594119895| 120596119894) 119875 (120596
119894)
sum119898
119896=1119875 (120594119895| 120596119896) 119875 (120596
119896)
(7)
119894= 119864 (120596
119894| 120594119895) =
119898
sum
119894=1
120596119894119875 (120594119895| 120596119894) (8)
4 Advances in Mathematical Physics
Input stock dataset interval duration of predictionOutput CDF of the prediction MSE on different dataset and different methods(1) Split dataset into training dataset and testing dataset(2) for (newdataset = dataset[ 119899])lowast 119899 is data number increasing by 40 lowastdo(3) for (interval = 2 4 8 16 32) do(4) Determine the feature of the mean and interval slope in every interval(5) Predict the mean price 120578
119894 using SVR or MMSE-BC method in training dataset
lowast Use the mean and interval slope as feature of the stock price trend lowast(6) end for(7) Segment transformation based on (2) 120578rarr 119897
(8) Calculate the MSE of this dataset(9) end for(10) Statistic 80 MSE of different dataset and plot the cumulative distribution function (CDF) of MSE
Algorithm 1 Stock price trend prediction model
It is important for the Bayes classifier to compute the priorprobability distribution 119875(120596
119894) for the target states based on
the samples and compute the joint probability distribution119875(120594119895| 120596119894) for each state 120596
119894 Then the posterior probability
119875(120596119894| 120594119895) was computed according to Formula (7)
24 Trend Prediction Model The following trend predictionmodel is proposed here as a way of preventing the gener-ation of cumulative errors The proposed stock price trendpredictionmodel has the following three steps first using theESP principle the estimated data segment is split into a set ofconsecutive segments whose lengths increase exponentiallyThe interval slope is used to describe the features of eachinterval Then the machine learning methods SVR andMMSE-BC were used to produce the transformmodel of thedata and by which the mean stock price was predicted in theprediction of the next interval
First the stock opening price data were selected Secondthe time series (stock opening price data) was split into a setof a future time interval segments (120578) whose lengths increaseexponentiallyThe length of following subinterval was 119905
0+2119894
where 119894 = 1 2 3 4 Third the mean and interval slopewere computed for every subinterval and the feature datasetwas split into training dataset and prediction dataset NextMMSE-BC and SVR were trained in order to produce themodel parameters For example it can compute the priorprobability 119875(120596
119894) and the conditional probability (119875(120594
119895| 120596119894)
in (8)) and produce a boundary that leads to the largestmargin from both sets of points in SVR and predict the meanstock price and the interval slope in prediction interval overa single interval The interval slope must transform into themean of the interval based on (5) because the mean valuesover consecutive future time intervals are used to express thelong-term trends in the time series Then the mean valuesover consecutive future time intervals l can be converted fromthe vector 120578 based on (2) At last the mean squared error ofthis dataset can be calculated
In order to evaluate the performance of MMSE-BC andSVR the entire dataset prediction mean squared error wascomputed For example the price over the first 1000 tradingdays was selected for training and the price over the next 32
1000 32Train Predict
Time axis
32Predict
32Predict
1040Train
1080Train
1120Train
4200Train
Time axis
Time axis
Time axis
32Predict
32Predict
Time axis
Figure 3 Setting the time window
days was selected for prediction The entire process followsthe procedure of the trend prediction model mentionedThen the first 1040 trading daysrsquo price can be learned and thenext 32 daysrsquo price can be predicted Next each process is tofind the mean squared error of the prediction process Theprocess with higher prices prediction performance continuesto predict the future stock price The method of setting thetime window is shown in Figure 3
Algorithm 1 gives the pseudocode of the stock price trendprediction model
3 Experiments and Comparison
This section presents experiments of the trend predictionmodel on the stock open price forecasting The trend predic-tionmodel was here shown to be able to capture the dynamicsof highly nonlinear nonstationary time series
31 Evaluation Indicator To evaluate the accuracy of thesepredictions the overall mean squared error (MSE) between
Advances in Mathematical Physics 5
Table 1 Optimized parameters for the method
Method Key parameters Valuescompute method
SVRKernel
Penalty parameter 119862 of the error termSlack variables 119871
120576
Sigmoidradial basis function100001
CDF
CDF
CDF
CDF
IBM Coca Cola
Microsoft Amazon
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
01
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
020 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 4 CDF of MSE for prediction of trends in stock price
the predicted stock price values and the true values in theprediction interval can be calculated as follows
mse (119904) = 1
119904
119899
sum
119894=1
119904119894(119897119894minus 119871119894)2 (9)
Here 119904119894= 2119894minus1 119894 ge 1 119904 = sum
119899
119894=1119904119894 119897119894is the predicted mean of
testing dataset 119871119894is the true mean of testing dataset and 119899 is
the total number of the segments in the prediction interval
32 Method of Training and Evaluation Eight openingstock price data samples were selected at random for theseexperiments IBM Coca Cola Microsoft Amazon Sony
Kimberly-Clark Bank of America andWalgreens in 199911ndash20141030
SVR and MMSE-BC were here used to predict the trendsin opening stock price and some key parameters are listed inTable 1
33 Experimental Results The results of MMSE-BC and SVRwere compared to the classic mean and the interval slopeEight stock opening price data samples IBM Coca ColaMicrosoft Amazon Sony Kimberly-Clark Bank of AmericaandWalgreens in 199911ndash20141030 were compared toMSEFigures 4 and 5 show the cumulative distribution function(CDF) of MSE of different prediction methods in whichSVRrsquos kernel is sigmoid
6 Advances in Mathematical Physics
Sony Kimberly-Clark
Bank of America Walgreen
CDF
CDF
CDF
CDF
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
01
0 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 5 CDF of MSE for prediction of trends in stock price
As shown in Figures 4 and 5 the interval slope curve isabove the interval mean curveThis indicates that the intervalslopersquos cumulative probability is greater than that of theinterval mean when the value of MSE is below a certainthreshold For example the IBM interval slopersquos cumulativeprobability was larger than the interval mean curve when theMSE value was less than 100Thatmeans that 88 of theMSEvalues using interval slope were below 100 and only 52 ofthe MSE values using interval mean were below 100
It is clear that interval slopersquos performancewas better thanthat of the mean as indicated by the MMSE-BC and SVRmethods In this way the interval slope can indicate morecomplex dynamics such as change trends In contrast themean can smooth out the dynamic fluctuations in stock price
As an example of prediction results Figure 6 shows IBMstock price trend prediction that is 119905 = 3440ndash3972 by SVRbased on interval slope in which SVRrsquos kernel is a radial basisfunction
Both the mean stock price over a future time interval andthe mean stock price over consecutive future time intervals
can be predicted accurately This shows that the predictionof long-term stock price can be performed precisely withoutgenerating cumulative errors The mean stock price overconsecutive future time intervals can express future trendssuch as sharp falls slight falls concussions slight increasessharp increases falls followed by increases and increases fol-lowed by falls According to the prediction of the fluctuationof opening price over a long-term period fund allocationmodels and trading strategies can be developed in advance
4 Conclusion and Future Work
In this paper ESP which does not generate cumulativeerrors was introduced and used to predict fluctuations in theopening prices of different stocks over a long period The useof a new feature of stock price in the evidence subintervalinterval slope was proposed to characterize the stock pricefluctuation over some time period It can be concluded thatthe interval slope can capture complex dynamics such astrends in the changes in stock price
Advances in Mathematical Physics 7
3500 3600 3700 3800
210
200
190
180
170
160
Stoc
k pr
ice
Time axis
True dataTrue mean stepPredict k SVR step
Figure 6 IBM stock price trend prediction
The premise of this method of trend prediction is thatfuture markets will change gradually rather than abruptlyThe complexities of changes in stock price can greatlyincrease the difficulty of prediction Future work shouldevaluate different learning methods and even combine dif-ferent learning methods Some new methods of evaluationshould be used to evaluate the interval slope the classic meanmethod and the rate of return
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by NSFC under Grantno 61273002 the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions noCITampTCD201304025 and the Key Science and TechnologyProject of BeijingMunicipal EducationCommission ofChinano KZ201510011012
References
[1] R S Tsay Analysis of Financial Time Series JohnWiley amp SonsHoboken NJ USA 2005
[2] G Sermpinis C Dunis J Laws and C Stasinakis ldquoForecastingand trading the EURUSD exchange rate with stochastic NeuralNetwork combination and time-varying leveragerdquo DecisionSupport Systems vol 54 no 1 pp 316ndash329 2012
[3] P-C Chang D-D Wang and C-L Zhou ldquoA novel modelby evolving partially connected neural network for stock pricetrend forecastingrdquo Expert Systems with Applications vol 39 no1 pp 611ndash620 2012
[4] E Kita M Harada and T Mizuno ldquoApplication of BayesianNetwork to stock price predictionrdquo Artificial IntelligenceResearch vol 1 no 2 2012
[5] Z Er-bo M Huan and H Zhan-Gang ldquoApplying geneticprogramming to analyze moving average and long amp mid-term
trends of stock pricesrdquo Application Research of Computers vol27 no 6 2010
[6] P Meesad and R I Rasel ldquoPredicting stock market priceusing support vector regressionrdquo in Proceedings of the 2ndInternational Conference on Informatics Electronics and Vision(ICIEV rsquo13) pp 1ndash6 IEEE Dhaka Bangladesh May 2013
[7] PDondio ldquoStockmarket predictionwithout sentiment analysisusing a web-traffic based classifier and user-level analysisrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 3137ndash3146 IEEE WaileaHawaii USA January 2013
[8] M Hagenau M Hauser M Liebmann and D NeumannldquoReading all the news at the same time predicting mid-term stock price developments based on news momentumrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 1279ndash1288 Wailea HawaiUSA January 2013
[9] D-Y Xu S-L Yang and R-P Liu ldquoA mixture of HMM GAand Elman network for load prediction in cloud-oriented datacentersrdquo Journal of Zhejiang University Science C vol 14 no 11pp 845ndash858 2013
[10] R Bisoi and P K Dash ldquoA hybrid evolutionary dynamic neuralnetwork for stock market trend analysis and prediction usingunscented Kalman filterrdquo Applied Soft Computing Journal vol19 pp 41ndash56 2014
[11] B Bican and Y Yaslan ldquoA hybrid method for time seriesprediction using EMD and SVRrdquo in Proceedings of the 6th Inter-national Symposium on Communications Control and SignalProcessing (ISCCSP rsquo14) pp 566ndash569 Athens GreeceMay 2014
[12] Z Huang and M-L Shyu ldquok-NN based LS-SVM frameworkfor long-term time series predictionrdquo in Proceedings of the IEEEInternational Conference on Information Reuse and Integration(IRI rsquo10) pp 69ndash74 IEEE Las Vegas Nev USA August 2010
[13] J Fan and Y Tang ldquoAn EMD-SVR method for non-stationarytime series predictionrdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance and SafetyEngineering (QR2MSE rsquo13) pp 1765ndash1770 IEEE ChengduChina July 2013
[14] T Fletcher and J Shawe-Taylor ldquoMultiple kernel learningwith fisher kernels for high frequency currency predictionrdquoComputational Economics vol 42 no 2 pp 217ndash240 2013
[15] Q Yang C Peng Y Yu et al ldquoHost load prediction basedon PSR and EA-GMDH for cloud computing systemrdquo inProceedings of the 3rd IEEE International Conference on Cloudand Green Computing (CGC rsquo13) pp 9ndash15 IEEE KarlsruheGermany October 2013
[16] S Di D Kondo and W Cirne ldquoGoogle hostload predictionbased on Bayesian model with optimized feature combinationrdquoJournal of Parallel and Distributed Computing vol 74 no 1 pp1820ndash1832 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Advances in Mathematical Physics
Input stock dataset interval duration of predictionOutput CDF of the prediction MSE on different dataset and different methods(1) Split dataset into training dataset and testing dataset(2) for (newdataset = dataset[ 119899])lowast 119899 is data number increasing by 40 lowastdo(3) for (interval = 2 4 8 16 32) do(4) Determine the feature of the mean and interval slope in every interval(5) Predict the mean price 120578
119894 using SVR or MMSE-BC method in training dataset
lowast Use the mean and interval slope as feature of the stock price trend lowast(6) end for(7) Segment transformation based on (2) 120578rarr 119897
(8) Calculate the MSE of this dataset(9) end for(10) Statistic 80 MSE of different dataset and plot the cumulative distribution function (CDF) of MSE
Algorithm 1 Stock price trend prediction model
It is important for the Bayes classifier to compute the priorprobability distribution 119875(120596
119894) for the target states based on
the samples and compute the joint probability distribution119875(120594119895| 120596119894) for each state 120596
119894 Then the posterior probability
119875(120596119894| 120594119895) was computed according to Formula (7)
24 Trend Prediction Model The following trend predictionmodel is proposed here as a way of preventing the gener-ation of cumulative errors The proposed stock price trendpredictionmodel has the following three steps first using theESP principle the estimated data segment is split into a set ofconsecutive segments whose lengths increase exponentiallyThe interval slope is used to describe the features of eachinterval Then the machine learning methods SVR andMMSE-BC were used to produce the transformmodel of thedata and by which the mean stock price was predicted in theprediction of the next interval
First the stock opening price data were selected Secondthe time series (stock opening price data) was split into a setof a future time interval segments (120578) whose lengths increaseexponentiallyThe length of following subinterval was 119905
0+2119894
where 119894 = 1 2 3 4 Third the mean and interval slopewere computed for every subinterval and the feature datasetwas split into training dataset and prediction dataset NextMMSE-BC and SVR were trained in order to produce themodel parameters For example it can compute the priorprobability 119875(120596
119894) and the conditional probability (119875(120594
119895| 120596119894)
in (8)) and produce a boundary that leads to the largestmargin from both sets of points in SVR and predict the meanstock price and the interval slope in prediction interval overa single interval The interval slope must transform into themean of the interval based on (5) because the mean valuesover consecutive future time intervals are used to express thelong-term trends in the time series Then the mean valuesover consecutive future time intervals l can be converted fromthe vector 120578 based on (2) At last the mean squared error ofthis dataset can be calculated
In order to evaluate the performance of MMSE-BC andSVR the entire dataset prediction mean squared error wascomputed For example the price over the first 1000 tradingdays was selected for training and the price over the next 32
1000 32Train Predict
Time axis
32Predict
32Predict
1040Train
1080Train
1120Train
4200Train
Time axis
Time axis
Time axis
32Predict
32Predict
Time axis
Figure 3 Setting the time window
days was selected for prediction The entire process followsthe procedure of the trend prediction model mentionedThen the first 1040 trading daysrsquo price can be learned and thenext 32 daysrsquo price can be predicted Next each process is tofind the mean squared error of the prediction process Theprocess with higher prices prediction performance continuesto predict the future stock price The method of setting thetime window is shown in Figure 3
Algorithm 1 gives the pseudocode of the stock price trendprediction model
3 Experiments and Comparison
This section presents experiments of the trend predictionmodel on the stock open price forecasting The trend predic-tionmodel was here shown to be able to capture the dynamicsof highly nonlinear nonstationary time series
31 Evaluation Indicator To evaluate the accuracy of thesepredictions the overall mean squared error (MSE) between
Advances in Mathematical Physics 5
Table 1 Optimized parameters for the method
Method Key parameters Valuescompute method
SVRKernel
Penalty parameter 119862 of the error termSlack variables 119871
120576
Sigmoidradial basis function100001
CDF
CDF
CDF
CDF
IBM Coca Cola
Microsoft Amazon
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
01
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
020 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 4 CDF of MSE for prediction of trends in stock price
the predicted stock price values and the true values in theprediction interval can be calculated as follows
mse (119904) = 1
119904
119899
sum
119894=1
119904119894(119897119894minus 119871119894)2 (9)
Here 119904119894= 2119894minus1 119894 ge 1 119904 = sum
119899
119894=1119904119894 119897119894is the predicted mean of
testing dataset 119871119894is the true mean of testing dataset and 119899 is
the total number of the segments in the prediction interval
32 Method of Training and Evaluation Eight openingstock price data samples were selected at random for theseexperiments IBM Coca Cola Microsoft Amazon Sony
Kimberly-Clark Bank of America andWalgreens in 199911ndash20141030
SVR and MMSE-BC were here used to predict the trendsin opening stock price and some key parameters are listed inTable 1
33 Experimental Results The results of MMSE-BC and SVRwere compared to the classic mean and the interval slopeEight stock opening price data samples IBM Coca ColaMicrosoft Amazon Sony Kimberly-Clark Bank of AmericaandWalgreens in 199911ndash20141030 were compared toMSEFigures 4 and 5 show the cumulative distribution function(CDF) of MSE of different prediction methods in whichSVRrsquos kernel is sigmoid
6 Advances in Mathematical Physics
Sony Kimberly-Clark
Bank of America Walgreen
CDF
CDF
CDF
CDF
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
01
0 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 5 CDF of MSE for prediction of trends in stock price
As shown in Figures 4 and 5 the interval slope curve isabove the interval mean curveThis indicates that the intervalslopersquos cumulative probability is greater than that of theinterval mean when the value of MSE is below a certainthreshold For example the IBM interval slopersquos cumulativeprobability was larger than the interval mean curve when theMSE value was less than 100Thatmeans that 88 of theMSEvalues using interval slope were below 100 and only 52 ofthe MSE values using interval mean were below 100
It is clear that interval slopersquos performancewas better thanthat of the mean as indicated by the MMSE-BC and SVRmethods In this way the interval slope can indicate morecomplex dynamics such as change trends In contrast themean can smooth out the dynamic fluctuations in stock price
As an example of prediction results Figure 6 shows IBMstock price trend prediction that is 119905 = 3440ndash3972 by SVRbased on interval slope in which SVRrsquos kernel is a radial basisfunction
Both the mean stock price over a future time interval andthe mean stock price over consecutive future time intervals
can be predicted accurately This shows that the predictionof long-term stock price can be performed precisely withoutgenerating cumulative errors The mean stock price overconsecutive future time intervals can express future trendssuch as sharp falls slight falls concussions slight increasessharp increases falls followed by increases and increases fol-lowed by falls According to the prediction of the fluctuationof opening price over a long-term period fund allocationmodels and trading strategies can be developed in advance
4 Conclusion and Future Work
In this paper ESP which does not generate cumulativeerrors was introduced and used to predict fluctuations in theopening prices of different stocks over a long period The useof a new feature of stock price in the evidence subintervalinterval slope was proposed to characterize the stock pricefluctuation over some time period It can be concluded thatthe interval slope can capture complex dynamics such astrends in the changes in stock price
Advances in Mathematical Physics 7
3500 3600 3700 3800
210
200
190
180
170
160
Stoc
k pr
ice
Time axis
True dataTrue mean stepPredict k SVR step
Figure 6 IBM stock price trend prediction
The premise of this method of trend prediction is thatfuture markets will change gradually rather than abruptlyThe complexities of changes in stock price can greatlyincrease the difficulty of prediction Future work shouldevaluate different learning methods and even combine dif-ferent learning methods Some new methods of evaluationshould be used to evaluate the interval slope the classic meanmethod and the rate of return
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by NSFC under Grantno 61273002 the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions noCITampTCD201304025 and the Key Science and TechnologyProject of BeijingMunicipal EducationCommission ofChinano KZ201510011012
References
[1] R S Tsay Analysis of Financial Time Series JohnWiley amp SonsHoboken NJ USA 2005
[2] G Sermpinis C Dunis J Laws and C Stasinakis ldquoForecastingand trading the EURUSD exchange rate with stochastic NeuralNetwork combination and time-varying leveragerdquo DecisionSupport Systems vol 54 no 1 pp 316ndash329 2012
[3] P-C Chang D-D Wang and C-L Zhou ldquoA novel modelby evolving partially connected neural network for stock pricetrend forecastingrdquo Expert Systems with Applications vol 39 no1 pp 611ndash620 2012
[4] E Kita M Harada and T Mizuno ldquoApplication of BayesianNetwork to stock price predictionrdquo Artificial IntelligenceResearch vol 1 no 2 2012
[5] Z Er-bo M Huan and H Zhan-Gang ldquoApplying geneticprogramming to analyze moving average and long amp mid-term
trends of stock pricesrdquo Application Research of Computers vol27 no 6 2010
[6] P Meesad and R I Rasel ldquoPredicting stock market priceusing support vector regressionrdquo in Proceedings of the 2ndInternational Conference on Informatics Electronics and Vision(ICIEV rsquo13) pp 1ndash6 IEEE Dhaka Bangladesh May 2013
[7] PDondio ldquoStockmarket predictionwithout sentiment analysisusing a web-traffic based classifier and user-level analysisrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 3137ndash3146 IEEE WaileaHawaii USA January 2013
[8] M Hagenau M Hauser M Liebmann and D NeumannldquoReading all the news at the same time predicting mid-term stock price developments based on news momentumrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 1279ndash1288 Wailea HawaiUSA January 2013
[9] D-Y Xu S-L Yang and R-P Liu ldquoA mixture of HMM GAand Elman network for load prediction in cloud-oriented datacentersrdquo Journal of Zhejiang University Science C vol 14 no 11pp 845ndash858 2013
[10] R Bisoi and P K Dash ldquoA hybrid evolutionary dynamic neuralnetwork for stock market trend analysis and prediction usingunscented Kalman filterrdquo Applied Soft Computing Journal vol19 pp 41ndash56 2014
[11] B Bican and Y Yaslan ldquoA hybrid method for time seriesprediction using EMD and SVRrdquo in Proceedings of the 6th Inter-national Symposium on Communications Control and SignalProcessing (ISCCSP rsquo14) pp 566ndash569 Athens GreeceMay 2014
[12] Z Huang and M-L Shyu ldquok-NN based LS-SVM frameworkfor long-term time series predictionrdquo in Proceedings of the IEEEInternational Conference on Information Reuse and Integration(IRI rsquo10) pp 69ndash74 IEEE Las Vegas Nev USA August 2010
[13] J Fan and Y Tang ldquoAn EMD-SVR method for non-stationarytime series predictionrdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance and SafetyEngineering (QR2MSE rsquo13) pp 1765ndash1770 IEEE ChengduChina July 2013
[14] T Fletcher and J Shawe-Taylor ldquoMultiple kernel learningwith fisher kernels for high frequency currency predictionrdquoComputational Economics vol 42 no 2 pp 217ndash240 2013
[15] Q Yang C Peng Y Yu et al ldquoHost load prediction basedon PSR and EA-GMDH for cloud computing systemrdquo inProceedings of the 3rd IEEE International Conference on Cloudand Green Computing (CGC rsquo13) pp 9ndash15 IEEE KarlsruheGermany October 2013
[16] S Di D Kondo and W Cirne ldquoGoogle hostload predictionbased on Bayesian model with optimized feature combinationrdquoJournal of Parallel and Distributed Computing vol 74 no 1 pp1820ndash1832 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 5
Table 1 Optimized parameters for the method
Method Key parameters Valuescompute method
SVRKernel
Penalty parameter 119862 of the error termSlack variables 119871
120576
Sigmoidradial basis function100001
CDF
CDF
CDF
CDF
IBM Coca Cola
Microsoft Amazon
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
01
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
020 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 4 CDF of MSE for prediction of trends in stock price
the predicted stock price values and the true values in theprediction interval can be calculated as follows
mse (119904) = 1
119904
119899
sum
119894=1
119904119894(119897119894minus 119871119894)2 (9)
Here 119904119894= 2119894minus1 119894 ge 1 119904 = sum
119899
119894=1119904119894 119897119894is the predicted mean of
testing dataset 119871119894is the true mean of testing dataset and 119899 is
the total number of the segments in the prediction interval
32 Method of Training and Evaluation Eight openingstock price data samples were selected at random for theseexperiments IBM Coca Cola Microsoft Amazon Sony
Kimberly-Clark Bank of America andWalgreens in 199911ndash20141030
SVR and MMSE-BC were here used to predict the trendsin opening stock price and some key parameters are listed inTable 1
33 Experimental Results The results of MMSE-BC and SVRwere compared to the classic mean and the interval slopeEight stock opening price data samples IBM Coca ColaMicrosoft Amazon Sony Kimberly-Clark Bank of AmericaandWalgreens in 199911ndash20141030 were compared toMSEFigures 4 and 5 show the cumulative distribution function(CDF) of MSE of different prediction methods in whichSVRrsquos kernel is sigmoid
6 Advances in Mathematical Physics
Sony Kimberly-Clark
Bank of America Walgreen
CDF
CDF
CDF
CDF
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
01
0 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 5 CDF of MSE for prediction of trends in stock price
As shown in Figures 4 and 5 the interval slope curve isabove the interval mean curveThis indicates that the intervalslopersquos cumulative probability is greater than that of theinterval mean when the value of MSE is below a certainthreshold For example the IBM interval slopersquos cumulativeprobability was larger than the interval mean curve when theMSE value was less than 100Thatmeans that 88 of theMSEvalues using interval slope were below 100 and only 52 ofthe MSE values using interval mean were below 100
It is clear that interval slopersquos performancewas better thanthat of the mean as indicated by the MMSE-BC and SVRmethods In this way the interval slope can indicate morecomplex dynamics such as change trends In contrast themean can smooth out the dynamic fluctuations in stock price
As an example of prediction results Figure 6 shows IBMstock price trend prediction that is 119905 = 3440ndash3972 by SVRbased on interval slope in which SVRrsquos kernel is a radial basisfunction
Both the mean stock price over a future time interval andthe mean stock price over consecutive future time intervals
can be predicted accurately This shows that the predictionof long-term stock price can be performed precisely withoutgenerating cumulative errors The mean stock price overconsecutive future time intervals can express future trendssuch as sharp falls slight falls concussions slight increasessharp increases falls followed by increases and increases fol-lowed by falls According to the prediction of the fluctuationof opening price over a long-term period fund allocationmodels and trading strategies can be developed in advance
4 Conclusion and Future Work
In this paper ESP which does not generate cumulativeerrors was introduced and used to predict fluctuations in theopening prices of different stocks over a long period The useof a new feature of stock price in the evidence subintervalinterval slope was proposed to characterize the stock pricefluctuation over some time period It can be concluded thatthe interval slope can capture complex dynamics such astrends in the changes in stock price
Advances in Mathematical Physics 7
3500 3600 3700 3800
210
200
190
180
170
160
Stoc
k pr
ice
Time axis
True dataTrue mean stepPredict k SVR step
Figure 6 IBM stock price trend prediction
The premise of this method of trend prediction is thatfuture markets will change gradually rather than abruptlyThe complexities of changes in stock price can greatlyincrease the difficulty of prediction Future work shouldevaluate different learning methods and even combine dif-ferent learning methods Some new methods of evaluationshould be used to evaluate the interval slope the classic meanmethod and the rate of return
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by NSFC under Grantno 61273002 the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions noCITampTCD201304025 and the Key Science and TechnologyProject of BeijingMunicipal EducationCommission ofChinano KZ201510011012
References
[1] R S Tsay Analysis of Financial Time Series JohnWiley amp SonsHoboken NJ USA 2005
[2] G Sermpinis C Dunis J Laws and C Stasinakis ldquoForecastingand trading the EURUSD exchange rate with stochastic NeuralNetwork combination and time-varying leveragerdquo DecisionSupport Systems vol 54 no 1 pp 316ndash329 2012
[3] P-C Chang D-D Wang and C-L Zhou ldquoA novel modelby evolving partially connected neural network for stock pricetrend forecastingrdquo Expert Systems with Applications vol 39 no1 pp 611ndash620 2012
[4] E Kita M Harada and T Mizuno ldquoApplication of BayesianNetwork to stock price predictionrdquo Artificial IntelligenceResearch vol 1 no 2 2012
[5] Z Er-bo M Huan and H Zhan-Gang ldquoApplying geneticprogramming to analyze moving average and long amp mid-term
trends of stock pricesrdquo Application Research of Computers vol27 no 6 2010
[6] P Meesad and R I Rasel ldquoPredicting stock market priceusing support vector regressionrdquo in Proceedings of the 2ndInternational Conference on Informatics Electronics and Vision(ICIEV rsquo13) pp 1ndash6 IEEE Dhaka Bangladesh May 2013
[7] PDondio ldquoStockmarket predictionwithout sentiment analysisusing a web-traffic based classifier and user-level analysisrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 3137ndash3146 IEEE WaileaHawaii USA January 2013
[8] M Hagenau M Hauser M Liebmann and D NeumannldquoReading all the news at the same time predicting mid-term stock price developments based on news momentumrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 1279ndash1288 Wailea HawaiUSA January 2013
[9] D-Y Xu S-L Yang and R-P Liu ldquoA mixture of HMM GAand Elman network for load prediction in cloud-oriented datacentersrdquo Journal of Zhejiang University Science C vol 14 no 11pp 845ndash858 2013
[10] R Bisoi and P K Dash ldquoA hybrid evolutionary dynamic neuralnetwork for stock market trend analysis and prediction usingunscented Kalman filterrdquo Applied Soft Computing Journal vol19 pp 41ndash56 2014
[11] B Bican and Y Yaslan ldquoA hybrid method for time seriesprediction using EMD and SVRrdquo in Proceedings of the 6th Inter-national Symposium on Communications Control and SignalProcessing (ISCCSP rsquo14) pp 566ndash569 Athens GreeceMay 2014
[12] Z Huang and M-L Shyu ldquok-NN based LS-SVM frameworkfor long-term time series predictionrdquo in Proceedings of the IEEEInternational Conference on Information Reuse and Integration(IRI rsquo10) pp 69ndash74 IEEE Las Vegas Nev USA August 2010
[13] J Fan and Y Tang ldquoAn EMD-SVR method for non-stationarytime series predictionrdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance and SafetyEngineering (QR2MSE rsquo13) pp 1765ndash1770 IEEE ChengduChina July 2013
[14] T Fletcher and J Shawe-Taylor ldquoMultiple kernel learningwith fisher kernels for high frequency currency predictionrdquoComputational Economics vol 42 no 2 pp 217ndash240 2013
[15] Q Yang C Peng Y Yu et al ldquoHost load prediction basedon PSR and EA-GMDH for cloud computing systemrdquo inProceedings of the 3rd IEEE International Conference on Cloudand Green Computing (CGC rsquo13) pp 9ndash15 IEEE KarlsruheGermany October 2013
[16] S Di D Kondo and W Cirne ldquoGoogle hostload predictionbased on Bayesian model with optimized feature combinationrdquoJournal of Parallel and Distributed Computing vol 74 no 1 pp1820ndash1832 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Advances in Mathematical Physics
Sony Kimberly-Clark
Bank of America Walgreen
CDF
CDF
CDF
CDF
MSE of stock predictionMSE of stock prediction
MSE of stock prediction MSE of stock prediction
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
10
09
08
07
06
05
04
03
02
01
0 100 200 300 400 5000 100 200 300 400 500
0 100 200 300 400 500 0 100 200 300 400 500
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Mean MMSE-BCInterval slope MMSE-BCInterval slope SVR sigmoid Mean SVR sigmoid
Figure 5 CDF of MSE for prediction of trends in stock price
As shown in Figures 4 and 5 the interval slope curve isabove the interval mean curveThis indicates that the intervalslopersquos cumulative probability is greater than that of theinterval mean when the value of MSE is below a certainthreshold For example the IBM interval slopersquos cumulativeprobability was larger than the interval mean curve when theMSE value was less than 100Thatmeans that 88 of theMSEvalues using interval slope were below 100 and only 52 ofthe MSE values using interval mean were below 100
It is clear that interval slopersquos performancewas better thanthat of the mean as indicated by the MMSE-BC and SVRmethods In this way the interval slope can indicate morecomplex dynamics such as change trends In contrast themean can smooth out the dynamic fluctuations in stock price
As an example of prediction results Figure 6 shows IBMstock price trend prediction that is 119905 = 3440ndash3972 by SVRbased on interval slope in which SVRrsquos kernel is a radial basisfunction
Both the mean stock price over a future time interval andthe mean stock price over consecutive future time intervals
can be predicted accurately This shows that the predictionof long-term stock price can be performed precisely withoutgenerating cumulative errors The mean stock price overconsecutive future time intervals can express future trendssuch as sharp falls slight falls concussions slight increasessharp increases falls followed by increases and increases fol-lowed by falls According to the prediction of the fluctuationof opening price over a long-term period fund allocationmodels and trading strategies can be developed in advance
4 Conclusion and Future Work
In this paper ESP which does not generate cumulativeerrors was introduced and used to predict fluctuations in theopening prices of different stocks over a long period The useof a new feature of stock price in the evidence subintervalinterval slope was proposed to characterize the stock pricefluctuation over some time period It can be concluded thatthe interval slope can capture complex dynamics such astrends in the changes in stock price
Advances in Mathematical Physics 7
3500 3600 3700 3800
210
200
190
180
170
160
Stoc
k pr
ice
Time axis
True dataTrue mean stepPredict k SVR step
Figure 6 IBM stock price trend prediction
The premise of this method of trend prediction is thatfuture markets will change gradually rather than abruptlyThe complexities of changes in stock price can greatlyincrease the difficulty of prediction Future work shouldevaluate different learning methods and even combine dif-ferent learning methods Some new methods of evaluationshould be used to evaluate the interval slope the classic meanmethod and the rate of return
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by NSFC under Grantno 61273002 the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions noCITampTCD201304025 and the Key Science and TechnologyProject of BeijingMunicipal EducationCommission ofChinano KZ201510011012
References
[1] R S Tsay Analysis of Financial Time Series JohnWiley amp SonsHoboken NJ USA 2005
[2] G Sermpinis C Dunis J Laws and C Stasinakis ldquoForecastingand trading the EURUSD exchange rate with stochastic NeuralNetwork combination and time-varying leveragerdquo DecisionSupport Systems vol 54 no 1 pp 316ndash329 2012
[3] P-C Chang D-D Wang and C-L Zhou ldquoA novel modelby evolving partially connected neural network for stock pricetrend forecastingrdquo Expert Systems with Applications vol 39 no1 pp 611ndash620 2012
[4] E Kita M Harada and T Mizuno ldquoApplication of BayesianNetwork to stock price predictionrdquo Artificial IntelligenceResearch vol 1 no 2 2012
[5] Z Er-bo M Huan and H Zhan-Gang ldquoApplying geneticprogramming to analyze moving average and long amp mid-term
trends of stock pricesrdquo Application Research of Computers vol27 no 6 2010
[6] P Meesad and R I Rasel ldquoPredicting stock market priceusing support vector regressionrdquo in Proceedings of the 2ndInternational Conference on Informatics Electronics and Vision(ICIEV rsquo13) pp 1ndash6 IEEE Dhaka Bangladesh May 2013
[7] PDondio ldquoStockmarket predictionwithout sentiment analysisusing a web-traffic based classifier and user-level analysisrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 3137ndash3146 IEEE WaileaHawaii USA January 2013
[8] M Hagenau M Hauser M Liebmann and D NeumannldquoReading all the news at the same time predicting mid-term stock price developments based on news momentumrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 1279ndash1288 Wailea HawaiUSA January 2013
[9] D-Y Xu S-L Yang and R-P Liu ldquoA mixture of HMM GAand Elman network for load prediction in cloud-oriented datacentersrdquo Journal of Zhejiang University Science C vol 14 no 11pp 845ndash858 2013
[10] R Bisoi and P K Dash ldquoA hybrid evolutionary dynamic neuralnetwork for stock market trend analysis and prediction usingunscented Kalman filterrdquo Applied Soft Computing Journal vol19 pp 41ndash56 2014
[11] B Bican and Y Yaslan ldquoA hybrid method for time seriesprediction using EMD and SVRrdquo in Proceedings of the 6th Inter-national Symposium on Communications Control and SignalProcessing (ISCCSP rsquo14) pp 566ndash569 Athens GreeceMay 2014
[12] Z Huang and M-L Shyu ldquok-NN based LS-SVM frameworkfor long-term time series predictionrdquo in Proceedings of the IEEEInternational Conference on Information Reuse and Integration(IRI rsquo10) pp 69ndash74 IEEE Las Vegas Nev USA August 2010
[13] J Fan and Y Tang ldquoAn EMD-SVR method for non-stationarytime series predictionrdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance and SafetyEngineering (QR2MSE rsquo13) pp 1765ndash1770 IEEE ChengduChina July 2013
[14] T Fletcher and J Shawe-Taylor ldquoMultiple kernel learningwith fisher kernels for high frequency currency predictionrdquoComputational Economics vol 42 no 2 pp 217ndash240 2013
[15] Q Yang C Peng Y Yu et al ldquoHost load prediction basedon PSR and EA-GMDH for cloud computing systemrdquo inProceedings of the 3rd IEEE International Conference on Cloudand Green Computing (CGC rsquo13) pp 9ndash15 IEEE KarlsruheGermany October 2013
[16] S Di D Kondo and W Cirne ldquoGoogle hostload predictionbased on Bayesian model with optimized feature combinationrdquoJournal of Parallel and Distributed Computing vol 74 no 1 pp1820ndash1832 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 7
3500 3600 3700 3800
210
200
190
180
170
160
Stoc
k pr
ice
Time axis
True dataTrue mean stepPredict k SVR step
Figure 6 IBM stock price trend prediction
The premise of this method of trend prediction is thatfuture markets will change gradually rather than abruptlyThe complexities of changes in stock price can greatlyincrease the difficulty of prediction Future work shouldevaluate different learning methods and even combine dif-ferent learning methods Some new methods of evaluationshould be used to evaluate the interval slope the classic meanmethod and the rate of return
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by NSFC under Grantno 61273002 the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions noCITampTCD201304025 and the Key Science and TechnologyProject of BeijingMunicipal EducationCommission ofChinano KZ201510011012
References
[1] R S Tsay Analysis of Financial Time Series JohnWiley amp SonsHoboken NJ USA 2005
[2] G Sermpinis C Dunis J Laws and C Stasinakis ldquoForecastingand trading the EURUSD exchange rate with stochastic NeuralNetwork combination and time-varying leveragerdquo DecisionSupport Systems vol 54 no 1 pp 316ndash329 2012
[3] P-C Chang D-D Wang and C-L Zhou ldquoA novel modelby evolving partially connected neural network for stock pricetrend forecastingrdquo Expert Systems with Applications vol 39 no1 pp 611ndash620 2012
[4] E Kita M Harada and T Mizuno ldquoApplication of BayesianNetwork to stock price predictionrdquo Artificial IntelligenceResearch vol 1 no 2 2012
[5] Z Er-bo M Huan and H Zhan-Gang ldquoApplying geneticprogramming to analyze moving average and long amp mid-term
trends of stock pricesrdquo Application Research of Computers vol27 no 6 2010
[6] P Meesad and R I Rasel ldquoPredicting stock market priceusing support vector regressionrdquo in Proceedings of the 2ndInternational Conference on Informatics Electronics and Vision(ICIEV rsquo13) pp 1ndash6 IEEE Dhaka Bangladesh May 2013
[7] PDondio ldquoStockmarket predictionwithout sentiment analysisusing a web-traffic based classifier and user-level analysisrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 3137ndash3146 IEEE WaileaHawaii USA January 2013
[8] M Hagenau M Hauser M Liebmann and D NeumannldquoReading all the news at the same time predicting mid-term stock price developments based on news momentumrdquo inProceedings of the 46th Annual Hawaii International Conferenceon System Sciences (HICSS rsquo13) pp 1279ndash1288 Wailea HawaiUSA January 2013
[9] D-Y Xu S-L Yang and R-P Liu ldquoA mixture of HMM GAand Elman network for load prediction in cloud-oriented datacentersrdquo Journal of Zhejiang University Science C vol 14 no 11pp 845ndash858 2013
[10] R Bisoi and P K Dash ldquoA hybrid evolutionary dynamic neuralnetwork for stock market trend analysis and prediction usingunscented Kalman filterrdquo Applied Soft Computing Journal vol19 pp 41ndash56 2014
[11] B Bican and Y Yaslan ldquoA hybrid method for time seriesprediction using EMD and SVRrdquo in Proceedings of the 6th Inter-national Symposium on Communications Control and SignalProcessing (ISCCSP rsquo14) pp 566ndash569 Athens GreeceMay 2014
[12] Z Huang and M-L Shyu ldquok-NN based LS-SVM frameworkfor long-term time series predictionrdquo in Proceedings of the IEEEInternational Conference on Information Reuse and Integration(IRI rsquo10) pp 69ndash74 IEEE Las Vegas Nev USA August 2010
[13] J Fan and Y Tang ldquoAn EMD-SVR method for non-stationarytime series predictionrdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance and SafetyEngineering (QR2MSE rsquo13) pp 1765ndash1770 IEEE ChengduChina July 2013
[14] T Fletcher and J Shawe-Taylor ldquoMultiple kernel learningwith fisher kernels for high frequency currency predictionrdquoComputational Economics vol 42 no 2 pp 217ndash240 2013
[15] Q Yang C Peng Y Yu et al ldquoHost load prediction basedon PSR and EA-GMDH for cloud computing systemrdquo inProceedings of the 3rd IEEE International Conference on Cloudand Green Computing (CGC rsquo13) pp 9ndash15 IEEE KarlsruheGermany October 2013
[16] S Di D Kondo and W Cirne ldquoGoogle hostload predictionbased on Bayesian model with optimized feature combinationrdquoJournal of Parallel and Distributed Computing vol 74 no 1 pp1820ndash1832 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of