Predicting Stock Prices Using Polynomial Classifiers: The Case of Dubai Financial Market

doi:10.4236/jilsa.2011.32010

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Journal of Intelligent Learning Systems and Applications, 2011, 3, 82-89

doi:10.4236/jilsa.2011.32010 Published Online May 2011 (http://www.SciRP.org/journal/jilsa)

Predicting Stock Prices Using Polynomial

Classifiers: The Case of Dubai Financial Market

Khaled Assaleh1, Hazim El-Baz2, Saeed Al-Salkhadi2

1Department of Electrical Engineering, American University of Sharjah, Sharjah, UAE; 2Engineering Systems Management Graduate

Program, American University of Sharjah, Sharjah, UAE.

Email: [email protected]

Received July 10th, 2010; revised October 1st, 2010; accepted December 1st, 2010.

ABSTRACT

Predicting stock price movements is a challenging task for academicians and practitioners. In particular, forecasting

price movements in emerging markets seems to be more elusive because they are usually more volatile often accompa-

nied by thin trading-volumes and they are susceptible to more manipulation compared to mature markets. Technical

analysis of stocks and commodities has become a science on its own; quantitative methods and techniques have been

applied by many practitioners to forecast price movements. Lagging and sometimes leading technical indicators pro-

vide rich quantitative tools for traders and investors in their attempt to gain advantage when making investment or

trading decisions. Artificial Neural Networks (ANN) have been used widely in predicting stock prices because of their

capability in capturing the non-linearity that often exists in price movements. Recently, Polynomial Classifiers (PC)

have been applied to various recognition and classification application and showed favorable results in terms of recog-

nition rates and computational complexity as compared to ANN. In this paper, we present two prediction models for

predicting securities’ prices. The first model was developed using back propagation feed forward neural networks. The

second model was developed using polynomial classifiers (PC), as a first time application for PC to be used in stock

prices prediction. The inputs to both models were identical, and both models were trained and tested on the same data.

The study was conducted on Dubai Financial Market as an emerging market and applied to two of the market’s leading

stocks. In general, both models achieved very good results in terms of mean absolute error percentage. Both models

show an average error around 1.5% predicting the next day price, an average error of 2.5% when predicting second

day price, and an average error of 4% when predicted the third day price.

Keywords: Dubai Financial Market, Polynomial Classifiers, Stock Market, Neural Networks

1. Introduction

Accurate prediction of stock prices presents a challeng-

ing task for traders and investors. Multitude of economi-

cal, social, political and psycho logical factors interacts in

a complex way to form stock movement patterns. The

early Efficient Market Theory (EMT) claims that prices

move in a random way and it is not possible to develop

an algorithm of any kind that predicts stock prices [1].

Other researchers contradicted this claim and presented

considerable evidence showing that stock prices are, to

some extent, predictable. Forecasting or predicting stock

prices may be done following one or a combination of

four approaches: fundamental analysis approach, techni-

cal analysis approach, time series forecasting and ma-

chine learning. Each approach has its own merits as well

as limitations.

In recent years, ANN has been successfully used for

modeling financial time series. For example, the back

propagation ANN was used to forecast future price

movements of Kuala Lumpur Stock Exchange which is

major emerging market [2]. Brownstone [3] reported the

prediction error of neural networks measured by the

mean square error and the root mean square error to pre-

dict daily market close 5 days and 25 days ahead for the

Financial Times Stock Exchange (FTSE) in the UK. Lee

and Jo [4] developed an expert system using the Japanese

Candlestick chart patterns to predict th e proper timing of

stocks and reported a success ratio of 72% of their sys-

tem. Template matching, from pattern recognition, and

the feed forward neural networks were combined and

used to forecast stock market activity in the New York

Stock Exchange Composite index [5]. Another learning

system that integrates genetic algorithm (GA) and Sup-

Predicting Stock Prices Using Polynomial Classifiers: The Case of Dubai Financial Market83

port Vector machines (SVM) for stock market prediction

was reported by Choudhry and Garg [6]. The next day

stock price of stock market indexes using a hybrid ap-

proach that integrates both GA and ANN was reported by

Armano et al. [7]. In a different but related application,

the NN was used to capture the underlying rules of

movements in currency exchange rates with reasonable

results in [8]. The use of Artificial Neural Networks to

predict stock market behavior in terms of its direction

was reported in [9] for Istanbul Stock Exchange where

the historical data was grouped in seven different predic-

tion systems models to which eight different ANN and

Logistics Regressions models were applied. Chena et al.

[10] used the probabilistic neural network (PNN) to

forecast the direction of Taiwan Stock Marekt Index re-

turn using historical data. Enke and Thawornwong [11]

combined information gain technique of data mining and

Neural Network to evaluate the predictive relationship of

financial and economic variables. Caoa et al. [12] com-

pared financial forecasting linear models to the predictive

power of univariate and multivariate Neural Networks

for the Chinese stock market. Kaastra and Boyd [13]

provided an eight-step procedure to design neural net-

work forecasting model with trade-offs in parameters

selection and common pitfalls. Kohzadi et al. [14] ex-

amined whether neural networks can outperform a tradi-

tional autoregressive integrated moving average (ARIMA)

model for forecasting commodity prices. Olsona Mossmanb

[15] compared the prediction accuracy of neural network

and logistic regression technique using historical data of

Canadian Stock exchange

Common to all the reviewed predictio n methods is the

use of a pattern classifier which in most cases is some

sort of a neural classifier. Whether used to model the

manifolds of each class or to discriminate the patterns of

different classes, neural classifiers can be divided into

relative density models and discriminative models. Ex-

amples of relative density models include linear mixture

models and auto-association networks. Whereas, dis-

criminative neural classifiers include the multi-layer per-

ceptron (MLP), the radial basis function (RBF) networks,

and the polynomial classifier (PC) [16].

PCs can be described as higher-order neural networks

which consist of a single-layer network with the polyno-

mial terms of patterns feature as inputs [16]. The poly-

nomial classifiers are learning algorithms proposed and

adopted in recent years for regression, classification and

recognition with significant properties and generalization

capability [16].

Due to their need for less training examples and far

less computational requirements, PCs have shown supe-

rior performance to multilayer neural networks. Polyno-

mial classifiers have been successfully used for various

pattern recognition and signal detection applications in-

cluding speech and speaker recognition [17,18]. Another

recent study on the recognition of sign language alphabet

using polynomial classifiers delivered superior recogni-

tion results [19] over Adaptive Neurofuzzy Inference

systems (ANFIS). More applications of PCs have been

reported in modeling of tool wear [20,21] and noninva-

sive fetal ECG extraction.

This paper describes the use of the Polynomial Classi-

fiers (PC) to predict stock prices in the Dubai Financial

Market in the United Arab Emirates and compares the

results with those obtained by using Artificial Neural

Networks ( ANN).

It is worth mentioning here that, at the time of this re-

search, literature review revealed that there is no reported

research that applied the polynomial classifiers for the

prediction of stock price movements. In addition, there

were no research found that reported using neural net-

work to predict stock price movements in Dubai Finan-

cial Market.

2. Dubai Financial Market (DFM)

In the last years of the previous century, the United Arab

Emirates (UAE) became one of the fastest developing

countries in the Middle East and South Asia. Dubai,

which is considered the commercial capital and the cen-

ter of international business in the UAE, has taken the

leadership in developing and modernizing both govern-

mental and private sectors with state-of-the-art strategies,

policies, technologies and infrastructu re. One of the fast-

est and most powerful growing sectors in the UAE is

finance.

On March 26th, 2000, Dubai Financial Market com-

menced operation with listing shares of seven companies

and ten brokers. DFM has grown rapidly and has scored

noticeable records in terms of trading volume and market

values. At the time of this study, more than 50 companies

are listed with the DFM and it is considered one of the

leading financial markets in the Middle East.

Two of DFM’s leading stocks were selected for this

research; Emaar Properties (EMAAR) and Dubai Islamic

Bank (DIB). EMAAR is the leading real estate developer

in the Middle East and DIB the world’s first fully-

fledged Islamic bank. These stocks were chosen because

they have sufficient historical data, actively traded, and

each to represent different sectors in the UAE economy.

The data used for both stocks is the closing prices cover-

ing the period of April 2000 to March 2006 (total of 2176

data points). These data points are the daily closing stock

prices in the currency of the United Arab Emirates Dir-

ham (AED). The AED is tied with the USD with a con-

version rate of approximately 1 USD = 3.67 AED.

Predicting Stock Prices Using Polynomial Classifiers: The Case of Dubai Financial Market

3. Background of Modeling Techniques

In this work we have used Artificial Neural Networks

(ANN) and Polynomial Classifiers (PC) as modeling

techniques to predict stock prices from historical price

data. The ANN that we used is the standard feed forward

architecture trained with the standard back propagation

method. Since this architecture is widely used and very

well known we felt th at it would be redundant to exp lain

it here. The reader can refer to [2] for more details about

feed forward back propagation ANN. On the other hand

polynomial classifier being used for the first time for this

application is explained here with adequate details.

3.1. Polynomial Classifiers

Polynomial Classifiers (PCs) provide an efficient method

for describing non-linear input/output relationships. To

model a sequence of input/output data using polynomial

classifiers, a set of parameters, also referred to as weights,

are determined such that the multi-dimensional feature

vectors, which represent the input sequence, are best

mapped to the corresponding output sequence. For the

problem at hand (i.e. stock price prediction) our input is a

sequence of feature vectors formed from the historical

prices of the stock being studied. For each of the input

feature vectors the corresponding output is the future

price of t he s tock.

Consequently, the training data is arranged into a N

d-dimensional feature vectors arranged in the N × d ma-

trix X, along with its corresponding N dimensional col-

umn vector tx representing the target vector (future

prices). For instance, to predict the next day price from

the previous d days we arrange the given training time

series of di mension N + d + 1 data as follows.

112

223

NN Nd

xx x





 

 



 

 























A Kth order polynomial classifier consists of Kth order

polynomial expansion of the d-dimensional feature vec-

tor, x, resulting in a higher dimensional vector, p(x). For

example, if





112

xx, then its second order polyno-

mial expansion results in the following 6th dimensional

vector.



121 212

xxxxx





Hence, for a sequence of N: d-dimensional feature vec-

tors, the Kth order polynomial expansion results in the

Nx6 matrix M; where,

 











N

Mpxpx px

The polynomial classifier is then trained using the ex-

panded v ectors, M, to approximate the target using mean

squared error as its decisive factor to determine its opti-

mum syste m parameters (weights), w; such as

argmin x

wMwt

which results in the well-known solution





wMMMt

On testing the trained model, an unknown feature

vector, z, is expanded to its polynomial terms, p(z) and

by using the previously obtained weights the corre-

sponding output (predicted future price) tz, is obtained as

follows.



ztwpz

4. Methodology

4.1. Training Scenarios

In this research, the neural network and polynomial clas-

sifiers prediction model are trained on three different

training scenarios. These scenarios refer to the number of

data points used for training and for validation. In sce-

nario #1, 1/3 of the data is used for training and the other

2/3 is used for validation, while in scenario #2, 1/2 of the

data is used for training and the other 1/2 is used for va-

lidation. In scenario #3, 2/3 of the data is used for train-

ing and the remaining 1/3 is used for validation.

The portion of data used for training in the three sce-

narios consists of data points captured from different

periods throughout the available historical data. For ex-

ample, in scenario #1, if five hundred data points to be

used for training and the remaining 1000 points for vali-

dation, then these points will not be taken from one block

of the historical data. Instead, several blocks from dif-

ferent periods will form the total five hundred points by

taking the first hundred from early data stream, the next

two hundred from some middle data, and the last two

hundred will be somewhere around most recent data.





This technique is used to assure that both the NN and

the PC prediction methods will learn different market

situations and various price patterns in the various mar-

ket conditions of bear market (period of decline), con-

solidating market (period of neither growth nor decline),

and bull market (per iod of growth).

Predicting Stock Prices Using Polynomial Classifiers: The Case of Dubai Financial Market85

4.2. Prediction Modes

Two modes of prediction will be used for predicting

stock prices in this study.

Mode 1: Predicting the next day given the five preced-

ing days:

In this mode, the inputs to the model are the closing

prices of the previous five trading days, and the desired

output (target) is closing price for the following trading

day.

51nn







Mode 2: Predicting the next three days given the

twelve preceding days:

Similar to the previous mode, however in this mode,

the inputs are increased to cover the previous twelve

trading days, while the desired output still be the closing

prices for the following three trading days.

111314 15

211415 16

1432 1nnnn

dddd

dd ddd







4.3. Implementation of Prediction Models

In this study, two techniques were used for predicting

stock prices. The first technique is Neural Networks and

the second technique is Polynomial Classifiers. The same

inputs are used for both techniques; namely, the histori-

cal stocks daily closing prices as explained in Section 4.2.

Likewise, the desired outputs (targets) for both tech-

niques are the corresponding future prices according to

the prediction modes described in Section 4.2.

The criteria used to measure the prediction accuracy

for both the NN and PC is related to the difference be-

tween the predicted closing prices and the actual closing

prices, as follows.

Let,

= Actual closing price for day nine AED

= Predicted closing price for day nine AED

= Number of days in validation



= prediction error for day nine AED

Then the prediction error for day n is given by





while the prediction error percentage relative to the ac-

tual closing price for day n is g i ven by

%100





Additionally, the mean absolute error (MAE) in AED

over N days is given b y





,

and the mean absolute error percen tage (MAEP) is given







4.3.1. N eu ral Network Predi ction Technique

The neural network architecture used here is the multi-

layer feed forward architecture with one input layer, one

hidden layer and one output layer. The training method

used here is the standard back propagation method. The

network is implemented via MATLAB and the training

parameters were set properly to insure convergence for

the two prediction modes explained in Section 4.2. For

each mode, different ANN models were developed for

the three training scenarios as explained in Section 4.3.

Consequently, for all the combinations of the three train-

ing scenarios and the two prediction modes, 6 different

ANNs we r e g enerated .

4.3.2. Polynomial Classifier Prediction Technique

The inputs and targets used in training the polynomial

classifier technique were identical to those used in the

neural network technique following the three training

scenarios and the two prediction modes. However, for

each one of the 6 combinations two PCs were created;

one is a 1st order PC and the other is a 2nd order PC. The

results obtained from the different ANN and PC combi-

nations are reported in th e following section.

5. Experimental Results and Discussion

5.1. Neural Network Prediction Model

The three training scenarios with two prediction modes

were applied to EMAAR and DIB stocks. Following are

the results obtained for Mode 1 and Mode 2 for both

stocks under study.

5.1.1. Mode 1: Predicting the Next Day Given the

Previous Five Days

5.1.1.1. Emaar Properties (EMAAR)

Figure 1 shows that ANN achieves a small mean abso-

lute error percentage (MAEP) in all three training sce-

narios. As expected, the MAEP decreases as the amount

of training data is increased. However, the MAEP is still

acceptable even with using only 1/3 of the data for train-

ing (i.e. training scenario #1).

It is worth noting here that the distribution of the pre-

diction error (in AED) is found to have a small spread

around its average which is close to zero. An example of

the distribution of prediction error is depicted in Figure 2

Predicting Stock Prices Using Polynomial Classifiers: The Case of Dubai Financial Market

Figure 1. MAEP vs. the amount of training data using ANN.

Figure 2. Prediction error histogram; ANN trained on 1/3

of data—Mode 1, Scenario #1 for EMAAR stock.

which shows the histogram of the prediction error over

the entire validation data of scenario #1.

Further analysis of the histogram of the prediction er-

ror in Figure 2 is summarized in Table 1. Ta b l e 1 shows

that the prediction error percentage εn% is bounded to

±1% for about one fifth of the validation data. It also

shows that 97.13% of the pr edicted prices are within εn%

= 5% of the actual prices over the entire validation data.

Moreover, 99.92% of the validation data was predicted

within εn% = 10%. Similar analyses were done for the

other two training scenarios and the results are summa-

rized in Table 1. Clearly, increasing the training data

makes results in a narrower distribution and hence better

price prediction.

5.1.1. 2 . Dubai Islamic Bank (D IB)

Same analyses were done to DIB for the case of predic-

tion Mode 1, and the results are summarized in Table 2.

The overall network performance on DIB was superior to

its performance on EMAAR especially for the interval

from –0.01 to 0.01 where there is a noticeable improve-

ment for all the three training methods.

The improved results shown in Table 2 for DIB could

be attributed to the less volatility o f DIB stock compared

to EMAAR.

Table 1. Intervals of εn% using ANN, prediction Mode 1 for

EMAAR stock.

Percentage of the validation data within the interval

εn% Scenario #1 Scenario #2 Scenario #3

–1% to 1% 21.52% 36.38% 49.61%

–5% to 5% 97.13% 97.02% 96.85%

–10% to 10%99.92% 99.68% 100.00%

Table 2. Intervals of εn% using ANN, prediction Mode 1 for

DIB stock.

Percentage of the validation data within the interval

εn% Scenario 1 Scenario 2 Scenario 3

–1% to 1% 58.14% 72.59% 67.40%

–5% to 5% 97.01% 97.04% 96.22%

10% to 10%99.35% 99.26% 99.37%

5.1.2. Mode 2: Predicting the Next Three Days Given

the Previous Twelve Days

In this mode the number of previous days used for pre-

diction is increased since the number of predicted days is

increased to 3 instead of 1. In this mode, two weeks of

historical closing prices plus two days in the third week

were used as input to the ANN. This means that we are

predicting the last three days of the third week given the

previous two weeks and the first two days of the third

week. The method was applied to the two stocks under

study.

Table 3 shows the results of the ANN for the three

training scenarios to predict the 13th, 14th, and 15th day

when given the closing prices of the previous 12 days for

EMAAR data. As expected, the ANN performs better as

the training data increases.

The results obtained in Table 3 suggest that increasing

the number of days in the historical data beyond 5 does

not yield a better prediction in the next day from the pre-

vious 12 days (Day 1). Mor eover, pr ed icting the pr ices of

the 14th and the 15th days (Day 2 and Day 3) yields sig-

nificantly poorer results than those of predicting Day 1.

Similar trends in the prediction results were observed

when the NN was applied to DIB stock.

5.2. Polynomial Classifiers Prediction Model

The inputs used in the NN model were used for the Po-

lynomial Classifiers model. The model was trained and

tested on EMAAR and DIB. Fir st and second order clas-

sifiers were examined. Below is a summary of the results

of the Polynomial Classifiers compared to the results of

Predicting Stock Prices Using Polynomial Classifiers: The Case of Dubai Financial Market

Neural Network.

5.2.1. Mode 1: Predicting the Next Day Given the

Previous Five Days

The MAEP results of the 1st order as well as the 2nd

order Polynomial Classifiers for predicting the closing

price of the 6th day given the closing prices of the pre-

vious five days (Mode 1) are shown in Table 4. The ta-

ble also shows the MAEP for the ANN. The table sug-

gests that the MAEP results for ANN, 1st order PC, and

2nd order P C are co mparable an d there seems no signifi-

cant advantage of the PC over ANN. However, it should

be noted that PC is far more computationally advanta-

geous over ANN since the training process is done

non-iteratively with no concern regarding convergence.

5.2.2. Mode 2: Predicting the Next Three Days Given

the Previous Twelve Days

In the case of Mode 2 the closing prices of three con-

secutive days ar e predicted from the closing prices of the

previous 12 days. As shown in Table 5, EMAAR’s re-

sults indicate that PC yields better results (lower MAEP)

than ANN. Once again the MAEP decreases as the train-

ing data increase and the MAEP is best for Day1 and

worst for Day 3 for all training scenarios.

Unlike EMAAR’s results shown in Table 5, the DIB

stock results show no advantage of PC over ANN as

shown in Table 6.

Table 3. Intervals of εn% using ANN, prediction Mode 2 for EMAAR stock.

Percentage of the validation data within the interval

Training Scenario #1 Training Scenario #2 Training Scenario #3

εn% Interval

Day 1 Day 2 Day 3 Day 1 Day 2 Day 3 Day 1 Day 2 Day 3

–1% to 1% 17.61% 7.23% 5.98% 29.63% 12.09% 7.63% 46.10% 20.85% 15.59%

–5% to 5% 95.10% 58.80% 28.41% 96.51% 83.55% 56.86% 96.95% 88.47% 78.47%

–10% to 10% 99.75% 96.84% 84.72% 99.78% 98.04% 94.99% 99.83% 98.31% 96.78%

Table 4. Mode 1—MAEP using NN and PC for EMAAR and DIB.

EMAAR DIB

Scenario #1 Scenario #2 Scenario #3 Scenario #1 Scenario #2 Scenario #3

NN 1.86% 1.56% 1.40% 1.22% 1.04% 1.18%

1st order PC 1.37% 1.17% 1.28% 1.25% 1.04% 1.18%

2nd order PC 1.74% 1.32% 1.32% 1.38% 1.33% 1.24%

Table 5. Mode 2—MAEP using NN and PC for EMAAR.

EMAAR

Scenario #1 Scenario #2 Scenario #3

day 1 day 2 day 3 day 1 day 2 day 3 day 1 day 2 day 3

NN 2.33% 4.68% 6.73% 1.73% 3.36% 4.81% 1.46% 2.61% 3.58%

1st order PC 1.38% 2.52% 3.32% 1.12% 2.01% 2.79% 1.32% 2.26% 2.91%

2nd order PC 1.71% 3.45% 4.73% 1.30% 2.42% 3.23% 1.37% 2.17% 2.72%

Table 6. Mode 2—MAEP using NN and PC for DIB.

DIB

1/3 1/2 2/3

day 1 day 2 day 3 day 1 day 2 day 3 day 1 day 2 day 3

NN 1.28% 2.24% 3.04% 1.05% 1.78% 2.32% 1.19% 1.92% 3.49%

PC 1st 1.29% 2.42% 3.11% 1.21% 1.82% 2.39% 1.26% 1.94% 2.49%

PC 2nd 1.38% 2.51% 3.18% 1.38% 2.44% 2.98% 1.34% 2.15% 2.77%

Predicting Stock Prices Using Polynomial Classifiers: The Case of Dubai Financial Market

6. Conclusions

Two prediction models developed in this study. The first

model was developed with the well-known back propa-

gation feed forward neural network. The second model

used here is based on polynomial classifiers which are

being used for the first time in stock prices prediction.

The inputs to both models were identical, and bo th mod-

els were trained and tested on the same data in three dif-

ferent training scenarios and two prediction modes. The

data used here is the historical p rices for two of the lead-

ing stocks in Dubai F i n ancial Market.

In general, both models achieved outstanding resu lts in

terms of mean absolute error percentage (MEAP). Both

models achieved around 1.5% MEAP in predicting the

next day, 2.5% MEAP in predicting the second day, and

around 4% MEAP in predicting the third day. The pre-

diction accuracy of the two models was certainly re-

markable, where around 60% of the predicted prices of

the first day, 50% of the predicted prices of the second

day, and 35% of the predicted prices of the third day,

were all within –1% to 1% of the actual prices of the

three days.

When comparing the neural network and polynomial

classifiers prediction models, it was found that first order

polynomial classifier performed comparable to or slight-

ly better than the neural network. Whereas the second

order polynomial classifier could barely achieve similar

results on the stocks used in this study. Further work can

be done using other stocks in similar emerging markets

and mature markets, to verify this conclusion.

On the other hand it should be noted that PC is a lot

more computationally efficient than ANN since its

weights can be obtained directly an d non-iteratively fro m

a closed formula as shown in Section 3.1.

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