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.
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 84
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
1
d
d1
N
NN Nd
xx x
xx x
xx x


 
 
 

 
 
 




x
x
X
x
1
2
1
d
d
x
Nd
x
x
x










t
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
x
xx, then its second order polyno-
mial expansion results in the following 6th dimensional
vector.

22
121 212
1
x
xxxxx

 px
Hence, for a sequence of N: d-dimensional feature vec-
tors, the Kth order polynomial expansion results in the
Nx6 matrix M; where,
 
T
12





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
2
argmin x
w
wMwt
which results in the well-known solution

1
TT
x

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
6
7
n
d
d
d
n
d
d
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.
15
26
51nn
dd
dd
dd








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
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,
n
P
ˆ
P
= Actual closing price for day nine AED
n
N
= Predicted closing price for day nine AED
= Number of days in validation
n

= prediction error for day nine AED
Then the prediction error for day n is given by
ˆ
nn
PP  
n
,
while the prediction error percentage relative to the ac-
tual closing price for day n is g i ven by
%100
n
n
n
P



Additionally, the mean absolute error (MAE) in AED
over N days is given b y
1
1N
n
n
N



,
and the mean absolute error percen tage (MAEP) is given
by
1
1
%%
N
n
n
N



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 86
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 87
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 88
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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