Time series prediction is the problem of determining the future values of a
system from the past and current data points. Time series prediction models
are broadly used in financial area, such as stock marketing, foreign currency
exchange rates and so on (Mukherjee et al., 2011a;
Raja and Selvam, 2011; Mukherjee
et al., 2011b). In many situations, complexity of incomplete information
has been taking placed. In other word some random factors such as time-varying
parameters, social and economic factors and so on, make it difficult to obtain
an accurate model of time series systems (Ekpenyong, 2008;
Lin et al., 2006; Uysal,
2007). In present study a grey prediction model has been proposed to improve
the above problem.
Grey system theory, developed originally by Ju-Deng (1982).
Main idea of this theory is to study uncertainty of system with small amount
of data and incomplete data. It avoids the inherent defects of conventional
methods such as probability theory and mainly works on poor, incomplete or uncertain
data to estimate the behavior of uncertain system or a time series (Li
et al., 2007).
In system theory, a system called a white system if its information is completely
known and it called black system if its information is completely unknown. A
grey system is a system with both known and unknown information (Kayacan
et al., 2010).
The grey system has been successfully applied to systems analysis, prediction,
data processing, modeling, decision making and control. The Grey Model (GM)
based on the grey system theory is a forecasting dynamic model and has been
widely used in many applications (Huang and Jane, 2009;
Hsu and Chen, 2003; Kayacan et
al., 2010; Kung, 2005; Li
and Wang, 2011). Grey prediction is able to consider as a curve fitting
approach that has extremely good performance for real world data.
One of most important time series is gold price. Gold is an essential traded
product and forecasting its price, has theoretical and practical significance
(Ismail et al., 2009). The information restricted
in this study is believed to be valuable for financial managers. In fact, the
knowledge of oncoming gold price allows financial actors to make best decision
on trading. The accurate prediction of gold price has been considered in this
study. For this purpose, two GM model include ordinary GM (1, 1) model and Fourier
series modified GM (1, 1) model have been presented. Also, the performance of
above GM models has been compared with ARIMA as a Conventional model.
FUNDAMENTAL OF GREY THEORY
The GM (1, 1) model: The grey prediction is based on GM (n, m) where n is the order of grey difference equation and m is the number of variables. Among the family of grey prediction model, most of the pervious researchers have focused on GM (1, 1) model in their predictions. GM (1, 1) model ensure a fine agree between simplicity and accuracy of the results.
A non-negative sequence of raw data as:
where, n is the sample size of data.
Accumulating Generation Operator (AGO) is used to smooth the randomness of primitive sequence. The AGO converting the original sequence into a monotonically increasing sequence. A new sequence X(1) is generated by AGO as:
The generated mean sequence of x(1) is defined as:
where, z(1) (k) is the mean value of adjacent data, i.e.,
The GM (1, 1) model can be constructed by establishing a first order differential equation for x(1) (k) as:
The solution, also known as time response function, of above equation is given by:
denotes the prediction x at time point k+1 and the coefficients [a,b]T
can be obtained by the Ordinary Least Squares (OLS) method:
Inverse AGO (IAGO) is used to find predicted values of primitive sequence. By using the IAGO:
Therefore, the fitted and predicted sequence
is given as:
are called the GM (1, 1) fitted sequence while:
are called the GM (1, 1) forecast values.
Improved grey prediction model: Residual modification model of GM (1,
1) developed as the difference between the real values, x(0) and
the model predicted values, .
The residual sequence has been denoted as:
The residual GM (1, 1) model has been developed to improve the modeling accuracy of original GM (1, 1) model. In this study the fourier series has been used to modify the grey system.
Fourier series modification of residual model: The error residual in Eq. 14 can be expressed in Fourier series as:
where, T = n-1,
and the result only take integer number (Guo et al.,
Eq. 15 can be rewritten as:
where, P and C matrixes are defined as:
By using ordinary least squares method, Eq. 16 can be solved and matrix C has been calculated as:
Therefore, the Fourier series modification can be denoted as follow:
Error analysis and validation: Prediction accuracy is a vital criterion
for evaluating forecasting authority. In present study Absolute Mean Percentage
Error (AMPE) criterion has been used to estimate model performances and reliability.
AMPE is a general accepted percentage measure of prediction accuracy (Ismail
et al., 2008). This indicator is calculated as:
To show the efficiency of proposed method, Gold Prices (GP) from the London PM Fix (Noon fixing time) has been used in this study. The prediction of the gold price is a very important topic in financial area. Accurate prediction of gold price is very difficult. In fact because of the time-varying statistical properties of the gold price, mathematical modeling of its behavior is not easy. Figure 1 shows gold price between the dates 01.02.2011 and 28.04.2011. According to Fig. 1, the data are highly nonlinear, noisy and nonstationary.
In order to compare with above models, the most widely used ARIMA (1, 0, 0)
model (Han, 1994) as a conventional model, has been
presented in this study. Table 1 shows the AMPE values for
original GM (1, 1) model (OGM), Fourier series modified GM (1, 1) model (FGM)
and ARIMA model prediction. The AMPE of the OGM model, FGM model and ARIMA model
for interpolation data are 0.3212, 0.1531 and 0.4380%, respectively. Also, the
AMPE of the OGM model, FGM model and ARIMA model for extrapolation data are
0.5621, 0.2101 and 0.5510%, respectively. So, FGM (1, 1) model has better performance
when compared to OGM model and ARIMA model as expected. Also, the performance
of interpolation is giving better in compared with extrapolation.
|| Gold price chart as data set
|| The accuracy of the studied models
Gold price prediction is an important factor in economic decision making. In the other hand, grey prediction model is a widely used forecasting model that has been applied to many forecasting fields. In present study the performance of the various prediction models include Original GM (1, 1) model, Fourier series modified GM (1, 1) model and conventional ARIMA method have been compared in gold prediction forecasting area. Results of the study demonstrate that modified GM(1,1) model has better performances not only on model fitting but also on model forecasting.