Ali Ebrahimi
Entekhab Industial Group, Iran, Islamic Republic of
E-mail: aliebrahimi64@gmail.com
Submission: 23/09/2018
Revision: 25/09/2018
Accept: 23/10/2018
ABSTRACT
Every player in the market has a
greater need to know about the smallest change in the market. Therefore, the
ability to see what is ahead is a valuable advantage. The purpose of this
research is to make an attempt to understand the behavioral patterns and try to
find a new hybrid forecasting approach based on ARIMA-ANN for estimating
styrene price. The time series analysis and forecasting is an essential tool
which could be widely useful for finding the significant characteristics for
making future decisions. In this study ARIMA, ANN and Hybrid ARIMA-ANN models
were applied to evaluate the previous behavior of a time series data, in order
to make interpretations about its future behavior for styrene price.
Experimental results with real data sets show that the combined model can be
most suitable to improve forecasting accurateness rather than traditional time
series forecasting methodologies. As a subset of the literature, small number
of studies has been done to realize the new forecasting methods for forecasting
styrene price.
Keywords: ARIMA; Hybrid ARIMA-ANN; Artificial neural networks; Time series
forecasting
1. INTRODUCTION
Styrene
is an unsaturated liquid hydrocarbon gained as a petroleum by-product. Also
known as ethylbenzene, vinyl benzene, and phenyl ethane that is an organic
compound with the chemical formula C6H5CH=CH2 (ENERGY,
2013). It is easily polymerized and is used to make plastics and resins (YOUSIF, 2013). A few of the most familiar uses
of styrene include: Solid and film polystyrene, used in rigid foodservice
containers, CD cases, appliance housings; Polystyrene foam, used in food
service products and building insulation; Composite products, used in tub and
shower enclosures and many other applications
(CIECIERSKA, 2015).
The
miscellaneous types of factors have been causing directly on the price of the
styrene move up and down; exchange rates, feedstock price increases, strong
demand, poor availability and even storms in the US (SHERMAN, 2010). These lead to an extensive risk and ambiguity in
the process of price forecasting. The key player needs reliable forecasts of
predictable styrene price in order to make the correct decision. This results
in increased price variability and accords prominence to reliable price
forecasting methods. The styrene price forecasts are so important for the
manufacturer in order to production planning and marketing planning decisions
on the predictable prices, which may have financial impacts in future months.
The
price data analysing has become more complex under the modern market today. The
main goal of the study is to find a technique that improves the forecasting
estimate. Therefore, numerous types of methods and methodologies can be used.
Time series forecasting is an imperative area of forecasting in which past
observations of the same variable are collected and analysed to develop a model
describing the underlying relationship (ZHANG,
2003).
Time
series based models like Autoregressive Integrated Moving Average (ARIMA) and
Artificial Neural Networks (ANN) are favoured when time series of historical
prices are used. The styrene price data contain both linear and nonlinear
patterns, no single model is capable to identify all the characteristics of
time series data. Thus, in this study, ARIMA, ANN time series models and hybrid
of both ARIMA and ANN models were used to model and forecast the price of
styrene.
These
methods undoubtedly lack the ability to catch all price spikes or price
decreases in order to changes in important drivers but offer a way to have a
constant expectation of tomorrow’s power prices
(Ozozen, 2016). By merging different methods, the problem of model
selection can be made better with little extra work. There is a long discussion
about which methods are more efficient. Although more than a few comparative
studies have been labelled in the literature, the findings do not suggest what
circumstances make a method better than another. Therefore, situations of
difficulty, seasonality, and perishability, require analytical studies about
the most suitable method for each condition.
This
paper is organized as follows. In the next section, we review the ARIMA and ANN
modelling approaches to time series forecasting. The new proposed combination
methodology explains in Section 3. Empirical results from real data sets are
reported in Section 4. Section 5 contains the concluding remarks and future
work respectively.
2. THEORETICAL BACKGROUND
2.1.
Styrene
Figure
1 shows the main method of making styrene from ethane
(derived from natural gas reserves) or naphtha (a product mainly derived from
crude oil). Ethane (or naphtha) with steam is fed into the cracker unit where
ethylene and coproducts (propylene, butadiene, benzene, etc.) are made. The
ethylene and benzene are then further processed (catalytic alkylation) to make
ethylbenzene from the cracker. This is then fed into a dehydrogenation reactor
to make styrene (with minor coproduct benzene and toluene). The styrene is then
typically piped to other chemical plants where it is further processed into
derivative products such as polystyrene (ICIS,
2015).
Figure 1: Manufacturing Process of Styrene
Monomer
Source:
Miyashita (2012)
2.2.
ARIMA
Model
The
ARIMA (Autoregressive Integrated Moving Average) model was popularized by
George Box and Gwilym Jenkins in the 1970s, with use in time series analysis
and forecasting. The underlying theories described by Box and Jenkins and later
by Box, Jenkins and Reinsell are sophisticated but easy to understand and apply
(DA VEIGA, 2014). The “I” in ARIMA implies that the dataset undergoes
differentiation and that, upon completion of the modelling, the results undergo
an integration process to produce final predictions and estimates (TULARAM, 2016). It is a
combination of three statistical models. It uses Autoregressive, Integrated and
Moving Average (ARIMA) model for statistical information (JAIN, 2017).
The
ARIMA model has been widely studied and applied in researches of forecast in
order to their attractive theoretical attributes
and because of the many empirical supportive pieces of evidence. In addition,
the ARIMA model has equivalence with most models of exponential smoothing (DA VEIGA, 2014). The ARIMA Model
analyse and Forecasts uniformly spaced univariate statistic information,
transmission of function data, and intercession information that is done by
using Autoregressive Integrated Moving Average (JAIN, 2017).
The
first thing to do in ARIMA is to determine the stationarity. It is a common
assumption in many time series techniques. The probability laws governing the
process do not change over time. The process is in statistical equilibrium.
Besides, a stationary process has the property that the mean, variance and auto
covariance structure do not change over time (RUSIMAN, 2017).
This
method has some interesting features that made it more desirable for
researchers. It eases the forecasting process allowing researchers to use only
single variable time data series while also allow multiple for more complex
cases (BARI, 2015). Also, the main advantage
of this class of models lies in its ability to quantify random variations
present in any economic time series (DAREKAR,
2016).
(NEWAZ, 2008) and (AHMAD, 2013) have applied ARIMA model based on which they
predict can provide well forecasts. In another contribution, (MADDEN, 2007) has addressed the forecasting of
gold prices through ARIMA model and had concluded by proposing that the gold
selling prices are in increasing trends and could be deliberated as a worthy
investment. (AS' AD, 2012) devised ARIMA
Models based past three, six, nine and twelve months of data and advised that
the ARIMA model build based on past three months' data is the best model in
terms of forecasting. (LIM, 2001) and (DA VEIGA, 2014) figure out that the ARIMA
methodology performs fairly well in their studies.
2.3.
ANN
Model
The
conception of artificial neural networks (ANN) has been used for almost 50
years, only in the late 1980s could one determine that it gained significant
use in scientific and technical performances
(HAMID, 2004). ANN is a mathematical model that has a greatly connected
structure similar to brain cells. A neural network is a machine that is
designed to model the way in which the brain does a specific task. It resembles
the brain in two respects: 1. Knowledge is attained by the network from its
environment through a learning procedure; 2. Interneuron connection strengths,
identified by synaptic weights, are applied to store the acquired knowledge (YADAV, 2017).
Artificial
neural network (ANN) methods have shown great capability in modelling and
forecasting nonlinear and complex time series. ANN offers an effective approach
for handling large amounts of dynamic, nonlinear, and noise data (SHABRI, 2014). Neural networks, with
their incredible ability to derive meaning from complex or imprecise data, can
be used to extract patterns and identify trends that are too complex to be
observed by either humans or other computational methods. Other advantages
include 1. Adaptive learning: An capability to learn how to do tasks based on the
data set as training or initial experience. 2. Self-Organization: An ANN can
create its own organization or representation of the information it receives
during learning time. 3. Real-Time Operation: ANN computations may be carried
out in parallel, and special hardware devices are being designed and
manufactured which take advantage of this capability (YADAV, 2017).
ANN
is developed based on biological neural networks that neurons are the
fundamental building blocks ones. An artificial neuron is a model of a
biological neuron as you see in Figure 2. An artificial neuron receives signals
from other neurons, gathers these signals, and when fired, transmits a signal
to all connected neurons (MOMBEINI, 2015).
Figure 2: Biological model of a neuron
Source:
neuron (2016)
An
ANN model frequently contains three layers: the first layer is the input layer
where the data are announced to the network, the network is executed by using
electric modules or in software on a digital computer simulation. The second
layer is the hidden layer where data are processed, and the last layer is the
output layer where the results of the given input are produced (SHABRI, 2014).
An
important step in the neural network is to train the model to learn the relationship
between input and output parameters. In multilayer perceptron (MLP), weights
are determined by Error Back-Propagation (EBP) algorithms which minimize a
quadratic cost function by a gradient descent method. The interconnecting
weights between the neurons are adjusted based on the inputs and desired output
during the training phase (GODARZI, 2014). The most
popular and successful model is the feed forward multilayer network. Figure 3
shows a three-layer feedforward neural network with a single output unit, hidden units, input units. is the connection weight from the input unit to the hidden unit, and is the connecting weight from the hidden unit to the output unit. In its
applications, the data series is usually divided into a training set (in sample
data) and a test set (out of sample). The training set is used for the
construction of the neural network, whereas the test set is used for measuring
the predictive ability of the model. The training process is used essentially
to find the connection weights of the networks (PAO,
2007).
Figure 3: Three-layer feedforward ANN model
Numerous
papers have already presented the successful application of ANN for modelling
and price forecasting. (MOVAGHARNEJAD, 2011) used ANN and a
time variable as a constant variable; thus the dynamic nature of the process
was not accounted for. (YU, 2008) stated that the
performance of the ANN is superior to various traditional statistical models.
ANN has a capability to learn complicated and nonlinear time series that is
problematic to model with conventional models. However, there are some
weaknesses of ANN. (YOUSEFI, 2015) concluded that
using wavelet transform can enhance the forecasting accuracy when it is
compared with a regular neural network prediction algorithm. (KHASHEI, 2011) concluded that Although ANN
has advantages of accurate forecasting, their performance in some specific
situation is inconsistent. (AHMAD, 2001)
and (PAO, 2007) and (PANELLA, 2012) show that artificial neural
network provided more accurate predictions.
2.4.
The
Hybrid Methodology
In
the past two decades, ANN and ARIMA (ANN-ARIMA) techniques widely applied to
achieve high accuracy forecasting's in linear and non-linear domains
respectively; especially to forecast financial data onto the different type of
economic and financial conditions (RATHNAYAKA,
2015).
Nevertheless,
none of them is a general model that is appropriate for all circumstances. The
estimate of ARIMA models to complex nonlinear problems might not be acceptable.
On the other hand, using ANNs to model linear problems have yielded mixed
results (ZHANG, 2003). For instance,
via simulated data, (DENTON, 1995) indicated that
when there are outliers or multi-collinearity in the data, neural networks can
significantly outperform linear regression models.
(ZHANG, 2003) stated that it is more
effective to combine individual forecasts that are based on different
information sets. Also (NAVEENA, 2017) concluded that
the hybrid method which combines linear and non-linear models can be an
effective way to improve forecasting performance. (RATHNAYAKA, 2015) suggested that the hybrid model is more
significant and gives the best solution for predicting future predictions under
the high volatility fluctuations than traditional forecasting approaches.
3. DATA AND METHODS
The
well-known datasets - Reed Business Information (ICIS) - is used in this study
to determine the effectiveness of the hybrid method. ICIS is the world's
biggest petrochemical market data provider with divisions spanning energy and
fertilizers. The data we collected from the official websites of ICIS contains
the daily number of styrene prices from 11/21/2005 to 9/19/2017, providing a
total of 3002 sample values. This data is selected from the Chinese market,
which is called "Styrene CFR China USD/tonne". The reason for
choosing this country is its deep impact on global styrene prices.
3.1.
Autoregressive
Integrated Moving Average process (ARIMA)
A time series forecasting method,
ARIMA, is preferred in such markets and fast results are achieved. ARIMA is
defined as follows:
An autoregressive
process with the order is predicted
value at t time, history data
at (t-p) time and is estimated parameter:
Moving Average models provide forecasts based on previous
forecasting errors where can be predicted at time t with approximate error at the time
(t-q) and is an estimated parameter :
[1]
Both of them are
combined together with an estimated stationary parameter of :
[2]
Then, Autoregressive
Integrated Moving Average (ARIMA) is attained by considering the differences in
time becomes:
[3]
Adding the
seasonality factor D and dependence on the average it becomes SARIMA by
assuming that the time series is distributed normally with:
[4]
3.2.
Artificial
Neural Network (ANN) Model
The
ANN model for a specific problem in time series prediction includes the
definition of the number of layers and the total number of nodes in each layer.
The one hidden layer feedforward ANN with one output node is most frequently
used in forecasting uses. The ANN
is defined under eight-steps as follows: Variable Selection - Data collection -
Data preprocessing - Training, testing, and validations - Define Network
paradigms (Hidden layers, Hidden neurons, Output neurons) – Evaluation -
Training (Number of iterations and learning rate) Implementation.
The
connection between the output () and the inputs has the following mathematical representation:
[5]
where and are the
model parameters often called the connection weights; p is the number of input
nodes and q is the number of hidden nodes.
[6]
where is a vector
of all factors and is a
function defined by the neural network structure and
connection weights. Therefore, the neural network is equal to a nonlinear
autoregressive model. Note that expression (6) implies one output node in the
output layer which is typically used for one-step-ahead forecasting.
3.3.
Hybrid
ARIMA-ANN model
It
might be rational to consider a time series to be consist of a linear
autocorrelation and a nonlinear component. That is,
[7]
Which represents the linear component and represents
the nonlinear component. These components have to be estimated from the data.
First, we allow ARIMA to model the linear component, in the following, the
residuals will contain the nonlinear relationship. Let represent the residual at time from the
linear model, then
[8]
where is the forecast value for time from the
estimated relationship. Residuals are imperative in the analysis of the
adequacy of linear models. A linear model is insufficient if there are linear
correlation structures left in the residuals. Any important nonlinear pattern
in the residuals will show the limitation of the ARIMA. The nonlinear relations
could be revealed by modeling residuals using ANNs. With input nodes,
the ANN model for the residuals is as follows:
[9]
where is a
nonlinear function determined by the neural network and is the
random error. A reminder that if the model is
inappropriate, the error is not necessarily random. Therefore, the correct
model identification is critical. The combined forecast will be:
[10]
The widespread forecasting evaluation methods like
root mean squared error (RMSE) and mean absolute percentage error (MAPE) were
applied to estimate the above models.
4. RESULT AND DISCUSSION
The
price series on styrene covered weekdays data from January 1995 to February
2016. Figure 4 shows the time series plot of weekdays price of styrene and the
pattern of the graph is an upward trend with a seasonal pattern. A perusal of
the figure reveals a positive trend that specifies the time series
non-stationary.
Figure 4: The Time Plot For Styrene Price of China FOB
An
ARIMA model was attempted using the XLSTAT. The model was then used to forecast
14 days out-of-sample set. Using XLSTAT add-ins in Excel 2016, the ARIMA model
was estimated. After several times trying, ARIMA (1,1,2) (0,0,0) model was
obtained to be the finest among the other family of ARIMA models. ARIMA Model
goodness and parameters are given in Table 1 and Table 2.
Table 1: Estimate of the ARIMA Model
Goodness for styrene price
Goodness of fit statistics: |
|
Observations |
2640 |
DF |
2636 |
SSE |
505525.9087 |
MSE |
191.4870866 |
RMSE |
13.83788592 |
WN Variance |
191.4870866 |
MAPE(Diff) |
98.23328068 |
MAPE |
0.76126304 |
-2Log(Like.) |
21364.81687 |
FPE |
191.6322076 |
AIC |
21372.81687 |
AICC |
21372.83205 |
SBC |
21396.331 |
Iterations |
501 |
Table 2: Estimate of the ARIMA Model
parameter for styrene price
Parameter |
Value |
Hessian standard error |
Lower bound (95%) |
Upper bound (95%) |
Asympt. standard error |
Lower bound (95%) |
Upper bound (95%) |
AR(1) |
0.652 |
0.171 |
0.316 |
0.987 |
0.049 |
0.555 |
0.748 |
MA(1) |
-0.410 |
0.177 |
-0.757 |
-0.063 |
0.059 |
-0.526 |
-0.295 |
MA(2) |
-0.028 |
0.060 |
-0.146 |
0.090 |
0.019 |
-0.066 |
0.010 |
This
model satisfies the inevitability condition and stationary condition and all
the coefficients were found to be statistically significant at the 1% level of
significance. As well as value, RMSE and MSE are 0.932, 10.837, 117.440
correspondingly at the model fitting stage. The appropriateness of the model
was also adjudicated through the values of Box-Pierce Q statistics (17660.656
i.e.) it found to be unimportant. Thus, generally, we could say ARIMA (1,1,2)
(0,0,0) model shown the acceptable result, among other ARIMA models.
The
information about the Neural network structure indicates that network has an
input layer with one input nodes, two hidden layers with 10,7 hidden nodes and
an output layer with one output node means (1,10,7,1) feedforward network. The
activation function is Sigmoidal at the hidden layer and the output layer. The
error is the total sum of squares error because identity, activation function
is used to the output layer. Figure 5 represents the Actual v/s ANN fitted plot
of styrene price time series.
Figure 5: Actual V/S ANN Fitted Plot of Styrene Price
Time Series
In
the next step, residuals are obtained from the fitted ARIMA model. Figure 6
reveals the ARIMA residuals plot of styrene price time series. The Dickey-Fuller
test and Phillips-Peron test were applied to test the presence of
non-linearity. The results of this test are brought in Table 3 that p-value is
greater than the significance level alpha=0.05, which show that the nonlinear
pattern is existent.
Figure 6: ARIMA Residuals Plot of Price Time
Series
Table 3: Non-Linearity Testing For ARIMA
of Styrene Price Time Series
Phillips-Peron |
Dickey-Fuller |
|
Tau (Observed value) |
-0.118 |
-2.956 |
Tau (Critical value) |
-1.941 |
-3.384 |
p-value (one-tailed) |
0.643 |
0.142 |
alpha |
0.05 |
0.05 |
ANN
model specification for ARIMA residuals of styrene price time series shows that
network has an input layer with one input nodes, two hidden layers with 10,7
hidden nodes and an output layer with one output node means (1,10,7,1)
feedforward network. The activation function applied is Sigmoidal at the hidden
layer and the output layer. The error is the sum of squares error because
identity, the activation function is applied to the output layer. Figure 7
represents ANN plot of residuals of styrene price time series.
Figure 7: ANN plot of residuals of styrene price time
series
Forecasting
performance of different models for styrene price time series in training
dataset as given in Table 4, shows minimum MAPE, MSE, and RMSE value.
Table 4: Forecasting performance of
different models for styrene price time series in training dataset
Criteria |
ARIMA |
ANN |
ARIMA-ANN |
MSE |
117.440 |
18.062 |
16.523 |
RMSE |
10.837 |
4.253 |
4.012 |
MAPE |
8.761 |
6.947 |
4.112 |
To calculate the forecasting
performance last 14 observations of the considered time series was forecasted
employing the offered approach. This approach was compared with the
conventional ARIMA and Zhang hybrid approach (ARIMA-ANN). The results are given
in the following Table 5 as a test set.
Table 5: Forecasting performance of
different models for styrene price time series in testing data set
Date |
Actual |
Forecast |
||
ARIMA |
ANN |
ARIMA-ANN |
||
9/20/2017 |
1365.0 |
1361.77 |
1367.98 |
1365.22 |
9/21/2017 |
1367.5 |
1359.25 |
1369.21 |
1367.86 |
9/22/2017 |
1332.5 |
1345.29 |
1335.47 |
1331.12 |
9/25/2017 |
1330.0 |
1338.74 |
1330.77 |
1330.45 |
9/26/2017 |
1337.5 |
1336.91 |
1335.46 |
1336.10 |
9/27/2017 |
1312.5 |
1330.57 |
1320.21 |
1311.51 |
9/28/2017 |
1277.5 |
1320.89 |
1285.41 |
1277.75 |
9/29/2017 |
1247.5 |
1290.26 |
1255.53 |
1248.23 |
10/2/2017 |
1247.5 |
1285.11 |
1249.72 |
1246.41 |
10/3/2017 |
1247.5 |
1252.31 |
1247.30 |
1248.66 |
10/4/2017 |
1247.5 |
1249.33 |
1250.47 |
1246.47 |
10/5/2017 |
1247.5 |
1248.19 |
1248.55 |
1247.53 |
10/6/2017 |
1232.5 |
1241.61 |
1235.67 |
1232.89 |
10/9/2017 |
1200.0 |
1223.23 |
1231.32 |
1201.55 |
9/20/2017 |
1365.0 |
1270.80 |
1260.44 |
1366.28 |
9/21/2017 |
1367.5 |
1301.55 |
1360.10 |
1367.59 |
Criteria |
MSE |
11.01 |
6.72 |
6.38 |
RMSE |
121.31 |
45.21 |
40.75 |
|
MAPE |
5.47 |
1.80 |
1.02 |
The comparative outcomes for
the best ARIMA, ANN and ARIMA-ANN models are given in the Table 5. MSE, RMSE
and MAPE statistic gives the indication of overall the superiority of ARIMA-ANN
for forecasting of styrene price. Figure 8 represents the Actual v/s ARIMA-ANN
fitted plot of styrene price time series.
Figure 8: Actual v/s ARIMA-ANN fitted plot of Styrene
price time series
5. CONCLUSION
The
price of styrene is important for the companies that produce styrene, as well
as the companies, use it as a raw material. Hence, insight into likely future behaviour
and patterns of styrene prices can help decrease the impacts of styrene price
movements and unpredicted market fluctuations. One of the advantages of price forecasting is to
efficiently manage the resources of an organization so that the organization
can maximize the potential of the products or services produced or offered by
the company.
The price of the styrene mostly
depends on the financial steadiness of the company, turn over value, share
volume and other variety of other financial and economic factors. Therefore,
find the proper forecasting method to predict long/short-term price predictions
is a large challenge today.
The traditional linear time
series models are not always suitable for time series that have both linear and
non-linear structures. As a result, this study generally concentrated attempted
to recognizing the suitable hybrid forecasting approach based on ANN and ARIMA.
The other purpose of predicting the future styrene price is because this
predicted value can be used for future planning. Furthermore, model accuracy
testing results of the mean absolute percentage error (MAPE) and (MAPE [ARIMA
(1, 1, 2)] < MAPE [ANN], MAPE [ ARIMA (1, 1, 2)] < MAPE [ANN]), proposed
that new hybrid method which combines linear and non-linear models can be
proper way to improve forecasting performance.
Lastly, we toughly believed that
the present study makes the important contribution to the manufacturer of
styrene. Achievements of this study also will lead to all the styrene market
role players in Iran. Furthermore, to provide protection for the organization
against the occurrence of negatively-valence events while allowing the
organization to benefit from the occurrence of positively-valence events.
There are certain limitations in
forecasting. It becomes difficult to capture the exact change in case of a
sudden change in the data set (when the variation is large) and in case of the
change in economic instability in the world. Also, some forecasting methods might
use the similar data but provide broadly different forecasts.
In further studies, one can
improve the forecasting accurateness by using some other methods like machine
learning techniques. It is recommended that the proposed method is applied to
different markets/different products and the results shall be compared.
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