IRJET-Forecasting of Time Series Data using Hybrid ARIMA Model with the Wavelet Transform.

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 05 Issue: 06 | June 2018 www.irjet.net p-ISSN: 2395-0072
© 2018, IRJET | Impact Factor value: 6.171 | ISO 9001:2008 Certified Journal | Page 1048
Forecasting Of Time Series Data Using Hybrid ARIMA Model With The
Wavelet Transform.
Jigneshkumar R. Valvi1, Prashant K. Shah2.
1M.Tech. Student, Department of Electronics Engineering, SVNIT Surat, Gujarat, India
2Associate Professor, Department of Electronics Engineering, SVNIT Surat, Gujarat, India
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Abstract - To do accurate forecasting of any process, we
should have to understand the process and according to
that, we can develop the model. This can help to decide the
best possible values of forecast. To do the best forecast, we
can use past values as well as present values. We can divide
available data information into data points with respect to
appropriate time, that data is called as time series data. For
predicting time series data various models are available
that can help to forecast, i.e Stochastic, Neural networks
and SVM based models. All these models have their own
merits and demerits. In this paper, we will talk about
stochastic model (ARIMA) modeling with help of wavelet
transform decomposition; the designed hybrid model is
applied in the application related to finance. For that we
have taken previous six months data (daily basis) of S&P
BSE information technology, based on this available data we
have predicted the 10 days step ahead future values, and
comparative analysis has been done between ordinary
model and proposed models.
Key Words: ARIMA Model, Wavelet transform, Time
series data analysis.
1. INTRODUCTION
The finance is the field that deals study of investments; it
is also called as management of money. A stock exchange
is one subset of finance, where stockbrokers and traders
can buy and sell their shares, in the stock exchange where
and when to invest is a most difficult task, because proper
investment may bring a huge profit, the same chances of
losses, so prediction of accurate future value is more
necessary.
There are various non-linear and linear models are
available in stochastic modeling where, ARCH
(autoregressive conditional heteroskedasticity) GARCH
(generalized autoregressive conditional
heteroskedasticity), are non-linear models and AR
(Autoregressive), MA (Moving Average) ARMA
(Autoregressive Moving Average), ARIMA (Autoregressive
Integrated Moving Average), are examples of some linear
models [1]. As discussed above forecasting models can be
divided into three categories Stochastic, ANN (Artificial
neural networks), SVM (Support vector machine) based.
Among all these models stochastic models have been used
more frequently due to their best forecasting capability
and modelings steps [2]. They help to analysis various
types of time series data. The ARIMA model is used in this
paper due to its fabulous performance for short-term
periods compare to others models [2]. To use these kinds
of linear regression models we have to stationary test.
Only stationary data can be used for the forecast, and if
data is not stationary, we should have to make stationary
by differencing it. In ARIMA ‘I’ suggest integrated which is
same as differencing. In stationary models the statistical
properties mean and variance remains constant while in
non-stationary, these properties are time-dependent.
Linear models are the models which focus on conditional
means while non-linear models focus on conditional
variance.
In the literature, these types of models have been used
not only for economics but it applies for various purposes,
Some of their applications i.e. [3] has used regression
forecasting models to improve the location information,
which helps the user to get accurate information provided
by mobile networks or wireless networks. [4] has used
ARMA model for biomedical signal applications. [5] has
used these models for network traffic modeling. [6]-[7]
has used ARMA model for application to audio coding.[8]
has used ARMA model to avoid disturbances of non-
Gaussian signals from the non-stationary vibration signals.
In [9] classification of a various signal has done by using
different stochastic models at different scale, ARMA model
models the signal at a different scale, and the features at
coarse scales are extracted from the model without
performing expensive filtering operation, which increases
the accuracy of radar signal classification by exploiting
features at different scales.
We have used hybrid ARIMA model with wavelet
decomposition technique in this paper. Wavelet
decomposition which has been used in many research
area such as image processing, signal processing etc.
Usually, all the time series data are not deterministic in
nature; we assume in many cases the series to be
stationary time series. One way to model any time series
to consider it as a deterministic function plus white noise.
Denoising is a method which helps to minimize unwanted
white noise in the time series data, where some
mathematical models such as Fourier transform and
wavelet transform can be applied. As discussed in [10]
wavelet transform is the best method for decomposing
any signal (data) in time series. The wavelet transform

decomposes the data in terms of both time and frequency,
which helps to do accurate and detailed filtering. There
are many more methods available to do prediction,
filtering, and smoothing, among them state space models
are also a good area to do modeling due to Kamal filtering
types approaches; these models are more accurate and
trustable. Later, which can be modeled as ARIMA, ARMA,
etc. in state space domain and various state space domain
features can be applied [11].
In this paper, the wavelet decomposing methods with
the hybrid ARIMA model. The data which we have used
are from S&P BSE information technology, data belongs to
last six months closed data (daily). Data belong to range
from 01-Dec-2017 to 31-May-2018 and analysis has been
done for that data. In between some days are missing to
close, total 124 days analysis has been done; and the best
model has been designed to forecast ten days ahead of
value, with 95% surety range.
2. WAVELET TRANSFORM
A wavelet-transform is transform which provides the time-
frequency representation.it was developed as an
alternative to the STFT (Short time Fourier Transform)
[12]. We pass the time-domain signal from the various high
pass and low pass filters, which filters out either high
frequency or low frequency portions of the signal. This
procedure is repeated, every time some portion of the
signal corresponding to some frequencies being removed
from the signal. Suppose we have a signal with 500 Hz
frequency, in the first stage we split up the signal into two
parts by passing a signal from the high pass and low pass
filter. (Filters should satisfy some certain conditions, so-
called admissibility condition) which results in two
different versions of the same signal, portion of the signal
corresponding to 0-250 Hz (low pass portion), and 250-
500 Hz (high pass portion). Then, we take either portion
(usually low pass portion) or both and do the same thing
again. This operation is called decomposition. Wavelet
transform gives higher frequencies are better resolved in
time, and lower frequencies are better resolved in
frequency. This means that a certain high-frequency
component can be located better in time (with less relative
error) than a low-frequency component. On the contrary, a
low-frequency component can be located better in
frequency compared to high-frequency component. The
wavelet transform was used to adopt a wavelet prototype
function (mother wavelet), which utilized the basic
function ) that is stretched and shifted to capture
features that are local in time and local in frequency. The
above function satisfied following property:
(I)
∫
| ̂ |
Where ̂ is the Fourier transform of this condition is
called admissibility conditions, ensures ̂ goes to zero
quickly as In fact to guarantee that < < , it is
necessary ̂ which is equivalent to
∫
(2.1)
(II) Wavelet function must have unit energy.
That is
∫ | |
(2.3)
Above both equations (2.1) and (2.2) imply that at least
some coefficient of wavelet transform must be different
from zero and these departures from zero must cancel out.
By consolidating a few mixes of moving and extending of
the mother wavelet, the wavelet change can catch all the
data in the time arrangement and connect it with
particular specific horizon time and location in time. [15]
Wavelet can be defined as pairs of filters here we will
define father wavelet and mother wavelet, where father
wavelet works as low pass filter and mother wavelet as
high pass filter as discussed above. father and mother
wavelet can be defined as below.
∫
And
∫
Where is mother wavelet and is father wavelet, by
given this family of basis function, we can define a
sequence of coefficients.
∫
And
∫
Where is the coefficient of father wavelet (smooth
coefficients), and are coefficient for mother wavelet
(detail coefficient).
So from the coefficients, the function can be
represented by
∑
∑
∑
of can be represented as
Where ∑ and ∑
.
The Discrete wavelet transform (DWT) represents time
series data in terms of the coefficient that are associated
with particular scales, so it is an effective tool for time
series analysis. DWT decomposed signal (data) into a

different scale of resolutions, by using inverse DWT we
can reconstruct a signal from its wavelet coefficient. We
have used MATLAB software to all the procedure for a
prediction here also we used MATLAB to do DWT on
signal means obtained wavelet coefficient. We will get two
parts of wavelet coefficient one is smoothed part (low
frequency) and other is detailed part (high frequency)
part. for decomposition of the signal, a specific wavelet is
required. There are various wavelets are available i.e
Daubechies, Symlet, Meyer, etc. choice of mother wavelet
is to depend on characteristics of data. we have used
Symet wavelet to do decomposition.
3. ARIMA MODEL
The ARIMA word made of three other names AR-auto
regressive, I–integrated, MA-moving average. When these
three models are combined simultaneously it called as
ARIMA model, this model is very famous due to its
applicability for stationary and non-stationary time series
data, in MATLAB ARIMA is defined as ARIMA (p,d,q) where
p is order (lag) for AR term (autoregression),d is order of
integrated term (difference), and q is for moving average
part.
above is for AR(1 order) which suggest number of lag
elements in equation. For p order it can as below
Where is data series and constant parameter which
we can estimate by different methods i.e Yule walker, least
square estimation etc., is white
Gaussian noise and p is call order of series.
Same for moving average
Moving average part depends on noise part of current
values as well as past values; it is working as low pass
filter. When we combine both models with integrated is
model ARIMA, Which can define as
where ∑ , ∑ , are
polynomial terms of B of degree p and q, ,
B=shift operator, i.e , mean.
ARIMA is model which use Box-Jenkins methodology. We
can determine order of AR,MA using ACF (autocorrelation
function) and PACF (Partial autocorrelation function) but
to determine ARMA and ARIMA order is challenging task
for researcher do to exact order to fit model (in some
cases), even though we can take help of AIC (Akaike
information criterion), BIC (Bayesian information
criterion) techniques to find optimum order for ARIMA.
Table 1 gives an idea to choose suitable models to among
AR and MA, ARMA as well as suggesting a best fit order to
forecast future value, suppose for AR (2), PACF plot will
give cuts off after 2 lags and ACF plot will tails of gradually,
same opposite for MA. Random distribution means ARMA.
Table 1
Model Autocorrelation Partial
Autocorrelation
AR(p) Tails off gradually Cuts off after p
lags
MA(p) Cuts off after q lags Tails off gradually
ARMA(p, q) Tails off gradually Tails off gradually
4. HYBRID ARIMA-WAVELET
The block diagram 1 gives the flow of hybridization using
two ARIMA model with wavelet decomposition method. In
this method, first of all, we had decomposed time series
using wavelet transform and reconstructed it in two parts
(after inverse wavelet) these two parts categorized as
approximated signal and detailed signal, we have applied
separate ARIMA model to these parts then forecasting was
applied. Moreover, we can use GARCH model as used in
[13], if residues of hybrid ARIMA is heteroscedastic, we
can use any vocality model for it
4. MODELING AND METHODOLOGY FOR S&P BSE
INFORMATION TECHNOLOGY
Basics steps for modeling
1. Plotting the data,
2. Possibly transforming the data,
3. Identifying the dependence orders of the model,
4. Parameter estimation,
5. Diagnostics, and
6. Model choice.
Start
Time series data,
𝑦𝑡 [𝑦 𝑦 𝑦𝑛]
Wavelet
decomposition
Approximated
Reconstructed Signal
Detailed
Reconstructed Signal
ARIMA Model ARIMA Model
Forecast
End
Block Diagram 1: Procedure for Hybrid ARIMA-Wavelet

We have taken as only-ARIMA model as reference and
explained the steps for it, first step in the modeling is plot
data to check whether time series data is stationary or not,
as well as to do various analysis if data is stationary we
can use AR, MA, or ARMA model as by using Box-Jenkins
methodology, which uses ACF and PACF to select order
and check stationarity, if ACF will not decay to zero fast as
graph of data increases. Thus, a slow decay in ACF is an
indication that differencing may be needed, or we can use
various stationary test i.e kpsstest, for which model is fit
for our data we can refer Table 1.
Figure 1: Original Data
Secondly, if data is not stationary we should have to make
it stationary by differencing it, up to it would be
stationary, a number of difference will be an order of ‘I’
term in ARIMA model, generally order 1 is enough for all
series to best fit [1].
Figure 2 is ACF and PACF for data of Figure 1, after
observing fig.2 and 1, we can say that data is not
stationary because as data increases in fig 1, there is not
decreasing order of lags in ACF (not meets zero quickly).
Moreover, we had stationary kpsstest, test for it which also
tells the series is not stationary, so after differencing we get
ACF and PACF as shown in figure 3, where not any lag is
crossing confidential area so this types of processes called
random walk [1], here we can simply take order as (0,1,0).
Moreover, according to [14]. as shown in figure 2, there is
lag one in PACF which tells the AR (1) process, AR (1)
coefficient (which is the height of the PACF spike at lag 1)
will be almost exactly equal to 1, so no need to be
differencing for that case. We have tried it but (0,1,0) gives
the best result.
The third step is to find out the optimum order for this
model which we can decide by the figure 3 is
ARIMA(0,1,0), where AR(0), I(1), and MA order is 0. Also,
we can use AIC/BIC methods to choose a suitable order for
the model we have used BIC for this case which also
suggests same values.
For estimating parameters which are constants i.e , to
choose best optimum values for it method like maximum
likelihood and Bayesian methods can be used, here
maximum likelihood is used by the software.
The fifth step is diagnostics, in this step, we will choose the
best model by seeing vocality is there or not. if there we
can use ARCH family model to remove it or select other
best model order which does not have vocality component,
also an important thing to check on Residues of series that
it should not serial correlated (insignificant
autocorrelation, therefore the residuals are uncorrelated in
time). Means it should be like random Gaussian noise. We
can see it by ACF plot of Residues and normal distribution
of it if residues are uncorrelated in time tells the best fit,
that order and other same types of order can compare with
AIC, BIC methods, also RMSE and MAE, TIC like
performance parameters may help to decide the optimum
order. For this case the Residues plot and probability
distribution plot are available in Figure 4, then the last step
after selecting proper order is to use that model for
forecast. Figure 5, gives forecasting for 10 days step ahead
of data, table 2 list gives estimated parameters for this case.
Forecasting can be written directly in difference equation
by
where is number of step ahead to forecast,
Figure 2: ACF and PACF of Data
Here, the software use the MMSE forecast which minimized
the forecasted error between new predicted forecast value
i.e suppose, at origin , that we are to make a forecast ̂ of
and then MMSE would be taken between them [1].
Forecasting Performance Measures:
We have use RMSE (Root Mean Squared Error), MAE
(Mean Absolute Error), TIC (Theil Inequality Coefficient).

Figure 3: First differenced ACF and PACF of Data
RMSE:
RMSE measure difference between predicted value and
true value, (a predicted value which does not step ahead of
real value its predicted value at the same time), this gives
accuracy about the model forecast. Minimum RMSE value
gives goodness of model.
RMSE=√
∑ ̂
MAE (Mean Absolute Error):
Mean Absolute Error is the absolute value of average
Residuals; here also minimum value is observed for the
best fit model.
MAE=∑ |
̂
|
Theil Inequality Coefficient (TIC):
TIC gives a value between 0 to 1, near to 0 value gives a
best model fit suggestion.
TIC=
√∑ ̂
√∑ ̂ √∑
Where ̂ =predicted value, =true time series values.
After completing these steps we can use this model for
forecast future value. The forecast was done for 10 days
ahead values.
Figure 4: Residues “s ACF and Normal distribution
Figure 5: 10 days ahead forecast for Only-ARIMA
5. COMPARATIVE ANALYSIS AND RESULTS
DISCUSSION
There are 124 days data are available based on that we
have predicted next 10 days data, which we can see as in
figure 5.
The same methodology was applied to other two methods
Method-1: In this method, we have directly applied ARIMA
model after adding inversed wavelet which contains
detailed and smoothed data.
Method-2: Hybrid method, which contains two separate
ARIMA model after Wavelet decomposition.
Method-1: In this method, we have added both parts
detailed part and approximated part and then we have
applied ARIMA model which gives better results than
ordinary ARIMA model. First of we have checked
stationarity, where we found that data is not stationary, so
we have differenced it once again checked it for
stationarity which tells that data is stationary, so here the
order of ‘I’ is selected as 1, Figure 6 tells about first
differenced ACF and PACF plot. For selecting order of
ARIMA model ACF and PACF observed where ACF cuts
down after lag 1 and PACF is random which telling MA
effect, so we can select order (0,1,1) but which failed to
residues test of using ACF and PACF as well as, Ljung-Box
Q-test, in that it is failed but we have checked its nearest
value,(0,1,2) which is successfully satisfied, so it is best fit
model, but the vocality test not satisfied for it so we can use
ARCH family model on residues to remove it. We can see in
Figure 7, which tells about a test of a perfect fit model is
satisfied. Because its normal distribution flows the cross
black line, which suggests residues are random in nature,
same in ACF, lags are not broken confident level, these
things tell that model is perfectly fit. Table 3 shows chosen
parameters for the selected model to perfectly fit.
Figure 8 gives forecated prices for our derived model with
95% surety range.

Figure 6: First differenced ACF and PACF of Method-1
(Approx+detailed parts)
Figure 7: Residues “s ACF and Normal distribution
Figure 8: Forecast using ARIMA-Wavelet (Method-1)
Method-2: In this method we have applied two different
ARIMA model for two different parts of wavelet data,
approximated and smoothed, to determine its order, first
of check stationarity test for both the part separately,
where gets that approximated part is stationary by two
times differencing, for that the ACF and PACF is given in
figure 9. The second part of wavelet called detail part is
already stationary process, hence do no go ahead, now in
both plot, we can see that it suggest ARMA process, where
an approximate part needs difference by two, and detail is
simple ARMA model. Here its seems too difficult to decide
order but using ACF and PACF observation as well as help
of BIC method we came to do select best-fitted model is
(3,1,3) for approximated part and (1,0,6) for detailed part,
here we have used only 1 differenced parameter instead of
using 2, because we get good result for 1 and [14] when
model is AR(1) or more than it, it makes one differenced by
itself so no need to choose higher difference order. After
selecting appropriate order, we have done Residues test
and vocality using Ljung-Box Q-test. Which suggest that
there is not any vocality available, and the model is
perfectly fit, for the data. The plot of resides
autocorrelation function and normal distribution of same is
available in figure 11. Where ACF tells that the values of the
correlation coefficient of residuals for various time lag is
not significantly different for zero, and its residual value
indicates that the there is no significant autocorrelation.
All the values are lied between confidential area which
presented with two blue line, which suggests residue is
normal, in the same figure probability distribution has
been plotted for residual, on the figure we can see that
distribution is following the blacked dotted line, which
shows normal distribution, because residues are random
and normally distributed, which brings to assume noise as
a white Gaussian noise, that helps to decide optimum
parameters of models for better forecast. Table 4 shows
estimated parameters for the selected model. Figure 12
shows the forecasted data for 10 days to the hybrid ARIMA
model (second model). moreover, table 5 shows
comparative analysis done by comparative parameters.
Figure 9: Second differenced ACF, PACF for method-
2(approximated part)

Figure 10: detailed part ACF&PACF for method-2
Figure 11: Residue’s ACF and normal Distribution
Figure 12: Hybrid ARIMA-Wavelet (Method-2) Forecast
Table 2: Parameter for only ARIMA
Parameters ARIMA(0,1,0)
Constant 22.698
Variance 17384.4
Table 3: Parameters for method-1
Parameters ARIMA-Wavelet (0,1,2)
MA [1.67, 0.752]
constant -10
variance 3040.33
Table 4: Parameters for method-2
Table 5: Forecasting Performance
Model RMSE MAE TIV
ARIMA 131.54 95.42 0.0054
ARIMA-with
Wavelet (First
method)
77.97 60.40 0.0032
Hybrid Arima-
wavelet(second
method)
62.76 47.76 0.0026
3. CONCLUSIONS
After doing analysis for S&P BSE Telecom data using
ordinary method as well as two unique methods using
wavelet transform with ARIMA, among three methods
Hybrid ARIMA method (method-2) which used two ARIMA
model, gives best results by all the tests as well as RMSE
performance is very less compared to both the methods
and also removes some vocality parameters due to control
on both parts, which suggest the supremacy of this
method. While wavelet with ARIMA also gives good
performance compared to normal ARIMA model. The
disadvantage of these hybrids methods is to find the best
order. There might be a chance of difficulty on that in
some cases due to filtering and two separated parts, it
might vary case by case but use ACF&PACF and AIC/BIC,
and trial and error methods help to overcome.
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Parameters ARIMA-Wavelet (3,1,3) ARIMA-Wavelet
(1,0,6)
AR [0.695,-0.33,0.26] 0.371
MA [0.968, 0.929, 0.963] [0.319, -1.5, -1, 0.615,
0.59 ,0.04]
constant 7.690 0.013
Variance 291.263 620.525

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[15] Jatinder kumar, Amandeep Kaur, and Pammy
Manchanda. "Forecasting the time Series Data Using
ARIMA with Wavelet." Journal of Computer and
Mathematical Sciences 6.8 (2015): 430-438.

IRJET-Forecasting of Time Series Data using Hybrid ARIMA Model with the Wavelet Transform.

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