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Vol. xx No.x
xx
A new first order fuzzy time series forecasting method based
on distance measures and modified weighted average
approach
Shafqat Iqbal 1, Chongqi Zhang1 , Munawar Hassan3 , Shuangyin Liu1, and
Shahbaz Gul Hassan 1,*
(1. College of information and Technology, Zhongkai University of Agriculture and Engineering
2. School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China ；
3. School of Economics and Management, Jilin Agriculture University, Changchun, China)
Abstract: The precise estimates about finance, atmospheric science, power sector, industries, agriculture and other
science help governments and institutions economically in making the relevant policies regarding import-export,
demand, consumption, storage, and local industries. Due to uncertainty and non-deterministic behavior of data series
with respect to time, the foremost challenge is to develop and identify the practical method to handle the above stated
complex issues. As an illustration, this study presents an analysis of a new fuzzy time series approach and comparison
with traditional forecasting models for prediction. Taking into consideration the theory of fuzzy sets, fuzzy time series,
triangular membership functions, distance measures and modified weighted average approach, a robust and effective
methodology is developed for prediction of time series data. Conventional statistical forecasting methods such as Holt’s
linear trend, Holt’s Exponential trend and Holt’s damped exponential trend models are also applied for comparison. To
identify the primacy of modeling and forecasting, the techniques of Root Mean Squared Error (RMSE) and Mean
Absolute Error (MAE) are used as a criterion. These findings demonstrate that proposed fuzzy time series approach is
robust for forecasting in the environment of uncertainty.
Keywords: Fuzzy sets; Fuzzy time series; Forecasting; Distance measures; weighted average approach
DOI:
Citation: Shahbaz GH, Shafqat I, Zhang cq, Munawar H, Liu sy, et al. A new first order fuzzy time series forecasting
method based on distance measures and modified weighted average approach.Int J Agric & Biol Eng, 201x; xx(x):
xx–xx
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1 Introduction
Time series forecasting is the extrapolation in
the statistical modeling of the chronologically ordered
data. The principal objectives of time series analysis are:
modeling the vague mechanisms which summarize the
observed series and forecasting the future values of that
time series data. It's application spans across the area of
finance, medicine, business, engineering and other
domains. The precise estimates about different types of
production assist governments in taking decisions about
the import-export volume, the trade at both national and
international level, the price stability and the
management.
Agriculture is the main sector with significant
contribution in Gross Domestic Product (GDP) of the
world [1]. It contributes to make the prosperous economy
of any country. Due to environmental changes, instability
in food markets, increase in population and food
consumption, pulses have an influential role in the
agriculture sector. Many vegetarian and meat diets
around the globe rely on pulses as part of a healthy diet.
In the last two decades, the world demand for pulses has
increased at every level because it meets the
requirements of the major part of the population [2]. In
this study, the focus is on forecasting the gram pulse
production whose importance is recognized by the Food
and Agriculture Organization of the United Nations [3].
The year 2016 has been declared as the International
Year of Pulses (IYP) (FAO,2016). India is the largest
gram producer of the world while Australia and Mexico
are leading in gram pulse production after India [4]. In
the current scenario, it is obligatory to have significant
information regarding production and cultivated area of
the gram and other agricultural commodities to compete
in the world food market.
Conventional statistical methods are adequate and
perform well, but in some situations of uncertainty, they
fail to produce precise results. In decision making
problems, matter of interest is finding an alternative with
the most effective or highest achievable performance.
Fuzzy forecasting methodologies emerged as a robust
approach for forecasting the time series data in the
circumstances where trends, outliers are not reviewed
and historical data consists of imprecise and vague
information. Researchers are focusing their efforts in
fuzzy time series methodologies because they are simpler
and without strict restrictions perform better than the
conventional statistical methods. First, the fuzzy theory
was introduced by Zadeh [5] to cope with areas of
uncertainties. After this development, many researchers
contributed in the extension of fuzzy set theory. The
theory of L-fuzzy sets was presented by Goguen [6],
which is the further extension and continuation of Zadeh
[5] work.
The fuzzy time series theory along with the
process of fuzzy logic relations was proposed by Song &
Chissom [7]. In a further attempt, they developed
time-variant forecasting models. Recently, many
researchers have contributed to the development of fuzzy
modeling after the innovation in fuzzy theory introduced
in [5] and [7]. Chen [8] developed a more comprehensive
method using the arithmetic operations. This model in
the field of fuzzy time series is projected as the milestone
for the development of new models. Further, Chen [9]
developed a two-factor time-variant model based on
fuzzy time series. Also, a novel method of forecasting
based on clustering methodology and fuzzy logical
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relationships to decrease forecasting error was presented
by Chen [10]. In [11], Kumar conducted a research in the
induction of fuzzy sets by using the Intuitionistic fuzzy
sets theory to develop a robust forecasting model.
Additionally, Chen [12] introduced the fuzzy time
series methodology by using the concepts of granular
computing and binning based partition. Singh [13, 14]
made significant improvisation to develop the forecasting
models based on the fuzzy theory. A competent fuzzy
time series method developed by [15] by using the fuzzy
theory to forecast rice production, Kumar [16] applied
the fuzzy-based time series to forecast rice production,
Xiao [17] proposed a fuzzy based combined approach in
relationship with Autoregressive Integrated Moving
Average (ARIMA) and Holt-Winters statistical models
for predicting the yield production. Intelligent systems
are fitted by adaptive neuro-fuzzy inference and artificial
neural networks to forecast the yield production of the
potato crop [17]. A competent forecasting model was
developed by Garg [18] using fuzzy theory and statistical
approach. In [14], Singh formulated a forecasting model
based on three order time-variant parameter to predict the
crops production. In [19] Tseng and Tzeng combined the
fuzzy and statistical approaches to develop a unique
model for accurate forecasting by using the concepts of
fuzzy regression and seasonal ARIMA.
The present study is focused on the analysis of
proposed fuzzy time series methodology, and
comparison with the fuzzy time series and conventional
statistical models in order to recommend forecasting
approach for yield production of a gram food crop, SBI
share prices and other domains. The proposed fuzzy time
series method accounts output by reducing the
fluctuations in time series data which is organized on the
basis of stages presented in Chen [8] work. For this
purpose, firstly, the proposed fuzzy methodology defines
the universe of discourse into seven equal lengths of
intervals, assigns the membership and non-membership
grades to the time series data for the purpose of
fuzzification. A triangular membership function is
applied to compute the fuzzified production and market
share prices. A technique of distance measures id
implemented to select the corresponding fuzzy set. Fuzzy
logical relationships are established and a unique
modified weighted average approach is implemented to
generate crisp forecasts. The motivation to use fuzzy
time series in crop production forecast is that the
estimation about agricultural commodities is one of the
real-life problems due to uncertainty in known and some
unknown parameters. Other reasons are: the uncertainty
in the crop production due to some uncontrolled
parameters and imprecision of primary data.
The rest of the study is structured as follows:
section 2 discusses the basic theory of fuzzy sets and
fuzzy based time series. Section 3 presents the fuzzy time
series approach and Holt’s smoothing methods. Section 4
treats the implementation and performance of the
proposed methodologies and presents the results and
discussion. The last section 5, provides the conclusion
and recommendations of the current study.
2
2.1
Basic
Theories
Fuzzy Sets
Fuzzy sets theory is proposed with the
fundamental concept of fuzzy logic, which is related to
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fuzzy sets and their characteristics [5]. Fuzzy sets
are
specified by a membership function which is associated
with elements of observed variable by assigning a real
number with value from 0 to 1.
Definition 1.
Fuzzy set on the universe of
discourse
can be defined
as
where
are linguistic values of U,
is the
membership grade of
in fuzzy set A with
and 1 ≤ k ≤ n
Definition 2. For a fuzzy set A, triangular
membership function can be defined as
where elements of Y are assigned values 0 to 1. The
assigned value is called degree of membership. A
triangular membership function for fuzzy sets can be
written as
(1)
Definition 3. If F(t-1) and F(t) are related to
each other in terms of fuzzy, then this relationship can be
presented as F(t) = F(t-1) ° R(t, t-1), and F(t-1) → F(t),
where R(t, t-1) describes the fuzzy relationship and ‶ º "
is the symbol for composition operator.
2.2
Fuzzy Time Series
Fuzzy set theory [5] provided a prudent avenue to
cope with the constraints of haziness in the desired
information. This theory is significant in the areas of
ambiguous information, complex historical data and
circumstances of uncertainty. Song & Chissom [7]
introduced the concept of fuzzy time series as a
significant process under the reference theory of fuzzy
sets and fuzzy logic relations.
Definition 4. Let Y (t)(t= …, 0,1,2,…), a
subset of real numbers, be the universe of discourse on
which fuzzy sets
(i=1,2, … ) are defined. If F(t) is a
collection of sets
(i=1,2, …), then F(t) is called a
fuzzy time series on Y(t) (t = 0,1,2, …). In prevalent time
series study, observations are termed as real numbers, but
in fuzzy time series, they are termed as fuzzy sets.
3 Methodologies
This section presents a stepwise description of the
proposed fuzzy based time series approach and the Holt’s
smoothing models to forecast time series data. For this
objective, time series data regarding gram production of
Pakistan is extracted from the website of the Food and
Agriculture Organization (FAO). The proposed fuzzy
and statistical forecasting models are explained as
follows.
3.1 Proposed Fuzzy Time series Method
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In this study, the fuzzy relationship will be
employed to develop a fuzzy time series model. The
proposed methodology is based on the following steps.
Step 1. Equal space data partitioning techniques
have been adopted in many studies [8, 20-23] with
different approaches of establishing fuzzy logical
relationships and defuzzification. Also, in literature of
fuzzy time series modeling, data series is partitioned
without partitioning or removing the trend component [7,
10, 24, 25]. But in our study first, stationarity will be
confirmed and then universe of discourse will be defined
and partitioned. So, in this step, need to confirm whether
data is stationary or not, because sometimes highly
fluctuate data affect the accuracy of forecasting. After
confirmation of stationarity, for the given time series
data, define the universe of discourse. If
&
are the minimum and maximum figures for given data set,
respectively then the universe of discourse is defined as,
, where
&
are two
positive integers. This process makes the universe of
discourse appropriate and fit for further evaluation of the
observed data series.
Step 2. This step involves the partitioning of the
universe of discourse into fixed equal spaced intervals
for production. If U is the universe of discourse then
,
, . . .
are equal length intervals for U. Where
universe of discourse U is portioned with equal space
intervals with length defined as
(2)
Now define the fuzzy sets in corresponding
intervals as fuzzy intervals. Each interval is associated
with the characteristics of the variable as
: poor yield production
: below average yield production
average yield production
:
: good yield production
: very good yield production
: excellent yield production
: bumper yield production
Step 3. This step considers the process of
assigning membership values to the fuzzy variables. The
procedure of assigning membership values with
correspondence to the probability density function to the
variable of interest can be done in many ways [26].
Membership functions are the key part of the fuzzy set
theory because the fuzziness in a fuzzy set on a suitable
domain can be computed by a membership function.
Trapezoidal membership function is employed in many
studies [27-29] in fuzzification process. Also, Bose [30]
combined trapezoidal and triangular function to improve
the forecast accuracy. In this study, triangular
membership function is used to assign membership
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values and it is defined in the previous section as
Definition 2.
Step 4. In this step calculate the distances
,
. . .,
between the y of the observed fuzzified historical
values of the time series data and the corresponding
centers , , . . .,
of the equal length of intervals
, ,...,
, where
It contains some well know properties.
≥ 0, for all y and c and
only
if y = c
=
for all values of y and c
Where
is the distance between
observed or actual value of time series data
and the
center of corresponding interval.
For Example: If two fuzzy sets
and
for
value of time series data y are constructed, then
distance measure for fuzzy set
and
can be
computed as
Where
&
are center
corresponding equal length of intervals.
points
of
Step 5. In this step, after assigning the
membership grades, construct the fuzzy logical
relationships (FLR’s) and fuzzy logical relationships
groups (FLRG’s), explained in Definition 3.
Step 6. This step contains a defuzzification
procedure to acquire forecasted value for fuzzy time
series data and the forecasts obtained are better than the
other methods in the literature. In literature principles of
defuzzification have been improved with time to get
accuracy in forecasts. Song and Chissom [7, 24]
defuzzification method contain three principles on the
basis of membership grades, Chen’s model [8] rely on
three rules for defuzzification: (1) if
has no
relationship then the defuzzification value is the center
point of that corresponding interval . (2) if ⟶
has the relationship then defuzzification value is the
center
point
of
interval.
(3)
if
⟶
establish the relationship group
then defuzzification value is the average of center values
of corresponding intervals
,…,
. Yu’s [31]
concentrate on repetitive fuzzy relationships and
weighted fuzzy relationships. A Trend-Weighted matrix
approach is also implemented in some studies [12, 32,
33] . Also, Li [34] employed deterministic forecasting
model, Huang [35] combined global and local
information in forecasting model. So, to improve
accuracy in forecasting a unique defuzzification model is
proposed in this study which is explained as
On the basis of FLR’s and FLRG’s, select the
significant weighted constant value corresponding to
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each observed value in historical data, and calculate the
crisp forecast using formula presented in Eq (3).
(3)
Where
is the smoothing or weighted constant
for the current year of observed data series,
is the
forecasted value for the current year, p identifies the
number of FLR’s and
is the center value of
corresponding fuzzy set. For this, following rules are
illustrated for the process of defuzzification to figure out
the forecasted output.
Rule 1. If the observed production value for the
year
(i = 1,2,…,n) has the membership value
and
(j = 1,2,…,q) is the fuzzy set corresponding to
value of time series data and there exists only one fuzzy
logical relationship such as ⟶ , then the forecast of
actual value for
time period is computed as
(4)
Where
is the weighted constant for the
corresponding year of observed data series
which
related to fuzzy set , whereas
is the forecasted
value for the time
and
is the center value of the
triangular fussy set corresponding to
.
Rule 2. If the observed production value for the
year
has the membership value
and
is the
fuzzy set corresponding to
value of time series data
and there exists a fuzzy logical relationship group such
as
⟶
, then the forecasted value for
time period will be given as
Where
is the weighted constant for the
corresponding year of observed data series
,
is
the forecasted value for the year ,
is the center
value of the triangular fuzzy set corresponding to
and p shows number of fuzzy logical relationships, for
example: if
⟶
which shows a relationship
of
with
then the value of p equal to 2.
3.2 Exponential Smoothing Approach
Exponential smoothing is a widely used approach
for forecasting the time series data. In the era of 1950s,
this conducive methodology was developed by [36] and
[37] and implemented in different empirical studies.
Gardner [38], explained about simple models related to
Holt winters terminology, general exponential smoothing
models with a brief discussion about properties,
parameters, the initialization process of modeling and
precise forecasting. Later, Holt [39], expressed the
development of forecasting equations for exponential
weighted moving averages covering the component of
non-seasonal with no trend. Some of the exponential
methods with trend are expressed as classification of an
exponential function [40].
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3.2.1 Holt’s Linear Trend Method
When the series express some form of the trend,
Holt [37] extended the Simple Exponential Smoothing
by introducing a term that accounts the possibility of a
trend. In this method, three equations are used, one for
forecasting and the other two for smoothing (level and
trend). The model for Holt’s Linear Trend Method can be
written as
(5)
(13)
3.3 Performance evaluation techniques
After the identification process, the next step is to
specify the model with parameters estimation. It can be
achieved by using some information criterion which is
explained a
3.3.1 Mean Absolute Error (MAE)
(6)
(k=
1,2,3,
…)
(7)
where α and
are known as smoothing parameters, and
their ranges are from 0 to 1. Again,
is used for a level
of the series and
for the best estimate of the trend,
where
is the point forecast for the period t and
k is for the number of next year forecasts.
3.2.2 Holt’s Exponential Trend Method
In contrast to Holt’s Linear Trend methodology,
in this method, level and slope are multiplied instead of
being added. Its equation can be written as
MAE is the performance measure used to assess the
forecasting error. It is an absolute difference between the
actual gram production and the forecasted production
which measures the error in the same units as the
observed series. The equation of the above stated
accuracy
measurement
can
be
written
as
(14)
3.3.2 Root Mean Squared Error (RMSE)
RMSE measures how residuals are spread out and how
time series data concentrated around the best fit of line.
The equation of the RMSE is expressed as
(8)
RMSE=
(9)
(15)
(10)
Here
shows the growth rate which is to be estimated.
3.2.3 Holt’s Damped Exponential Trend Method
where
is the forecasted value and
observed historical data series.
shows the
3.4 Comparison of the Models
The predicted values from Holt's Linear Method show a
constant trend while Exponential Method shows an
exponential trend. The preceding studies revealed that
these previously stated methods in some situations tend
to over-forecast. A damping parameter is introduced by
[41] in the Exponential Trend Method that dampens the
trend to a smooth line. Its equation is presented as
(11)
In the time series analysis, the most important
step is a selection of the robust methodology after fitting
all the proposed methods to the observed data series. In
this study, the fuzzy based time series and Holt’s
smoothing methods are employed to the historical gram
production data and SBI share prices at SBE, India. At
the end, the best-fitted model is selected by the minimum
values of accuracy measures, RMSE and MAE.
4 Forecasting and Model Evaluation
4.1 Crop production data
(12)
The proposed model is implemented on the one
of the food crops production data. The inspiration to
utilize crop yield production data is that the estimation
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about food is one of the genuine issues because of
vagueness in known and some obscure parameters. For
this purpose, time series data of gram (chickpea) of
Pakistan is taken from the website of Food and
Agriculture Organization of the United Nations
(http://www.fao.org/faostat/en/#data/TP).
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xx
4.1.1 The proposed Fuzzy time series Method
Description of the data regarding production of gram
pulse of Pakistan is presented in Table 1. The maximum
figure for production of gram pulse in Pakistan was
868.3 thousand tons. The minimum value for production
was 284.304 thousand tons in the period 1987-2013. The
time series data of gram pulse is plotted in Figure 2.
Table 1. Descriptive statistics regarding production of Gram pulse.
Variable
Mean
S.E Mean
Standard
Deviation
Median
Kurtosis
skewness
Production (000)
tons
488.9
29.6
153.8
456.4
-0.83
0.36
Figure 2. Plot of total production
Step 1. After the initial process of computing
maximum and minimum quantities of the observed time
series data gram production, the universe of discourse is
defined as
. Setting
and
21.7, the universe of discourse is,
. Where the values of production are
expressed in thousand tones in whole study.
: poor yield production
: below average yield production
Step 2. The universe of discourse is partitioned
into seven intervals of equal lengths using the Eq.1.
= [260, 350],
: average yield production
= [350, 440],
: good yield production
= [440, 530],
= [620, 710],
= [530, 620],
: very good yield production
= [710, 800],
: excellent yield production
= [800, 890]
The fuzzy sets Ai are defined into seven intervals of
equal length for the gram production, such as
: bumper yield production
Step 3. In this step triangular fuzzy sets are
established and the values of membership grades and
non-membership grades or parameters are calculated by
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the triangular membership function. The formula for
triangular membership function is given in Eq.2. Seven
triangular fuzzy sets on the basis of equal length of
intervals defined on universe of discourse are given as
follows:
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xx
= 530 – 493/ 530-440
= 37/90 = 0.411
= 493 – 440/ 530-440
=
= [260, 350, 440],
= [440, 530, 620],
[620, 710, 800],
= [800, 890, 890]
= [350, 440, 530],
= [530, 620, 710],
= [710, 800, 890],
A sample calculation for the year 1987 production is as
follows:
For the year 1987, calculated the membership and
non-membership grades. The production value of 493
falls in fuzzy sets F2 & F3. If F2 = [350, 440, 530], then
using a prescribed membership function, calculated the
membership and non-membership grades for the
corresponding market price.
From Eq.1 the value of
= 493 satisfies the condition
of {
} for triangular fuzzy set F3 =
[350, 440, 530], and also, satisfies the condition of
{
} for triangular fuzzy set F2. then
the value of membership grade
calculated as
and
are
= 53/90 = 0.589
Similarly, the further calculations of membership degrees
and non-membership degrees regarding given dataset of
gram production are computed. On this basis of this
information, fuzzy sets for both membership and
non-membership degrees are computed as
Actual value of gram yield production for the time 1987
was 493, which lies in triangular fuzzy sets F2 and F3.
Therefore, two fuzzy sets
&
are constructed for
the production value 493 using the triangular
membership function with membership information of
0.411 and 0.589, respectively.
Step 4. In this step calculated the distances
,
. . .,
between the y of the observed fuzzified historical
values of the gram production and the corresponding
centers , , . . .,
of the equal length of intervals
,
, . . . ,
. On the basis of minimum value
between the values of
, choose the fuzzy set
. The process of computing distance measures and
selected fuzzy sets is presented in Table 2.
Table 2. Fuzzified Production
Year
Gram production
Fuzzy sets
1987
1988
493
371.5
(
(
)
)
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
456
561.9
531
512.8
347.3
410.7
558.5
679.6
594.4
767.1
697.9
564.5
397
362.1
675.2
611.1
868.3
479.5
838
(
(
(
(
)
)
)
)
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
Distance measures
set
(
)
Selected Fuzzy
sets
(k = 1,2, . . ., 7)
(98, 8)
(66.5, 23.5)
(0.589, 0.411)
(0.233,0.767)
(61, 29)
(76.9, 13.1)
(46, 44)
(117.8, 27.8)
--(105.7, 15.7)
(73.5, 16.5)
(104.6, 14.6)
(109.4, 19.4)
(102.1, 12.1)
(122.9, 32.9)
(79.5, 10.5)
(92, 2)
(57.1, 32.9)
(100.2, 10.2)
(126.1, 36.1)
(113.3, 23.3)
(84.5, 5.5)
(83, 7)
(0.178, 0.822)
(0.345, 0.655)
(0.012, 0.988)
(0.809, 0.191)
(0.97, 0.03)
(0.675, 0.325)
(0.317, 0.683)
(0.663, 0.337)
(0.716, 0.284)
(0.635, 0.365)
(0.866, 0.134)
(0.383, 0.617)
(0.523, 0.477)
(0.135, 0.865)
(0.614, 0.386)
(0.902, 0.098)
(0.759, 0.241)
(0.439, 0.561)
(0.423, 0.577)
Fuzzified
Values
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2008
2009
2010
2011
2012
2013
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474.6
740.5
561.5
496
284.304
751.223
(
(
(
(
(
xx
(79.6, 10.4)
(0.385, 0.615)
(75.5, 14.5)
(0.339, 0.661)
(76.5, 13.5)
(0.35, 0.65)
(101, 11)
(0.623, 0.377)
--(0.27, 0.73)
(86.223, 3.77)
(0.458, 0.512)
optimize length of intervals ,
,
&
are
305, 395, 485, 575, 755 and 845 respectively, then
forecast production using Eq.3 is calculated as
)
)
)
)
)
(
Vol. xx No.x
)
Step 5. After selection of fuzzy sets with
membership and non-membership information in respect
of time series data of share prices, established the FLR’s
and FLRG’s shown in Table 3 & Table 4.
Step 6. In this step, on the basis of fuzzy FLR’s
and FLRG’s corresponding membership information is
taken as weights to each share price. The formula for this
purpose is presented in Eq.3. For example, calculation
for the time 1987 are computed as
= 534.1
Other forecasts are also computed in a similar
way and presented in table 5. The comparison graph
(Fig.3) shows that forecasted values are very close to the
actual values.
Table 3. Fuzzy logical relationships of the fuzzified production
For gram production of time 1987, fuzzy set is
=˂
493, {0.589, 0.411} ˃ and FLRG corresponding to
is
⟶ ,
,
. Since the midpoints of the
FLR’s of the fuzzified data
⟶ ,
⟶
⟶ ,
⟶ ,
⟶ ,
⟶
,
⟶
⟶ ,
⟶
,
⟶ ,
⟶ ,
⟶
⟶
⟶
,
⟶
⟶
⟶ ,
,
⟶
,
⟶ ,
,
,
⟶ ,
⟶
⟶
,
⟶
,
⟶
,
⟶
,
⟶
Table 4. Fuzzy logical relationships Groups of the fuzzified production
FLRG’s of the fuzzified data
⟶
⟶ ,
⟶
⟶
⟶
⟶
⟶
⟶ ,
⟶ ,
⟶
⟶ ,
⟶
,
⟶
,
⟶ ,
⟶ ,
⟶
,
,
⟶ ,
Table 5. Forecasted Production (000 tonnes)
⟶
⟶
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Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Actual
Production
493
371.5
456
561.9
531
512.8
347.3
410.7
558.5
679.6
594.4
767.1
697.9
564.5
Forecasted
production
534.1
531.91
558.98
610.37
628.352
502.19
313.1
452.925
611.882
634.67
590.336
705.725
652.94
608.318
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Year
Actual
Production
Forecasted
production
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
397
362.1
675.2
611.1
868.3
479.5
838
474.6
740.5
561.5
496
284.304
751.223
480.86
550.125
630.26
580.292
758.24
535.429
637.28
540.35
665.765
610.1
518.93
502.1
663.23
xx
Figure 3: Plot of the Actual and forecasted values for fuzzy time series method
For visual presentation, forecasted values plotted in
Figure 3, significantly indicates that lines of actual
production and proposed method pass each other very
closely.
4.1.2
Holt’ Smoothing Models
(16)
where
and
. Then using
the Eq.16, the point forecasts are calculated with the
process given below.
(t = 0, k = 1
1. Holt’s Method (Linear Trend)
In this method, the value of trend is smoothed by
the weights computed by R software. Here the most
important concern is optimizing the combination of
smoothing parameters which is complex procedure as
compared to one parameter estimation. The value of
alpha (α = 0.5) is fitted on a subjective basis and the
value of 𝛽 is determined by the application designed for
this study. The initial values of the level and growth rates
are estimated by the R software and are fitted in the least
square trend line which is presented as
In order to determine the next years forecasts, the
values
and
are estimated by putting the values of
smoothing parameters, and 𝛽 in Eq.5 & Eq.6. Then
the estimated ,
and the forecasts for next years are
computed as
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,
, α = 1e-04, beta =
0.0188 and = 0.9287, then the point forecast of
of
time = 0 is computed as
In this study the time series data is used for the
period between 1987 and 2013 and the forecasts are
computed by estimating the smoothing parameters for all
given years. Now for future prediction of the gram
production, k ahead forecasts are calculated by putting
the values in Eq.7. If k = 1 then for the year 2015
forecasting equation is given as
This process continued and was used for
computing the forecasts for all the years. Similarly, the
forecasts of the years following 2015 are calculated by
putting the information in Eq.11, Eq.12 & Eq.13 as
follows
(17)
To compute three years ahead production value,
the value of k = 3 is put in Eq.7. The pattern and trend of
the next few years forecast by this method can be seen in
Figure 4 (a).
2. Holt’s Method (Exponential Trend)
In this method, the local growth rate
is
modeled by the smoothing ratios
, of the level.
This methodology models the trend in a multiplicative
way and forecasts are formed by the multiplication of the
level and slope. The software generated the values of
level, the growth rate, the smoothing parameters and the
forecasting accuracy measures, where
,
, α = 0.0358 and 𝛽 = 0.0358. After fitting
the Holt’s exponential method on the dataset, the actual
and forecasted trend lines are shown in Figure 4 (b). The
procedure of calculations is repeated here as it was
considered in Holt’s Linear method.
(18)
The plot of Holt’s damped method with an
exponential trend is shown in Figure 4 (c), which
illustrates that the forecasted series is damped with the
decrease of the forecasting errors as compared to the
previous Holt's exponential method. But on the basis of
minimum values of AIC and BIC, Holt’s method with
exponential trend is found to be the best fit method
among all the Holt’s smoothing methods within this
study. The values of the accuracy measures are given in
Table 4.
= 470.9348,
In order to compute the point forecast, the values
of l1, b1 by using Eq.8 and Eq.9, were put in Eq.10, then
the forecasted value for the observation
was
The forecasts for the following years are calculated by
putting the values of the growth rate and level in Eq.10.
3. Holt’s Method (Damped Exponential Trend)
Pegels [40] suggested that multiplicative trend
methods are more applicable than the additive trend
methods in real life applications. Gardner [38] introduced
the damping parameter
which can be used in Holt's
methods to manage the trend extrapolation. In this
method the growth rate further undergoes in the process
of damping for each future value. The value of the lies
between 0 and 1, which indicates that multiplicative
trend is damped. The increase or decrease in the value of
phi indicates the degree of damping. In the present study,
the computed value of damping parameter is 0.9287
which shows that the degree of damping is decreased.
The R software estimated the damping, smoothing and
other parameters. Using these estimated values, the same
procedure and calculations are repeated here by using the
relevant equations (9,10 &11) as it was done in Holt’s
Linear and exponential trend methods, where
(a)
(b)
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4.1.3 Comparison of the proposed Fuzzy time series with
the other models
(c)
Figure
4. Plot of Holt’s Methods for gram production
In this part of the section, the performances of
fuzzy time series method, Holt’s Linear trend method,
Holt’s exponential trend and Holt’s damped exponential
trend method are evaluated by using the forecasting
accuracy measures, RMSE and MAE. The computed
values of the above-mentioned accuracy measures are
presented in Table 6. The findings demonstrated that the
value of MAE for proposed fuzzy based time series
method is 57.65837, which is significantly lower than all
the values of Holt’s smoothing models. The computed
value of RMSE is 81.18085 which is also meaningfully
lower than RMSE values of the Holt’s Linear,
exponential and damped exponential methods.
Forecasting measures showed that forecasts with
proposed fuzzy time series method are the closest to the
actual observations with minimum error as compared to
other methods.
Table 6: Model Selection for production
Model
MAE
RMSE
Rank
Proposed Fuzzy time series method
74.8897
93.1683
1
126.2541
156.1272
3
Holt’s Method (Exponential Trend)
128.7694
157.9536
4
Holt’s Method (Damped Exponential
Trend)
120.0311
146.8991
2
Holt’s Method (Linear Trend)
4.2 Proposed method for prediction of SBI shares prices
In this part of the section proposed model is fitted on the benchmark dataset of SBI share prices at SBE, India presented in
Table 7.
Table 7. Actual SBI share price at BSE, India
Months
Months
Actual
SBI
shares
Actual SBI
shares
04/2008
1819.95
04/2009
1355
05/2008
1840
05/2009
1891
06/2008
1496.7
06/2009
1935
07/2008
1567.5
07/2009
1840
08/2008
1638.9
08/2009
1886
09/2008
1618
09/2009
2235
10/2008
1569.9
10/2009
2500
11/2008
1375
11/2009
2394
12/2008
1325
12/2009
2374
01/2009
1376.4
01/2010
2315
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Vol. xx No.x
02/2009
1205.9
02/2010
2059.95
03/2009
1132.25
03/2010
2120.05
Description of the data regarding market prices of SBI
share, at SBE, India is presented in Table 8. The
xx
maximum value for market price was 2500 and the
minimum value was 1132. The time series data of market
prices of SBI share is plotted in Figure 5.
Table 8. Descriptive statistics regarding SBI share prices.
Variable
Mean
SBI share Prices
1786
Minimum
1132
Standard
Deviation
Median
Maximum
399.4764
1786
2500
(a)
Figure 5. Plot of SBI share prices
Step.1 After the initial process of computing
maximum and minimum quantities of the observed time
series data regarding SBI share price, the universe of
discourse is defined as
.
Setting
and
35, the universe of
discourse is,
.
Step 2. The universe of discourse is partitioned into
seven intervals of equal lengths using the Eq.1.
:
very low share prices
: below average share prices
: average share prices
: good share prices
= [1100, 1305],
= [1305, 1510],
= [1510, 1715],
= [1715, 1920],
: very good share prices
= [1920, 2125],
= [2125, 2330],
: Excellent share prices
= [2330, 2535]
The fuzzy sets Ai are defined into linguistic terms
corresponding to seven length of intervals, such as.
: Rich share prices
Step 3. In this step triangular fuzzy sets are
established and the values of membership grades and
non-membership grades or parameters are calculated by
the triangular membership function. The formula for
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triangular membership function is given in Eq.2. Seven
triangular fuzzy sets defined on universe of discourse are
given as follows:
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xx
[1100, 1305, 1510], and also, satisfies the condition of
{
} for triangular fuzzy set F2. The
value of membership grade
sets are calculated as
and
for both fuzzy
= [1100, 1305, 1510],
= 1510 – 1375/ 1510-1305
= [1305, 1510, 1715],
= 135/205 = 0.658
= [1510, 1715, 1920],
= 1375-1305/ 1510-1305
= 170/205 = 0.342
= [1715, 1920, 2125],
Similarly, the further calculations of membership degrees
and non-membership degrees regarding given dataset of
market prices of SBI share are computed. On this basis
of this information, fuzzy sets for both membership and
non-membership degrees are computed as
= [1920, 2125, 2330],
= [2125, 2330, 2535],
= [2330, 2535, 2535]
A sample calculation for the month 11/2008 is as follows:
For this month, computed the membership and
non-membership grades using a triangular membership
function. The market price value of SBI share “1375”
falls in fuzzy set F1 = [1100, 1305, 1510], then using a
prescribed membership function, calculated the
membership and non-membership grades for the
corresponding market price.
From Eq.1 the value of
= 1375 satisfies the condition
of {
} for triangular fuzzy set F1 =
Actual SBI share price for the time Nov-2008 was 1375,
which lies in triangular fuzzy sets F1 and F2. Therefore,
two fuzzy sets
&
are constructed for the price
1375 using the triangular membership function with
membership information of 0.658 and 0.342,
respectively.
Step 4. In this step calculated the distances
,
. . .,
between the y of the observed fuzzified historical
values of the SBI share prices and the corresponding
centers , , . . .,
of the equal length of intervals
, ,...,
. On the basis of minimum value
between the values of
, choose the fuzzy set
. The process of computing distance measures and
selected fuzzy sets is presented in Table 9.
Table 9. Distance measures and selected fuzzy sets for SBI share prices
Fuzzy sets
Distance measures
(
)
(k = 1,2, . . ., 7)
Actual SBI shares
Selected Fuzzy sets
(
)
(207.45, 2.45)
(0.512, 0.488)
(
)
(227.5, 22.5)
(0.610, 0.390)
(
)
(294.2, 89.2)
(0.932, 0.068)
(
)
(160, 45)
(0.281, 0.719)
(
)
(231.4, 26.4)
(0.629, 0.371)
(
)
(210.5, 5.5)
(0.527, 0.473)
(
)
(162.4, 42.6)
(0.293, 0.707)
(
)
(172.5, 32.5)
(0.342, 0.658)
(
)
(122.5, 82.5)
(0.098, 0.902)
(
)
(173.9, 31.1)
(0.342, 0.652)
1819.95
1840
1496.7
1567.5
1638.9
1618
1569.9
1375
1325
1376.4
1205.9
1132.25
(
)
---
(0.512, 0.488)
(
)
---
(0.157, 0.843)
Fuzzified Values
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1355
1891
1935
1840
1886
2235
2500
2394
2374
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Vol. xx No.x
(
)
(152.5, 52.5)
(0.756, 0.244)
(
)
(278.5, 73.5)
(0.858, 0.142)
(
)
(117.5, 87.5)
(0.073, 0.927)
(
)
(227.5, 22.5)
(0.610, 0.390)
(
)
(273.5, 68.5)
(0.834, 0.166)
(
)
(212.5, 7.5)
(0.537, 0.463)
(
)
(272.5, 67.5)
(0.829, 0.171)
(
)
(166.5, 38.5)
(0.312, 0.688)
(
)
(166.5, 38.5)
(0.215, 0.785)
(
)
(292.5, 87.5)
(0.927, 0.073)
(
)
(242.45, 37.45)
(0.683, 0.317)
(
)
(302.55, 97.55)
(0.976, 0.024)
Step 5. After selection of fuzzy sets with
membership and non-membership information in respect
of time series data of share prices, established the FLR’s
and FLRG’s shown in Table 10 & Table 11.
1612.5 and 1817.5 respectively, then forecast share price
for the month 11/2008 using Eq.3 is calculated as
Step 6. In this step, on the basis of fuzzy FLR’s
and FLRG’s corresponding membership information is
taken as weights to each share price. The formula for this
purpose is presented in Eq.3. For example, calculation
for the time 11/2008 are computed as
For share price of time 11/2008, fuzzy set is
= ˂
1375, {0.342, 0.658} ˃ and FLRG corresponding to
is
⟶ ,
. Since the midpoints of the optimize
length of intervals , ,
&
are 1205.5, 1407.5,
= 1495.88
Other forecasts on the basis of Rule 1 and Rule 2
are also computed in a similar way and presented in table
12. The comparison graph (Fig.3) shows that forecasted
values are very close to the actual values.
Table 10. FLR’s of the fuzzified SBI shares
FLR’s of the fuzzified data
⟶ ,
⟶ ,
⟶
xx
⟶ ,
⟶
,
⟶ ,
⟶
,
⟶ ,
⟶ ,
⟶ ,
⟶ ,
⟶
,
⟶ ,
⟶ ,
⟶ ,
⟶ ,
⟶
,
⟶ ,
⟶ ,
⟶ ,
⟶
,
⟶ ,
⟶ ,
Table 11. FLRG’s of the fuzzified SBI shares
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Vol. xx No.x
FLRG’s of the fuzzified data
⟶
⟶
⟶ ,
⟶
⟶
⟶
⟶ ,
⟶
,
,
⟶
,
⟶ ,
⟶
⟶ ,
⟶ ,
⟶ ,
⟶ ,
⟶ ,
⟶ ,
⟶
Table 12. Forecasts and comparison for SBI share prices
Months
Chen
(1996)
Huarng
(2001)
Kumar and
Gangwar
(2016)
Gupta
and
Kumar
(2018)
Proposed
Method
--
--
--
1852.73
Joshi and
Kumar
(2012)
Actual SBI
shares
04/2008
05/2008
06/2008
07/2008
08/2008
09/2008
10/2008
11/2008
12/2008
01/2009
02/2009
03/2009
04/2009
05/2009
06/2009
07/2009
08/2009
09/2009
10/2009
11/2009
12/2009
01/2010
1819.95
1840
1496.7
1567.5
1638.9
1618
1569.9
1375
1325
1376.4
1205.9
1132.25
1355
1891
1935
1840
1886
2235
2500
2394
2374
2315
--
--
1900
1855
1777.8
1725.98
1860.08
1844.15
1900
1855
1865.7
1725.98
1860.08
1418.34
1500
1575
1531.5
1512.39
1452.59
1465.11
1500
1505
1531.5
1512.39
1452.59
1536.45
1600
1610
1777.8
1574.35
1544.29
1515.54
1600
1505
1531.5
1574.35
1544.29
1467.57
1500
1482
1531.5
1512.39
1452.59
1495.88
1433
1155
1504.23
1305.52
1682.31
1532.32
1433
1365
1504.23
1665.9
1682.31
1486.62
1433
1482
1504.23
1305.52
1682.31
1302.54
1433
1155
1258.23
1294.27
1264.98
1375.32
1300
1365
1258.23
1294.27
1264.98
1434.67
1433
1482
1504.23
1665.9
1682.31
1825.32
1900
1890
1865.71
2006.51
2138.31
1832.46
1900
1890
1883.93
2006.51
1853.54
1844.15
1900
1855
1865.71
1725.98
1860.08
1826.96
1900
1855
1865.71
2006.51
2138.21
2227.5
2300
2485
2142.04
2520
2466.99
2397.45
2300
2415
2245.65
2420
2328.48
2291.46
2300
2345
2191.75
2365.99
2321.66
2271.58
2300
2205
2191.75
2365.99
2321.66
2227.50
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2300
2205
2142.04
2020
2070.4
1957.52
2100
2135
1883.93
2120
2138.21
2017.58
xx
2120.05
Figure 6. Plot of the Actual and forecasted values
For visual presentation, forecasted values are plotted in
Figure 6, which significantly indicates that lines of actual
share prices and proposed method pass each other very
closely.
4.2.1 Comparison of proposed fuzzy time series method
with the Existing models
various existing benchmark models [8, 42-45]. The
computed values of forecasting accuracy in terms of
RMSE are depicted in Table 13. The computed values of
accuracy measures for proposed model are 106.51
regarding RMSE. These findings illustrate that proposed
model outer-perform several time series models with
minimum error. Moreover, the actual SBI share prices
and proposed values with previous competing models are
shown in Table 12 &
In this part of the section, primacy and performance of
the proposed fuzzy time series model is compared with
Fig.7, which illustrates that testing results are more similar to the actual values as compared to the other models.
Figure 7. Comparison with the Existing Models
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Table 13. Accuracy Measures for Evaluation
Model
RMSE
Rank
Chen (1996)
187.26
5
Huarng (2001)
164.04
3
Joshi and Kumar (2012)
200.17
6
Kumar and Gangwar (2015)
134.24
2
Gupta and Kumar (2018)
182.98
4
Proposed Method
106.51
1
5.
Conclusion
In this paper, a robust fuzzy time series
methodology for forecasting the production and SBI share
prices has been proposed in order to manage the prediction
situations when uncertainty, hesitancy and decision-making
problems occur in practice. Different studies have been
conducted regarding the Fuzzy time series and
conventional statistical methods in order to develop
methodologies for prediction but insufficient work is done
for developing the methodologies regarding food crops like
gram, wheat, corn, etc. In this study, the proposed fuzzy
time series methodology is found to be robust for the
accurate forecasts. A Fuzzy time series methodology is
proposed based on emphatic membership fuzzy logic
relationship, which determines the membership and
non-membership grades. Distance measuring technique and
modifies weighted average approach is integrated with the
model. Additionally, statistical Holt’s Smoothing approach
for prediction is also applied to the same data series and the
results are analyzed and compared on the basis of accuracy
measures. The calculations revealed that proposed fuzzy
time series approach is much better and more accurate than
the aforementioned fuzzy time series and Holt's smoothing
models for the prediction of time series data. This fruitful
procedure can be employed as a precise computational
method to forecast the time series data in other domains.
This achievement will help the policy makers in planning,
management, import-export and other areas. Future work is
underway to extend the proposed model to tackle with
problems of multivariate time series forecasting.
6.
References
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xx