Post on 28-Mar-2023
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Enhanced Accuracy of Ordered Weighted Averaging
based Fuzzy Predictor using Genetic Algorithm
Bindu Garg
Department of Computer Eng.
Jamia Millia Islamia, New Delhi-110025, India
bindugarg80@gmail.com
M.M. Sufyan Beg
Department of Computer Eng.
Jamia Millia Islamia, New Delhi-110025, India
mmsbeg@cs.berkeley.edu
A.Q. Ansari
Department of Electrical Eng.
Jamia Millia Islamia, New Delhi-110025, India
aqansari@ieee.org
Abstract—Accuracy is one of the most important aspects while designing forecasting model. Particularly, accuracy in
fuzzy prediction system considerably depends on subjectively
decided parameters such as membership function and relative
weight of past observations. In this regard, we have proposed
a novel concept to optimize ordered weighted aggregation
(OWA) based fuzzy time series predictor (FTSP) using
genetic algorithm (GA). Firstly, accurateness of FTSP is
enhanced by applying effective method of aggregation on past
observations using OWA weights. These weights are
determined on the basis of importance of fuzzy set in the
system by employing regularly increasing monotonic (RIM) quantifiers. Subsequently, GA is used to optimize membership
functions of FTSP by generating its wide range of parameters
in the region of time series. Lastly, this model is capable of
controlling its performance by varying GA parameters. To
assess proposed method, we used dataset of enrollments and
outpatient visits, as used by almost all previous research in this
domain. Evaluation results indicate coalescing OWA and GA
for FTSP significantly reduced mean square error (MSE) and
average forecasting error rate (AFER).
Keywords—Time Series analysis; Fuzzy Logic; Genetic
algorithm; Optimization; Ordered Weighted Aggregation
I. INTRODUCTION
Prediction plays a vital role in business, finance and other industries to reckon conclusion. However, accuracy of
prediction is a challenging task. Selection of pertinent and
efficient forecasting technique is very momentous factor in all
problem domains. Time series analysis has been active
research area since many years for forecasting purpose. We
encounter chronological and uncertain data in several realms
of application like: annual crop yielding of sugar-beets and
their price, weekly interest rates, daily stock races, monthly
rates of unemployment, meteorology records of hourly wind
speeds, maximum and minimum value of temperatures per
day, annual rainfall, possibly impending earthquakes, an electrocardiogram traces heart waves, annual death & birth
rates, number of outpatient visits & average length of stay in
health care domain, etc. There are clearly numerous reasons to
record and analyze these types of time series data.
Consequently, sound and robust mechanism for interpretation
of time series is very much demanded. This strong mechanism
should be proficient in doing thorough analysis with clear
understanding of time series. In other words, purpose of time
series analysis is to design a predictive model such that forecasting error between the predicted value and actual value
is as small as possible. Key difference between time series
models and other predictive models is: in time series models
lag values of the target variable is used as predictor variable,
whereas traditional predictive methods utilize other variables
as predictors and does not apply concept of lag value because
past observations don’t represent a chronological sequence. In
general, there are two basic ways for prediction using time
series: linear approach and non-linear approach. In linear
approach, structure of system is pre-known and linear models
such as Auto regression, Moving average model, ARMA method, Box Jenkins and Multiple linear regression model, etc
are categorized [1-2]. Although, these techniques can
maneuver problems arising from new trends, however these
failed to forecast for problems having nonlinearity and had
inaccuracy in results. To address this, principles and practice
of soft computing techniques have been poised in the analysis,
design, and interpretation of time series. These are called non-
linear approach, where very few assumptions about the
internal structure of the system are made. In particular, this
capitalize on the important facets of learning, structural design
and interpretability along with human-centricity; all of these
aspects are vigorously supported by the leading technologies of soft computing. Soft computing techniques rely on the
input-output relationships to describe the behaviour of time
series and thus can better manage nonlinearity and complexity
of historical time series data by creating training rules or fuzzy
relationships that increase accuracy in forecasting model to a
considerable factor. Higher forecasting accuracy is measured
in terms of least values of mean square error (MSE), average
forecasting error rate (AFER). Challenging task is to design
such a forecasting model that is efficient as well as accurate
and can process imprecise data of real world problems. In past
few years, soft computing techniques are widely utilized in various domains and have already been proved a powerful tool
for accurate forecasting. Artificial Neural Network is being
used in designing of prediction models due to vast
development in the area of artificial intelligence. Garg et al [3]
performed a vast and logical survey on implementation of
forecasting method using artificial neural network. However,
Artificial Neural Network could not generate efficient
predictors because of its some drawbacks (1) it has large
training time (2) it can only utilize numerical data pairs (3) It
traps in local minima that can deviates from optimal
performance. Another soft computing technique which has
recently received attention is fuzzy based approach.
II. RELATED WORK
Initial work of Zadeh [4-5] concerning fuzzy set theory has been applied in several diversified areas. Song and Chissom
[6-7] proposed first forecasting method that laid the
foundation of fuzzy time series. This model was tested on the
time series data of University of Alabama to forecast
enrollments. Predominant reason of fuzzy time series
popularity is: it can utilize historical data more effectively in
fuzzy logical relationship. Chen [8] presented simplified
arithmetic operations and considered high order fuzzy time
series model. Following that concept of fuzzy stochastic time
series was proposed to extend time series [9]. Hwang et al [10]
improved the prediction accuracy by using more simplified
form of arithmetic operation. Further, Huarng [11-12] proposed distribution and average based length of intervals to
improve the fuzzy time series forecast. After while, Chen [13]
proposed two phase partitioned first order time-variant
method. Lee et al [14] proposed two factors based fuzzy time
series forecasting model and Jilani et al [15] developed high
order fuzzy time series model by using frequency density
based partitioning. Meanwhile, Singh [16] presented
forecasting model using computational algorithm. Chen et al
[17] presented forecasting model based on fuzzy time series
and fuzzy variation group to forecast daily Taiwan stock
exchange index (TAIEX). Garg et al [18-19] improved forecasting accuracy to a considerable factor by introducing
the concept of event discretization function and novel
approach of weighted frequency density based partitioning.
Aforementioned research on fuzzy time series for
forecasting problems treated fuzzy relationship equally
important which might not have properly reflected the
importance of each individual fuzzy relationship in
forecasting. In direction of acquiring more accuracy, Yu [20]
used concept of recurrent relationship to generate forecasting
model and recommended that different weight must be
assigned to fuzzy relationships. Expectation and Grade-
Selection method based on transitional weight was developed for calculating weights [21]. Yager [22] introduced a class of
function to generate weights and concept of ordered weighted
aggregation (OWA) operators. Flexible aggregation
characteristic of OWA makes it widely accepted in various
domains. Sufyan [23] applied OWA to aggregate user
feedback to enhance web search quality of a document. Sadiq
[24] utilized OWA to find overall status of environmental
system by taking data aggregation of environmental indices.
Subsequently, Yager [25] introduced the concept of smoothing
of time series by OWA operators. Consequently, Garg [26-28]
designed fuzzy and OWA based forecasting model to determine future value. However, accuracy can be improved
further by exploring other design aspects of model.
All of the above fuzzy predictors generated only one set of
parameter values for each membership function at time t But
accuracy of predictor can be further enhanced by employing
wide range of parameters for membership function. As seen in
previous literature, GA is best suited meta-heuristic approach
to solve such type of problems. GA has been used to optimize
various fuzzy based real time control applications [29-31] by
adjusting the shape of used membership function in the
system. Nevertheless, it is observed that limited research has
been carried out to optimize fuzzy predictor using GA. Kim
and Dim [32] proposed a genetic fuzzy predictor ensemble
(GFPE) to improve prediction of future. Implementation of
this model was done on non stationary time series. Wang et al
[33] presented fuzzy knowledge integration framework based
on genetic algorithm. Laribi et al [34] employed hybrid model based on genetic algorithm and fuzzy logic to generate path in
mechanism synthesis. Subsequently, Kang [35] developed an
efficient non linear time series prediction system using genetic
algorithm and fuzzy time series. Chen and Chung [36] used
genetic algorithm to tune length of interval in universe of
discourse. Fuzzy subtractive clustering and weighted fuzzy
short time series method were also employed for better trend
prediction [37]. A statistical genetic interval-value based fuzzy
system [38] was developed to predict survival time of patients.
Fowler [39] gave concept of evolved fuzzy system to predict
forest fire size. Almost all research in this direction assigned
equal weight to past fuzzy observations. Moreover, only few of them were designed for time series data.
In continuation of our previous research work and direction
of attaining optimal performance, [3, 18, 19, 26, 27, 28, 40,
41], in this study we utilized GA to optimize OWA-Fuzzy
based time series forecasting model. OWA is employed to
generate weight of past fuzzy observations and their
aggregation in order to predict next value. Weights are
generated on basis of importance of each fuzzy set in the
system. Further, GA search strategy is applied to set wide
parameter range of membership function at time t rather than
single parameter value, so that best prediction results for (t+1) can be realized [40]. Another novelty introduced is that only
those chromosomes are being considered which produce
forecasts within range of interval at time t+1. It is proved that
the coupling of GA with OWA for fuzzy time series can
control and certainly improve the efficiency and accuracy of
predictor to a considerable factor.
Preliminary experiment: an experiment & comparative
study of proposed method is carried out on enrollment data of
University of Alabama to demonstrate the performance of our
model. Same data is used by most of existing soft computing
prediction models in literature.
Extensive experiment: Due to significance of forecasting of outpatient visit in health care domain, which not only
influence patient waiting time but also improves the
coordination of care, we considered this application domain to
emphasize on potential of our proposed model. We used same
historical outpatient data that has been used in previous
research to evaluate proposed model.
This paper is organized as follows: Section 1 is introduction
of forecasting and section 2 is related research in this field.
Section 3 briefly illustrates ―what is meant by time series
analysis‖. Section 4 describes theory of fuzzy time series used
in algorithm. Section 5 highlights the concept of OWA and Section 6 elucidates GA. Section 7 explains the proposed
algorithm in detail, Section 8 exemplify experimental results
of OWA-FTSP-GA on enrollment data of University of
Alabama and comparative study of results with existing time
series analysis models (linear and non-linear). In Section 9,
simulation and performance comparison of OWA-FTSP-GA
on number of outpatient visits is performed. Section 10
concludes this research and addresses the future perspective.
III. TIME SERIES ANALYSIS
Statistical/ approach vs. fuzzy approach
Time series is sequence of observations generated
sequentially through equal space time period. Objective of
time series analysis can be stated succinctly as follows: given
a sequence up to time t, x(1), x(2), ... x(t), find the
continuation x(t+1), x(t+2) … x(t+n). Observations can either
be spatiotemporal or temporal data that is usually stumbled upon in variety of areas. A distinctive characteristic of time
series is a fact that data cannot be generated independently;
their diffusion varies with time which is controlled by trend
and cyclic components. Time series analysis consists of two
steps: (1) building a model that represents a time series, and
(2) using the model to predict (forecast) future values. A time
series is represented by a mathematical model Y(t)=F(t)+R(t).
Here, F(t) represents a ordered or systematic part known as
signal component and R(t) presents a random part know as
noise component. Nevertheless, observation of these two
components cannot be done separately since, involvement of
several parameters take place. Generally, stochastic processes
{Rt, t ∈ I} are used to deal with the random component of
time series that is random function of time and depends upon
structure of Rt. It becomes purely random process if occurred
noise is independent of time (E{Xt}=m),. Systematic part has
non random nature that is deterministic function of time series.
Numerous functions like Fourier series, low degree
polynomial and periodic function are used to analyze and
study the characteristics of the deterministic part of time series
such as trend, cyclic and seasonal component. Structure of Rt
classifies time series model in stationary or non-stationary
time series. Markov process, Moving average (MA),
Autoregressive (AR) and Autoregressive moving average (ARMA) process are used to analyze and handle stationary
time series [2]. Box-Jerkin method based on class of model;
also called Integrated autoregressive moving average
(ARIMA) model is used to deal with non-stationary time
series [1]. Box–Jenkins extended their research by modifying
ARIMA model in general Seasonal integrated autoregressive
moving average (SARIMA) model. On the other hand, fuzzy
time series does designing of time series model under the
fuzzy environment. Time series data in fuzzy environment
contains uncertainty, vagueness and imprecision, which makes
these two approaches act differently in their philosophy.
Concepts of stationary and non-stationary time series in statistical time series are here being dealt as time invariant
fuzzy time series and time variant fuzzy time series. Further,
fuzzy time series analysis deals with fuzzy logical relations in
time series data rather than random and non-random functions
in case of usual time series analysis. In many real life
situations it is hard to harvest a trend or a cycle component in
time series observations and thus can be analyzed by using the
fuzzy time series methods.
IV. FUZZY TIME SERIES
Fuzzy set theory is an intellectual quest in which philosophy
of mathematics: abstraction and idealization are combined.
Fuzzy set theory provides a strict mathematical framework in
which vague conceptual phenomena can be precisely and
rigorously studied. Fuzzy time series is applicable when
process is dynamic and historical data is fuzzy sets or
linguistic values. This section summarizes basic definitions of
fuzzy time series referred in subsequent text.
Fuzzy Set: Fuzzy set is a class of objects in which transition from membership to non-membership is gradual rather then
abrupt. Such class is characterized by a membership function
that assigns a grade or degree of membership between 0 and 1
to an element. Fuzzy set A is defined as set of ordered pairs A
= {(x, µA(x), xϵX} where µA(x) is grade of membership of
element x in set A and X is universal set. Greater µA(x) signify
strength of the statement ―element x belongs to set A‖.
Fuzzy Time Series: Imprecise data in chronological
sequence is considered as Fuzzy Time Series. Let Fi(t) for i=1,
2,… be the fuzzy sets defined on universe of discourse X(t)
than fuzzy time series F(t) consisting of Fi(t), i=1,2,…, is
called as fuzzy time series on X(t). At that, F(t) can be
understood as a fuzzy variable, whereas Fi(t), i=1,2.. are fuzzy
values of F(t).
Fuzzy Relationship : If there exists a fuzzy relationship
R(t+1, t), such that F(t+1)=F(t)*R(t+1,t), where symbol * is an
operator, signifying F(t+1) is derived by F(t). The relationship
between F(t+1) and F(t) can be denoted by expression F(t)
F(t+1). Denoting F(t+1) by At+1 and F(t) by At, the
relationship F(t+1) and F(t) can be defined by logical
relationship At At+1.
Fuzzy Time Series Order : Order of fuzzy time series can be
determined from its definition itself. Suppose, F(t+1) is caused
only by F(t) or F(t-1) or F(t-2) or F(t-m) (m>0).
This relation can be expressed by
0)m)(mt1,(tRm)1)...F(tF(tF(t)1)F(t (1)
Eq. (1) is called first order series of F(t+1). Now, if F(t+1)
is caused by F(t) and F(t-1) .. and F(t-m) simultaneously. This
relation can be expressed as:
0)m)(mt1,(tRm)1)...F(tF(tF(t)1)F(t (2)
Eq. (2) is call mth order model of F(t).
Basic steps of Fuzzy Time Series Predictor(FTSP)
Define Universe of Discourse (U )
Partition U in equal length of intervals.
Define fuzzy sets and fuzzy membership function
Compute fuzzy relationships.
Compute the fuzzified forecasted value.
Defuzzification of fuzzy output for crisp forecasted value.
V. ORDERED WEIGHTED AGGREGATION (OWA)
OWA is eminent aggregation operator that has been proved
effective in various domains. At times, formulation of multiple
parameters based decision problems cannot be done accurately
either by aggregation of pure ―ANDing‖ or pure ―ORing‖ of
parameters. These types of problems require aggregation that
lies between these two extremes. OWA is one such type of
operator that gives flexibility to adjust degree of aggregation.
This generalized averaging operator bounds aggregation range
by maximum (oring) and minimum (anding) operators [42].
OWA operators were first introduced by Yager [22]. They can
be defined as
n
1i ibiwnX,...,1XF (3)
Here, F is an OWA operator of n dimension having X1, X2,
…, Xn as its n arguments. bi is ith largest element from the
collection of arguments X1, X2, …, Xn . wi is ith collection of
weight of arguments such that and . The wi
weights are associated with ordered position of arguments Xi.
Subsequent examples briefly illustrate this concept.
Example 1: Let F be an OWA operator of three dimensions
X1=0.6, X2=0.3, X3=0.7 and associated weights be (0.45, 0.3,
0.25) implying w1=0.45, w2=0.3 and w3=0.25.
Since, X3 is largest among X1, X2 and X3. Hence, b1=X3 i.e
0.7. Similarly, b2=0.6 and b3=0.3. Using eq. (3) F(0.6,0. 3, 0.7)
can be evaluated as:
F (0.6, 0.3, 0.7)) = 0.7*0.45 + 0.6*0.3 + 0.3*0.25 = 0.57
Determination of OWA weights
In order to use OWA, we must determine weight of
arguments of problem domain. Various methods of weight
generation are available in the literature [22, 25,42 and 43].
Our problem domain is fuzzy time series. Therefore, we took
up Regularly Increasing Monotonic (RIM) quantifiers [22]
denoted by Q(r) to determine weights of fuzzy arguments of
fuzzy time series.
RIM quantifiers
Yager [22] in 1988 introduced a class of function to
generate weights using RIM quantifiers. RIM quantifiers accommodate situations involving qualitative statements in
form of fuzzy sets. RIM was proposed to determine OWA
weight by using linguistic quantifiers ―there exists‖ Q*(r)(OR)
and ―for all‖ Q*(r)(AND). Thus for any RIM quantifier Q(r),
the limit Q*(r) <= Q(r)<= Q*(r) holds true [23]. Also, Q(r)
must satisfy two properties: (a) Q(0)=0 and (b) ]1,0[r .
OWA weights with m number of criterion can be determined
using RIM quantifier as [42]
m
1iQ
m
iQiw i=1,2....m (4)
RIM quantifiers can continuously change its values between
Q*(r) and Q*(r). This generates family of RIM quantifiers.
This family of RIM quantifiers can be defined by
parameterized class of fuzzy subsets [43]. It can be further
defined as: Q(r) =rβ where, r≥0. Eq.(4) can be redefined as β
m
1iQ
β
m
iQiw
i=1,2......m (5)
Here, β is a degree of polynomial. By changing β one can
generate different types of quantifiers and associated operators
between two extreme cases of ―all‖ and ―at least one‖
quantifiers. Table I lists various RIM quantifiers for different β.
TABLE I. RIM(Q) VS Β
β Quantifiers Q β Quantifiers Q
β =0 At least one β =1 Half
β =0.1 At least few β =2 Most
β =0.5 A few β =∞ All
For β=1, uniform distribution of weights take place. It
means equal weight is assigned to each criterion i.e. wi =1/n,
where n is number of criterion. For β<1, the RIM quantifier
acts like ―or-type‖ operator. For β>1, the RIM quantifier acts
like ―and-type‖ operator. In fuzzy time series, usually most of the criterion must be satisfied to obtain a solution. Here
―criterion‖ is past observation and ―solution‖ is predicted
value. Since, most of the criterion must be satisfied, ―most‖
RIM quantifiers is best suited for aggregation of past n fuzzy
values. RIM ―most‖ quantifier is defined as Q(r) = r2 (ß=2)
[43].
VI. GENETIC ALGORITHM
Genetic algorithm (GA) is parallel, adaptive and iterative search procedure based on natural selection and mimics
evolution. Holland [44] invented the concept of GA and its
theoretical foundation in 1975. Main advantage of using GA is
that it requires only fitness information, not internal
knowledge of problem or gradient information. GA perform
search from many points simultaneously to find exact or
approximate solution to optimize solution. Pioneered work of
GA is most suitable for non-linear models where finding of
the global optimum is a complex task. Fitness checking
characteristics of GA for each variable makes it optimal for
complex non-linear problems. Also, one of its application areas is membership function optimization in fuzzy logic. Due
to this applicability of GA, it can predict and optimize data
more efficiently than other approaches. Fig. 1 represents basic
GA cycle. GA process starts with construction of population at
time t. Fitness function does evaluation of population.
Selection procedure performs selection of optimal values. A
set of genetic operators (reproduction, crossover and mutation)
creates new population called offspring’s and these offspring’s
are added back to the pool of initial populations.
Figure 1. Basic GA (t) cycle
Some terminology of GA can be defined as [45-46]:
Chromosome : Chromosome is a set of parameters that GA
ascertains as proposed solution of the problem. The
chromosome can be represented either as a binary string, real
number, permutation of elements, list of rules, program
elements or in complex data structure.
Gene : Every variable sequenced to construct chromosome
is called gene
Initial Population : GA has the potential solution set that is
either generated randomly or heuristically. Population size
Initial Population
(t)
Evaluation (t) Selection (t) Operation (t)
Offspring
∑ (At – Ft)
2
t=1
n
ring(t)
11
ni wi]1,0[
iw
depends on nature of the problem. Each individual of
population is called chromosome.
Evaluation : Through evaluation only, GA and problem it is
solving are linked. Each chromosome is decoded in to chosen
representation and assigned a fitness measure.
Selection : In Selection, chromosome for next generation on basis of fitness measure is selected. It selects chromosome of
high fitness values and deletes chromosome of low fitness
measure. Roulette and tournament selection are popular
methods for selection procedure.
Crossover & Mutation: Purpose of Crossover & Mutation
operators is to preserve and introduce diversity in population.
Crossover can be one point, two points, cut & slice and
uniform crossover. In Fig 2, P1 & P2 are parents. C1 and C2
are offspring’s with two point crossover.
Figure 2. Crossover
Crossover can be applied with certain probability (pc) in order to combine genes from two parent chromosomes and
create new children sub-chromosome. pc is random number
between 0 and 1. Crossover points can be randomly chosen. pc
probability means, child will be having pc % genes from
parent1 and pc % genes from parent2.
Mutation operator is to change chromosome from its initial
state by altering one or more of its gene values. The better
solution can be derived from these new gene values. Mutation
happens during evolvement on basis of user-definable
mutation probability (pm). pm of a bits is 1/L, where L is the
length of the binary vector.
Figure 3. Mutation
Terminating Condition : Genetic algorithm will typically
run forever if this stop condition is not mentioned. Common
terminating conditions are: fixed number of generations or
computation time is reached. Sometime, it could be a solution
that satisfies minimum criteria or having highest ranking.
Selection of termination condition depends on type of
problem. In some problems, if successive iterations no longer
produce better results than reaching at such plateau is
considered as a stopping point. Combination of more than one
terminating condition can be possible.
VII. PROPOSED OWA-FTSP-GA ALGORITHM
Principal approach is to design prediction model by using
rate of change of time series data that capture increasing and
decreasing rate of time series data instead of data itself. Rate
of change can better highlight the trend in data and is valid for
all type of application domains. Second advantage of doing so,
is to reduce complexity in GA by cutting down population
size. Since less number of bits are required to represent
percentage than data. Thereafter, weights are generated for
each past observation that contributes in prediction of next value. Weights are calculated on basis of importance of
corresponding fuzzy set in the system. Importance of each
fuzzy set is defined by introducing novel concept of priority
matrix. Subsequently, GA search strategy based on natural
selection is used to set wide parameter range of membership
function at time t rather than single parameter value of
membership function. Additionally, only selective population
is used for fitness checking. Nevertheless, aim is minimization
of squared error for each predicted value along with mean
square error. Fundamental concept underneath OWA-FTSP-
GA is coupling of OWA based FTSP with GA such that
forecasting error can be reduced.
A. Conceptualization of Fuzzy theory for OWA-FTSP-GA
1) Calculation of Rate change of data (RoC) This step is usually carried out as a first step toward making
universe of discourse suitable for numerical calculations by
associating events of different times. Rate of change (RoC) of
time series data can be defined as
100)tX \ )tX - 1t((X 1tRoC (6)
Here, Xt+1 and Xt are values at time t+1 and time t respectively.
RoC is the rate of change of value from time t to t+1.
2) Representation of Mf Fuzzy membership functions can be created randomly or
they can be created as random perturbations around some
nominal functions. Presented algorithm is appropriate for all
membership functions in fuzzy logic. However, in this study,
triangular membership function (Fig. 4) is used because of its
simplicity and importance in forecasting [47].
Figure 4. Triangular Membership Function
Each triangular Mf has three parameters a, b and c. To
represent each parameter of Mf, binary coded scheme is used
due to its simplicity [44-45]. Assuming data range is 1 to 2n-1.
It means that each parameter of triangular Mf requires n bits.
Entire triangular Mf is represented by 3n bits as in Fig. 5.
Each substring is base points of Mf.
Figure 5. Encoded Membership function (t)
Right Limit
{ n bits}
Substring1
Substring2
Substring3
Encoded Mf (t)
Left Limit
{n bits}
Peak Value
{n bits}
Reflected base values B1, B2 & B3 of Substring1, Substring2
& Substring3 respectively are calculated [31] as:
minmaxmin12
DD)(
dDB
Lk
k
(7)
Here, Bk is calculated base values of kth substring, Dmin and
Dmax is lower and upper bound of time series data region to be
transformed. dk is decimal value of kth substring. L is length of
binary string. Base values are reflected in domain interval for
fuzzy mf in range of [Dmax -Dmin].
3) FLR generation for OWA-FTSP-GA We used 3rd order model to forecast value at time t+1. Intent
of using 3rd order fuzzy time series model is to obtain FLR,
free from ambiguities while ensuring low complexity. In this
context, an ambiguity exists if two or more FLR have same
fuzzy sets. This implies that these FLR are not unique. This decreases the forecasting accuracy of model. Use of high-
order fuzzy time series can overcome drawback of ambiguities
that generally exist in low order fuzzy time series model [13].
However, higher order time series induces complexity and
reduces efficiency of model. In our problem domain, to avoid
ambiguities, at least three past fuzzy time series observations
are required to forecast value at time t+1. For simplicity and
appropriateness, we took 3rd order time series in our study.
FLR is derived after fuzzy sets for data have been defined. If,
n is total number of time series data than n-3 FLR in form of
rules will be generated by Ft, Ft-1, Ft-2→Ft+1.
4) OWA weights determination In order to predict Ft+1, we must aggregate Ft, Ft-1, Ft-2 (3
rd order fuzzy time series). In this study, we used OWA to aggregate. In order to use OWA, we must calculate weights of past values. We used importance of past fuzzy set in RIM quantifiers to calculate OWA weight as described.
Priority matrix is created to define importance of each fuzzy set in fuzzy time series like importance of each criterion in multiple criterion based decision problems as:
I. Count Number of RoC those fall in each fuzzy set.
II. Arrange all fuzzy set in ascending order on basis of number of RoC as Count (Fp) < Count (Fq) ...<Count (Fr). Here 1 => p, q, r <= n and n is number of fuzzy set. Assuming that fuzzy set Fr has highest count of RoC and Fp has lowest number of RoC
III. Importance µ (discussed in OWA section) is assigned to each fuzzy set on basis of count of RoC that falls in fuzzy interval of this set as fuzzy set having maximum number of RoC gets highest importance µ. Since this fuzzy set contributes more in overall fuzzy prediction system.
Assign µ(Fi)=1/n, µ(Fk)=2/n …. µ(Fn)=n/n. Now create a priority matrix as
TABLE II. PRIORITY MATRIX
µ(Fi) µ(Fk) µ(Fj) … µ(Fn)
1/n 2/7 3/n … n/n
Let F1, F2,... Fn be n fuzzy variables and importance
associated with these fuzzy variable be µ1, µ2, … µn as determined in previous steps. Eq. (5) can be redefined for
calculation of OWA weights on basis of importance of past fuzzy observation as:
β/T)1jQ(S
β/T)jQ(Sjw j=1, 2, … n (8)
Here, β=2, j
1k kμjS and n
1j jμT
Subsequent example illustrate the steps for calculation of weights using RIM quantifiers
Example 2: Let A1, A2 and A3 be three fuzzy sets. Importance associated with these fuzzy set is µ(A1)=0.5, µ(A2)=0.7, µ(A3)=0.4 and fuzzy quantifiers is ―most‖ (β=2)
n
jT
16.1)4.07.05.0()iµ(A
0976.0)6.1/0()6.1/5.0()6.1/()6.1/)( 2220
211 SQSQw
4649.0)6.1/5.0()6.1/2.1()6.1/()6.1/)( 2221
222 SQSQw
4375.0)6.1/2.1()6.1/6.1()6.1/()6.1/)( 2222
233 SQSQw
14375.01
4649.00976.0 n
j iw
5) Defuzzification Method It is wide area of research which method should be selected
for defuzzification. Way of defuzzification of fuzzy values also affects the accuracy of prediction. In 3rd order fuzzy time series forecasting model, Ft+1 is most effected by Ft follow by Ft-1 and then Ft-2. Hence, ordered sequence of fuzzy set is {Ft, Ft-1, Ft-2}. Let Bt, Bt-1, Bt-2 be base vales of fuzzy intervals Ft, Ft-1 and Ft-2 respectively. Defuzzified value of RoC at t+1 (DRoCt+1) can be calculated as [20]:
T]kw,jw,i[w]2t,1t,t[)1t( BBBRoCD (9)
Here wi, wj and wk are weights generated on basis of
importance of fuzzy sets at time t, t-1 and t-2 respectively in
the system. This forecasting formula fulfils the axioms of
fuzzy set like monotonicity, boundary conditions, continuity
and idempotency.
B. Redefining GA terminology for OWA-FTSP-GA
As triangular membership function is constrained by their
left and right limit of interval. So only, base length of
membership function is unknown variable. Objective of GA is
to generate base length of membership function [31].
Membership function can shrink, move or expand with change
of its parameter values at time t. GA keeps changing the value
of parameters (Substring1, Substring2 and Substring3) at time t
until accurate predicted value for time t+1 is obtained. In this
context, terminology of GA is re-defined for clarity of
proposed concept:
Chromosome: Chromosome in OWA-FTSP-GA is a set of
parameters that GA ascertains as proposed solution of the problem. Chromosome can be represented either as a binary
string or in complex data structure. Our algorithm uses binary
string representation that is encoded form of fuzzy triangular
Mf as in Fig. 5. Each chromosome should have enough
"genetic" information in its binary string to effectively
represent three parameters of membership function.
Representing chromosome in form of Sunstring1 Sunstring2
Substring3 contains the solution of problem. Since, each
substring can be denoted by n bits; this implies that each
chromosome can be characterized by 3n bits.
Chromosome=Base1 Base2 Base3
Gene: Parameters Base1, Base2 and Base3 are genes.
Initialization of Population: In OWA-FTSP-GA,
Population P(initial set of chromosomes) is randomly
generated.
Fitness technique: Fitness of chromosome can be judged
from the value of squared error (SE). Minimum value of SE leads to maximum fitness value
SEt = (FVt - AVt)2 (10)
Here, FVt is forecasted and AVt is actual value at time t.
Crossover & Mutation: In OWA-FTSP-GA, we used
uniform crossover and mutation with probability of crossover
as ―pc‖ and probability of mutation as ―pm‖.
Selection: Selection in OWA-FTSP-GA is selection of
offspring’s for next generation on basis of fitness technique
that minimizes the SE. In this algorithm, Roulette wheel
selection is used.
Stop Criteria : Stop criteria in OWA-FTSP-GA depends
upon how much accuracy is expected from forecasting system. In general, major factors that affects accuracy in prediction are
number of generation, pc & pm. In this algorithm, population
count = maximum generation is taken as stopping condition.
C. Design of OWA-FTSP-GA Algorithm
Step 1: Calculate RoC for time series data using eq. (6). This step is usually carried out as a first step towards making of Universe of Discourse.
Step 2: Define Universe of Discourse U= [Dmin-D1, Dmax+D2]. Here Dmax and Dmin are maximum and minimum values of RoC respectively. D1 and D2 are positive real values to partition U in intervals say; u1, u2, u3...un of equal lengths. Define fuzzy set Fi(i=1,..,n) for each interval and fuzzify the time series data such that every RoC is represented by some fuzzy set Fi defined as:
F1=1/ u1 + 0.5/ u2 + 0/ u3 + …+0/un-1 + 0/ un
F2=0.5/ u1 + 1/ u2 + 0.5/ u3 + …+0/un-1 + 0/ un
F3=0/ u1 + 0.5/ u2 + 1/ u3 + 0.5/ u4 …+0/un-1 + 0/ un
.
.
Fn-1=0/ u1 + 0/ u2 + 0/ u3 + …0.5/ un-2 +1/un-1 + 0.5/ un
Fn= 0/ u1 + 0/ u2 + 0/ u3 + …+0.5/un-1 + 1/ un
Now, define 3rd order fuzzy logical relationship (FLR) on these sets as discussed.
Step 3: Create Priority matrix (PM) to define importance of each fuzzy set in fuzzy time series as explained.
Step 4: In order to use OWA, we must calculate weight of past observations. We used RIM quantifiers to calculate OWA weight using eq. (8).
Step 5: To unify GA with OWA-Fuzzy, initialize GA parameters: P, pc, pm, termination generation G and number of offspring’s O to be added in each iteration.
Step 6: Randomly generate initial population P & initialize t=5.
Step 7: Compute defuzzified (forecasted) value of RoC using eq. (9)
Step 8: Apply fitness technique to evaluate each chromosome in generation at time t. Fitness of chromosome can be judged from the value of squared error (SE) defined by eq. (10). Minimum value of SE leads to maximum fitness value.
Step 9: Generate new O number of offspring by performing crossover and mutation on P. Till count of generation is less than stopping criterion repeat steps 7 to 9.
Step 10: Once stopping criteria is met, calculate final forecasted value as:
tX100))tX1)RoC(t((D1tFore. (11)
Here, Xt is actual value at time t and DRoC(t+1) is forecasted RoC at time t+1.
D. Algorithm Performance Parameters
There are number of factors which must be taken into consideration while formulating OWA-FTSP-GA algorithm. Accuracy and efficiency are correlated directly with its careful selection of parameters. Most influential point in simulation of OWA-FTSP-GA is how to firm probability of crossover, mutation rate, population size and count of terminating generation. These parameters interact with each other during algorithm execution and derive efficiency of the algorithm & accuracy of the predicted value. Selections of these parameters change with problem and are determined experimentally.
VIII. PRELIMINARY EXPERIMENT
As with most of cited papers, historical enrollment data of University of Alabama is used in this study to illustrate the
new forecasting process. Stepwise results obtained are:
Step 1: Time Series Data is given year wise. Calculated
RoC of year 1971 and onwards is shown in TABLE III.
TABLE III. ROC(ENROLL)
Year Enrollment RoC Year Enrollment RoC
1971 13055 1982 15433 -5.83%
1972 13563 3.89% 1983 15497 0.41%
1973 13867 2.24% 1984 15145 -2.27%
1974 14696 5.98% 1985 15163 0.12%
1975 15460 5.20% 1986 15984 5.41%
1976 15311 -0.96% 1987 16859 5.47%
1977 15603 1.91% 1988 18150 7.66%
1978 15861 1.65% 1989 18970 4.52%
1979 16807 5.96% 1990 19328 1.89%
1980 16919 0.67% 1991 19337 0.05%
1981 16388 -3.14% 1992 18876 -2.38%
Step 2: Obtain Dmin= -5.83%, Dmax=7.66% from TABLE III. Define universe of discourse U and partition it into intervals
u1, u2… un of equal length. Thus universe of discourse will be
U = [-6, 8]. Partitioning U into seven equal intervals as u1=[-6
-4], u2=[-4 -2], u3=[-2 0], u4=[0 2], u5=[2 4], u6=[4 6], u7=[6
8]. Fuzzification of data & FLR is shown TABLE V.
Step 3: Priority Matrix is created for enrollment data as:
number of RoC that fall in each fuzzy interval is calculated:
Count(F1)=1, Count(F2)=4, Count(F3)=1, Count(F4)=7,
Count(F5)=2, Count(F6)=5, Count(F7)=1. Further arranging
fuzzy set in ascending order on basis of RoC count i.e. F1≤ F3≤
F7<F5<F2<F6<F4 and assigning importance for priority matrix
defined in TABLE IV.
TABLE IV. PRIORITY MATRIX (ENROLL)
µ(F1) µ(F3) µ(F7) µ(F5) µ(F2) µ(F6) µ(F4)
1/7 2/7 3/7 4/7 5/7 6/7 7/7
Step 4: OWA weights are determined for each FLR. FLR at
year 1976 is F2, F6, F5 → F3. Hence weights w1, w2 and w3 are
determined for F2, F6 and F5 respectively. Importance of µ(F2),
µ(F6) and µ(F5) is 4/7,5/7 and 2/7 respectively from priority
matrix (TABLE IV. ).
{b1, b2, b3} = {µ(F2), µ(F6), µ(F5)} = {0.57, 0.71, .29}
w1 = (0.57/1.57)2 - (0/1.57)2 = 0.0625,
w2 = (1.28/2.28)2 - (0.57/2.28)2 = 0.25390625,
w3 = (2.28/2.28)2 - (1.28/2.28)2 = 0.6835,
Similarly, weights for all FLR are calculated and given in
TABLE V.
TABLE V. OWA WEIGHTS (ENROLL)
Year Fuzzy FLR OWA Weights
1971 -
1972 F5 -
1973 F5 -
1974 F6 -
1975 F2 F6,F5,F5 → F2 w1=0.18, w2=0.33, w3=0.49
1976 F3 F2,F6,F5 → F3 w1=0.11, w2=0.43, w3=0.46
1977 F4 F3,F2,F6 → F4 w1=0.02, w2=0.27, w3=0.71
1978 F4 F4,F3,F2 → F4 w1=0.25, w2=0.16, w3=0.59
1979 F6 F4,F4,F3 → F6 w1=0.19, w2=0.57, w3=0.23
1980 F4 F6,F4,F4 → F4 w1=0.09, w2=0.33, w3=0.58
1981 F2 F4,F6,F4 → F2 w1=0.12, w2=0.30, w3=0.58
1982 F1 F2,F4,F6 → F1 w1=0.08, w2=0.37, w3=0.56
1983 F4 F1,F2,F4 → F4 w1=0.01, w2=0.21, w3=0.79
1984 F2 F4,F1,F2 → F2 w1=0.29, w2=0.09, w3=0.62
1985 F4 F2,F4,F1 → F4 w1=0.15, w2=0.70, w3=0.15
1986 F6 F4,F2,F4 → F6 w1=0.14, w2=0.26, w3=0.60
1987 F6 F6,F4,F2 → F6 w1=0.11, w2=0.41, w3=0.48
1988 F7 F6,F6,F4 → F7 w1=0.10, w2=0.30, w3=0.60
1989 F6 F7,F6,F6 → F6 w1=0.04, w2=0.32, w3=0.64
1990 F4 F6,F7,F6 → F4 w1=0.16, w2=0.2, w3=0.64
1991 F4 F4,F6,F7 →F4 w1=0.19, w2=0.47, w3=0.34
1992 F2 F4,F4,F6 → F2 w1=0.12, w2=0.37, w3=0.51
Step 5: From TABLE III. , it can be observed that RoC
range is 1 to 24. It implies each parameter of triangular mf requires 4 bits to represent its value. Since, chromosome
constitutes three parameters of triangular mf; it will be
represented by 12 bits. For simulation of proposed model,
following combination of parameters has been used to achieve
better performance: initial P = 20, pc = 0.33, pm = 0.083,
Termination Generation G = 100 and O = 4.
Step 6 & 7: Randomly generated initial population P,
calculated reflected base values and computed forecasted RoC
for year 1975 (t=5) are listed in TABLE VI.
TABLE VI. INITIAL 20 CHROMOSOME AND DROC (ENROLL)
No Chromosome B1 B2 B3 DRoC
1 0010 0110 0000 4.031 -0.434 -5.83 -3.738
2 0100 1101 1001 2.233 5.861 2.264 2.612
3 0011 1101 1110 -3.132 5.861 6.761 4.65
4 1010 0100 1111 3.163 2.233 7.66 3.604
5 1110 1100 1000 6.761 4.962 1.365 3.53
6 0000 0010 0110 -5.83 4.031 -0.434 -2.6
7 0000 0101 0100 -5.83 1.333 2.233 -2.6
8 0010 1010 0100 4.031 3.163 2.233 -0.801
9 0010 1011 1101 4.031 4.063 5.861 3.457
10 1111 1101 0110 7.66 5.861 -0.434 3.108
11 1000 1111 1111 1.365 7.66 7.66 6.504
12 0101 0101 1011 1.333 1.333 4.063 1.31
13 1101 0110 1101 5.861 -0.434 5.861 3.806
14 0101 0100 1001 1.333 2.233 2.264 0.135
15 0001 1000 1001 4.931 1.365 2.264 0.649
16 1010 0110 1010 3.163 -0.434 3.163 1.989
17 1011 1011 1010 0.465 4.063 5.861 4.283
18 0010 1100 1111 -4.03 4.962 7.66 -1.168
19 1010 0000 0001 3.163 -5.83 4.931 1.254
20 0000 1010 1010 -5.83 3.163 3.163 3.016
TABLE VII. SELECTED POPULATION (ENROLL)
Chromosome B1 B2 B3 DRoC Fitness
0011 1101 1110 -3.132 5.861 6.761 4.65 0.266
0111 1011 1101 0.465 4.063 5.861 4.283 0.814
TABLE VIII. FORECASTED & SE (ENROLL)
Year Enroll. RoC DRoC Fore. SE
1971 13055 - -
1972 13563 3.891 -
1973 13867 2.241 -
1974 14696 5.978 -
1975 15460 5.199 4.65 15379.328 6507.927
1976 15311 -0.96 -1.207 15273.42 1412.28
1977 15603 1.907 1.842 15593.021 99.574
1978 15861 1.654 1.748 15875.817 219.553
1979 16807 5.964 6.455 16884.84 6059.092
1980 16919 0.666 0.609 16909.392 92.316
1981 16388 -3.14 -2.404 16512.344 15461.307
1982 15433 -5.83 -3.71 15780.05 120443.41
1983 15497 0.415 0.918 15574.6 6021.684
1984 15145 -2.27 -2.275 15144.451 0.301
1985 15163 0.119 0.028 15149.316 187.248
1986 15984 5.415 5.182 15948.677 1247.736
1987 16859 5.474 5.292 16829.94 844.484
1988 18150 7.658 7.032 18044.585 11112.252
1989 18970 4.518 4.314 18933.08 1363.064
1990 19328 1.887 2.12 19372.183 1952.135
1991 19337 0.047 -0.047 19318.856 329.188
1992 18876 -2.38 -2.417 18869.621 40.688
MSE=9633.0136
Step 8: Now fitness of each individual is evaluated as
shown in TABLE VII. Best forecasted value with least error is
4.65 at year 1975 (t=5). Similarity, forecasted value of
remaining time series is calculated. Forecasted value of each
time with least squared error for 1st iteration is shown in
TABLE VIII.
Step 9: Till stop criterion is met, generate offspring by
doing crossover & mutation with pc=0.330357 & pm=0.083
and repeat steps 7 to 9.
Step 10: At G=100, forecasted enrollment is shown in
TABLE IX.
TABLE IX. ACTUAL VS FORECASTED (:ENROLL)
Year Actual Fore.-Enroll Year Actual Fore. Enroll
1971 13055 1982 15433 15500.847
1972 13563 1983 15497 15494.119
1973 13867 1984 15145 15144.451
1974 14696 1985 15163 15170.38
1975 15460 15519.598 1986 15984 15948.677
1976 15311 15312.541 1987 16859 16829.94
1977 15603 15607.037 1988 18150 18044.585
1978 15861 15875.817 1989 18970 18965.722
1979 16807 16763.363 1990 19328 19317.588
1980 16919 16915.819 1991 19337 19341.288
1981 16388 16402.793 1992 18876 18869.621
E. Study of MSE & AFER with GA performance factors
Forecasting accuracy of whole system is measured by
average forecasting error rate (AFER) and mean squared error
(MSE) defined [49] by eq. (12) and eq. (13).
n
5t100%))/nt/A|tFtA(|(AFER (12)
n
tntFtAMSE
5/2)( (13)
Here, n is total number of time series data, Ft is forecasted
value and At is actual value of time series data at time t.
MSE & AFER varies with count of generation, pc and pm.
Forecasting of enrollment is done on initial P=20, pc=0.33 &
pm=0.083 and after every iteration population is increased by
4 i.e. 4 offspring’s added. TABLE X. lists MSE & AFER for
each generation count. Here, mutation rate and crossover
probability remains same at each G.
TABLE X. MSE & AFER VS G (ENROLL)
G MSE AFER (%) G MSE AFER (%)
24 8581.738 0.32 64 8036.338 0.275
28 8470.498 0.309 68 8036.338 0.275
32 8409.728 0.301 72 1482.672 0.161
36 8395.666 0.298 76 1482.672 0.161
40 8395.666 0.298 80 1478.046 0.159
44 8367.624 0.295 84 1387.401 0.152
48 8364.289 0.294 88 1387.401 0.152
52 8096.739 0.282 92 1337.872 0.142
56 8082.976 0.279 96 1332.218 0.139
60 8078.409 0.277 100 1332.218 0.139
Sometime, proposed algorithm has slow convergence speed
i.e. consecutively generated MSE & AFER are same due to
absence of diversity in the population, since pc & pm are same
for all iterations. Tuning probabilities of crossover ―pc‖ and
mutation rate ―pm‖ depends upon fitness value solution [35].
If value of G, pc and pm are increased, it will increase complexity and execution time of GA. Careful selection of G,
pc & pm depends on how much accuracy is expected from
system. TABLE XI. shows variation of MSE & AFER with
different values of G at fixed pc= 0.5, pm=0.083. TABLE
XII. shows variation of MSE & AFER with different values of
G at fixed pc=0.8, pm=0.16.
TABLE XI. PC=0.5, PM =0.083 & G=100 (ENROLL)
G MSE AFER (%) G MSE AFER (%)
24 6829.862 0.372 64 947.974 0.127
28 6217.147 0.324 68 925.172 0.123
32 5756.787 0.298 72 894.348 0.116
36 5756.138 0.298 76 894.348 0.116
40 5725.834 0.289 80 894.348 0.116
44 5594.094 0.272 84 894.348 0.116
48 5585.99 0.269 88 894.348 0.116
52 3012.586 0.19 92 894.348 0.116
56 2206.476 0.159 96 894.348 0.116
60 2206.476 0.159 100 894.348 0.116
TABLE XII. PC=0.8, PM =0.16 & G=100 (ENROLL)
G MSE AFER (%) G MSE AFER (%)
24 5232.382 0.324 64 2321.236 0.14
28 3733.519 0.258 68 2321.236 0.14
32 3166.93 0.219 72 2320.837 0.139
36 3147.176 0.213 76 2310.551 0.138
40 3137.824 0.211 80 2310.551 0.138
44 2717.25 0.184 84 2310.551 0.138
48 2530.918 0.163 88 2310.551 0.138
52 2514.997 0.162 92 601.616 0.096
56 2507.939 0.159 96 585.29 0.093
60 2343.469 0.146 100 585.29 0.093
It can be seen from TABLE XIII. that MSE & AFER tends
to zero at O=10, G=200, pc=0.8 and pm=0.16.
TABLE XIII. G=200, PC=0.8, PM =0.16 (ENROLL)
G MSE AFER (%) G MSE AFER (%)
20 9633 0.365 120 417.28 0.066
30 3071.5 0.204 130 389.13 0.06
40 2831.1 0.174 140 389.13 0.06
50 1589.4 0.135 150 388.31 0.059
60 1575 0.13 160 387.99 0.059
70 1383.4 0.105 170 384.66 0.057
80 1370.1 0.103 180 384.66 0.057
90 449.44 0.074 190 384.66 0.057
100 420.31 0.069 200 306.08 0.053
110 418.1 0.067
If past fuzzy observations are assigned equal weight, accuracy
of forecasting system gets degraded. From TABLE XIV. it can
be seen that utilizing OWA weights for past observations
reduced MSE by almost 50% of forecasting done by equal
weights. Thus it is proved that accuracy of system can be
improved by determining relative weight of past observations.
TABLE XIV. COMPARISON OF SE (W1=W2=W3) & SE-OWA (ENROLL)
Year Enroll
Fore. Enroll
w1=w2= w3)
SE (w1=w2
=w3) Fore.OWA
Proposed-SE
(OWA weights)
1971 13055
1972 13563
1973 13867 13910.72 1911.76
1974 14696 14679.79 262.729
1975 15460 15425.22 1209.96 15379.328 6507.927
1976 15311 15346.55 1264.124 15273.42 1412.28
1977 15603 15565.84 1380.762 15593.021 99.574
1978 15861 15862.7 2.895 15875.817 219.553
1979 16807 16790.67 266.797 16884.84 6059.092
1980 16919 16935.59 275.24 16909.392 92.316
1981 16388 16439.81 2684.736 16512.344 15461.307
1982 15433 15923.85 240937.3 15780.05 120443.41
1983 15497 15504.82 61.153 15574.6 6021.684
1984 15145 15104.54 1636.717 15144.451 0.301
1985 15163 15170.07 49.985 15149.316 187.248
1986 15984 16051.75 4590.601 15948.677 1247.736
1987 16859 16920.88 3828.58 16829.94 844.484
1988 18150 17897.71 63652.53 18044.585 11112.252
1989 18970 19050.6 6496.844 18933.08 1363.064
1990 19328 19285.74 1785.654 19372.183 1952.135
1991 19337 19359.99 528.737 19318.856 329.188
1992 18876 18847.3 823.867 18869.621 40.688
MSE=18415.36 MSE=9633.01
F. Performance evaluation and comparison
We evaluated our proposed model on enrollment data of
University of Alabama and compared the results with previous
selective prediction models [6,7,8,10,16,11,13,15,18,36] on
same data to showcase the performance of our method.
TABLE XV. COMPARISON OF MSE AND AFER OF MODELS
Method MSE AFER
Song Chissom[6] 775687 4.38
Song [7] Chissom - adv 407507 3.11
Chen[8] 321418 3.12
Hwang chen & lee[10] 226611 2.45
Singh[16] 90997 1.53
Hurang[11] 86694 1.53
Chen adv[13] 86694 1.53
Jilani, Burney & Ardi –adv[15] 41426 1.02
B.Garg, Sufyan & Ansari[18] 9917.16 0.34
Chen and Chung [36] (G=1000) 1101 0.12
Proposed method -G=100, pc=0.5, pm=0.083 894.348 0.116
Proposed method -G=100, pc=0.8, pm=0.16 585.29 0.093
Figure 6. and Figure 7. reveal MSE and AFER of all
selected models for enrollment data graphically.
Figure 6. Comparison of MSE (Enroll)
Figure 7. Comparison of AFER (Enroll)
G. Discussion
Least value of MSE and AFER are very crucial and important factors in evaluation of forecasting models. It is also
considered that lower value of MSE and AFER are measure of
higher forecasting accuracy. From TABLE XV. , it can be
observed that MSE and AFER are lowest in case of our
proposed model, clearly indicating its superiority over other
existing soft computing models. Previous forecasting models
neither focused on importance of each fuzzy set nor
considered the relative weights of past fuzzy observation in
designing of forecasting model. Most of the forecasting
models did prediction only on basis of one parameter set of
membership function. However, our model calculated importance of each fuzzy set in system and did aggregation of
past fuzzy observations on basis of their relative weights. Each
fuzzy set was assigned weight on basis of its contribution in
forecasting. Thereafter, we did coupling of GA with OWA
based FTSP in such a way that prediction can be done on basis
of wide range of parameters and accuracy can be improved for
each time series data along with the accuracy of whole system.
Although some models [36] also did optimization of MSE by
coupling of fuzzy time series with GA. This model was able to
reduce MSE to a certain point but with excessive complexity;
since this model used nine chromosomes and 1000 generation
to obtain least MSE. But in our model, first we tried to keep chromosome size as minimum as possible by taking rate
change of data and usage of binary coding. Second, we used
3rd order time series model to reduce complexity and
execution time. Hence, RoC instead of data, weight
aggregation of past observation and generating wide range of
parameter of mf have improved accuracy of forecasting model
& reduced complexity in the system.
IX. EXTENSIVE EXPERIMENT
Extensive experiment of our proposed model is performed on health care domain. In developing countries, health care is being considered as fast growing industry with 26% compounded annual growth rate [51, 52]. The quality of health care delivery is urged in this competitive environment. Quality in health care is directly proportional to distribution of hospital material resources suitably and schedule for human resources and finances reasonably. Consequently, efficient distribution and scheduling of resources is required in various departments of hospital. Among all departments, managing and planning of resources in the outpatient department is very important. This department provides patient diagnoses, treatment and health care protection. Prior knowledge of number of outpatient visit can help health care administrator to make strategic decision, manage operation and distribute resources adequately. Accurate forecasting of outpatient visits can help heath care administrators to determine best plan for work, supply and demand for health services. Furthermore forecasting of outpatient visit does not only influence patient waiting time but also improve the coordination of care. Therefore, we considered this application domain to demonstrate potential of our proposed model. We used same historical outpatient data, dataset for year 2004 and 2005 of outpatient visits [21] collected from the department of internal medicine of a hospital, that has been used in previous research [20, 21, 26] to evaluate proposed model. Step wise results obtained are:
Step 1: Calculated RoC of number of outpatient visit in
hospital for each month is shown in TABLE XVI.
TABLE XVI. ROC (OUTPATIENT)
Month Outpatient RoC Month Outpatient RoC
2004/01 6519 2005/01 5920 -1.53
2004/02 5979 -8.283 2005/02 5512 -6.892
2004/03 6322 5.737 2005/03 6548 18.795
2004/04 5666 -10.377 2005/04 5987 -8.568
2004/05 5318 -6.142 2005/05 5638 -5.829
2004/06 5364 0.865 2005/06 5851 3.778
2004/07 5513 2.778 2005/07 5514 -5.76
2004/08 5895 6.929 2005/08 5395 -2.158
2004/09 6682 13.35 2005/09 5598 3.763
2004/10 6254 -6.405 2005/10 5284 -5.609
2004/11 5681 -9.162 2005/11 4545 -13.986
2004/12 6012 5.826 2005/12 4624 1.738
Step 2: Obtain Dmin=-13.98, Dmax=18.79 from TABLE XVI.
Define Universe of Discourse U = [-15, 20] where D1=1.02
and D2=1.21. Equal intervals are u1=[-15 -10], u2=[-10 -5],
u3=[-5 0], u4=[0 5], u5=[5 10], u6=[10 15], u7=[15 20].
Fuzzification of data & FLR is shown in TABLE XVIII.
Step 3: Priority matrix is created as: number of RoC that fall
in each fuzzy interval is calculated: Count(F1)=2,
Count(F2)=8, Count(F3)=2, Count(F4)=5, Count(F5)=4,
Count(F6)=1, Count(F7)=1. Further arranging fuzzy sets in
ascending order on basis of RoC count i.e.
F7<F6<F3<F1<F5<F4<F2 and assigning importance in Priority
Matrix defined in TABLE XVII.
TABLE XVII. PRIORITY MATRIX (OUTPATIENT)
µ(F7) µ(F6) µ(F3) µ(F1) µ(F5) µ(F4) µ(F2)
1/7 2/7 3/7 4/7 5/7 6/7 7/7
Step 4: OWA weights are determined for each FLR. FLR at
time 2004/05 is F1, F5, F2 → F2. Hence weights w1, w2 and w3
are determined for F1, F5 and F2 respectively. Importance of
µ(F1), µ(F5) and µ(F2) is 4/7, 5/7 and 7/7 respectively from
Priority Matrix (TABLE XVII. ).
{µ(F1), µ(F5), µ(F2)} = {0.57, 0.71, 1}
w1 = (0.57/2.28)2 - (0/2.28)2 = 0.0625,
w2 = (1.28/2.28)2 - (0.57/2.28)2 = 0.25390625,
w3 = (2.28/2.28)2 - (1.28/2.28)2 = 0.6835,
Similarly, weights for all FLR are calculated and given in
TABLE XVIII.
TABLE XVIII. OWA WEIGHTS (OUTPATIENT)
Month Fuzzy FLR OWA Weights
2004/01 -
2004/02 F2 -
2004/03 F5 -
2004/04 F1 -
2004/05 F2 F1,F5,F2 → F2 w1=0.06, w2=0.26, w3=0.68
2004/06 F4 F2,F1,F5 → F4 w1=0.19, w2=0.28, w3=0.53
2004/07 F4 F4,F2,F1 → F4 w1=0.12, w2=0.46, w3=0.42
2004/08 F5 F4,F4,F2 → F5 w1=0.1, w2=0.3, w3=0.6
2004/09 F6 F5,F4,F4 → F6 w1=0.09, w2=0.33, w3=0.58
2004/10 F2 F6,F5,F4 → F2 w1=0.02, w2=0.27, w3=0.71
2004/11 F2 F2,F6,F5 → F2 w1=0.25, w2=0.16, w3=0.59
2004/12 F5 F2,F2,F6 → F5 w1=0.19, w2=0.58, w3=0.23
2005/01 F3 F5,F2,F2 → F3 w1=0.07, w2=0.33, w3=0.6
2005/02 F2 F3,F5,F2 → F2 w1=0.04, w2=0.24, w3=0.72
2005/03 F7 F2,F3,F5 → F5 w1=0.22, w2=0.22, w3=0.56
2005/04 F2 F7,F2,F3 → F2 w1=0.01, w2=0.52, w3=0.47
2005/05 F2 F2,F7,F2 → F2 w1=0.22, w2=0.06, w3=0.72
2005/06 F4 F2,F2,F7 → F4 w1=0.22, w2=0.65, w3=0.13
2005/07 F2 F4,F2,F2 → F2 w1=0.09, w2=0.33, w3=0.58
2005/08 F3 F2,F4,F2 → F3 w1=0.12, w2=0.3, w3=0.58
2005/09 F4 F3,F2,F4 → F4 w1=0.04, w2=0.35, w3=0.61
2005/10 F5 F4,F3,F2 →F5 w1=0.14, w2=0.18, w3=0.68
2005/11 F1 F5,F4,F3 → F1 w1=0.13, w2=0.49, w3=0.38
2005/12 F4 F1,F5,F4 → F4 w1=0.07, w2=0.29, w3=0.64
Step 5: From TABLE XVI. , it can be observed that RoC
range is 1 to 25. It implies each parameter of triangular mf
requires 5 bits to represent its value. Since, chromosome
constitutes three parameters of triangular mf; it will be
represented by 15 bits. For simulation of proposed model, following combination of parameters has been used to achieve
better performance: initial P = 20, pc = 0.33, pm = 0.067,
Termination Generation G = 100 and O = 4.
Step 6 & 7: Randomly generated initial population P,
calculated reflected base values and computed forecasted RoC
for month 2004/05 (t=5) are listed in TABLE XIX.
TABLE XIX. INITIAL 20 CHROMOSOME AND DROC (OUTPATIENT)
No Chromosome B1 B2 B3 DRoC
1 10010 01011 01101 5.049 -2.35 -0.24 -0.445
2 11010 01001 11011 13.51 -4.47 14.57 9.667
3 11100 00001 00101 15.62 -12.9 -8.7 -8.252
4 00111 10110 11001 -6.583 9.28 12.45 10.456
5 11011 10001 01010 14.57 3.99 -3.41 -0.408
6 11110 11111 11100 17.74 18.8 15.62 16.561
7 11111 00000 11110 18.8 -14 17.74 9.749
8 11110 00100 11100 17.74 -9.76 15.62 9.312
9 11100 00110 00110 15.62 -7.64 -7.64 -6.187
10 00100 00100 10110 -9.756 -9.76 9.278 3.256
11 00100 11001 01001 -9.756 12.5 -4.47 -0.503
12 01001 00100 11011 -4.469 -9.76 14.57 7.201
13 00110 10100 11011 -7.641 7.16 14.57 11.298
14 00101 10110 11010 -8.698 9.28 13.51 11.046
15 00001 01100 01111 -12.93 -1.3 1.876 0.145
16 00110 10000 01101 -7.641 2.93 -0.24 0.104
17 11111 10000 10011 18.8 2.93 6.106 6.094
18 01000 01011 11000 -5.526 -2.35 11.39 6.845
19 01001 10010 00111 -4.469 5.05 -6.58 -3.498
20 11100 10100 00101 15.62 7.16 -8.7 -3.151
TABLE XX. FITTNESS (OUTPATIENT)
No Chromosome B1 B2 B3 DRoC Fitness
3 11100 00001 00101 15.62 -13 -8.7 -8.252 4.453
9 11100 00110 00110 15.62 -7.6 -7.6 -6.187 0.002
TABLE XXI. FORECASTED & SE (OUTPATIENT)
Month Outpatient RoC DRoC Fore-Out SE
2004/01 6519 - -
2004/02 5979 -8.283 -
2004/03 6322 5.737 -
2004/04 5666 -10.377 -
2004/05 5318 -6.142 -6.187 5315.45 6.503
2004/06 5364 0.865 0.419 5340.266 563.298
2004/07 5513 2.778 2.659 5506.624 40.649
2004/08 5895 6.929 6.422 5867.042 781.641
2004/09 6682 13.35 10.186 6495.449 34801.424
2004/10 6254 -6.405 -7.09 6208.277 2090.633
2004/11 5681 -9.162 -7.56 5781.205 10040.977
2004/12 6012 5.826 6.065 6025.549 183.571
2005/01 5920 -1.53 -1.785 5904.697 234.189
2005/02 5512 -6.892 -6.71 5522.748 115.512
2005/03 6548 18.795 16.803 6438.153 12066.296
2005/04 5987 -8.568 -7.448 6060.317 5375.445
2005/05 5638 -5.829 -5.348 5666.81 830.007
2005/06 5851 3.778 4.893 5913.877 3953.487
2005/07 5514 -5.76 -5.547 5526.436 154.662
2005/08 5395 -2.158 -2.835 5357.692 1391.915
2005/09 5598 3.763 5.252 5678.333 6453.425
2005/10 5284 -5.609 -6.022 5260.87 534.983
2005/11 4545 -13.986 -7.666 4878.916 111499.98
2005/12 4624 1.738 2.426 4655.263 977.362
MSE= 9604.79
Step 8: Now fitness of each individual is evaluated as
shown in TABLE XX. The best forecasted value with least
error is -6.187 for month 2004/05 (t=5). Similarity, forecasted
value of remaining time series is calculated. Forecasted value
of each time with least squared error for 1st iteration is shown
in TABLE XXI.
Step 9: Till stop criterion is met, generate offsprings by
doing crossover & mutation with pc=0.33 & pm=0.083 and
repeat steps 7 to 9.
Step 10: At G=100, forecasted outpatient is shown in
TABLE XXII.
TABLE XXII. ACTUAL VS FORECASTED (OUTPATIENT)
Month Actual Fore-Out Month Actual Fore-Out
2004/01 6519 2005/01 5920 5922.643
2004/02 5979 2005/02 5512 5520.82
2004/03 6322 2005/03 6548 6456.819
2004/04 5666 2005/04 5987 5988.736
2004/05 5318 5318.728 2005/05 5638 5645.941
2004/06 5364 5364.205 2005/06 5851 5857.186
2004/07 5513 5525.875 2005/07 5514 5509.422
2004/08 5895 5897.881 2005/08 5395 5381.305
2004/09 6682 6712.213 2005/09 5598 5600.381
2004/10 6254 6256.672 2005/10 5284 5288.88
2004/11 5681 5708.67 2005/11 4545 4792.229
2004/12 6012 6013.8 2005/12 4624 4627.715
H. Study of MSE & AFER with GA performance factors
Forecasting accuracy is measured by average forecasting
error rate (AFER) and mean squared error (MSE) defined by
eq. (12) and eq. (13). Calculation of MSE and AFER are
shown in TABLE XXIII. for various generation G counts at
fixed pc=0.33, pm=0.083 & O=4.
Selection of P, pc, pm and G depends upon desired
accuracy from system. Variation in different values of pc, pm
& G can be studied in order to get least desired MSE & AFER
of the system. TABLE XXIV. shows variation of MSE &
AFER with different values of G at fixed pc= 0.5, pm=0.083.
TABLE XXV. shows variation of MSE & AFER with different values of G at fixed pc= 0.8, pm=0.13. Impact on
MSE & AFER for increased values of pc and pm can be
observed.
TABLE XXIII. MSE & AFER VS GENERATION (G) (OUTPATIENT)
G MSE AFER (%) G MSE AFER (%)
20 9604.798 1.117 64 6529.746 0.627
24 9339.589 0.928 68 6523.362 0.618
28 8776.186 0.868 72 4464.24 0.517
32 8562.679 0.804 76 4463.931 0.516
36 8540.506 0.788 80 3616.76 0.481
40 7644.029 0.73 84 3604 0.473
44 7644.029 0.73 88 3588.866 0.46
48 7577.033 0.689 92 3588.866 0.46
52 7558.111 0.683 96 3587.02 0.459
56 7516.965 0.664 100 3587.02 0.459
60 7516.965 0.664
TABLE XXIV. PC=0.5,PM=0.083 & G=100 (OUTPATIENT)
G MSE AFER (%) G MSE AFER (%)
20 9604.798 1.117 64 4618.228 0.589
24 7764.041 0.957 68 3911.135 0.501
28 7717.867 0.931 72 3911.135 0.501
32 7537.732 0.894 76 3911.135 0.501
36 7518.321 0.882 80 2041.643 0.402
40 7446.475 0.852 84 2041.643 0.402
44 6293.054 0.762 88 2039.509 0.397
48 4687.663 0.634 92 2039.509 0.397
52 4657.381 0.614 96 2037.923 0.395
56 4650.416 0.611 100 2037.923 0.395
60 4618.228 0.589
TABLE XXV. PC=0.8,PM=0.13 & G=100 (OWA-FTSP-GA:OUTPATIENT)
G MSE AFER (%) G MSE AFER (%)
24 6630.065 0.865 64 1272.215 0.345
28 6226.5 0.765 68 973.474 0.294
32 4225.714 0.63 72 973.474 0.294
36 4205.109 0.62 76 973.474 0.294
40 4189.203 0.614 80 969.104 0.29
44 3907.148 0.546 84 969.104 0.29
48 3852.394 0.513 88 968.335 0.289
52 3852.394 0.513 92 968.335 0.289
56 1282.891 0.35 96 962.285 0.283
60 1282.891 0.35 100 962.285 0.283
It can be seen from TABLE XXVI. that MSE & AFER
tends to zero at O=10, G=200, pc=0.8 and pm=0.13.
TABLE XXVI. G=200, PC=0.8, PM =0.13 (OUTPATIENT)
G MSE AFER (%) G MSE AFER (%)
20 9604.791 1.172 120 1472.215 0.327
30 5238.843 0.677 130 573.433 0.174
40 4137.614 0.608 140 573.433 0.174
50 4137.614 0.608 150 573.433 0.274
60 3565.74 0.535 160 371.745 0.099
70 3565.74 0.535 170 369.745 0.085
80 2752.864 0.524 180 369.745 0.085
90 2385.939 0.474 190 369.745 0.085
100 1472.215 0.327 200 369.745 0.085
110 1472.215 0.327
If all past fuzzy observations are assigned equal weight,
accuracy of forecasting system gets degraded. From TABLE
XXVII. , it can be seen that utilizing OWA weights for past
observations reduced MSE by almost 50% of forecasting done
by equal weights. Thus it is proved that accuracy of system can be improved by determining relative weight of past
observations.
TABLE XXVII. COMPARISON OF SE-W1=W2=W3 & SE-OWA(OUTPATIENT)
Month
Out-
patie
nt
Fore.
Out
( w1=w2
=w3)
SE
(w1=w2
=w3)
Fore.Out
Proposed-
SE
(OWA
weights)
2004/01 6519
2004/02 5979
2004/03 6322 6280.85 1693.17
2004/04 5666 5972.65 94033.23
2004/05 5318 5352.9 1217.92 5315.45 6.503
2004/06 5364 5361.54 6.05 5340.266 563.298
2004/07 5513 5483.55 867.56 5506.624 40.649
2004/08 5895 5869.06 673.13 5867.042 781.641
2004/09 6682 6608.19 5448.08 6495.449 34801.424
2004/10 6254 6312.75 3452.14 6208.277 2090.633
2004/11 5681 5908.41 51713.51 5781.205 10040.977
2004/12 6012 6047.91 1289.2 6025.549 183.571
2005/01 5920 5891.69 801.35 5904.697 234.189
2005/02 5512 5592.86 6538.79 5522.748 115.512
2005/03 6548 6470.29 6039.61 6438.153 12066.296
2005/04 5987 6186.16 39664.59 6060.317 5375.445
2005/05 5638 5656.16 329.8 5666.81 830.007
2005/06 5851 5922.64 5131.76 5913.877 3953.487
2005/07 5514 5527.68 187.02 5526.436 154.662
2005/08 5395 5403.66 74.95 5357.692 1391.915
2005/09 5598 5667.37 4811.97 5678.333 6453.425
2005/10 5284 5288.66 21.68 5260.87 534.983
2005/11 4545 4992.01 199816 4878.916 111499.98
2005/12 4624 4646.29 496.97 4655.263 977.362
MSE = 16429.11 MSE =
9604.79
I. Performance Evaluation and Comparison
We have compared the MSE &AFER of our designed model on outpatient data with previous selective prediction
models [26, 21(a), 21(b), 21(c), 21(d), 21,13] on same data to
highlight the performance of our method.
TABLE XXVIII. COMPARISON OF MSE & AFER OF MODELS (OUTPATIENT)
Name of Method Concept used AFER MSE
Proposed Method G=100, pc=0.8, pm=0.13 0.282 962.285
Proposed Method G=100, pc=0.5, pm=0.083 0.39 2037.92
Proposed Method G=20, pc=0.33, pm=0.083 1.17 9604.79
Garg et al [26] Fuzzy time series & OWA 3.06 165755
Cheng et all-a[21] Equal triangle G-S Method 7.79 308823
Cheng et all-b [21] Equal triangle Exp-Method 8.49 339317
Cheng et all-c [21] TFA G-S Method 7.52 294147
Cheng et all-d [21] TFA Exp Method 7.48 269189
Chen [13] High Order FTS 9.77 388571
Yu[20] Fuzzy-Recurrence 8.6 322971
Figure 8. and Figure 9. reveal MSE and AFER of all
selected models for outpatient visits graphically.
Figure 8. Comparison of MSE (Outpatient)
Figure 9. Comparison of AFER (Outpatient)
J. Discussion
Lower value of MSE and AFER are measure of higher
forecasting accuracy. From the results of TABLE XXVIII. ,
MSE and AFER are lowest in case of proposed model, clearly
indicating its superiority over Garg, Cheng, Yu’s, Chen model
and other existing soft computing models. Proposed model
generates weights for each past observation that contributes in
prediction of next value on basis of its importance in the
system. Existing fuzzy time series model did not count the
importance of each past fuzzy observation to predict next value. They treated all past fuzzy observation equally critical.
Furthermore, GA is applied to generate wide range of
membership function for past observation so that more
optimized predicted value can be generated.
X. CONCLUSION AND FUTURE WORK
In this study, we formulated a novel concept to optimize
Ordered Weighted Aggregation (OWA) based fuzzy time series forecasting model by Genetic Algorithm. It was
identified that accuracy of fuzzy time series model
considerably depends on subjectively decided parameters such
as fuzzy membership function. We determined OWA weights
on the basis of importance of fuzzy set in the system by
employing regularly increasing monotonic (RIM) quantifiers.
Subsequently, genetic algorithm was employed to generate
wide range of parameters for fuzzy membership functions
(mf) in the region of time series. Obtained values of MSE and
AFER indicated use of GA & OWA together can enhance
prediction accuracy significantly. This work illustrates the
dominance of OWA-FTSP-GA algorithm over existing prediction models. Due to least forecasting error, forecasting
model can be used to deal with future trends in every domain
where prediction is based on historical time series data such
as health care domain, information management etc. This
research can be further extended to generalize proposed model
to handle complex multidimensional time series data existing
in health care. Previous studies of hospital systems clearly
exhibit that forecasting related decision were made only on
basis of strong assumptions about the given statistical
multidimensional time series data. However, the ratio of
similarity in fuzzy logical relationships for forecasting
becomes lower when more dimensions are taken in to considerations. Plus, other aspects of proposed model can be
explored by memetic algorithm.
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