ARID ZONE JOURNAL OF ENGINEERING, TECHNOLOGY & ENVIRONMENT
AZOJETE, March, 2019. Vol. 15(1):40-54

Published by the Faculty of Engineering, University of Maiduguri, Maiduguri, Nigeria.
Print ISSN: 1596-2490, Electronic ISSN: 2545-5818

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40

ORIGINAL RESEARCH ARTICLE
CAT SWARM OPTIMIZATION BASED CLUSTERING ALGORITHM FOR FUZZY TIME

SERIES FORECASTING
A. M. Abubakar and S. Man-Yahya

(Department of Electrical and Computer Engineering, Ahmadu Bello University Zaria, Kaduna State, Nigeria)
*Corresponding Author: muktyx5@yahoo.com

ARTICLE INFORMATION

Submitted 31 March, 2018
Revised 11 August, 2018
Accepted 15 August, 2018

Keywords:
Cat Swarm Optimization
Clustering
fuzzy time series
forecasting

ABSTRACT
This paper presents the development of an improved Fuzzy Time Series
(FTS) forecasting model with Cat Swarm Optimization based Clustering
(CSO-C) algorithm. FTS forecasting is affected by some of the
subjective decisions made at the fuzzification stage like size of interval
length or universe of discourse, and approaches such as clustering,
trend-mapping, have been adopted to address this. Traditional
clustering techniques, such as fuzzy C-means (FCM) and K-means used
to tackle the problems associated with the fuzzification stage
encountered problems such as handling high dimensional data,
sensitivity to noise and outliers and pre-mature convergence. In this
paper, CSO-C algorithm was developed using MATLAB 2015a to
address pre-mature convergence with a view to improving forecast
accuracy in FTS forecasting. The developed model was tested on
student enrolment of University of Alabama and Taiwan Future
Exchange (TAIFEX). Also applied to forecast the student enrolment of
University of Maiduguri (UniMaid) data. In all cases, Mean Square Error
(MSE), Root Mean Square Error (RMSE) and Mean Absolute Percentage
Error (MAPE) were used as the performance metrics. The developed
model presents itself superior, with reduced prediction error when
compared with previously existing models in the literature.

© 2019 Faculty of Engineering, University of Maiduguri, Nigeria. All rights reserved.

1.0 Introduction
The field of time series analysis and forecasting is useful in a variety of scientific, engineering and
general applications (e.g., weather forecasting, predictive control, signal processing, population
forecasting, enrolment and finance among others) (Panagiotakis et al., 2016).

Linear, non-linear and clustering models are some of the model types for analysing and
forecasting time series. Traditionally, time series forecasting problems are solved using linear
Moving Average (MA) models, auto-regressive (AR) models and linear Auto-Regressive
Integrated Moving Average (ARIMA) models (Smith and Wunsch, 2015). The traditional
forecasting methods require larger observations and are unable to deal with forecasting
problems in which the historical data need to be represented by linguistic values (Huang et al.,,
2011; Shah, 2012; Song and Chissom, 1993a). It is also confined to linearity assumptions only
(Shah, 2012), which introduces large errors in the predicted values.

To overcome these limitations of traditional methods, many researchers used soft computing
techniques such as fuzzy logic introduced by Zadeh (1965) to handle the uncertainty and

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vagueness available in linguistic terms. The concepts of fuzzy sets (Zadeh, 1978) have been
successfully adopted to time series by Song and Chissom (1993a). Fuzzy time series methods
consist of three steps. These are fuzzification, identification of fuzzified relations and
defuzzification. Many studies on these three steps have been reported in literature, because they
influence the forecasting accuracy of a model. Using these three steps, Song and Chissom
(1993a) developed the benchmark fuzzy time series forecasting models, the time-invariant fuzzy
time series model and the time-variant fuzzy time series model, to forecast the student of
enrolment of the University of Alabama (Song and Chissom, 1993b, 1994). The min-max
composition adopted by Song and Chissom (1993a) is computationally tasking when the fuzzy
matrix is large. Chen (1996), improved on it by introducing a simple arithmetic operation to
replace the complicated operations in Song and Chissom’s (1993) models. Due to simplicity of
Chen’s (1996) model calculation, it was easy and straightforward to integrate heuristic
knowledge and forecast better (Huarng, 2001).

The model has since been widely extended and adapted, with researchers (Huang et al., 2011; Yu,
2005) Hu seeking to improve forecasting accuracy by adjusting the length of linguistic intervals
or by changing the weighting approach or incorporating clustering techniques to tackle the
interval problem. There are two major categories of FTS algorithms: FTS algorithms based on
intervals of the universal set (Askari et al., 2015) and FTS algorithms based on fuzzy clustering
(Askari et al., 2015). Determination of interval length and fuzzy relations are two important
factors affecting performance of fuzzy time series approaches. Many studies established the
efficacy of determining an objective interval length and universe of discourse which is an issue of
concern that affect forecast accuracy (Huang et al., 2011; Yu, 2005).

So many researchers in the last three decades have used various techniques in order to address
problems of subjective decisions associated with the fuzzification stage. Such techniques include;
Genetic Algorithm (GA) (Chen and Chung, 2006), Particle Swarm Optimization (PSO) (Huang et
al., 2011; Kuo et al., 2009), Artificial Bee Colony (ABC), Ant Colony Optimization (ACO) (Cai et al.,
2015), Fuzzy C-Means (FCM) (Egrioglu et al., 2013), Gustafson-kessel clustering algorithm
(Egrioglu et al., 2011) and hierarchical clustering (Chen and Tanuwijaya, 2011).

Even though some of these approaches try avoiding subjective judgements in determining
intervals, membership values were still chosen arbitrarily. Fuzzy clustering approaches have been
developed to overcome this problem. FCM clustering technique was used by Cheng et al. (2008)
and Li et al. (2008), Gustafson Kessel fuzzy clustering technique was used by Egrioglu et al. (2011)
in order to eliminate the problem of subjective determination of membership values.

In most of these studies, fuzzy sets having the highest membership values are considered and
other values are ignored in the determination of fuzzy relations. Yu and Huarng (2010)
suggested an approach in which all membership values are considered in the determination of
fuzzy relations with feed forward artificial neural network (FFANN) but again the membership
values are specified subjectively. Aladag et al., (2012) and Yolcu et al., (2013) considered all the
membership values which were determined objectively using FCM technique for first order FTS
forecasting models. However, most of the time series encountered in real life problems involve
more complicated relations, which makes first order FTS forecasting models insufficient to
forecast these kind of time series (Cagcag et al., 2016).

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A method considering all the membership values at the defuzzification step was introduced by
Yolcu et al. (2013), which Bas et al. (2015) and Cagcag et al. (2016) adopted and developed a
high order model. However, the FCM used has been associated with sensitivity to noise, difficulty
in handling outlier points and high dimensional data and pre-mature convergence. This,
consequently, affects and reduces the forecasting accuracy. Premature convergence is typically
caused by loss of diversity, a measure of the amount variety, i.e. number of different solutions in
the population, and how different they are (distance between alternative solutions).
This loss of diversity could be as a result of too strong selective pressure towards best solution
i.e. too much exploitation of existing building blocks from current population. As such, Cat
Swarm Optimization based Clustering (CSO-C) algorithm was developed to overcome the
problem of pre-mature convergence, in order to improve forecast accuracy.

2.0 Materials and Methods
2.1 Data Collection
In this paper, the benchmark fuzzy time series data of student enrolment of University of
Alabama and Taiwan Future Exchange (TAIFEX) data were used for testing the model and
UniMaid enrolment data was used for the model application.
The time series data of Student enrolment of University of Alabama between 1971 and 1992 was
obtained from the work of Song and Chissom (1993a) as can be found in Table 1.

Table 1: Student Enrolment Data of University of Alabama
Years Actual Years Actual
1971 13055 1982 15433
1972 13563 1983 15497
1973 13867 1984 15145
1974 14696 1985 15163
1975 15460 1986 15984
1976 15311 1987 16859
1977 15603 1988 18150
1978 15861 1989 18970
1979 16807 1990 19328
1980 16919 1991 19337
1981 16388 1992 18872
Table 2 shows the time series data of TAIFEX obtained from the work of Bas et al., (2015). The
time series has 47 daily observations between 03.08.1998 and 30.09.1998. The last 16
observations between 10.09.1998 and 30.09.1998 were used as the test set to evaluate the
model’s performance.

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Table 2: TAIFEX Data
Date TAIFEX Date TAIFEX Date TAIFEX
03.08.1998 7552.0 24.08.1998 695.5.00 11.09.1998 6726.50
04.08.1998 7560.00 25.08.1998 6949.00 12.09.1998 6774.55
05.08.1998 7487.00 26.08.1998 6790.00 15.09.1998 6762.00
06.08.1998 7462.00 27.08.1998 6835.00 16.09.1998 6952.75
07.08.1998 7515.00 28.08.1998 6695.00 17.09.1998 6906.00
10.08.1998 7365.00 29.08.1998 6728.00 18.09.1998 6842.00
11.08.1998 7360.00 31.08.1998 6566.00 19.09.1998 7039.00
12.08.1998 7330.00 01.09.1998 6409.00 21.09.1998 6861.00
13.08.1998 7291.00 02.09.1998 6430.00 22.09.1998 6926.00
14.08.1998 7320.00 03.09.1998 6200.00 23.09.1998 6852.00
15.08.1998 7320.00 04.09.1998 6403.20 24.09.1998 6890.00
17.08.1998 7219.00 05.09.1998 6697.50 25.09.1998 6871.00
18.08.1998 7220.00 07.09.1998 6722.30 28.09.1998 6840.00
19.08.1998 7285.00 08.09.1998 6859.40 29.09.1998 6806.00
20.08.1998 7274.00 09.09.1998 6769.60 30.09.1998 6787.00
21.08.1998 7225.00 10.09.1998 6709.75

The time series enrolment data of University of Maiduguri (UniMaid) shown in Table 3 was
obtained from the Academic planning Unit of the University. The time series has 18 yearly
enrolments between 1976 and 1993.

Table 3: UniMaid Enrolment Data
Year Enrolment Year Enrolment
1976 743 1985 5800
1977 1128 1986 6168
1978 1882 1987 6711
1979 2500 1988 7238
1980 2925 1989 7687
1981 3251 1990 7960
1982 4561 1991 8302
1983 5329 1992 9884
1984 5719 1993 11410

2.2 Fuzzy Time Series and its concept
The definition of fuzzy time series was firstly introduced by Song and Chissom (1993a). Basic
definitions of fuzzy time series not including constraints such as linear model and observation
number can be given as follows (Song and Chissom, 1993a).

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Definition 1 Fuzzy time series

Let   ...,0,1,2,...Y t t  , a subset of real numbers, be the universe of discourse by which fuzzy

sets  jf t are defined. If  F t is a collection of    1 1, ,...f t f t , then  F t is called a fuzzy

time series defined on  .Y t

Definition 2 Suppose  F t is caused by  1F t  only, that is,    .F t m F t  Then this

relation can be expressed as  ( ) 1 ( , t 1)F t F t R t  

where ( , 1)R t t  is the fuzzy relationship

between ( 1)F t  and ( )F t , and  ( ) 1 ( , t 1)F t F t R t  

is called the first order model of ( ).F t

Definition 3 Suppose ( , 1)R t t  is a first order model of  .F t If for any , ( , 1)t R t t  is

independent of t, that is, for any , ( , 1) R(t 1)(t 2),t R t t     then,  F t is called a time invariant
fuzzy time series otherwise, it is called a time variant fuzzy time series.
Definition 4 High order fuzzy time series forecasting model.

Let  F t be a fuzzy time series. If  F t is caused by      1 ,F 2 ,..., ,F t t F t n   then this
fuzzy logical relationship is represented by
       ,..., 2 , 1F t n F t F t F t    and it’s called the nth order fuzzy time series forecasting

model.

2.3 Cat swarm optimization clustering
In contrast to most of the existing clustering techniques, the CSO-C algorithm requires
no prior knowledge of the data to be classified. Rather, it determines the optimal number of
partitions of the Data“on the run” (Santosa and Ningrum, 2009).
The CSO Clustering consists of two main parts, clustering data and searching for the best cluster
center using the CSO algorithm (Chu and Tsai, 2007). The inputs for clustering CSO will be the
population of data that are going to be clustered, number of cluster k, and number of copy (will
be used in seeking mode). Steps of clustering CSO are described below (Santosa and Ningrum,
2009).
Step 1: Defining the initial cluster center
Randomly choose k point from the data to be the initial cluster centre.
Step 2: Grouping data into clusters
Put the data into cluster using Euclidean distance as follows (Santosa and Ningrum, 2009).

   
n

2
i i

i = 1

d x , y = x - y
(1)

Step 3: Calculating the Sum of Squared-Error (SSE)
SSE is the fitness function in this algorithm. The value of SSE are calculated, subject to the
following constraint, using (Santosa and Ningrum, 2009).

 
2

1
min , i

i
i

i D

k

x
SSE u x m



 
(2)

0 1iu  (3)

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In solving the minimization problem, the values of im are updated using the following formula:

 
 

 
2

1

1

1

,
,

ij
kk

j i

k j k

u
d x v
d x v






 
 
 
 


(4)

Where; x is data, mi the cluster centeri, k the number of cluster and ui the membership i.
Step 4: Clustering optimization with CSO
In this algorithm, cluster centre represents the cat and the solution set will be the new cluster
centre which is expected to come up with the smaller SSE value than before.
Step 4.1: Seeking mode
Seeking mode is intended to look for points in an area around the cluster centre which have
possibilities in resulting in more than an optimal fitness value.
Step 4.2: Updating SSE and cluster center
The value of SSE obtained from seeking mode will be compared with the previous value of SSE,
if seeking SSE < earlier SSE then it becomes the new cluster centre. Otherwise, if seeking SSE ≥
earlier SSE, the previous cluster centre will be used.
Step 4.3: Tracing Mode
Tracing mode is intended to shift the point so it will be concentrated to a better position with a
more optimal fitness value.
Step 4.4: Repeat step 4.2 for tracing SSE and cluster center
The value of SSE obtained from tracing mode will be compared with the previous value of SSE, if
tracing SSE < earlier SSE then it becomes the new cluster center. If tracing SSE ≥ earlier SSE, the
previous cluster center will be used.
Step 5: Repeat step 4 until it reach the stopping criteria.

2.4 Artificial Neural Network
Artificial neural networks is a mathematical algorithm inspired by biological neural networks,
which can learn from examples and generalize what is learnt (Bas et al., 2015). Architecture
structure, learning algorithm and activation function are the three basic operational component
of the artificial neural networks (Bas et al., 2015).

2.5 Developed Model Procedure
The flowchart of the developed model presented in figure 1 shows the steps carried out in the
model implementation aimed at improving forecast accuracy.

2.5.1 Training/testing time series data
The data’s of tables 1, 2 and 3 were used as the input data for the developed model.

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Figure 1: Flowchart of the developed model

2.5.2 Fuzzify time series observation by using CSO-C algorithm
At this stage, the CSO-C algorithm was used in fuzzifying the time series data specified in section
2.5.1 as a means of determining an optimized cluster center and membership values for fuzzy
time series forecasting. The population of data to be clustered used comprised of data shown in
Table’s 1, 2 and 3. The number of clusters chosen was seven as used in the work of Bas et al.,
(2015) as specified in Table 4

Table 4: Input Parameter to the CSO-C Algorithm
Parameters Specifications
D Data
K 7

The initial cluster center has been chosen randomly from the data set, the data were then
grouped into clusters by their closeness using Euclidean distance formula of equation 1 to
calculate the distance between the data and the cluster center. The sum of squared error
between the data and the cluster center was then calculated using equation 2.
In clustering with the CSO-C algorithm, the cluster center represents the cat and the solution set
will be the new cluster center which is expected to come up with the smaller SSE value than what
was obtained. Here, all the cats go through the process of tracing and seeking. The CSO-C
parameters initialization were specified in table 5, which include count of dimension to change
(CDC), seeking memory pool (SMP), seeking range of the selected dimension (SRD) and self-
position consideration (SPC). The CDC in seeking mode were assumed 100% value, so every
dimension of cat’s copy changes, SMP represents how many copy a cluster center has, SRD
declares a mutation ratio with a value between [0,1] and SPC is a Boolean random value{0,1}.

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Table 5: CSO Parameters (Santosa and Ningrum, 2009)
Parameters Specifications
SMP 5
CDC 100%
SRD 0.2
Const1 2
r1 [0,1]
Velmax 0.9

The simulation parameters of table 5 were selected based on the optimal performance achieved
as reported in the work of Santosa and Ningrum (2009).
The membership values were computed and updated using equations 2 and 4. The next step is
activating tracing mode which looks for an area around the cluster center with possibilities of
generating a more optimal fitness value. The distance between the data and the new cluster
center where been calculated using equation 1, the new cluster center calculated using 2 and the
membership values computed with equation 4.
The SSE of the tracing mode computed were then compared with the previous values of SSE and
if the tracing SSE is less than the previous SSE, then the membership value and cluster center
resulting from the tracing becomes the new membership value and cluster center. Conversely, if
the value of the tracing SSE was greater than or equal the earlier SSE, then the previous
membership values and cluster centers would be used. The process continues until the number
of iterations is met, after which, the number of cluster K, the membership values and the best
cluster center were then obtained.

2.5.3 Determination of high order fuzzy relation using FFANN
The feed forward artificial neural nework (FFANN) was used in establishing fuzzy relationship.
That is, the produced degree are employed as the inputs and the target values of FFNNs. As
such, this prevents arbitrarily degrees of membership determination, hard matrix operations or
complex group tables need not be used. The input to the FFNN consists of membership values
obtained from section 2.5.2 and the activation function used in all layers of the FFNN of whose
architecture is given as follows:
     1

1 expf x x


   (5)
Optimal weights were obtained by training using Lavenberg-Marquardt learning algorithm.

2.5.4 Defuzzify forecasted values
In defuzzifying the forecasted values, weighted averages were used instead of the central
method, and so, the fuzzy sets with a less degree of membership were considered. This helps in
avoiding possible knowledge loss. Using the following expression, the degrees of membership of
the fuzzy forecast obtained from the previous section were transformed into weights.

1 2

ˆ
ˆ ˆ ˆ...

it
it

t t kt

uw
u u u


   (6)

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The defuzzified forecasts of the observation at time t are calculated using the following
expression

1

k

t it i
i

X w m





(7)

Where, ˆ , 1,2,..., ,itu i k denotes the membership degrees for the fuzzy forecasting value of the
time series data at t obtained from the output of the FFANN. , 1,2,..., ,itw t k denotes the
weights used for defuzzifying the forecasts. , 1,2,..., ,im i k denotes the cluster center values.

2.5.5 Performance evaluation
In evaluating the performance of the developed model, mean absolute performance error (MAPE)
and root mean square error (RMSE) were used as metrics as presented in the following
expressions respectively.

 n
i i

i=1 i

actual - forecast1MAPE = ×100
n actual

(8)

 
n

2
i i

i=1

1RMSE = actual - forecast
n (9)

The percentage improvement of the developed model over other models reported in the
literature was computed using the expression as follows:

100%Initial FinalPercentage Improvement
Initial


 

(10)
Where initial stands for RMSE or MAPE values computed from the approaches of the Benchmark
FTS or FCM based, and final for the values obtained from the developed CSO-C based model.

3.0 Results and Discussion
In this section, the performance of the developed fuzzy time series forecasting is evaluated and
analyzed using two different data sets that were previously studied with other methods in the
literature and a new data set that has not been previously used

3.1 Obtained Results for Student Enrolment Data of University of Alabama
The optimized cluster center values of fuzzy sets and the membership values obtained from the
developed model on the training data set of Table 1 as earlier explained in section 2.5 were
presented in Table 6 and Table 7 respectively. The defuzzified forecast obtained were given in
Table 8.
Table 6: Obtained Cluster Centre Values for University of Alabama Data
m1 m2 m3 m4 m5 m6 m7
13490.51 15117.90 15621.22 16790.81 18145.80 18921.80 19331.84

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Table 7: Obtained Membership Values for University of Alabama Data
Year k1 k2 k3 k4 k5 k6 k7
1971 0.9053 0.0403 0.0261 0.0123 0.0066 0.0050 0.0044
1972 0.9955 0.0022 0.0012 0.0005 0.0002 0.0002 0.0002
1973 0.8538 0.0773 0.0393 0.0142 0.0066 0.0047 0.0041
1974 0.0872 0.7121 0.1481 0.0289 0.0107 0.0071 0.0059
1975 0.0054 0.1776 0.7993 0.0117 0.0029 0.0017 0.0014
1976 0.0079 0.7014 0.2719 0.0120 0.0033 0.0020 0.0016
1977 0.0000 0.0014 0.9981 0.0002 0.0000 0.0000 0.0000
1978 0.0085 0.0866 0.8314 0.0553 0.0092 0.0051 0.0040
1979 0.0000 0.0000 0.0002 0.9994 0.0002 0.0000 0.0000
1980 0.0014 0.0049 0.0094 0.9671 0.0106 0.0040 0.0027
1981 0.0129 0.0674 0.1849 0.6701 0.0352 0.0169 0.0125
1982 0.0067 0.2557 0.7166 0.0138 0.0035 0.0021 0.0018
1983 0.0034 0.0955 0.8888 0.0082 0.0020 0.0012 0.0009
1984 0.0003 0.9960 0.0033 0.0003 0.0000 0.0000 0.0000
1985 0.0007 0.9884 0.0096 0.0008 0.0002 0.0001 0.0001
1986 0.0146 0.1207 0.6877 0.1391 0.0194 0.0105 0.0081
1987 0.0004 0.0015 0.0030 0.9904 0.0028 0.0011 0.0008
1988 0.0000 0.0000 0.0000 0.0000 0.9999 0.0000 0.0000
1989 0.0000 0.0002 0.0002 0.0005 0.0033 0.9784 0.0174
1990 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9999
1991 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.9998
1992 0.0000 0.0001 0.0002 0.0005 0.0039 0.9853 0.0100

Table 8 shows the results of forecast obtained from the developed model, where the
performance criteria explained in section 2.5 were used in computing the model performance.
The obtained MSE, RMSE and MAPE were 785.17, 88.46 and 0.0046 respectively. From the results
comparison of Table 9, it can be seen that the developed model has the lowest prediction error
in terms of MSE, this is an indication of better forecast exhibited by the developed model. It
shows that the developed model recorded 98.10% over Song and Chissom (1993a), 86.98%, over
Egrioglu et al., (2011), 83.14% over Aladag et al., (2012), 70.02% over Bas et al., (2015) and
28.77% over Cagcag et al., (2016) computed with equation 10.

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Table 8: Defuzzified Forecast for University of Alabama Data
Year Data Forecast
1973 13867 13945
1974 14696 14702
1975 15460 15397
1976 15311 15386
1977 15603 15546
1978 15861 15934
1979 16807 16834
1980 16919 16828
1981 16388 16301
1982 15433 15389
1983 15497 15416
1984 15145 15295
1985 15163 15316
1986 15984 15950
1987 16859 16834
1988 18150 18161
1989 18970 19056
1990 19328 19186
1991 19337 19200
1992 18876 18990

MSE 7825.17
RMSE 88.46
MAPE 0.0046

Table 9: The Results Comparison
Methods MSE
Song and Chissom (1993a) 412499
Egrioglu et al., (2011) 60140
Aladag et al., (2012) 46422
Bas et al., (2015) 26108
Cagcag et al., (2016) 10987
The Developed 7825.17

3.2 Obtained Results for TAIFEX Data
Table 10 shows results of forecast obtained from TAIFEX data of table 2 following methods
discussed in section 2.5.
The results further shows the forecasts obtained from the model of Bas et al., (2015) and Cagcag
et al., (2016) on the same data set. After close examination, it can be seen that the developed

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model recorded the least prediction error with a significant improvement in terms of MSE, RMSE
and MAPE.
The performance improvement computed with equation 10 was used to obtain the performance
improvement of the developed model over some methods reported in literature. The model
developed recorded an RMSE improvement of 65.57% over that of Bas et al., (2015) and 64.22%
over that of Cagcag et al., (2016).

Table 10: Forecast for TAIFEX data
Date TAIFEX Bas et al., (2015) Cagcag et al., (2016) Developed
10.09.1998 6709.75 6804.31 6719.92 6721.21
11.09.1998 6726.50 6804.31 6719.92 6721.92
14.09.1998 6774.55 6761.17 6843.75 6773.72
15.09.1998 6762.00 6758.51 6719.92 6774.30
16.09.1998 6952.75 6856.13 6843.75 6946.40
17.09.1998 6906.00 6856.13 6843.75 6901.86
18.09.1998 6842.00 6856.13 6843.75 6841.70
19.09.1998 7039.00 6856.13 6843.75 7038.93
21.09.1998 6861.00 6856.13 6843.75 6861.63
22.09.1998 6926.00 6856.13 6843.75 6919.85
23.09.1998 6852.00 6856.13 6843.75 6853.61
24.09.1998 6890.00 6856.13 6843.75 6893.58
25.09.1998 6871.00 6807.48 6843.75 6863.90
28.09.1998 6840.00 6793.07 6843.75 6840.41
29.09.1998 6806.00 6793.07 6843.75 6819.25
30.09.1998 6787.00 6856.13 6778.28 6879.66

MSE 4869.25 4507.78 576.96
RMSE 69.78 67.14 24.02
MAPE 0.00761 0.0066 0.00152

3.3 Obtained Results for UniMaid Enrolment Data
The UniMaid enrolment data of table 3 was not previously applied for any fuzzy time series
forecasting research. In view of that, the developed model was applied to obtain the forecast as
shown in table 11. It can be seen that the forecast produced has an MSE of 1395543.21, RMSE 0f
373.55 and MAPE 0.0667.

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Table 11: Forecast for UniMaid Data
Year Enrolment Forecast
1978 1882 2702.84
1979 2500 2680.11
1980 2925 2737.43
1981 3251 2837.88
1982 4561 4497.60
1983 5329 5550.53
1984 5719 5849.53
1985 5800 5965.73
1986 6168 6457.61
1987 6711 6992.32
1988 7238 7372.57
1989 7687 7591.29
1990 7960 7807.23
1991 8302 8120.24
1992 9884 10031.38
1993 11410 10432.95

MAPE 0.0667
MSE 139543.21
RMSE 373.55

4.0 Conclusion
In minimizing the subjective decisions made at the fuzzification stage, traditional clustering
techniques such as fuzzy C-means (FCM) and K-means, have been previously used, but problems
such as handling high dimensional data, sensitivity to noise, outliers and pre-mature
convergence still persist. In this research, CSO-C algorithm was developed to address this issue
of pre-mature convergence with a view to improving forecast accuracy in fuzzy time series
forecasting. In the implementation, the developed model was tested and compared with other
models forecast reported in literature for the student enrolment of University of Alabama and
TAIFEX data.
The forecast results of enrolment of University of Alabama data produced an MSE improvement
of 98.10% over Song and Chissom (1993a), 86.98%, over Egrioglu et al., (2011), 83.14% over
Aladag et al., (2012), 70.02% over Bas et al., (2015) and 28.77% over Cagcag et al., (2016). Also,
the forecast results of TAIFEX data recorded an RMSE improvement of 65.57% over that of Bas et
al., (2015) and 64.22% over that of Cagcag et al., (2016). This clearly shows that the model
developed has better forecast with reduced prediction error.
The developed model was further to UniMaid enrolment data, which was not previously
reported to have been used in literature with respect the research area of fuzzy time series
forecasting. The forecast produced has an MSE of 1395543.21, RMSE 0f 373.55 and MAPE 0.0667.
It is expected that future researchers will forecast with the UniMaid enrolment data and see how
there model suit into the data.

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