Multivariate high-order-fuzzy-time-series-forecasting-for-car-road-accidents

World Academy of Science, Engineering and Technology
International Journal of Computer, Information Science and Engineering Vol:2 No:6, 2008

Multivariate High Order Fuzzy Time Series
Forecasting for Car Road Accidents
Tahseen A. Jilani, S. M. Aqil Burney, and C. Ardil
`

International Science Index 18, 2008 waset.org/publications/15658

Abstract—In this paper, we have presented a new multivariate
fuzzy time series forecasting method. This method assumes mfactors with one main factor of interest. History of past three years is
used for making new forecasts. This new method is applied in
forecasting total number of car accidents in Belgium using four
secondary factors. We also make comparison of our proposed
method with existing methods of fuzzy time series forecasting.
Experimentally, it is shown that our proposed method perform better
than existing fuzzy time series forecasting methods. Practically,
actuaries are interested in analysis of the patterns of causalities in
road accidents. Thus using fuzzy time series, actuaries can define
fuzzy premium and fuzzy underwriting of car insurance and life
insurance for car insurance. National Institute of Statistics, Belgium
provides region of risk classification for each road. Thus using this
risk classification, we can predict premium rate and underwriting of
insurance policy holders.

Keywords—Average forecasting error rate (AFER), Fuzziness of
fuzzy sets Fuzzy, If-Then rules, Multivariate fuzzy time series.
I. INTRODUCTION

I

N our daily life, people often use forecasting techniques to
model and predict economy, population growth, stocks,
insurance/ re-insurance, portfolio analysis and etc. However,
in the real world, an event can be affected by many factors.
Therefore, if we consider more factors for prediction, with
higher complexity then we can get better forecasting results.
During last few decades, various approaches have been
developed for time series forecasting. Among them ARMA
models and Box-Jenkins model building approaches are
highly famous.
In recent years, many researchers used fuzzy time series to
handle prediction problems. Song and Chissom [8] presented
the concept of fuzzy time series based on the concepts of
fuzzy set theory to forecast the historical enrollments of the
University of Alabama. Huarng [4] presented the definition of
two kinds of intervals in the universe of discourse to forecast
the TAIFEX. Chen [10] presented a method for forecasting
based on high-order fuzzy time series. Lee [6] presented a
Manuscript received November 25, 2006. This work was supported in part
by the Higher Education Commission of Pakistan.
T. A. Jilani is Lecturer in the Department of Statistics, University of
Karachi, Karachi-75100, Pakistan (phone: +923333040963; e-mail:
tahseenjilani@ieee.org)
S. M. Aqil Burney is Professor in the Department of Computer Science,
University
of
Karachi,
Karachi-75100,
Pakistan
(e-mail:
burney@computer.org).
C. Ardil is with National Academy of Aviation, AZ1045, Baku,
Azerbaijan, Bina, 25th km, NAA (e-mail: cemalardil@gmail.com).

method for temperature prediction based on two-factor highorder fuzzy time series. Melike [7] proposed forecasting
method using first order fuzzy time series for forecasting
enrollments in University of Alabama. Lee [6] Presented
handling of forecasting problems using two-factor high order
fuzzy time series for TAIFEX and daily temperature in
Taipei, Taiwan.
The rest of this paper is organized as follows. In section 2,
brief review of fuzzy time series is given. In section 3, we
present the new method for fuzzy time series modeling.
Experimental results are performed in section 4. The
conclusions are discussed in section 5.
In this paper, we present a new modified method to predict
total number of annual car road accidents based on the mfactors high-order fuzzy time series. This method provides a
general framework for forecasting that can be increased by
increasing the stochastic fuzzy dependence [3]. For simplicity
of computation, we have used triangular membership
function. The proposed method constructs m-factor high-order
fuzzy logical relationships based on the historical data to
increase the forecasting accuracy rate. Our proposed
forecasting method for fuzzy time series gives better results as
compared to [4], [5] and [10].
II. FUZZY TIME SERIES
Time series analysis plays vital role in most of the actuarial
related problems. As most of the actuarial issues are born with
uncertainty, therefore, each observation of a fuzzy time series
is assumed to be a fuzzy variable along with associated
membership function. Based on fuzzy relation, and fuzzy
inference rules, efficient modeling and forecasting of fuzzy
time series is possible, see [1] and [9]. This field of fuzzy
time series analysis is not very mature due to the time and
space complexities in most of the actuarial related issue, thus
we can extend this concept for many antecedents and single
consequent. For example, in designing two-factor kth-order
fuzzy time series model with X be the primary and Y be
second fact. We assume that there are k antecedent
( ( X1,Y1 ), ( X 2 ,Y2 ), . . ., ( X k ,Yk ) ) and one consequent X k +1 .
If ( X1= x1,Y1= y1) , ( X 2 = x2 ,Y2 = y2 ) ,
..., ( X k = xk ,Yk = yk ) →( X k +1= xk +1)

(1)

In the similar way, we can define m-factor i=1,2,...,m and kth
order fuzzy time series as

24


TABLE I
YEARLY CAR ACCIDENTS MORTALITIES AND VICTIMS FROM 1974 TO 2004


Year

Killed
(X)

Mortally
wounded
(Y1)

Died 30
days
(Y2)

Severely
wounded
(Y3)

Light
casualties
(Y4)

2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974

953
1,035
1,145
1,288
1,253
1,173
1,224
1,150
1,122
1,228
1,415
1,346
1,380
1,471
1,574
1,488
1,432
1,390
1,456
1,308
1,369
1,479
1,464
1,564
1,616
1,572
1,644
1,597
1,536
1,460
1,574

141
101
118
90
103
126
121
105
115
109
149
171
173
209
190
312
339
380
330
352
363
412
406
454
557
544
728
701
728
701
819

1,094
1,136
1,263
1,378
1,356
1,299
1,345
1,255
1,237
1,337
1,564
1,517
1,553
1,680
1,764
1,800
1,771
1,770
1,786
1,660
1,732
1,891
1,870
2,018
2,173
2,116
2,372
2,298
2,264
2,161
2,393

5,949
6,898
6,834
7,319
7,990
8,461
8,784
9,229
9,123
10,267
11,160
11,680
12,113
12,965
13,864
14,515
14,029
13,809
13,764
13,287
14,471
14,864
14,601
15,091
15,915
15,750
16,645
15,830
16,057
15,792
16,506

41,627
42,445
39,522
38,747
39,719
41,841
41,038
39,594
38,390
39,140
40,294
41,736
41,772
43,578
46,818
46,667
45,956
44,090
42,965
39,879
42,456
42,023
40,936
41,915
42,670
42,346
44,797
44,995
44,227
42,423
44,640

“ o ” is the max–min composition operator, then F ( t ) is

caused by F ( t−1) where F ( t ) and F ( t−1) are fuzzy sets. For
forecasting purpose, we can define relationship among present
and future state of a time series with the help of fuzzy sets.
Assume the fuzzified data of the i th and ( i +1) th day are A j

and

Ak ,

respectively, where A j , Ak ∈ U , then

A j → Ak

represented the fuzzy logical relationship between A j and Ak .
Let F ( t ) be a fuzzy time series. If F ( t ) is caused by
F ( t−1) , F ( t− 2 ) ,…, F ( t − n ) , then the fuzzy logical relationship

is represented by

F ( t −n ),..., F ( t −2), F ( t −1) →F ( t )

(3)

is called the one-factor nth order fuzzy time series forecasting
model. Let F ( t ) be a fuzzy time series. If F ( t ) is caused
by ( F1( t −1), F2 ( t −1) ) , ( F1( t − 2 ), F2 ( t − 2 ) ) ,..., ( F1( t − n ), F2 ( t − n ) ) ,
then this fuzzy logical relationship is represented by

( F1(t −n),F2 (t −n)) ,..., ( F1(t −2),F2 (t −2)) ,

(

)

F1( t −1), F2 ( t −1) → F ( t )

(4)

is called the two-factors nth order fuzzy time series
forecasting model, where F1 ( t ) and F2 ( t ) are called the main
factor and the secondary factor fuzzy time series' respectively.
In the similar way, we can define m-factor nth order fuzzy
logical relationship as

( F1( t − n ), F2 ( t − n ),..., Fm ( t − n ) ) , ...,
If ( X11 = x11 , X12 = x12 ,..., X1k = x1k ) ,

( F1( t − 2 ), F2 ( t − 2 ),..., Fm ( t − 2 ) ),

( X 21 = x21 , X 22 = x22 ,..., X 2 k = x2 k ),...,

( F1( t −1), F2 ( t −1),..., Fm ( t −1) ) → F ( t )

( X m1 = xm1 , X m 2 = xm 2 ,..., X mk = xmk )

(2)

then ( X m +1 ,k +1 = xm +1k +1 )
for i = 1, 2, ..., m , j = 1, 2, ..., k

Now, we formally give details of proposed method in
section 3.
III. NEW METHOD OF FORECASTING USING FUZZY TIME
SERIES
Let Y( t ), ( t=...,0,1,2,...) be the universe of discourse and

Y ( t ) ⊆ R . Assume that fi ( t ) , i =1,2,... is defined in the
universe of discourse Y( t ) and F ( t ) is a collection of
f ( t i ), ( i=...,0,1,2,...) , then F ( t ) is called a fuzzy time series of
Y( t ) , i=1,2,... ...

Using

fuzzy

relation,

we

define

F ( t ) = F ( t −1) o R( t ,t −1) , where R( t ,t−1) is a fuzzy relation and

(5)

Here F1( t ) is called the main factor and F2 ( t ), F3 ( t ),..., Fm ( t )
are called secondary factor fuzzy time series'. Here we can
implement any of the fuzzy membership function to define the
fuzzy time series in above equations. Comparative study by
using different membership functions is also possible. We
have used triangular membership function due to low
computational cost. Using fuzzy composition rules, we
establish a fuzzy inference system for fuzzy time series
forecasting with higher accuracy. The accuracy of forecast can
be improved by considering higher number of factors and
higher dependence on history.
Now we present an extended method for handling
forecasting problems based on m-factors high-order fuzzy
time series. The proposed method is now presented as follows.
Step 1) Define the universe of discourse U of the main
factor U =[ Dmin − D1, Dmax − D2 ] , where Dmin and Dmax are
the minimum and the maximum values of the main factor of
the known historical data, respectively, and D1 , D2 are two

25


TABLE II
FUZZIFIED YEARLY DATA FOR MORTALITY ACCIDENTS FROM 1974 TO 2004
Mortality
Year
Fuzzified Mortality Accidents
Accidents
2004
953
0.5/A1 + 1.0/A2 + 0.5/A3 (X2)
2003
1,035
0.5/A1 + 1.0/A2 + 0.5/A3 (X2)
2002
1,145
0.5/A2 + 1.0/A3 + 0.5/A4 (X3)
2001
1,288
0.5/A4 + 1.0/A5 + 0.5/A6 (X5)
2000
1,253
0.5/A4 + 1.0/A5 + 0.5/A6 (X5)
1999
1,173
0.5/A3 + 1.0/A4 + 0.5/A5 (X4)
1998
1,224
0.5/A3 + 1.0/A4 + 0.5/A5 (X4)
1997
1,150
0.5/A3 + 1.0/A4 + 0.5/A5 (X4)
1996
1,122
0.5/A2 + 1.0/A3 + 0.5/A4 (X3)
1995
1,228
0.5/A3 + 1.0/A4 + 0.5/A5 (X4)
1994
1,415
0.5/A5 + 1.0/A6 + 0.5/A7 (X6)
1993
1,346
0.5/A4 + 1.0/A5 + 0.5/A6 (X5)
1992
1,380
0.5/A5 + 1.0/A6 + 0.5/A7 (X6)
1991
1,471
0.5/A6 + 1.0/A7 + 0.5/A8 (X7)
1990
1,574
0.0/A6 + 0.5/A7 + 1.0/A8 (X8)
1989
1,488
0.5/A6 + 1.0/A7 + 0.5/A8 (X7)
1988
1,432
0.5/A5 + 1.0/A6 + 0.5/A7 (X6)
1987
1,390
0.5/A5 + 1.0/A6 + 0.5/A7 (X6)
1986
1,456
0.5/A6 + 1.0/A7 + 0.5/A8 (X7)
1985
1,308
0.5/A4 + 1.0/A5 + 0.5/A6 (X5)
1984
1,369
0.5/A5 + 1.0/A6 + 0.5/A7 (X6)
1983
1,479
0.5/A6 + 1.0/A7 + 0.5/A8 (X7)
1982
1,464
0.5/A6 + 1.0/A7 + 0.5/A8 (X7)
1981
1,564
0.0/A6 + 0.5/A7 + 1.0/A8 (X8)
1980
1,616
0.0/A6 + 0.5/A7 + 1.0/A8 (X8)
1979
1,572
0.0/A6 + 0.5/A7 + 1.0/A8 (X8)
1978
1,644
0.0/A6 + 0.5/A7 + 1.0/A8 (X8)
1977
1,597
0.0/A6 + 0.5/A7 + 1.0/A8 (X8)
1976
1,536
0.5/A6 + 1.0/A7 + 0.5/A8 (X7)
1975
1,460
0.5/A6 + 1.0/A7 + 0.5/A8 (X7)
1974
1,574
0.0/A6 + 0.5/A7 + 1.0/A8 (X8)

proper positive real numbers to divide the universe of
discourse into n equal length intervals u1,u2, ...,ul . Define the
universes of discourse Vi , i =1,2,...,m −1 of the secondaryfactors

Vi = ⎡ ( Ei )
−E , E
−E ⎤ ,
min i1 ( i )max i 2 ⎦
⎣

where

( Ei )min = ⎡( E1 )min , ( E2 )min ,...,( Em )min ⎤
⎣
⎦

and

( Ei )max = ⎡( E1 )max , ( E2 )max ,...,( Em )max ⎤ are the minimum
⎣
⎦
and maximum values of the secondary-factors of the known
historical data, respectively, and Ei1 , Ei2 are vectors of
proper positive numbers to divide each of the universe of
discourse Vi , i =1,2,...,m −1 into equal length intervals termed
as v1,l , v 2,l ,..., v m−1,l , l =1,2,..., p , where v1,l = ⎡ v1,1, v1,2 ,...v1, p ⎤
⎣
⎦
represents n intervals of equal length of universe of discourse
V1 for first secondary-factor fuzzy time series. Thus we have
( m−1)× l matrix of intervals for secondary-factors.
Step 2) Define the linguistic term Ai represented by fuzzy

sets of the main factor shown as follows:
A = 1u + 0.5 u + 0 u + 0 u + ... + 0 u
+ 0u + 0u
1
1
2
3
4
l −2
l −1
l
0.5 + 1 +0.5 + 0 +...+ 0
0
0
+
+
A2 =
u1 u2
u3 u4
ul −2 ul −1 ul
+0
+0
A3 = 0 +0.5 + 1 +0.5 +...+ 0
u1
u2 u3
u4
ul −2 ul −1 ul
.

(6)

.
.
+ 0.5 u + 1u
A = 0 u + 0 u + 0 u + 0 u + ... + 0 u
n
1
2
3
4
l −2
l −1
l

Similarly, for ith secondary fuzzy time series, we define the
linguistic term Bi , j , i =1,2,...,m−1, j =1,2,...,n represented by

Case 2) If the value of the main factor belongs to
ul , l = 2,3,..., p −1 then the value of the main factor is fuzzified

fuzzy sets of the secondary-factors,

into 0.5 A + 1 A + 0.5 A , denoted by X i .
i −1
i
i +1
Case 3) If the value of the main factor belongs to u p , then the

B = 1v + 0.5 v + 0 v + 0 v + ... + 0 v
+ 0v
+ 0v
i,1
i,1
i,2
i,3
i,4
i,l −2
i,l −1
i,l

value

Bi,2 =0.5
+1
+0.5
+0
+...+ 0
+0
+0
vi,1 vi,2
vi,3 vi,4
vi,l −2
vi,l −1 vi,l
Bi,3= 0
+0.5
+1
+0.5
+...+ 0
+0
+0
vi,1
vi,2
vi,3
vi,4
vi,l −2
vi,l −1 vi,l

0

(7)

.
.
.
B = 0 v + 0 v + 0 v + 0 v + ... + 0 v
+ 0.5 v
+ 1v
i,n
i,1
i,2
i,3
i,4
i,l −2
i,l −1
i,l

Step 3) Fuzzify the historical data described as follows. Find
out the interval ul ,l =1,2,..., p to which the value of the main
factor belongs
Case 1) If the value of the main factor belongs to u1 , then the
value of the main factor is fuzzified into 1 A + 0.5 A +0.0 A ,
1
2
3
denoted by X1 .

An − 2

of
+ 0.5

the
An −1

+1

main
factor
is
, denoted by X n .
A

fuzzified

into

n

Now, for ith secondary-factor, find out the interval Vi ,l to
which the value of the secondary-factor belongs.
Case 1) If the value of the ith secondary-factor belongs to v i ,1 ,
then the value of the secondary-factor is fuzzified
into 1 B + 0.5 B + 0 B ,
denoted
by
i ,1
i ,2
i ,3
Yi ,1 = ⎡Y1,1,Y2,1 ,...,Ym −1,1 ⎤ .
⎣
⎦

Case 2) If the value of the ith secondary-factor belongs to
vi ,l , l = 2,3,.., p −1 , then the value of the ith secondary-factor is
fuzzified

into

0.5

Bi , j −1

+ 1

Bi , j

+ 0.5

denoted by Yi , j , where j = 2,3,....,n −1 .

26

Bi , j +1

, j =i = 2,3,...,n −1


TABLE III
FORECASTED YEARLY CAR ACCIDENT CAUALITIES FROM 1974-2004

Case 3) If the value of the ith secondary-factor belongs
to v i , p , then the value of the secondary-factor is fuzzified into
0

Bi ,n − 2

+ 0.5

Bi ,n −1

+ 1

Bi ,n

, denoted by Yi ,n .

Fi − Ai

Fi − Ai
Ai

Actual
Killed Ai

Forecasted
Kills Fi

2004

953

995

-42

0.04404

2003

1,035

995

40

0.038676

Year

Step 4) Get the m-factors kth-order fuzzy logical relationships
based on the fuzzified main and secondary factors from the
fuzzified historical data obtained in Step 3). If the fuzzified
historical data of the main-factor of ith day is X i , then

2002


j +1
∑ w
j −1
tj=
w j −1 w j w j +1
+ +
a j −1 a j a j +1

1296

-43

0.034397

1,173

1196

-23

0.019437

1,224

1196

28

0.023039

1,150

1196

-46

0.039826

1,122

1095

27

0.023708

1995

1,228

1396

-168

0.137134

1994

1,415

1296

119

0.084028

1993

1,346

1396

-50

0.037444

1992

1,380

1497

-117

0.084565

1991

1,471

1497

-26

0.017471

1990

1,574

1497

77

0.049111

1989

1,488

1396

92

0.061559

1988

1,432

1396

36

0.02486

1987

1,390

1497

-107

0.076763

1986

0.109821

-88

0.067584

1,369

1497

-128

0.09328

1,479

1497

-18

0.011968

1982

1,464

1497

-33

0.022336

1981

1,564

1497

67

0.043031

1980

1,616

1497

119

0.073824

1979

1,572

1497

75

0.047901

1978

fulfills the axioms of fuzzy sets like monotonicity, boundary
conditions, continuity and idempotency. For measurement of
accuracy of forecasting for fuzzy time series forecasting, we
use average forecasting error rate (AFER) as the performance
criteria, defined as

160

1396

1983

1,644

1497

147

0.089599

1977

1,597

1497

100

0.062805

1976

1,536

1497

39

0.025586

1975

1,460

1497

-37

0.025137

1974

ul −1, ul and ul +1 respectively. Above forecasting formula

1296

1,308

1984

(9)

1,456

1985

Where al −1, al and al+1 are the midpoints of the intervals

n
∑ ( Forecasted value of Day j − ActualValue of Day j )
×100%
AFER = i =1
n

1,253

1997

of main factor X j , Yi , j − k , i =1,...,m−1, j =1,...,k .Then, divide
the derived fuzzy logical relationships into fuzzy logical
relationship groups based on the current states of the fuzzy
logical relationships. The secondary factors acts like a
secondary component to the m-dimensional state vector and is
used in Step 5).
Step 5) For m-factor kth order fuzzy logical relationship, the
forecasted value of day j based on history of third order is
calculated as follows,

0.006289

1998

where j > k . X j − k shows the k-step dependence of jth value

0.043319

-8

1999

(8)

50

1296

1996

( X j−k;Y2, j−k,...,Ym−1, j−k ) ,..., ( X j−2;Y2, j−2,...,Ym−1, j−2) ,
( X j−1;Y1, j−1,Y2, j−1,...,Ym−1, j−1) , → X j

1095

1,288

2000

construct the m-factors kth-order fuzzy logical relationships,

1,145

2001

1,574

1497

77

0.0489199

We assumed eight intervals of equal length for the main and
secondary fuzzy time series'. For main factor, we assume
Dmin =953 and Dmax =1644, thus for main factor time series
we get U = [850,1650] . Similarly for secondary factors
Y1,Y2 ,Y3 and Y4 , we assumed that Emin =[90,1094,5949,38390]

(10)

IV. EXPERIMENT
In this experiment, our goal is to extend the work of [6].
We have applied this new technique on car road accident data
taken from National Institute of Statistics, Belgium for the
period of 1974-2005. In this data, the main factor of interest is
the yearly road accident causalities and secondary factors are
mortally wounded, died within one month, severely wounded
and light casualties.

and Emax =[819, 2393,16645, 46818] to determine v1, v 2 , v3 , v 4 .
Selection of Dmin , Dmax , Emin and Emax have significant
effects on the accuracy of this new method. We can introduce
learning to stabilize the heuristic selection of these constants.
Using (6) and (7), we formed fuzzy times series from main
and secondary factors. Therefore, each observation of a time
series is now represented by a combination of fuzzy sets.
Using (10), we calculated the forecasted values corresponding
to each actual value of the main factor time series in III. Using
equation (11) for AFER, we formed table IV, showing a
comparison of actual and forecasted values. Finally, we have
compared proposed method with [6].

27


TABLE IV
COMPARISON OF PROPOSED METHOD AND LEE L. W. (2006). METHOD FOR
YEALRY CAR ACCIDENT CAUALITIES IN BELGIUM FROM 1974-2004

Year

Forecaste
d
Actual
Killed (Ai) causalities
(Fi)

Fi − Ai
Ai

Forecasted
causalities
(Fi)

Fi − Ai
Ai

type-II defuzzified forecasted values ( t j ) may also be
calculated using some other method, e.g. learning rules from
fuzzy time series.
1700

1600

995

0.044040

1000

0.049318

1035

995

0.038676

1000

0.033816

2002

1145

1095

0.043319

1100

0.039301

2001

1288

1296

0.006289

1300

0.009317

2000

1253

1296

0.034397

1300

0.037510

1999

1173

1196

0.019437

1200

0.023018

1998

1224

1196

0.023039

1200

0.019608

1997

1150

1196

0.039826

1200

0.043478

1996

1122

1095

0.023708

1100

0.019608

1995

1228

1396

0.137134

1400

0.140065

1994

1415

1296

0.084028

1300

0.081272

1993

1346

1396

0.037444

1400

0.040119

1992

1380

1497

0.084565

1500

0.086957

1991

1471

1497

0.017471

1500

0.019714

1990

1574

1497

0.049111

1500

0.047014

1989

1488

1396

0.061559

1400

0.059140

1988

1432

1396

0.024860

1400

0.022346

1987

1390

1497

0.076763

1500

0.079137

1986

1456

1296

0.109821

1300

1308

1396

0.067584

1400

1369

1497

0.093280

1500

1479

1497

0.011968

1500

0.014199

1982

1464

1497

0.022336

1500

0.024590

1981

1564

1497

0.043031

1500

0.040921

1980

1616

1497

0.073824

1500

0.071782

1979

1572

1497

0.047901

1500

0.045802

1978

1644

1497

0.089599

1500

0.087591

1977

1597

1497

0.062805

1500

0.060739

1976

1536

1497

0.025586

1500

0.023438

1975

1460

1497

0.025137

1500

0.027397

1974

1574

1497

0.048920

1500

0.025413

× 100%

= 5.061793%

1200

1100

1000

900
1970

1975

1980

1985

1990

1995

2000

2005

Years

Fig. 1 A Comparison of proposed and Lee L. W. et. al. (2006) [6]
Methods

ACKNOWLEDGMENT
The authors are very thankful for the kind support of
National Institute of Statistics, Belgium.

0.095690

1983

1300

0.070336

1984

1400

0.107143

1985

1500
Annual car road accidents

953

2003


2004

n

AFER =

∑
i =1

Fi − Ai
Ai
31

= 5.067887%

V. CONCLUSION
From Table IV, we can see that our proposed method is
better that [6]. As the work of Lee et. al. [6] outperformed the
work of [4], [5] and [10], so, indirectly we can conclude that
our general class of methods for fuzzy time series modeling
and forecasting.
Furthermore, we have shown fuzziness of fuzzy
observations by presenting each datum of the main series as
composed of many fuzzy sets. Thus, fuzzy time series
modeling extends to type-II fuzzy time series modeling. The

REFERENCES
[1]

G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and
Applications, Prentice Hall, India, 2005, Ch. 4.
[2] H. Ishibuchi, R. Fujioka and H. Tanaka, "Neural Networks that Learn
from Fuzzy If-Then Rules", IEEE Transactions on Fuzzy Systems, Vol.
1, No. 1, pp.85-97, 1993.
[3] H. J. Zimmerman, Fuzzy set theory and its applications, Kluwer
Publishers, Boston, MA, 2001.
[4] K. Huarng, “Heuristic models of fuzzy time series for forecasting,”
Fuzzy Sets Systems, vol. 123, no. 3, pp. 369–386, 2001a.
[5] K. Huarng, “Effective Lengths of Intervals to Improve Forecasting in
Fuzzy Time Series,” Fuzzy Sets System, Vol. 123, No. 3, pp. 387–394,
2001b.
[6] L. W. Lee, L. W. Wang, S. M. Chen, “Handling Forecasting Problems
Based on Two-Factors High-Order Time Series,” IEEE Transactions on
Fuzzy Systems, Vol. 14, No. 3, pp.468-477, Jun. 2006.
[7] Melike Sah and Y. D. Konstsntin, “Forecasting Enrollment Model based
on first-order fuzzy time series,” Published in proc., International
Conference on Computational Intelligence, Istanbul, Turkey, 2004.
[8] Q. Song and B. S. Chissom, “Forecasting Enrollments with Fuzzy Time
Series—Part I,” Fuzzy Sets and System, Vol. 54, No. 1, pp. 1–9, 1993 a.
[9] R. R. Yager and P. P. D. Filev, Essentials of FUZZY MODELING and
Control, John Wiley and Sons, Inc. 2002.
[10] S. M. Chen, “Forecasting Enrollments Based on High-Order Fuzzy Time
Series,” Cybernetic Systems, Vol. 33, No. 1, pp. 1–16, 2002.
[11] S. Park and T. Han, “Iterative Inversion of Fuzzified Neural Networks,”
IEEE Transactions on Fuzzy Systems, Vol. 8, No. 3,pp. 266- 280, 2000.

28



S. M. Aqil Burney received the B.Sc.(Physics,
Mathematics, Statistics) from D. J. College affiliated
with the University of Karachi in 1970; First class first
M.Sc. (Statistics,) from Karachi University in 1972
with M.Phil. in 1983 with specialization in
Computing, Simulation and stochastic modeling in
from risk management. Dr. Burney received Ph.D.
degree in Mathematics from Strathclyde University,
Glasgow with specialization in estimation, modeling
and simulation of multivariate Time series models
using algorithmic approach with software development.
He is currently professor and approved supervisor in Computer Science and
Statistics by the High education Commission Government of Pakistan. He is
also the project director of Umair Basha Institute of Information technology
(UBIT). He is also member of various higher academic boards of different
universities of Pakistan. His research interest includes AI, Soft computing
neural networks, fuzzy logic, data mining, statistics, simulation and stochastic
modeling of mobile communication system and networks and network
security. He is author of three books, various technical reports and supervised
more than 100software/Information technology projects of Masters level
degree programs and project director of various projects funded by
Government of Pakistan.
He is member of IEEE (USA), ACM (USA) and fellow of Royal Statistical
Society United Kingdom and also a member of Islamic society of Statistical
Sciences. He is teaching since 1973 in various universities in the field of
Econometric, Bio-Statistics, Statistics, Mathematic and Computer Science He
has vast education management experience at the University level. Dr. Burney
has received appreciations and awards for his research and as educationist.

Tahseen A. Jilani received the B.Sc.(Computer
Science, Mathematics, Statistics), First class second
M.Sc. (Statistics) and MA(Economics) in 1998,2001
and 2003 respectively. Since 2003, he is Ph.D.
research fellow in the Department of Computer
Science, University of Karachi.
His research interest includes AI, neural networks, soft
computing, fuzzy logic, Statistical data mining and
simulation. He is teaching since 2002 in the fields of
Statistics, Mathematic and Computer Science.

29

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