Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. VI (2011), No. 4 (December), pp. 603-614

Fuzzy Local Trend Transform based Fuzzy Time Series
Forecasting Model

J. Dan, F. Dong, K. Hirota

Jingpei Dan, Fangyan Dong, Kaoru Hirota
Tokyo Institute of Technology
Japan, 226-8502 Yokohama, 4259 Nagatsuta, Midori-ku,
E-mail: {dan,tou,hirota}@hrt.dis.titech.ac.jp

Jingpei Dan
Chongqing University
P.R.China, 400044 Chongqing, 174 Shazhengjie, Shapingba
E-mail: danjingpei@hotmail.com

Abstract: A fuzzy local trend transform based fuzzy time series forecasting
model is proposed to improve practicability and forecast accuracy by provid-
ing forecast of local trend variation based on the linguistic representation of
ratios between any two consecutive points in original time series. Local trend
variation satisfies a wide range of real applications for the forecast, the practi-
cability is thereby improved. Specific values based on the forecasted local trend
variations that reflect fluctuations in historical data are calculated accordingly
to enhance the forecast accuracy. Compared with conventional models, the
proposed model is validated by about 50% and 60% average improvement in
terms of MLTE (mean local trend error) and RMSE (root mean squared error),
respectively, for three typical forecasting applications. The MLTE results in-
dicate that the proposed model outperforms conventional models significantly
in reflecting fluctuations in historical data, and the improved RMSE results
confirm an inherent enhancement of reflection of fluctuations in historical data
and hence a better forecast accuracy. The potential applications of the pro-
posed fuzzy local trend transform include time series clustering, classification,
and indexing.
Keywords: time series forecasting, fuzzy time series, trend, transform.

1 Introduction

Based on Zadeh’s works (see [1] and [2]), the concept of fuzzy time series and its models for
forecasting have been proposed to solve the forecasting problems where the historical data are
linguistic values (see [3]- [5]). Conventional fuzzy time series forecasting models that are based
on fuzzy time series of original data are limited to forecasting specific values that do not reflect
fluctuations in historical data. Local trend variations, however, are mainly concerned with real
applications. For example, forecast of changing direction of stock price are more important for
stock investors to make reasonable determinations than specific forecast values of stock price. In
addition, the forecasted specific demand values are unreliable since historical data is distorted
when it transfers along the supply chain due to the bullwhip effect, so local trend variations
of demand that are not suffered from the bullwhip effect are more valuable and practical for
supply chain managers. Therefore the practicability of conventional fuzzy time series forecasting
methods suffers from the limitation of forecasting specific values. This study aims to improve the

Copyright c⃝ 2006-2011 by CCC Publications



604 J. Dan, F. Dong, K. Hirota

practicability and forecast accuracy by forecasting local trend variations that reflect fluctuations
in historical data. It should be noted that the word "trend" usually refers to the long-term trend
in statistics, whereas as used in this paper word "trend" means local trend variation in short
term or during one period.

Recently, some trend involved fuzzy time series models have been proposed to improve fore-
casting. Huarng has proposed heuristic models by integrating problem-specific heuristic knowl-
edge with Chen’s model [6] to improve forecasting by reflecting the fluctuations in fuzzy time
series [7]. A trend-weighted fuzzy time series model for forecasting Taiwan Stock Exchange
Capitalization Weighted Stock Index (TAIEX) has been proposed in [8]. Chen and Wang have
proposed a method to predict TAIEX based on fuzzy-trend logical relationship groups to improve
forecast accuracy [9]. The interval rearranged method has been proposed to reflect fluctuations
in historical data and improve forecast accuracy of fuzzy time series in [10]. Although all these
methods are involved with trends, they are intrinsically conventional fuzzy time series forecasting
methods since they are all based on original data and their forecasting targets are specific values.
Fuzzy local trend transform is proposed to provide a different forecasting basis by transforming
original data into a linguistic representation of local trend variations called fuzzy local trend time
series. In contrast to conventional fuzzy time series forecasting models, local trend variations
are forecasted based on the transformed fuzzy local trend time series in the proposed model.
Forecast accuracy of specific values is hence enhanced by forecasted local trend variations that
reflect fluctuations in historical data.

Three typical forecasting targets, enrollment forecasting, stock index forecasting, and in-
ventory demand forecasting are used to validate the proposed model. To make an effective
evaluation, forecasts are evaluated by two measures from different aspects. Root mean squared
error (RMSE) is used to evaluate forecast accuracy of specific values while mean local trend
error (MLTE) measure [11] is used to evaluate how accurately forecast reflects fluctuations in
historical data. For enrollment forecasting, the proposed model outperforms typical fuzzy time
series models in terms of RMSE and MLTE. Especially, comparing to Chen’s model [6], Huarng’s
model [12], and Cheng et al.’ model [13], MLTE results show an improvement of 73.3%, 55.6%,
and 66.7%, respectively. For TAIEX forecasting, the proposed model gets the smallest RMSE re-
sult while the second rank in terms of MLTE compared to Chen’s model [6], Yu’s model [14], and
Cheng’s model [8]. For inventory demand forecasting, compared to Huarng and Yu’ model [15],
Cheng et al.’ model [13], and Chen and Wang’ model [9], the proposed model yields about
50%, 50%, and 33.3% improvement in terms of MLTE and 48.3%, 73.3%, and 49.1% improve-
ment in terms of RMSE. The MLTE results demonstrate that the proposed model outperforms
conventional fuzzy time series models in significantly reflecting fluctuations in historical data,
and the improved RMSE results confirm an inherent enhancement of reflection of fluctuations in
historical data and hence a better forecast accuracy.

The rest of the paper is organized as follows: in section 2, fuzzy time series and fuzzy c-means
clustering are briefly reviewed. The proposed fuzzy local trend transform based fuzzy time series
forecasting model is elaborated in section 3. Empirical analyses on three forecasting targets to
demonstrate the proposed model are illustrated in section 4.

2 Fuzzy Time Series and Fuzzy C-means Clustering: a Brief Re-
view

The proposed model is based on conventional fuzzy time series forecasting model and the
fuzzy c-means clustering method is applied in the proposed fuzzy local trend transform, so fuzzy
time series and fuzzy c-means clustering are briefly reviewed by adjusting the notations.



Fuzzy Local Trend Transform based Fuzzy Time Series Forecasting Model 605

2.1 Fuzzy time series

Let U be the universe of discourse, where U = {u1, u2, · · · , ub}. A fuzzy set Ai of U is defined
as Ai =

∑b
j=1 fAi(uj)/uj, where fAi is the membership function of the fuzzy set Ai; fAi : U →

[0, 1], ua is a generic element of fuzzy set Ai and fAi(ua) is the degree of belongingness of ua to
Ai; fAi(ua) ∈ [0, 1] and a ∈ [1, b]. In [3], the general definitions of fuzzy time series are given as
follows:

Definition 1. Let a subset of real numbers Y (t)(t = · · · , 0, 1, 2, · · · ) be the universe of
discourse by which fuzzy sets fj(t) are defined. If F(t) is a collection of f1(t), f2(t), · · · , then
F(t) is called a fuzzy time series defined on Y (t).

Definition 2. If fuzzy time series relationships assume that F(t) is caused only by F(t − 1),
then the relationship can be expressed as: F(t) = F(t − 1) ∗ R(t, t − 1), which is the fuzzy
relationship between F(t) and F(t − 1), where ∗ represents as an operator. (Note that the
operator can be either max-min [4], min-max [5], or arithmetic operator [6].)

To sum up, let F(t − 1) = Ai, and F(t) = Aj, the fuzzy logical relationship between F(t)
and F(t − 1) can be denoted as Ai → Aj, where Ai refers to the left-hand side and Aj refers to
the right-hand side of the FLR. Furthermore, these fuzzy logical relationships can be grouped to
establish different fuzzy relationship. These groups are called fuzzy logical relationship groups
(FLRGs). On account of its simplicity, FLR method is chosen by most researchers.

The procedure for forecasting using conventional fuzzy time series models has four main steps:
(1) Define universe of discourse and intervals; (2) Define fuzzy sets and fuzzify observations in
the original time series; (3) Establish fuzzy relationships; (4) Forecast and defuzzify the outcome.
Assume that the current state of F(t) is Ai, Fdef(t + 1) can be forecasted and defuzzified by the
following rules:

Rule 1: If there is Ai → Aj in fuzzy logical relationship groups, then F(t + 1) = Aj and
defuzzified as Fdef(t+1) = centerj, where centerj is the center of cluster j to which Aj belongs.

Rule 2: If there is Ai → # in fuzzy logical relationship groups, then F(t + 1) = Ai and
defuzzified as Fdef(t + 1) = centeri, where # represents null value and centeri is the center of
cluster i to which Ai belongs.

Rule 3: If there is AiA1, A2, · · · , Aj in fuzzy logical relationship groups, then the forecast at
t + 1 is calculated as Fdef(t + 1) = (center1 + center2 + · · · + centerj)/j, where cj is the center
of cluster to which Aj belongs.

2.2 Fuzzy c-means clustering

Fuzzy c-means (FCM) clustering is a method of clustering which allows one piece of data to
belong to two or more clusters, and is frequently used in pattern recognition [16].

The FCM is based on minimization of the objective function

Jm =

N∑
i=1

C∑
j=1

umij ∥xi − cj∥
2, 1 ≤ m ≤ ∞, (1)

where m is any real number greater than 1, uij is the degree of membership of xi in the cluster
j, xi is the i-th of d-dimensional measured data, cj is the d-dimension center of the cluster, and
∥ ∗ ∥ is any norm expressing the similarity between any measured data and the center.

Fuzzy partitioning is carried out through an iterative optimization of the objective function
(1), with the update of membership uij = 1/

∑C
k=1 (∥xi − cj∥/∥xi − ck∥)

2/m−1 and the cluster
centers cj =

∑N
i=1 u

m
ij (xi/

∑N
i=1 u

m
ij ).



606 J. Dan, F. Dong, K. Hirota

This iteration will stop when Maxij|uk+1ij − u
k
ij| < ε, where ε is a termination criterion

between 0 and 1, whereas k is the iteration step. This procedure converges to a local minimum
or a saddle point of Jm.

3 Proposed Fuzzy Local Trend Transform based Fuzzy Time Se-
ries Forecasting Model

In literature, trends are widely represented by using absolute variations, slopes, or relative
variations between two consecutive points in literature. In section 3.1, trends or local trend
variations in study are defined as relative variations, or ratios. The algorithm of fuzzy local
trend transform based on local trend variations is then elaborated in section 3.2. The fuzzy time
series forecasting model based on fuzzy local trend transform is presented in section 3.3.

3.1 Local trend variations

To address the limitation of absolute variations [17] and slopes( [18]- [20])for representing
local trend variations, relative variations, which are the ratios between two consecutive data
points in a given historical time series, are adopted to indicate local trend variations in this
study.

Assuming that for any given time series P(t), t = 1, 2, · · · , n, n ∈ N, local trend variation
between time t and t − 1 is defined as

rt = (P(t) − P(t − 1))/P(t − 1), t = 2, 3, · · · , n, (2)

then the time series of local trend variations for P(t) is defined as T(t) = rt, t = 2, · · · , n.
The reasons for forecasting based on local trend variations instead of original time series data

are explained as follows. First, the forecasts based on original time series data may not reflect
the fluctuations in historical data properly. In most previous studies, the forecasts are equal at
some consecutive points which indicated that forecasting based on original time series are not
appropriate for reflecting fluctuations [15].

Second, original time series data varies dramatically in different contexts while the time series
of local trend variations varies slightly.

Finally, forecasting based on local trend variations are more suitable for reflecting fluctuations
in historical data since directions and variation degrees of local trend variations can be indicated
by signs and magnitudes of ratios easily. Ratios are preferable in terms of demonstrating the
differences in various contexts, ratios-based lengths of intervals are hence adopted to improve
fuzzy time series in [15].

Forecasting based on local trend variations are therefore considered more suitable for reflect-
ing fluctuations in historical data and forecast accuracy should be further improved inherently.

3.2 Fuzzy local trend transform

To forecast local trend variations, original time series is represented by linguistic local trend
variations of original data firstly, the algorithm of fuzzy local trend transform is hence proposed
as follows:

Step 1 : Obtain the local trend variation time series T(t) by calculating ratios between each
two consecutive data points in the original time series P(t) in terms of equation (2).

Step 2: Divide T(t) into three basic clusters in terms of local trend changing direction, i.e.,
decreasing cluster Td, unchanged cluster Tu, and increasing cluster Ti. It is easy to determine Td
and Ti in terms of sign of ratios after Tu is determined. Assume that the interval for unchanged



Fuzzy Local Trend Transform based Fuzzy Time Series Forecasting Model 607

Table 1: Parameter ŚÁ for determining the interval of unchanged cluster

Max(|T(t)|)(×10−2) Max(|T(t)|) ≤ 1 Max(|T(t)|) ≤ 10 Max(|T(t)|) ≤ 20 · · ·
α(×10−2) 0.01 0.1 0.2 · · ·

cluster is [−α, α], α is determined by Max(|T(t)|) according to Table 1 since it is possible that
the definition of unchanged cluster varies from problem to problem. The observations in T(t),
of which the values are greater than α, are then assigned to Ti, while the observations of which
the values less than −α are assigned to Td.

Step 3: Divide Ti and Td into ci and cd clusters by applying FCM, respectively. Assume
that the number of observations in Ti and Td are ni and nd, respectively, then the number of
clusters for Ti, Td and T(t) are predefined by users as ci (2 ≤ ci ≤ ni), cd (2 ≤ cd ≤ nd), and
c (c = cd + ci + 1), respectively. Ti and Td are then divided into ci and cd clusters by FCM,
respectively, as described in section 2.2. Consequently, the cluster centers and memberships with
respect to the clusters are obtained.

Step 4: Fuzzify the local trend time series T(t) as fuzzy local trend time series FT (t). First,
the clusters of T(t) are achieved by combining the clustering results obtained in Step 2 and Step
3. Then the linguistic terms Ai (i = 1, 2, · · · , c) are defined corresponding to the clusters. T(t)
is finally fuzzified into FT (t) by assigning Ai to T(t) when the maximum membership of T(t)
occurs at the cluster to which Ai belongs.

3.3 Fuzzy local trend transform based fuzzy time series forecasting model

Theoretically, the proposed fuzzy local trend transform can be integrated with any conven-
tional fuzzy time series forecasting model. Because of simplicity, the proposed model is integrated
with Chen’s model [6] as stated in section 2.1. The proposed model differs from Chen’ model in
Step 1 and Step 4 as described in the following:

Step 1: Transform original time series into fuzzy local trend time series by the proposed fuzzy
local trend transform as stated in section 3.2.

Step 2: Establish fuzzy logical relationships and fuzzy logical relationship groups based on
fuzzy local trend time series obtained in Step 1 as described in section 2.1.

Step 3: Forecast and defuzzify the possible outcomes of local trend variations, which is
denoted as Tdef(t), t = 1, 2, · · · , n − 1, based on fuzzy logical relationship groups as described in
section 2.1.

Step 4: Calculate specific values based on forecasted local trend variations obtained in Step
3 in terms of equation(3) that is defined as

Ppre(t) = P(t) × Tdef(t), t = 1, 2, · · · , n − 1, (3)

where Ppre(t) indicates predicted specific values.

4 Empirical Analyses on Forecasting based on Fuzzy Local Trend
Time Series

To validate the proposed model, three applications are used in the empirical analyses, in-
cluding student enrollment forecasting (the enrollments of the University of Alabama [4]), stock
index forecasting (Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)), and
inventory demand forecasting [15]. The first two data sets are algebraic growth data and widely
used to validate fuzzy time series models in many relevant studies while the inventory demand
data set is exponential growth data and typical in supply chain management application. We



608 J. Dan, F. Dong, K. Hirota

compare the proposed model with typical fuzzy time series models in terms of two forecast accu-
racy measures. One is conventional measure RMSE (root mean squared error) that is commonly
used to measure forecast accuracy of specific values based on quantitative error in fuzzy time
series forecasting, while the other is MLTE (mean local trend error), which is proposed to mea-
sure how accurately forecasts reflect fluctuations in actual data based on local trend error [11].
It is more effective and proper for comparing the models by evaluating them from two different
aspects than using one or more conventional measures that based on quantitative error. For a
given time series yt, t = 1, · · · , n, the prediction of yt is ft, t = 1, · · · , n, RMSE is defined as

RMSE =

√√√√1
n

n∑
t=1

(ft − yt)2, t = 1, 2, · · · , n, n ∈ N, (4)

ĄĄ while MLTE is defined as

MLTE=
1

n−1

n−1∑
i=1

Ei×100%, Ei =

{
1 sign(yt+1−yt) ̸=sign(ft+1−ft)
0 sign(yt+1−yt)=sign(ft+1−ft)

t=1, 2,· · ·, n, n ∈ N, (5)

where Ei indicates the number of local trend change errors, sign indicates the operator for
outputting the sign of the operand. When the predicted local trend variation is inconsistent
with the original one in the same interval, Ei is equal to 1, otherwise Ei equals to 0 [11].

The experiments are implemented using Matlab R2010a. In section 4.1, the forecasting
algorithm is illustrated step by step with the example of enrollment forecasting. Performance
of the proposed model for stock index and inventory demand forecasting are analyzed in section
4.2 and section 4.3, respectively.

4.1 Forecasting enrollments

The yearly data of student enrollments of the University of Alabama from 1971 to 1992 are
commonly used in previous studies on fuzzy time series to validate fuzzy time series models. To
make proper comparison with other models, the same data set is used in this study to validate the
proposed model. The procedure of forecasting enrollments by the proposed model is as follows:

Step 1: Transform original time series into fuzzy local trend time series by the proposed fuzzy
local trend transform.

(1) Obtain the local trend time series T(t) as shown in the third column of Table 2 by
calculating the local trend variations of the enrollments between each two consecutive years in
the original time series P(t) in terms of equation(2) .

(2) Divide T(t) into decreasing cluster Td, unchanged cluster Tu, and increasing cluster Ti.
Since none of the observations in T(t) is equal to 0 and maximum absolute value of the obser-
vations in T(t) is 7.6675%, α is determined to be 0.1% according to Table 1. Consequently,
Td = {−5.8274, −3.1385, −2.3840, −2.2714, −0.9638}, Tu = {0.0466}, and Ti = {0.1189, 0.4147,
0.6664, 1.6535, 1.8872, 1.9071, 2.2414, 3.8912, 4.5179, 5.1987, 5.4145, 5.4742, 5.9643, 5.9782,
7.6576}.

(3) Divide Ti and Td into ci into cd clusters by applying FCM, respectively. For FCM, the
predefined number of clusters c = cd + ci + 1. To make fair comparison, the total number of
clusters is predefined to be 7 in this study as the same as used in the previous studies. Assume
that cd = ci = 3, the cluster centers of Td and Ti and the membership grades of Td and Ti are
shown in Table 3 and Table 4, respectively. Naturally, the center of unchanged cluster Ti is 0%.

(4) Fuzzify the local trend time series T(t) into fuzzy local trend time series FT (t) as shown
in the fourth column of Table 2. Combine the clustering results obtained in Step (2) and Step



Fuzzy Local Trend Transform based Fuzzy Time Series Forecasting Model 609
Table 2: Forecasting local trend variations and specific values of the enrollments

Year Actual Local trend Fuzzified local Forecasted local trend Forecasted
enrollment variation trend variation variation (×10−2) enrollment

1971 13055 - - - -
1972 13563 3.8912 A6 - -
1973 13867 2.2414 A5 0.1233 13580
1974 14696 5.9782 A7 1.1623 14028
1975 15460 5.1987 A6 3.8752 15266
1976 15311 -0.96378 A3 0.1233 15479
1977 15603 1.9071 A5 1.2224 15498
1978 15861 1.6535 A5 1.1623 15784
1979 16807 5.9643 A7 1.1623 16045
1980 16919 0.66639 A5 3.8752 17458
1981 16388 -3.1385 A2 1.1623 17116
1982 15433 -5.8274 A1 -2.3003 16011
1983 15497 0.4147 A5 1.2224 15622
1984 15145 -2.2714 A2 1.1623 15677
1985 15163 0.11885 A5 -2.3003 14797
1986 15984 5.4145 A7 1.1623 15339
1987 16859 5.4742 A7 3.8752 16603
1988 18150 7.6576 A7 3.8752 17512
1989 18970 4.5179 A6 3.8752 18853
1990 19328 1.8572 A5 0.1233 18993
1991 19337 0.046565 A4 1.1623 19553
1992 18876 -2.384 A2 -2.5770 18839

RMSE 438.18
MLTE 21.0526%

(3) to achieve the clusters of T(t), the linguistic variables are then defined as shown in Table 5
according to the clusters. Each local trend variation in T(t) is fuzzified by the linguistic variable
to which the maximum membership belongs in terms of the results in Table 3 and Table 4.

Table 3: Membership grades of decreasing cluster for each linguistic variable

clusters Cluster Centers Linguistic Td(1) Td(2) Td(3) Td(4) Td(5)
variables

1 -2.5770 A2 0.0000018 0.9 0.9785 0.9407 0.000055
2 -5.8231 A1 0.9999974 0.0393 0.0031 0.0070 0.000006
3 -0.9758 A3 0.0000008 0.0607 0.0184 0.0523 0.999939

FT (t) A1 A2 A2 A2 A3

Step 2: The FLRs of FT (t) are established as shown in Table 6 according to Definition 3 in
2.1. Then, the FLRs are rearranged into FLRGs as shown in Table 7.

Step 3: The possible outcomes of local trend variations from 1973 to 1992 are forecasted and
defuzzified as shown in the fifth column of Table 2.

Table 4: Membership grades of increasing cluster for each linguistic variable

clusters Cluster Linguistic Ti(1) Ti(2) Ti(3) Ti(4) Ti(5) Ti(6) Ti(7) Ti(8)
Centers variables

1 4.3997 A6 0.0603 0.0387 0.0215 0.0238 0.0639 0.0684 0.1719 0.9139
2 1.2224 A5 0.9077 0.9416 0.9680 0.9667 0.9123 0.9063 0.7715 0.0332
3 6.0036 A7 0.0319 0.0197 0.0105 0.0095 0.0238 0.0253 0.0566 0.0530

FT (t) A5 A5 A5 A5 A5 A5 A5 A6
Ti(9) Ti(10) Ti(11) Ti(12) Ti(13) Ti(14) Ti(15)

1 4.3997 A6 0.9924 0.4936 0.2486 0.1929 0.00063 0.00027 0.1947
2 1.2224 A5 0.0013 0.0199 0.0146 0.0123 0.00007 0.00003 0.0499
3 6.0036 A7 0.0063 0.4864 0.7371 0.7948 0.9993 0.9997 0.7554

FT (t) A6 A6 A7 A7 A7 A7 A7

Step 4: The enrollment values from the year 1973 to 1992 are calculated based on the fore-
casted local trend variations obtained in Step 3 as shown in the last column of Table 2. The num-
ber of intervals of the universe of discourse affects forecasting results [21]. When analyzing the
sensitivity of c, which varies in {7, 9, 11, 13}, the RMSE and MLTE results {502.401, 440.3566, 380.8953, 313.2307}



610 J. Dan, F. Dong, K. Hirota

Table 5: Cluster centers of local trend variations of enrollments

Cluster Centers Linguistic variables
(×10−2)
-5.8231 A1(big decrease)
-2.5770 A2(decrease)
-0.9758 A3(small decrease)

0 A4(almost unchanged)
1.2224 A5(small increase)
4.3997 A6(increase)
6.0036 A7(big increase)

and {23.1579%, 24.2105%, 20%, 13.6842%} respectively. The average RMSE and MLTE results
indicate that forecast error decrease as the number of clusters increases, in other words, the
bigger the number of clusters the higher the forecast accuracy.

Table 6: Fuzzy logical relationships of local trend variations of enrollments

A6→A5 A5→A7 A7→A6 A6→A3 A3→A5
A5→A5 A5→A7 A7→A5 A5→A2 A2→A1
A1→A5 A5→A2 A2→A5 A5→A7 A7→A7
A7→A7 A7→A6 A6→A5 A5→A4 A4→A2 A2→#

Table 7: Fuzzy logical relationship groups of local trend variations of enrollments

Group 1: A1→A5 Group 2: A2→A1,A5,# Group 3: A3→A5 Group 4: A4→A2
Group 5: A5→A2, A4, A5, A7 Group 6: A6→A3, A5 Group 7: A7→A5, A6, A7

Table 8: Comparisons for enrollment forecasting under the same conditions

Year Actual Chen Huarng Cheng et al. The proposed
enrollment [6] [12] [13] model

1971 13055 - - - -ĄĄ
1972 13563 14000 14000 14242 -ĄĄ
1973 13867 14000 14000 14242 13580
1974 14696 14000 14000 14242 14028
1975 15460 15500 15500 15474.3 15266
1976 15311 16000 15500 15474.3 15479
1977 15603 16000 16000 15474.3 15498
1978 15861 16000 16000 15474.3 15784
1979 16807 16000 16000 16146.5 16045
1980 16919 16833 17500 16988.3 17458
1981 16388 16833 16000 16988.3 17116
1982 15433 16833 16000 16146.5 16011
1983 15497 16000 16000 15474.3 15622
1984 15145 16000 15500 15474.3 15677
1985 15163 16000 16000 15474.3 14797
1986 15984 16000 16000 15474.3 15339
1987 16859 16000 16000 16146.5 16603
1988 18150 16833 17500 16988.3 17512
1989 18970 19000 19000 19144 18853
1990 19328 19000 19000 19144 18993
1991 19337 19000 19500 19144 19553
1992 18876 19000 19000 19144 18839

RMSE ĄĄ 646.79 477.91 466.17 438.18
MLTE 78.9474% 47.3684% 63.1579% 21.0526%

The proposed method is compared with typical models for enrollment forecasting, including
Chen’s model [6], Huarng’s model [12], and Cheng et al.’ model [13]. To make fair comparison,
seven linguistic variables are defined in all the compared models and the RMSEs and MLTEs
are computed with reference to the year 1973. As shown in Table 8, the comparative results
show that the proposed method outperforms the other models in terms of RMSE and MLTE.



Fuzzy Local Trend Transform based Fuzzy Time Series Forecasting Model 611

Especially, comparing to Chen’s model, Huarng’s model, and Cheng et al.’ model, the proposed
model makes about 73.3%, 55.6%, and 66.7% improvements in terms of MLTE, respectively.

4.2 Stock index forecasting

Table 9: Comparisons for TAIEX forecasting

Date Actual Actual local Chen Yu Cheng The proposed Forecasted local
Index trend (×10−2) [6] [14] [8] model trend (×10−2)

00/11/02 5,626.08 1.477 5300 5340 5463.85 5524.81 -0.35
00/11/03 5,796.08 2.933 5750 5721.67 5644.8 5611.08 -0.27
00/11/04 5,677.30 -2.0922 5450 5435 5797.8 5815.39 0.33
00/11/06 5,657.48 -0.35033 5750 5721.67 5690.9 5657.47 -0.35
00/11/07 5,877.77 3.7478 5750 5721.67 5673.06 5648.99 -0.15
00/11/08 6,067.94 3.134 5750 5760 5871.32 5897.36 0.33
00/11/09 6,089.55 0.35487 6075 6062.5 6042.47 6088.09 0.33
00/11/10 6,088.74 -0.013303 6075 6062.5 6061.92 6093.84 0.07
00/11/13 5,793.52 -5.0957 6075 6062.5 6061.19 6079.54 -0.15
00/11/14 5,772.51 -0.36397 5450 5435 5795.5 5685.36 -1.87
00/11/15 5,737.02 -0.61861 5450 5435 5776.59 5763.81 -0.15
00/11/16 5,454.13 -5.1867 5450 5435 5744.65 5728.37 -0.15
00/11/17 5,351.36 -1.9204 5300 5340 5409.92 5352.29 -1.87
00/11/18 5,167.35 -3.561 5350 5350 5317.42 5304.12 -0.88
00/11/20 4,845.21 -6.6486 5150 5150 5151.81 5070.94 -1.87
00/11/21 5,103.00 5.0517 4850 4850 4861.89 4953.18 2.23
00/11/22 5,130.61 0.53814 5150 5150 5093.9 5120.92 0.35
00/11/23 5,146.92 0.31689 5150 5150 5118.75 5134.17 0.07
00/11/24 5,419.99 5.0382 5150 5150 5213.56 5150.48 0.07
00/11/27 5,433.78 0.25378 5300 5340 5459.32 5439.04 0.35
00/11/28 5,362.26 -1.3338 5300 5340 5391.6 5425.62 -0.15
00/11/29 5,319.46 -0.80459 5350 5350 5327.23 5314.92 -0.88
00/11/30 5,256.93 -1.1895 5350 5350 5288.71 5311.49 -0.15
00/12/01 5,342.06 1.5936 5250 5250 5232.44 5210.46 -0.88
00/12/02 5,277.35 -1.2262 5350 5350 5309.05 5327.84 -0.27
00/12/04 5,174.02 -1.9971 5250 5250 5250.81 5230.78 -0.88
00/12/05 5,199.20 0.48431 5150 5150 5157.82 5128.29 -0.88
00/12/06 5,170.62 -0.55274 5150 5150 5180.48 5202.82 0.07
00/12/07 5,212.73 0.80783 5150 5150 5154.76 5162.82 -0.15
00/12/08 5,252.83 0.7634 5250 5250 5192.66 5216.33 0.07
00/12/11 5,284.41 0.59761 5250 5250 5228.75 5256.46 0.07
00/12/12 5,380.09 1.7784 5250 5250 5257.17 5288.08 0.07
00/12/13 5,384.36 0.079304 5350 5350 5343.28 5398.00 0.33
00/12/14 5,320.16 -1.2067 5350 5350 5347.12 5376.30 -0.15
00/12/15 5,224.74 -1.8263 5350 5350 5289.34 5273.20 -0.88
00/12/16 5,134.10 -1.7655 5250 5250 5203.46 5178.54 -0.88
00/12/18 5,055.20 -1.5608 5150 5150 5121.89 5088.74 -0.88
00/12/19 5,040.25 -0.29661 5450 5405 5050.88 5010.54 -0.88
00/12/20 4,947.89 -1.8667 5450 5405 5037.42 5032.72 -0.15
00/12/21 4,817.22 -2.7126 4950 4950 4954.3 4904.19 -0.88
00/12/22 4,811.22 -0.12471 4850 4850 4836.7 4800.37 -0.35
00/12/26 4,721.36 -1.9033 4850 4850 4831.3 4803.96 -0.15
00/12/27 4,614.63 -2.3129 4750 4750 4750.42 4679.69 -0.88
00/12/28 4,797.14 3.8046 4650 4650 4654.37 4598.48 -0.35
00/12/29 4,743.94 -1.1214 4750 4750 4818.62 4813.06 0.33
00/12/30 4,739.09 -0.10234 4750 4750 4770.74 4736.76 -0.15
RMSE 176.32 170.27 121.47 114.63
MLTE 64.44% 62.22% 26.67% 31.11%

The daily stock index, TAIEX (Taiwan Stock Exchange Capitalization Weighted Stock In-
dex), which is the other widely used data set in fuzzy time series studies, is used to further
validate the proposed model for out-sample forecasting. The TAIEX data during 2000/01/01 -
2000/10/31 are used as training data set, and the data during 2000/11/01 - 2000/12/31 are used
as testing data set.



612 J. Dan, F. Dong, K. Hirota

The proposed model is compared with Chen’s model [6], Yu’s model [14], and Cheng’s model
[8]. The comparison of the forecasting results is shown in Table 9. The proposed model gets the
smallest RMSE result while the second rank in terms of MLTE among the compared models.
Cheng’s model gets the smallest MLTE since it incorporates trend-weighting into Chen’s model
for TAIEX forecasting. The proposed model, however, improves forecast accuracy of specific
value about 6% compared to Cheng’s model by reflecting fluctuations in historical data.

4.3 Inventory demand forecasting

Demand forecasting plays a very important role in supply chain management. To further
validate the applicability of the proposed method for demand forecasting, an inventory demand
data set [15] is used in this study. This data set has been used in several previous studies, so it
is proper for fair comparison. Inventory demand data from 1 to 19 are used as training set while
data from 20 to 24 are used as testing set.

Table 10: Comparisons for inventory demand forecasting

Time Actual Huarng and Yu Cheng et al. Chen and Wang The proposed
inventory demand [15] [13] [9] model

20 227 206 205.5290 209.945 215.5516
21 223 228 216.4187 224.055 231.8985
22 242 228 216.4187 224.055 243.5307
23 239 244 216.4187 234.11 246.5969
24 266 244 216.4187 244.33 261.0037

RMSE 15.1 28.7295 14.88 7.6845
MLTE 100% 100% 75% 50%

The proposed model is compared with Huarng and Yu’ model [15], Cheng et al.’ model [13],
and Chen and Wang’ model [9]. As shown in Table 10 comparing with Huarng and Yu’ model,
Cheng et al.’ model, and Chen and Wang’ model, the MLTE results are improved by the
proposed model about 50%, 50%, and 33.3%, respectively. This indicates that the proposed
model outperforms the comparative models significantly in reflecting fluctuations in historical
data. Consequently, the forecast accuracy of specific value is improved by the proposed model
about 48.3%, 73.3%, and 49.1%, respectively, in terms of RMSE, which indicates that forecast
accuracy is inherently improved by the proposed model by reflecting fluctuations in historical
data.

5 Conclusion

In contrast to conventional fuzzy time series forecasting models that are based on original
fuzzy time series, a different forecasting basis, fuzzy local trend time series, which is the linguistic
representation of local trend variations of original data, is provided by the proposed fuzzy local
trend transform. Local trend variations, which are defined as ratios between any two consecutive
data points in original time series, are thereby forecasted based on fuzzy local trend time series
and the specific values are are calculated accordingly. Therefore the practicability and forecast
accuracy are improved by reflecting fluctuations in historical data.

The proposed model is validated by using three typical forecasting targets. The results
are evaluated by two measures from different aspects. Compared to conventional fuzzy time
series models, the proposed model yields about 50% and 60% average improvement in terms of
MLTE and RMSE, respectively for the three application areas. The MLTE results indicate that
the proposed model outperforms conventional fuzzy time series models significantly in reflecting
fluctuations in historical data, and the improved RMSE results confirm an inherent enhancement
of reflection of fluctuations in historical data and hence a better forecast accuracy.



Fuzzy Local Trend Transform based Fuzzy Time Series Forecasting Model 613

Theoretically, the proposed fuzzy local trend transform can be integrated with any fuzzy
time series model. Optimization of the parameters predefined in the proposed fuzzy local trend
transform is being considered. A linguistic representation of time series data based on the
proposed fuzzy local trend transform for the application to time series data mining is ongoing.

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