A new hybrid fuzzy time series forecasting model based on combing fuzzy C-means clustering and particle swam optimization

Journal of Computer Science and Cybernetics, V.35, N.3 (2019), 267–292

DOI 10.15625/1813-9663/35/3/13496

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

BASED ON COMBING FUZZY C-MEANS CLUSTERING AND

PARTICLE SWAM OPTIMIZATION

1,∗

2

NGHIEM VAN TINH , NGUYEN CONG DIEU

¹Thai Nguyen University of Technology, Thai Nguyen University

²Thang Long University, Ha Noi, Viet Nam

^∗nghiemvantinh@tnut.edu.vn

ꢀ

Abstract. Fuzzy time series (FTS) model is one of the eﬀective tools that can be used to identify

factors in order to solve the complex process and uncertainty. Nowadays, it has been widely used in

many forecasting problems. However, establishing eﬀective fuzzy relationships groups, ﬁnding proper

length of each interval, and building defuzziﬁcation rule are three issues that exist in FTS model.

Therefore, in this paper, a novel FTS forecasting model based on fuzzy C-means (FCM) clustering

and particle swarm optimization (PSO) was developed to enhance the forecasting accuracy. Firstly,

the FCM clustering is used to divide the historical data into intervals with diﬀerent lengths. After

generating interval, the historical data is fuzziﬁed into fuzzy sets. Then, fuzzy relationship groups

were established according to chronological order of the fuzzy sets on the right-hand side of the

fuzzy logical relationships with the aim to serve for calculating the forecasting output. Finally, the

proposed model combined with PSO algorithm has been applied to optimize interval lengths in the

universe of discourse for achieving the best predictive accuracy. The proposed model is applied to

forecast three numerical datasets (enrollments data of the University of Alabama, the Taiwan futures

exchange(TAIFEX) data and yearly deaths in car road accidents in Belgium). Computational results

indicate that the forecasting accuracy of proposed model is better than that of other existing models

for both ﬁrst - order and high - order fuzzy logical relationship.

Keywords. Enrollments; Forecasting; FTS; Time - Variant Fuzzy Relationship Groups; PSO;

FCM.

1. INTRODUCTION

Advance forecasting of events in our daily life like temperature, stock market, popu-

lation growth, car fatalities, economy growth and crop productions are main scientiﬁc is-

sues in the forecasting ﬁeld. To make a forecast for these kinds of problems with 100%

accuracy may not be possible, but obtaining results with the smallest forecasting error

is possible. Previously, many classical forecasting models were developed to resolve dif-

ferent problems such as regression analysis, moving average, exponential moving average

and ARIMA model. These approaches require having the linearity assumption and needing

a large amount of historical data. The FTS forecasting models which were proposed by

Song and Chrissom [32, 33] even don’t need a limitation of the observations and the line-

arity assumption either. To forecast the enrollments of the University of Alabama, their

c

ꢀ 2019 Vietnam Academy of Science & Technology

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NGHIEM VAN TINH, NGUYEN CONG DIEU

approaches apply the max-min operations to handle uncertainty and imprecise data. Howe-

ver, the limitations in their scheme are not convincing to determine the length of intervals

and whenever the fuzzy logical relation matrix becomes larger, more amount of compu-

tation they face. To overcome those drawbacks and be more accurate in forecasting, the

ﬁrst-order FTS approach suggested by Chen [6] uses simple arithmetic calculations rather

than max-min composition operations [32]. Since then, fuzzy time series model is more

discovered by many researchers. They presented various improvements from Chen’s model

[6] in terms of determining the lengths of intervals including the static length of inter-

vals [7, 17, 18, 37, 38] and dynamic length of intervals [3, 4, 9, 14, 22, 26, 27, 35], con-

structing fuzzy relationship groups [4, 9, 10, 11, 15, 16, 22, 23, 26, 36] and defuzzication

process [23, 30, 31, 35]. Speciﬁcally, Huarng [16] suggested an eﬀective computational met-

hod to determine the appropriate intervals. He stated that the result of forecasting model

is greatly inﬂuenced by diﬀerent lengths of intervals in the universe of discourse. Other

research works [3, 5, 7, 4, 9, 14, 15, 24, 25] oﬀered diﬀerent computational approaches in fo-

recasting based on high-order FTS models to defeat the downsides of ﬁrst-order forecasting

models [6, 17]. Singh [31] introduced a new forecasting model for objective of decreasing

amount of computations of fuzzy relational matrices or ﬁnding out a suitable defuzziﬁcation

process for prediction enrollments of University of Alabama and crop production.

Recently, many authors have hybridized the intelligent computation with various FTS

models to deal with complicated problems in forecasting. For example, Lee et al. [25] re-

viewed the high order FTS model for forecasting the temperature and the TAIFEX based

on genetic algorithm. Furthermore, they also applied simulated annealing technique [24] in

determining the length of each interval to achieve better forecasting accuracy. By introdu-

cing genetic algorithm(GA) for partitioning intervals in the universe of discourse, Chen &

Chung introduced two ﬁrst-order [4] and high - order forecasting models for forecasting the

enrollments of University of Alabama. Moreover, to receive optimal intervals and avoid the

harmful results of the mutation operation in GA. Eren Bas et al. [1] proposed a new GA

called MGA for forecasting “killed in car accidents” in Belgium and the enrollments in the

University of Alabama. At present, the application of PSO in selecting the proper intervals

in FTS forecasting model has attracted many attentions of researcher. They demonstrate

that suitable selection of intervals by using PSO also increases the performance of forecasting

model, as can be seen in the works [5, 11, 16, 22, 23, 28, 39, 40]. Speciﬁcally, Kuo et al.

proposed a novel forecasting model by hybridizing PSO with FTS model to improve fore-

casting accuracy. Kuo et al. [23] also based on PSO to suggest a new model for forecasting

TAIFEX by proposing new defuzziﬁcation rule. Hsu et al. [15] provided a new two-factor

high-order model for forecasting temperature and TAIFEX. With the same goal of using PSO

in selection of appropriate intervals, Park et al. [28] considered a two-factor high-order FTS

model combined with PSO to achieve more appropriate forecasting results. Huang et al. [16]

presented the hybrid forecasting model which combined PSO and the reﬁnement in the fo-

recasting output rule for forecasting enrollments . In addition, Dieu N.C & Tinh N.V [11]

introduced the time-variant fuzzy relationship groups concept (TV-FRGs) and combined it

with PSO in ﬁnding optimal intervals to get better forecasting results. Except for this study,

the forecasting model [36] is also based on PSO and TV-FRGs, but extended in the two

cases of ﬁrst- order and high- order FRGs to forecast stock market indices of TAIFEX and

enrollments. Chen and Bui [8] use the PSO technique not only to bring optimal intervals

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

269

but also to obtain optimal weight vectors. They proposed the forecasting model which used

optimal partition of intervals and optimal weight vectors to predict the TAIFEX and the

NTD/USD exchange rates. Cheng et al. [10] produced a FTS model to predict the TAIFEX

based on use the PSO for obtaining the appropriate lengths of intervals and the K-means

algorithm for partitioning the subscripts of the fuzzy sets into cluster center of each clus-

ter. One another of the methods for determining the optimal intervals can be mentioned

as clustering techniques which have been advanced for minimizing error in forecasting. The

methods such as Rough Fuzzy C- means [3], automatic clustering [9], fuzzy C-means [13, 39],

K-means [34, 35] are introduced in recent works. Some other FTS models use neural network

for forecasting oil demand [29] and adaptive neuro-fuzzy inference systems to forecast the

daily temperature of Taipei [30].

As already mentioned in researches above, determining the appropriate length of inter-

vals, establishing fuzzy relationships and making the forecasting rules are considered to be

challenging tasks and critically inﬂuence the accuracy of forecasting model. In spite of sig-

niﬁcant achievements in using the length of each interval as well as discovering forecasting

output rules, these problems still raise attention of researchers. Up to now, there are still

rather many ways to determine the length of intervals in the universe of discourse and cal-

culate crisp output values from fuzziﬁed values. Therefore, the objective of this study is

to propose a new hybrid forecasting FTS model using high-order TV-FRGs [11], combining

FCM clustering with PSO for selecting optimal length of intervals and reﬁnement of forecas-

ting values by new defuzziﬁcation rules. To verify eﬀectiveness of the proposed model, three

following real-world data sets are used for experimenting: (1) dataset of enrollments at the

University of Alabama [6]; (2) Historical data of the TAIFEX [25] in Taipei, Taiwan; and (3)

car road accident data in Belgium [1]. The experimental study shows that the performance

of proposed model is better than those of any existing models. The remaining content of

this paper is organized as follows.

In Section 2, the basic concepts of FTS and algorithms are brieﬂy introduced. Section 3

presents a hybrid FTS forecasting model which combines with the FCM and PSO algorithm.

Section 4 makes a comparison of forecasting results of the proposed model with the existing

models from three real life data sets. Conclusion and future work are discussed in Section 5.

2. BASIC CONCEPTS OF FTS AND ALGORITHMS

2.1. Basic concepts of FTS

The idea of FTS was ﬁrst introduced and deﬁned by Song and Chissom [33, 34]. Let

U = {u₁, u₂, ..., u_n} be an universe of discourse; a fuzzy set A of U can be deﬁned as

A = {f_A(u₁)/u₁+ f_A(u₂)/u₂+ ... + f_A(u_n)/u_n} ,

where f_Ais a membership function of a given set A : U → [0, 1], f_A(u_i) indicates the grade

of membership of u_iin the fuzzy set A. f_A(u_i) ∈ [0, 1] and 1 ≤ i ≤ n. The basic deﬁnitions

of FTS are as below.

Deﬁnition 1. (Fuzzy time series [32, 33]) Let Y (t), (t = 0, 1, 2, ...) a subset of R, be the

universe of discourse on which fuzzy sets f_i(t), (i = 1, 2, ...) are deﬁned and if F(t) is a

collection of f₁(t), f₂(t), · · · then F(t) is called a FTS deﬁnition on Y (t), (t = 0, 1, 2, ...).

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NGHIEM VAN TINH, NGUYEN CONG DIEU

Deﬁnition 2. (Fuzzy logical relationships(FLRs) [32, 33]) The relationship between F(t)

and F(t − 1) can be presented as F(t − 1) → F(t). If let A_i= F(t) and A_j= F(t − 1), the

relationship between F(t) and F(t − 1) is represented by FLR A_i→ A_j, where A_iand A_j

refer to the left - hand side and the right-hand side of FTS.

Deﬁnition 3. (m - order fuzzy logical relationships [33]) Let F(t) be a FTS. If F(t) is

caused by F(t − 1), F(t − 2), · · · , F(t − m + 1), F(t − m) then this fuzzy logical relationship

is represented by F(t − m), · · · , F(t − 2), F(t − 1) → F(t) and is called an m - order FTS.

Deﬁnition 4. (Fuzzy relationship groups (FRGs) [6]) The fuzzy logical relationships having

the same left- hand side can be further grouped into a FRG. Assume there are exists FLRs

as follows: A_i→ A_k1, A_i→ A_k2, · · · , A_i→ A_km; these FLRs can be put into the same FRG

as A_i→ A_k1, A_k2, · · · , A_km

.

Deﬁnition 5. (Time-variant fuzzy relationship groups(TV-FRGs) [11]) The fuzzy logical

relationship is determined by the relationship F(t − 1) → F(t). Let F(t) = A_i(t) and

F(t − 1) = A_j(t − 1), the FLR between F(t − 1) and F(t) can be denoted as A_j(t − 1) →

A_i(t). Also at time t, we have the following fuzzy logical relationships A_j(t − 1) → A_i(t);

A_j(t1 − 1) → A_i1(t1);...; A_j(tp − 1) → A_ip(tp) with t1, t2, .., tp ≤ t. It is noted that A_i(t1)

and A_i(t2) are the same fuzzy A_ibut appear at diﬀerent times t1 and t2, respectively. It

means that if these FLRs occur before A_j(t−1) → A_i(t), we can group the FLRs having the

same left - hand side into a FRG as A_j(t − 1) → A_i1(t1), A_i2(t2), A_in(tn), A_i(t). It is called

ﬁrst- order TV-FRGs.

2.2. Algorithms

2.2.1. Fuzzy C- means clustering

Fuzzy C-Means is a method of clustering proposed by Bezdek [2]. The basic idea of

the fuzzy C-means clustering is described as follows. From a raw data set of input vectors

X = {x₁, x₂, ..., x_n}, the FCM employs fuzzy partitioning such that a data object can belong

to two or more clusters with diﬀerent membership grades between 0 and 1. It is based on

the minimization of the following objective function

C

n

X X

J(U, V ) =

u^m_ijd²_ij(x_j, v_i),

(1)

i=1 j=1

where, m is fuzziness parameter which is a weighting exponent on each fuzzy membership,

C is the number of clusters (2 ≤ C ≤ n), n is the number of objects in the data set X, v_i

is the prototype of the center of cluster i, u_ijis the grade of membership of x_jbelonging

to cluster i and d²_ij(x_j, v_i) or d_ijis the distance between object x_jand cluster center v_i, U

is the membership function matrix, V is the cluster center vector. The FCM focused on

minimizing J(U, V ), subject to the constrains on U by Eq. (2) as follows

n

X

u_ij∈ [0, 1];

u_ij= 1;

u_ij≤ n.

(2)

j=1

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

271

Algorithmic steps for Fuzzy C-Means clustering is presented as follows

Step 1. Fix the number of clusters C, initialize the cluster center matrix V (0) by using a

random generator from the original dataset. Record the cluster centers set t = 0, m = 2,

and decided by ꢀ, where ꢀ is a small positive constant (e.g., ꢀ = 0.0001).

Step 2. Initialize the membership matrix U(0) by using Eq. (3)

1

u_ij(t) =

,

(3)

2

ꢀ

ꢁ

C

P

d_ij(t)

m−1

i=1

2

where d_ij= kx_j− v_ik is the distance between object x_jand cluster center v_i. If d_ij(t) = 0

then u_ij= 1 and u_rj= 0 (r = j).

Step 3. Increase t = t + 1. Compute the new cluster center matrix V_ijusing Eq. (4)

P

n

_j=1u^m_ij(t) × x_j

P

v_i(t + 1) =

.

(4)

n

_j=1u^m_ij(t)

Step 4. Compute the new membership matrix U_ijby using Eq. (3).

Step 5. If max {|u_ij(t + 1) − u_ij(t)|} ≤ ꢀ then stop, otherwise go to Step 3 and continue to

iterative optimization.

2.2.2.

Particle swarm optimization

PSO algorithm is an intelligent optimization algorithm, which was ﬁrstly proposed by

Eberhart and Kannedy [21] for ﬁnding the global optimal solution. In PSO, a set of particles

which is called a swarm; each particle indicates a potential solution and always moves through

the search space (d-dimensional space) for searching the optimal solution. In the movement

process of particles (i.e, N particles), all particles have ﬁtness values to evaluate their perfor-

mance. Each particle id (i = 1, · · · , N) has a position vector X_id= [x_i,1, x_i,2, · · · , x_i,d] and a

velocity vector V_id= [v_i,1, v_i,2, · · · , v_i,d] to indicate its current state in the search space. The

position of the best particle of total number of particles found so far is saved and each parti-

cle retains its personal best position which has passed previously. The position X_idand the

velocity V_idare updated by the best position P_{best id}= [p_id,1, p_id,2, · · · , p_id,n] encountered by

the particle so far and the best position G_best= min(P_b^t_{est id}) found by the whole population

of particles according to formulas of velocity and position as follows

t+1

id

V

= ω^t× V_i^t_d+ C₁× Rand() × (P_{best id}− X_i^t_d) + C₂× Rand() × (G_best− X_i^t_d),

(5)

(6)

t+1

id

X

= X_i^t_d+ V_i^t_d⁺¹

,

t × (ω_max− ω_min

)

ω^t= ω_max

−

.

(7)

iter max

In this paper, we combine the standard PSO [21] with Constrained Particle Swarm Optimi-

zation CPSO [12] by using the following Eq. (8) to replace Eq. (5) as follows

t+1

id

V

= K × [ω^t× V_i^t_d+ C₁× Rand() × (P_{best id}− X_i^t_d) + C₂× Rand() × (G_best− X_i^t_d)], (8)

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NGHIEM VAN TINH, NGUYEN CONG DIEU

2

p

K =

.

(9)

|2 − ϕ − (ϕ²− 4 × ϕ)|

The new position of the particle id is changed by adding a velocity to the current position

as follows

t+1

id

X

= X_i^t_d+ V_i^t_d⁺¹

,

(10)

where X_i^t_dis the current position of the particle id at time step t; V_i^t_dis the velocity of the

particle id at time step t, and is limited to [-V_max, V_max], where V_maxis a constant predeﬁned

by user; ω is the time-varying inertia weight, which is the same as the ones presented in [22];

iter max is the total number of iterations; c₁and c₂are two learning factors which control

the inﬂuence of the cognitive and social components, respectively, c₁= c₂= 2.05 which are

the same as the ones presented in [12], such that φ = c₁+ c₂= 4.1 and the constriction

factor K= 0.7298.

Algorithm 1 brieﬂy summarizes steps of the PSO algorithm for minimizing a ﬁtness

function (f) value.

Algorithm 1. A brieﬂy description of the PSO

- Input: Population of N particles, the maximum number of iterations(iter max)

- Output: G best value

1. Initialize: Set K = 0.7298, ω_min, ω_max, V_max

for each particle id, (1 ≤ i ≤ N ) do

- Random positions x_id, Random velocities v_idin d dimensional space

- Set P_bⁱ_{est id}= x_id;

if f(P_bⁱ_{est id}) ≤ f(G_best) then G_best= P_bⁱ_{est id}; end if

end for

2. while (t ≤ iter max) do

2.1. for each particle id, (1 ≤ i ≤ N ) do

• calculate the ﬁtness value of particle id: f(x_id)

- if f(x^t_id⁺¹) < f(P_b^t_{est id}) then P_b^t_e⁺_s¹_{t id}= f(x_i^t_d⁺¹

)

- if f(x_i^t_d⁺¹) > f(P_b^t_{est id}) then P_b^t_e⁺_s¹_{t id}= f(P_b^t_{est id}

end for

)

2.2. Update the f(G_best) position of all particles according to the ﬁtness value.

2.3. for each particle id, (1 ≤ i ≤ N) do

• update the velocity vector using Eq. (8)

• update the position vector using Eq. (10)

end for

• Update ω^taccording to Eq. (7)

end while

return G_bestvalue and corresponding position

3. A PROPOSED FTS FORECASTING MODEL BASED ON FCM

AND PSO

In this section, a novel FTS forecasting model is suggested by incorporating FCM with

PSO to increase forecasting accuracy. The outline of proposed model is presented in Figure 1,

which consists of three stages; the ﬁrst stage is to partition the historical data into intervals

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

273

based on FCM algorithm in Subsection 3.1, the second stage is to build the FTS forecasting

model which is presented details in Subsection 3.2 and uses PSO algorithm for ﬁnding optimal

lengths of intervals in the third stage which is introduced Subsection 3.3. To handle these

stages, all historical enrollments data [6] are utilized for illustrating forecasted process. The

three stages of proposed model are described as follows.

3.1. Using FCM algorithm for generating intervals from a raw time series data

In this section, FCM clustering algorithm is applied to classify the collected data into

clusters and adjusted these clusters into contiguous intervals. All historical enrollments

data [6] from 1971s to 1992s are utilized to present in the stage of generating intervals. The

algorithm composed of two main steps is introduced as follows:

Step 1. Apply the FCM clustering algorithm to partition the historical data into C clusters.

For simplicity we partition enrollments dataset into 7 clusters as shown in the second

column 2 of Table 1. Similarly, we can change the number of clusters C from 5 to 21.

Step 2. Adjust the clusters into intervals.

In this step, we adjust the clusters into intervals based on cluster centers as follows:

Suppose that V_iand V_i+1are adjacent cluster centers and each cluster Cluster_iis assigned

as an interval interval_i, then the upper bound Interval UB_iof interval_iand the lower bound

Interval LB_i+1of interval_i+1can be calculated according Eqs. (11) and (12) as below

V_i+ V_i+1

Inteval UB_i=

,

(11)

(12)

2

Interval LB_i+1= Interval UB_i,

where i = 1, · · · , C − 1. Because of lacking intervals before the ﬁrst interval and lacking

intervals after the last interval, the lower bound Interval LB₁of the ﬁrst interval and the

upper bound Interval UB_Cof the last interval can be computed according to Eqs. (13)

and (14) as below.

Table 1. The completed result of clusters from the enrollments dataset

STT

Data in cluster

{13055, 13563}

{13867}

Cluster center (V_i)

13309

13867

1

2

3

4

5

6

7

{14696}

14696

{15145, 15163, 15311, 15433, 15460, 15497, 15603 } 15373.14

{15861, 16807, 16388, 15984 }

{16919, 16859 }

16260

16889

18932.2

{18150, 18970, 19328, 19337, 18876 }

Interval LB₁= V₁− (Interval UB₁− V₁),

Interval UB_C= V_C+ (V_C− Interval LB_C).

(13)

(14)

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NGHIEM VAN TINH, NGUYEN CONG DIEU

Figure 1. Flowchart of the proposed FTS forecasting model

Compute midpoint value Mid value_iof the interval Interval_ias follows

Interval LB_i+ Interval UB_i

Mid value_i=

,

(15)

2

where Interval LB_iand Interval UB_iare the lower bound and the upper bound of the

interval Interval_i, respectively. Based on the rules in Step 2, we obtain 7 intervals corre-

sponding to the clusters in Step 1, named u_i(1 ≤ i ≤ 7) and compute midpoint values of

the intervals as listed in Table 2.

Table 2. The completed results of intervals

No

1

Interval

Mid value

13309

u₁= [13030, 13588)

2

3

4

5

6

7

u₂= [13588, 14281.5)

u₃= [14281.5, 15034.57)

u₄= [15034.57, 15816.57)

u₅= [15816.57, 16574.5)

u₆= [16574.5, 17910.6)

u₇= [17910.6, 19953.8)

13934.75

14658.04

15425.57

16195.54

17242.55

18932.2

3.2. Establish FTS forecasted model based on the ﬁrst order and high order

TV-FRGs

The details of next steps of the forecasting model are established as follows:

Step 3. Determine linguistic terms for each of interval obtained in Step 2.

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

275

After creating the intervals in Step 2, linguistic terms are deﬁned for each interval which

the historical data is distributed among these intervals. For seven intervals, we get seven

linguistic values which are the same as the ones in [6] i.e., {A₁, A₂, A₃, A₄, A₅, A₆, A₇} which

can be represented by fuzzy sets A_i, as below

a_i1a_i2a_i3

a_i7

u₇

A_i=

+

+ ... +

,

(16)

u₁

u₂

u₃

where a_ij∈ [0, 1] is the membership grade of u_jbelonging to A_i, which is deﬁned by Eq. (17),

the symbol ‘+’ denotes the set union operator and the symbol ‘/’ denotes the membership

of u_jwhich belongs to A_i.



1

if i == j



a_ij=

(17)

0.5 if j == i − 1 or j = i + 1





0

otherwise.

From Eq. (16), each fuzzy set contains 7 intervals, and each interval belongs to all fuzzy

sets with diﬀerent grade of membership values presented in Eq. (17)). For instance, u₁

corresponds to linguistic variables A₁and A₂with degree of membership values 1 and 0.5 re-

spectively, and remaining fuzzy sets with membership grade 0. The descriptions of remaining

intervals, i.e., u₂, u₃, · · · , u₇can be explained in a similar way.

Step 4. Fuzzify all historical data.

Each of interval contains one or more historical data value of time series. To fuzzy all

historical data, the common way is to map historical data into a fuzzy set which has the

highest membership value in the interval containing this historical data. For instance, the

historical data of year 1973 is 13867, and it belongs to interval u₂= [13588, 14281.5). So, we

allocate the linguistic value A₂corresponding to interval u₂to it. According to Eq.(16), the

fuzzy set A₂with the highest membership value occurs at interval u₂. Hence, the fuzziﬁed

value for year 1973 is considered as A₂. With a similar explanation for remaining years, we

can obtain the results of fuzziﬁcation of enrollments data for all years which are shown in

Table 3.

Table 3. The results of fuzziﬁcation for enrollments data under seven intervals

Year Actual data Fuzzy sets Maximum membership value

Linguistic value

not many

1971

1972

1973

—–

13055

13563

13867

—–

A₁

A₂

—

[1 0.5 0 0 0 0 0]

[0.5 1 0.5 0 0 0 0]

——————

not many

not too many

—————-

too many many

1991

1992

19337

18876

A₇

[0 0 0 0 0 0.5 1]

Step 5. Create all m^th- order fuzzy logical relationships (m ≥ 2 ).

The m^th- order FLR is constructed based on two or many consecutive fuzzy sets in time

series. After transforming historical data into fuzzy sets, then m^th- order FLRs can be

created based on Deﬁnition 3. That means, we need to ﬁnd any relationship which has the

type F(t−m), F(t−m+1), ..., F(t−1) → F(t), where F(t−m), F(t−m+1), · · · , F(t−1)

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NGHIEM VAN TINH, NGUYEN CONG DIEU

and F(t) are called the left-hand side and the right-hand side of FLR, respectively. Then,

the m^th- order FLR is obtained by substituting the corresponding fuzzy sets as follows:

A_im, A_i(m−1), · · · , A_i2, A_i1→ A_k. For instance, suppose m = 1, we need to point out

all ﬁrst-order FLRs having the form F(t − 1) → F(t). Based on Table 3, a fuzzy logical

relationship A₁→ A₂is created by substituting the historical data of F(1972) and F(1973)

with fuzzy set as A₁and A₂, respectively. From this viewpoint, all ﬁrst-order FLRs of

historical time series are shown in column 2 of Table 4. Similarly, we can generate high-

order fuzzy logical relationships. Suppose that there is a 2^nd- order FLR which is expressed as

F(1972), F(1973) → F(1974). Based on Table 3, F(1972) = A₁, F(1973) = A₂and F(1974)

= A₃are obtained, then a 2^ndFLR A₁, A₂→ A₃is created by substituting the historical

data of F(1972), F(1973) and F(1974) to A₁, A₂and A₃, respectively. By a similar manner,

we can establish the 2^ndFLRs from the fuzziﬁed data values, which are shown in column 4 of

Table 4, where, the symbol # within the last relationship is used to represent the unknown

linguistic value.

Table 4. The complete ﬁrst - order and second - order fuzzy logical relationships

Year 1st-order FLR

1st-order F(t)

———-

2nd-order FLR

——–

2nd-order F(t)

———–

1971

1972

1973

—-

——

A₁→ A₁

———

F(1971) → F(1972)

——-

————

F(1972) → F(1973) A₁, A₁→ A₂F(1971), F(1972) → F(1973)

—————–

F(1991) → F(1992) A₇, A₇→ A₇F(1990), F(1991) → F(1992)

F(1992) → F(1993) A₇, A₇→ # F(1991), F(1992) → F(1993)

————–

————————-

1992

1993

A₇→ A₇

A₇→ #

Step 6. Generate all m^th-order time-variant FRGs.

Each fuzzy relationship group may include one or more fuzzy logic relationships with the

same left - hand side. In previous studies, the repeated FLR were simply ignored and it can

be only counted one time [7, 6, 22] or the recurrent FLRs are taken into account but were

not interested in chronological order [38] when fuzzy relationship groups were established. In

this study, we rely on a concept of TV-FRGs [11] and it is mentioned in Deﬁnition 5 to create

FRGs. In this approach, the TV-FRGs are determined by seeing the history of appearance

of the fuzzy sets on the right-hand side of the FLRs. This means, only the fuzzy sets on

the right - hand side appearing before the fuzzy sets on the left-hand side of the FLRs

at forecasting time is grouped into a FRG. To explain this, two examples are described

as below. Firstly, considering the three ﬁrst -order FLRs at three diﬀerent time functions,

F(t = 1972, 1973, 1974) in Table 4 as follows F(t = 1972) : A₁→ A₁; F(t = 1973) : A₁→ A₂;

F(t = 1974) : A₂→ A₃; where, there are two FLRs at time F(1972) and F(1973) with the

same fuzzy set A₁on the left hand side. If considering at forecasting time t = 1992, we

obtain a ﬁrst-order FRG (i.e., G1) as follows A₁→ A₁. If considering at forecasting time

t = 1993, before that there are two FLRs with the same on left - hand side, these FLRs

can be grouped into a FRG as G2 : A₁→ A₁, A₂. If we consider the forecasting time t =

1994, then the group G3 is expressed as follows A₂→ A₃. The column 3 of Table 5 shows

the ﬁrst-order FRGs, where there are 21 groups in training phase and one group in testing

phase. Similarly, the second-order FRGs can be established and listed in column 5 of Table 5

including 20 groups in training phase and one group in testing phase.

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

277

Table 5. The complete ﬁrst - order and second - order TV- FRGs

Year 1st-order FLR

1st-order F(t)

——-

2nd-order FLR

2nd-order F(t)

——-

1971

1972

1973

1974

—-

G1

——

G1

A₁→ A₁

——-

G2

A₁→ A₁, A₂

A₂→ A₃

A₁, A₁→ A₂

A₁, A₂→ A₃

——–

G3

G2

—

——

—–

1992

1993

G21

G22

A₇→ A₇, A₇, A₇, A₇

A₇→ #

G20

G21

A₇, A₇→ A₇, A₇, A₇

A₇, A₇→ #

Step 7. Defuzzify and calculate the forecasting output value for all TV-FRGs.

To defuzzify the fuzziﬁed time series values and obtain the crisp output values. First,

the new defuzziﬁcation rules is developed here to compute the forecasted value for all ﬁrst

- order and high - order time variant FRGs in training phase. Second, we use the master

voting (MV) scheme [22] to calculate forecasted value for fuzzy relationship groups with the

untrained pattern in testing phase. The forecasting principles is presented as follows:

Principle 1: Using for the ﬁrst - order TV-FRGs.

For calculating forecasted value based on information of each group, we investigate all in-

formation which appear on the right-hand side of each FRG, which is called Global_inf, then

combine with the local information of the same FRG which is presented as follows.

Forecasted value = 0.5 × (Global inf + Local inf),

(18)

where: - Global inf is the global information which can be determined based on all the fuzzy

sets on the right-hand side of FRG.

- Local inf is the local information which is determined by the fuzzy set appearing at

forecasting time on the right-hand side and the latest past in the left - hand side of FRG.

Suppose that there is a ﬁrst - order FRG at forecasting time t is presented as: A_t−1

A_t1, A_t2, · · · , A_tn. Based on research [11], the value of Global inf is calculated as follows

→

1 × m_t1+ 2 × m_t2+ · · · + n × m_tn

Global inf =

,

(19)

1 + 2 + · · · + n

where m_t1, m_t2, · · · , m_tnare the midpoint values of intervals u₁, u₂, · · · , u_nwith respect to

n fuzzy sets existing on the right-hand side of FRG, respectively. By accounting into the

variation of latest time on the left-hand side as a forecasting factor, the Local inf value is

expressed as follows

Ub_ti− Lb_tim_ti− m_t−1

Local inf = Lb_ti+

×

,

(20)

2

m_ti+ m_t−1

where A_t−1is the lastest fuzzy set on left-hand side of the ﬁrstorder FRG; A_ti(1 ≤ i ≤ n)

is the i^thfuzzy set in right - hand side of the ﬁrst - order FRG. Here, m_t−1and m_tiare

middle values of intervals u_t−1and u_tiwith respect to A_t−1and A_ti. Lb_ti, Ub_tidenote the

lower bound and upper bound value of interval u_ti= [Lb_ti, Ub_ti), t is forecasting time with

respect to i^thfuzzy set on right - hand side of the ﬁrst - order FRG.

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NGHIEM VAN TINH, NGUYEN CONG DIEU

For example, suppose that we want to forecast the enrollment of year 1973. Based on

column 3 of Table 5, the ﬁrst - order FRG (G2: A₁→ A₁, A₂) is formed from two FLRs

having next state respectively as A₁→ A₁, A₁→ A₂. The highest membership grade of the

fuzzy sets A₁and A₂appear at intervals u₁and u₂, respectively, where u₁= [Lb_t1, Ub_t1) and

u₂= [Lb_t2, Ub_t2). From Table 2, u₁=[13030, 13588) and u₂=[13588, 14281.5). The midpoints

of the intervals u₁and u₂are m_t1= 13309 and m_t2= 13934.75. From Eq. (19), the value of

m_t1+ 2 × m_t2

Global inf =

= 13726.2. Based on Eq. (20), by setting u_t−1= u₁, u_t= u₂,

3

then Lb_t2= 13588, Ub_t2= 14281.5 and the value of the Local inf on the enrollment of year

t = 1973 can be calculated as follows

14281.5 − 13588 13934.75 − 13309

= 13595.97.

Local inf = 13588 +

×

2

13934.75 + 1330

From values of Global inf and Local inf obtained above, based on Eq. (18), the forecasting

output value of year 1973 is calculated as Forecasted value = 0.5 × (13726.2 + 13595.97) =

13661.09.

Principle 2: Using the m^thorder TV-FRGs (m ≥ 2).

For getting the forecasted results of proposed model based on the high order TV-FRGs,

we compute all forecasted values for these groups based on fuzzy sets on the right-hand

side within the same group. The viewpoint of this rule is described as follows: For each

high - order FRG, we partition each corresponding interval of each linguistic value on the

right-hand side into four sub-intervals which have the same length, and compute forecasted

output for each group according to Eq. (21).

n

X

1

Forecasted value =

(Subm_ik+ V al Lu_ik),

(21)

2 × n

i=1

where n is the sum of fuzzy sets on the right-hand side of FRG; Subm_ikis the midpoint value

of one of four sub-intervals (1 ≤ k ≤ 4) with respect to i^thfuzzy set on the right-hand side of

fuzzy relation group, in which the actual data at forecasting time belong to this sub-interval;

V al Lu_ikis one of two values belonging to the lower bound and upper bound value of one

of four sub-intervals which has the actual data at forecasting time falling within sub-interval

u_ik(i.e., u_ik= [L_ik, U_ik].

• If the actual data at forecasting time is smaller than middle value of sub-interval u_ik

V al Lu_ikis assigned by the lower bound of sub-interval u_ik.

• If the actual data at forecasting time is larger than middle value of sub-interval u_ik

V al Lu_ikis assigned by the upper bound of sub-interval u_ik.

For instance, assume that we want to forecast the enrollment of year 1973. From column

5 of Table 5, it is seen that the second - order FRG (G1:A₁, A₁→ A₂) is formed from

a FLR with next state A₂which occurs at year 1973, where the maximum membership

grade of A₂belongs to interval u_2.2= [13588, 14281.5). Hence, we partition the interval

u₂into four sub-intervals which are u_2.1=[13588, 13761.38), u_2.2= [13761.38, 13934.75),

u_2.3= [13934.75, 14108.13) and u_2.4=[14108.13,14281.5), respectively. The group G1 as

A₁, A₁→ A₂achieve from relation F(1971), F(1972) → F(1973), where the historical data of

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

279

year 1973 is 13867 and it is within sub-interval u_2.2=[13761.38,13934.75) and then the middle

value subm_2.2of sub-interval u_2.2is 13848.06. Then, we ﬁnd out the value of V al Lu_ikby

comparing the historical data of year 1973 with the middle value of sub-interval u_2.2. From

this viewpoint, we obtain the value of V al Lu_ik(V al Lu_2.2) is 13934.75 (the historical data

of year 1973 of 13867 is larger than middle value of sub-interval u_2.2). Finally, forecasted

value of year 1973 can be calculated according to Eq. (21) as follows

1

Forecasted value = (13848.06 + 13934.75) = 13891.4.

2

Principle 3: Calculate forecasting value in the testing phase.

For testing phase, we calculate forecasted value for a group of fuzzy relationship which has

the unidentiﬁed linguistic value on the right-hand side based on the master vote scheme [22],

and the forecasting value is estimated based on Eq. (22), where the symbol w_his the highest

votes predeﬁned by user for each other problem, m is the order of the FLRs, the symbols

M_t1, M_t2, · · · , M_ti, · · · are the middle values of the corresponding intervals which are related

to the latest fuzzy set and other fuzzy sets on the left-hand side of fuzzy logical relationship

group, respectively with the maximum membership values of A_t1, A_t2, · · · , A_ti, · · · and u_tm

occur at intervals u_t1, u_t2, · · · , u_ti, · · · and u_tm, respectively

m_t1× w_h+ m_t2+ · · · + m_ti+ · · · + m_tm

Forecasted value =

.

(22)

w_h+ (m − 1)

For instance, assume that we want to forecast the enrollment of year 1993 by using ﬁrst-

order fuzzy relationship. As shown in column 3 of Table 5, the group G22 has a ﬁrst order

fuzzy logical relationship as A₇→ # which is created by the fuzzy relationship F(1992) →

F(1993); since the linguistic value of F(1993) is unknown within the historical data, and this

unknown right-hand side state is symbolized by the sign #. Then, the forecasted enrollment

of year 1993 is calculated by Eq. (22). Similarly, we can forecast the enrollment of year 1993

by using high-order fuzzy logical relationships. Based on the three forecasted rules above and

from Table 3 and Table 5, we complete forecasted results for the enrollments in the period

from 1971 to 1992 based on ﬁrst-order and high order TV-FRGs under seven intervals as

shown in Table 6.

Table 6. The complete forecasted output values based on the ﬁrst order and high - order FTS

Year

1971

1972

1973

—-

Actual data Fuzzy sets 1st -order forecasted value 2nd-order forecasted value

13055

13563

13867

——

18876

N/A

A₁

Not forecasted

13169.5

Not forecasted

13891.4

A₂

13661.09

——-

18421.6

—-

A₇

——–

19147.62

1992

1993

MSE

N/A

18932.2

140045.4

49873.7

To verify the forecasting accuracy of proposed model, two evaluation indices are used, the

mean square error (MSE) and the root mean square error (MAPE). The formulas of both

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NGHIEM VAN TINH, NGUYEN CONG DIEU

indices are listed as follows:

n

X

1

MSE =

(F_i− R_i)²,

(23)

(24)

n _i=m

v

u

t

n

X

1

RMSE =

(F_i− R_i)²,

n _i=m

where R_i, F_idenotes actual data and forecasting value at year i, respectively; n is number

of the forecasted data; m is order of the fuzzy logical relationships.

3.3. A hybrid FTS forecasting model based on combining the FCM and PSO

algorithm

The goal of this subsection is that we present the hybrid FTS forecasting model by

combining FCM algorithm for partition data set into the unequal lengths of intervals with

Algorithm 1 in Subsection 2.2.2. The main purpose of this algorithm is to adjust the initial

intervals length with an intent to obtain the optimal intervals that do not increase the

number of intervals in the model. The detailed descriptions of the hybrid forecasting model

are given as follows. In proposed model, each particle represents the partitioning of historical

time series data into intervals. The number of intervals are determined by FCM (e.g., n

intervals). Let the lower bound and upper bound of the universe of discourse U be x₀and x_n,

respectively. Each particle denotes a vector consisting of n − 1 elements are x₁, x₂, ..., x_n−2

and x_n−1, where (1 ≤ i ≤ n − 1) and x_i≤ x_i+1. From these n − 1 elements, deﬁne

the n intervals as u₁= [x₀, x₁], u₂= [x₁, x₂], · · · , u_i= [x_i−1, x_i], · · · and u_n= [x_n−1, x_n],

respectively. When a particle moves from one position to another position, the elements of

the corresponding new array need to be sorted to ensure that each element x_iarranges in

an ascending order such that x₁≤ x₂≤ · · · ≤ x_n−1. In the processing of the training phase,

the hybrid forecasting model permits each particle to move from current position to other

position by Eqs. (8) and (10), and repeat the steps until the stopping criterion is satisﬁed.

If the stopping criterion is satisﬁed, then all the FRGs obtained by the global best position

(Gbest) among all personal best positions (Pbest) of all particles which used to forecast

the new testing data in testing phase. Here, the function MSE (23) is used to evaluate

the forecasting accuracy of each particle. The complete steps of the proposed model are

presented in Algorithm 2.

Algorithm 2: The FCM-FTS-PSO algorithm

1. Input: Historical time series data

2. Output: The forecasting results and the MSE value (MSE = Gbest = min(Pbest))

Begin

3. Select the initial set of intervals by applying FCM algorithm and use forecasting steps in

Subsection 3.2 to get the initial forecasting accuracy (MSE).

4. Initialize: a population of N particles

• The initial position X_idof all particles be limited by: x₀+Rand( )×(x_n− x₀); where,

x₀and x_nare the lower bound and upper bound of the universe of discourse U which

is created by FCM; the intervals created by particle 1 are identical to the one created

by FCM in Subsection 3.1.

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

281

The velocity V_idof all particles be exceeded by

v_min+ Rand() × (v_max− v_min); v_min= −v_max

• The initial personal best positions are set as the initial positions of all particles and

ﬁnd Gbest

5. Repeat

5.1. for particle id, (1 ≤ i ≤ N) do

• Deﬁne linguistic terms according to all intervals deﬁned by the current position of

particle id based on Step 3 in Subsection 3.2

• Fuzzify all historical data according to the linguistic terms deﬁned above by Step 4 in

Subsection 3.2

• Create all m- order fuzzy logical relationships by Step 5 in Subsection 3.2

• Build all m- order time -variant fuzzy relationship groups by Step 6 in Subsection 3.2

• Forecast and defuzzify output values by Step 7 in Subsection 3.2

• Calculate the MSE values for particle id based on Eqs. (23) and (24)

• The new Pbest of particle id is saved according to the MSE values.

end for

5.2. The new Gbest of all particles is saved according to the MSE values

6. for particle id, (1 ≤ i ≤ N) do

• The particle id is moved to another position according to Eqs. (8) and (10) end for

• Change ω according to Eq. (7)

until (the stop condition (the maximal moving steps or minimum MSE criteria are

reached) is true);

End.

4. EXPERIMENTAL RESULTS

4.1. Setup parameters for forecasting problems

In this study, the performance of the proposed model is evaluated based on three diﬀe-

rent data sets consisting of enrollments data of University of Alabama [6], Taiwan futures

exchange dataset (TAIFEX) [25] and vehicle road accidents dataset [1]. These datasets are

utilized to illustrate the proposed model’s application in one-step-ahead prediction and the

forecasting results got from the proposed model are compared to other forecasting models.

For implementing the forecasting model on these datasets, we have coded the proposed mo-

del by the C sharp programming language on an Intel Core i7 PC with 8GB RAM. In the

proposed model we use parameters of PSO, but there are no common principle to determine

these parameter values. For ease of comparison with other forecast models using PSO. In

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NGHIEM VAN TINH, NGUYEN CONG DIEU

the proposed model, we choose the maximum number of iterations (the stop condition of the

optimal algorithm) is 150. Like the previous articles [16, 22, 23, 28] the maximum number of

iterations have been generally deﬁned intuitively due to the data in most of the applications

and is usually set within range from 100 to 500 to achieve the best solution. This has been

demonstrated through experimental results in articles such as: the model [22] set number of

iterations to 100, the model [23] has number of iterations of 100, and the models in [28] use

number of iterations is 500. Therefore, the parameters of PSO used in this research were

intuitively determined like in other studies available in the literature. The parameter values

of proposed model are determined for each dataset which are listed in Table 7. With the

parameters describled in Table 7 the proposed model runs 30 times for each experiment, and

takes the best value as the forecasting output value.

(1) The enrollments data of University of Alabama

The enrollments dataset contains 22 observations during the period from 1971 to 1992,

see Figure 2(a). This data set has been selected to simulate with the great amount of study

works published in the literatures [1, 3, 4, 6, 7, 9, 8, 11, 16, 18, 22, 26, 27, 32, 35]. The

results of them will be utilized for comparing with that of the proposed model in this paper.

(2) The TAIFEX time series dataset

The dataset including daily values of the Taiwan futures exchange between August 3, 1998

and September 30, 1998, which has 47 observations is shown in Figure 2(b). This dataset

is handled in the literatures [23, 24, 28, 25, 36]. In this study, the historical observations of

the TAIFEX between 8/3/1998 and 9/23/1998 are used as the training data set. The last

ﬁve observations between 9/24/1998 and 9/30/1998 are used as the testing dataset.

(3) The vehicle road accidents dataset in Belgium

The dataset of “killed in car road accidents” consists of 31 observations from 1974 to 2004

that were taken from National Institute of Statistics, Belgium. The plot of yearly deaths

in car road accidents is shown in Figure 2(c). This dataset is published in the previuos

works [1, 19, 20, 39] .These results are also referred to campare with that of the proposed

model in this study.

Table 7. Parameters of the proposed model are setup for forecasting enrollments, TAIFEX and car

road accidents

Description for the parameters

Values

of enroll-

ments

Values of TAIFEX Values of car

road accident

Number of particles

30

150

30

150

30

150

The max iteration number is set

The inertial weigh limit from

The acceleration coeﬃcient C₁= C₂

The velocity in search range

The position in search range

1.4 to 0.4

2

1.4 to 0.4

2

1.4 to 0.4

2

[-100,100

By FCM

[-50,50]

By FCM

[-50, 50]

By FCM

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

283

7600

7400

7200

7000

6800

6600

6400

6200

1700

1600

20000

19000

18000

17000

16000

15000

14000

13000

Testing data set:

24/9 -30/9/2018

1400

1200

Training data set:

3/8 - 23/9/1998

1000

900

1971 1974

979

1984 1988 1992

1974 1980 1985 1990 19951999 2004

Years

(a) The enrollment time series dataset

(c) The car road accidents time series dataset

Dates

(b) The TAIFEX time series dataset

Figure 2. The value of change of historical time series

4.2. Forecasting enrollments of University of Alabama

In this subsection, the proposed forecasting model is applied for forecasting enrollments

from yearly observations [6]. To show the performance of the proposed forecasting model

based on the ﬁrst order FTS under diﬀerent number of intervals, four forecasting models

presented in articles [4, 22, 26, 27] are selected for the purpose of comparison. Table 8 shows

a comparison of the MSE and RMSE values for diﬀerent forecasting models. To be easily

visualized, Figure 3 depicts the trend of actual data compared to the trend of forecasted

value between the proposed model and other models. From this ﬁgure, it can be seen that

the curve of proposed model is closest to the actual data among ﬁve compared models. Based

on forecasting results in Table 8, the proposed model gets the smallest MSE value of 4070

and RMSE value of 63.8 among all the compared models with diﬀerent number of intervals.

This can be seen that the proposed model gives the most accurate forecasting results for

enrollments of University of Alabama. Diﬀerences between the proposed model and models

mentioned above accord to the way that the fuzzy relationship group and methods of par-

titioning the universe of discourse are applied to the structuring process of model. Four

forecasting models [4, 22, 26, 27] are constructed based on Chen’s model to forecast diﬀerent

problems and perform various methods of interval partitioning such as, the unequal-sized

intervals partitioning by using GA algorithm, by using PSO algorithm, the diﬀerent intervals

partitioning based on hedge algebras and intervals partitioning based on interval information

granules to improve forecasting accuracy while the proposed model uses an approach that

beneﬁts from the concept of time-variant FRG [11] to establish the forecasting model and

combine FCM clustering with PSO algorithm for ﬁnding optimal interval lengths with an

intent to reach better forecasting accuracy.

Next, in order to test the accuracy in the proposed forecasting model according to various

number of intervals, ﬁve FTS models in papers [4, 11, 16, 22, 36] are referred for comparing

in terms of the MSE value . The MSE value is obtained from the proposed forecasting model,

as listed in Table 9 is far smaller than that of all the existing forecasting models mentioned

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NGHIEM VAN TINH, NGUYEN CONG DIEU

Figure 3. Flowchart of the proposed FTS forecasting model

Table 8. A comparison of the forecasting results between the proposed model and its counterparts

based on the ﬁrst - order FTS using 14 intervals

Year

Actual data Model [4] Model [23] Model[28], Model [27] Proposed model

h=17

1971

13055

13563

13867

14696

—–

——

13714

14880

—–

13555

13994

14711

—–

13678

14602

—–

——

13558.75

13868

1972

1973

13582

14457

—–

1974

14783.75

—–

1990

19328

19337

18876

19300

19149

19014

35324

187.9468

19340

19014

22965

151.5421

19574

19146

65689

256.2987

19297

19059

46699

216.1

19325.5

18960.835

4070

1991

1992

MSE

RMSE

63.8

above based on ﬁrst-order FLRs for all intervals. In Table 9, all forecasting models use

fuzzy relationship group to service for computing the forecasting output values. But three

models [4, 16, 22] are designed based on establishing FRGs from Chen’s model [6]. The

remaining three models such as the model [11], the model [36] and the proposed model all

use TV - FRGs. In addition, the proposed model is diﬀerent from the model [4] in the way

that the optimization approaches are utilized. The former employs the PSO, while the latter

utilizes the GA for obtaining the proper lengths of intervals, respectively. From Table 9, it

is obvious that the optimal performance of the proposed model using PSO is better than

the model [4] using GA. This conclusion is also remarked in previous papers. Comparing

with four models presented in articles [11, 16, 22, 36], the proposed model is able to generate

forecasting values with better accuracy than the three compared models. It can be easily

seen that the combination of the FCM algorithm with the PSO in the proposed model yields

more optimal interval lengths.

In addition, the forecasting results of the proposed model are also compared with each model

A NEW HYBRID FUZZY TIME SERIES FORECASTING MODEL

285

Table 9. A comparison of MSE value between the proposed model and the models [4, 12, 17, 23,37]

based on ﬁrst - order FTS with diﬀerent number of intervals

forecasting models

Number of intervals

8

9

10

11

12

13

14

Model [4]

Model [23]

132963

119962

27435

34457

33983

96244

90527

24860

25855

25841

85486

60722

19698

20533

20322

55742

49257

19040

15625

15472

54248 42497 35324

34709 24687 22965

16995 11589 8224

14630 10004 7475

Model [17]

Model [12]

Model [37]

12588

7078

5396

Proposed model

28681 22076.4 14603 10243.7 8337.6 6096.4 4070

which is introduced in articles [4, 7, 16, 22, 31, 36] based on the various high - order FTS

with diﬀerent number of intervals. A comparison of these models is shown in Table 10, where

four models, namely, the model [22], the model [16], the model [36] and the proposed model

use 9th-order FLR and number of intervals is 14 for forecasting the enrollments.

Table 10 shows that proposed model bears the lowest MSE value of 5.08 and far exceeds

compared to its counterparts. The major diﬀerence among all the high - order FTS models

mentioned above is that the defuzziﬁcation rules is used to forecast output results and

optimization technique is handled to get the proper intervals. The diﬀerent parameters of the

model [31] were used as fuzzy relation in forecasting years for calculating output value. Three

forecasting models [4, 7, 22] apply Chen’s [6] defuzziﬁcation rules for computing forecasting

value. The model [16] gets the forecasting value by combining the global information of fuzzy

logical relationships with the local information of latest fuzzy ﬂuctuation. Meanwhile, the

proposed model shows that the forecasting accuracy can be improved by considering more

information of sub-intervals within all next states of all fuzzy relationships which has the

actual data at forecasting time belonging to these sub-intervals. Among forecasting models

above, there are three models using the PSO algorithm as the HPSO model [22], the AFPSO

model [16] and the model [36], but the proposed model still obtains far lower MSE value

from 9^th- order fuzzy logical relationship.

Table 10. A comparison of the results obtained between the proposed model and its counterparts

from high - order of the FTS with diﬀerent number of intervals

Years

Actual data Model

[32]

Model

[7]

Model

[8]

Model

[23]

Model

[17]

Model

[37]

Proposed

model

1971

—

13055

—

N/A

—

N/A

—

N/A

—

N/A

—

N/A

—

N/A

—

N/A

—

1979

1980

1981

—–

16807

16919

16388

—–

16500

16361

16362

—–

16500

—–

16846

16420

—–

N/A

16890

16395

—–

N/A

16920

16388

—–

N/A

16919

16390

—–

N/A

16920

16388

—–

1991

1992

MSE

19337

18876

19487

18744

133700

19500

18500

86694

19334

18910

1101

19337

18882

234

19335

18882

173

19334

18872

9.23

19332

18876

5.08

286

NGHIEM VAN TINH, NGUYEN CONG DIEU

Table 11. The comparison of the MSE value between the proposed model and its counterparts with

various number of orders under 7 intervals

Models

Number orders of FLRs

2

3

4

5

6

7

8

9

Model [7]

Model [8]

Model [23

Model [17]

Model [12]

Model [37]

89093

67834

67123

19594

19868

86694 89376 94539 98215 104056 102179 102789

31123 32009 24984 26980

31644 23271 23534 23671

31189 20155 20366 22276

31307 23288 23552 23684

26969

20651

18482

20669

618.9

370.6

22387

17106

14778

17116

358.43

18734

17971

15251

17987

617.8

8836.2 822.47 686.39 658.18 659.14

Proposed model 8551.81 600.32 447.67 387.12 495.62

319.86 463.46

Figure 4. Flowchart of the proposed FTS forecasting model

For more detail, we also perform experiment for each order of proposed forecasting model

under seven intervals to compare with the existing models [4, 7, 11, 16, 22, 36], as listed in

Table 11. From Table 11, it is obvious that the forecasting error of the proposed model

decreases signiﬁcantly for all orders from 2 to 9. Particularly, the proposed model gets the

lowest MSE value of 319.86 with 8^th-order fuzzy logical relationship. For easily visualizing,

from curves in Figure 4, it can clearly see that the proposed model gives remarkably better

forecasting accuracy compared with its counterparts based on high - order FTS. From the

above analyses, it can be concluded that the proposed forecasting model outperforms any

existing methods for forecasting the enrollments of the University of Alabama.

4.3. Forecasting TAIFEX

In this subsection, the proposed model is applied to forecast the TAIFEX [25] between

8/3/1998 and 9/30/1998. All historical data of TAIFEX are partitioned into two phases

to implement comparison results of the proposed model with the existing models based on

various orders and diﬀerent intervals. The performance of the proposed model is evaluated

using the MSE (21).