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802 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 6, NOVEMBER 2008
A Hybrid System Integrating a Wavelet and TSK
Fuzzy Rules for Stock Price Forecasting
PeiChann Chang and ChinYuan Fan
Abstract—The prediction of future time series values based on
past and present information is very useful and necessary for var
ious industrial and ﬁnancial applications. In this study, a novel
approach that integrates the wavelet and Takagi–Sugeno–Kang
(TSK)fuzzyrulebased systems for stock price prediction is devel
oped. A wavelet transform using the Haar wavelet will be applied
to decompose the time series in the Haar basis. From the hierar
chical scalewise decomposition provided by the wavelet transform,
we will next select a number of interesting representations of the
time series for further analysis. Then, the TSK fuzzyrulebased
system is employed to predict the stock price based on a set of
selected technical indices. To avoid rule explosion, the kmeans al
gorithm is applied to cluster the data and a fuzzy rule is generated
in each cluster. Finally, a K nearest neighbor (KNN) is applied as a
sliding window to further ﬁnetune the forecasted result from the
TSK model. Simulation results show that the model has success
fully forecasted the price variation for stocks with accuracy up to
99.1% in Taiwan Stock Exchange index. Comparative studies with
existing prediction models indicate that the proposed model is very
promising and can be implemented in a realtime trading system
for stock price prediction.
Index Terms—Fuzzy ruled system, Kmean clustering, multi
ple regression analysis (MRA), simulated annealing (SA), wavelet
preprocessing.
I. INTRODUCTION
M
INING stock market tendencies is a challenging task
due to its high volatility and noisy environment. Many
factors inﬂuence the performance of a stock market including
political events, general economic conditions, and traders’ ex
pectations. Though stocks and futures traders have relied heav
ily upon various types of intelligent systems to make trading
decisions, the success so far is quite limited [4].
Many attempts have been made to predict the ﬁnancial mar
kets, ranging from traditional time series approaches to artiﬁcial
intelligence techniques such as fuzzy systems, and especially, ar
tiﬁcial neural network (ANN) methodologies [1]. However, the
main drawback with ANNs, and other blackbox techniques,
is the tremendous difﬁculty in interpreting the results. They
do not provide an insight into the nature of the interactions
Manuscript received March 9, 2007; revised August 5, 2007 and January 3,
2008. First published September 26, 2008; current version published October
20, 2008. This paper was recommended by Associate Editor G. Papadimitriou.
P.C. Chang is with the Department of Information Management, Yuan
Ze University, Taoyuan 32026, Taiwan, R.O.C. (email: iepchang@saturn.
yzu.edu.tw).
C.Y. Fan is with the Department of Industrial Engineering and Manage
ment, Yuan Ze University, Taoyuan 32026, Taiwan, R.O.C. (email: S948906@
mail.yzu.edu.tw).
Color versions of one or more of the ﬁgures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identiﬁer 10.1109/TSMCC.2008.2001694
between the technical indicators and the stock market ﬂuctua
tions. Thus, there is a need to develop methodologies that facili
tate an increased understanding of market processes, in addition
to providing temporally accurate predictions [17], [35], [65].
Another issue to be dealt with is that the dimensionality of ﬁ
nancial time series data also creates another challenge in ANN
approaches.
The development of a timely and accurate trading decision
making tool is a key for stock traders to make proﬁts. Since
the stock price series is affected by a mixture of deterministic
and random factors [17], new tools and techniques are needed
in dealing with noise and nonlinearity in stock price prediction.
Data mining, aimed at ﬁnding rules hidden in very large amount
of data, is a new and efﬁcient approach for time series analysis.
Data mining on time series needs to translate the continuous
time series into discrete symbol sequences ﬁrst. In this work,
the wavelet transform using the Haar wavelet will be applied to
decompose the time series in the Haar basis. From the hierarchi
cal scalewise decomposition provided by the wavelet transform,
we will next select a number of interesting representations of the
time series for further analysis. In addition, statistical analysis
is employed to select important factors that affect the perfor
mance of the stock market the most. These factors are chosen as
the inputs of the Takagi–Sugeno–Kang (TSK) fuzzyrulebased
system to predict the future stock price. The reason for choosing
the TSK fuzzy system is owing to its universal approximation
capability [54] and the possibility to gain insights into the data,
which is of particular interest for stock price prediction.
The proposed framework combines several soft computing
(SC) techniques such as a wavelet transform, TSK fuzzy sys
tem, data clustering, simulated annealing (SA), and K nearest
neighbor (KNN). In addition to waveletbased data representa
tion, the kmeans clustering algorithm is applied to cluster the
data before the TSK fuzzy rules are generated. A fuzzy rule is
then generated for each cluster, which enables us to determine
the membership functions of the fuzzy subsets, and the optimal
number of fuzzy rules as well. Finally, a KNN is applied as a
sliding window to further ﬁne tune the forecasted result from
the TSK model.
The remainder of the paper is organized as follows. Section II
reviews the different methods for stock forecasting using SC
techniques such as neural networks (NNs) and fuzzy systems.
Section III describes the proposed hybrid approach to stock
price prediction by integrating a wavelet with the TSK fuzzy
rulebased system. Section IV presents empirical results of the
hybrid approach and compare it with three other approaches.
Finally, conclusions and future directions of the research are
discussed in Section V.
10946977/$25.00 © 2008 IEEE
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CHANG AND FAN: HYBRID SYSTEM INTEGRATING A WAVELET AND TSK FUZZY RULES 803
II. LITERATURE SURVEY
Conventional research addressing the stock forecasting prob
lem has generally relied on time series analysis techniques, i.e.,
mixed autoregression moving average (ARMA) as well as mul
tiple regression models (MRMs). However, the assumptions of
these methods may come out with ineffective results since a
number of missing factors such as macroeconomical or political
effects may seriously inﬂuence stock tendencies.
White [59] was the ﬁrst to use NNs for market forecasting. He
used a feedforward NN (FFNN) to study the IBM daily common
stock returns and he found that his training results were overop
timistic, being the result of overﬁtting or of learning irrelevant
features. In general, there are two different methodologies for
stock price prediction in using ANN as a research tool [68]. The
ﬁrst methodology is to consider the stock price variations as a
time series and predict the future price based on its past values. In
this approach, ANNs have been employed as the predictor, see,
e.g., [5], [10], [14], [16], [17], [27], [32], [37], [42], [48], [63],
and [64]. These prediction models, however, have their limita
tions owing to the tremendous noise and high dimensionality
of stock price data. Therefore, the performances of the existing
models are not satisfactory [65].
The second approach takes the technical indices and qualita
tive factors such as political effects into account in stock market
forecasting and trend analysis.
Yao and Poh [63] use technical indicators (%K and %D)
along with price information to predict future price values. They
achieved good returns, and found that their models performed
better using daily data rather than weekly data. Hobbs and Bour
bakis [26] predict prices of stocks based on the ﬂuctuations in
the rest of the market for the same day. They show consistently
high rates of return, although the investment is done in a fric
tionless environment. Paying commissions on the large number
of trades instigated would certainly erode much of the beneﬁt
from the trading strategy proposed. Austin and Looney [8] de
velop an NN that predicts the proper time to move money into
and out of the stock market. They used two valuation indicators,
two monetary policy indicators, and four technical indicators
to predict the four week forward excess return on the dividend
adjusted S&P 500 stock index. The results signiﬁcantly out
performed the buyandhold strategy. Backpropagation ANNs
are applied to predict future elements in the price time series
in the Korea composite stock price index (KOSPI) [30]. L
´
opez
et al. [38] use time delay connections in enhanced NNs (that is,
the addition of timedependant information in each weight) to
forecast IBEX35 (Spanish stock index) index close prices one
day ahead. Stochastic NNs is applied for forecast the volatil
ity of index returns in the TUNINDEX (Tunisian stock index),
and ﬁnds that the outof sample NN results are superior to tra
ditional generalized autoregressive conditional heteroskedastic
(GARCH) models [52]. Nenortaite and Simutis [41] present
a trading approach based on onestep ahead proﬁt estimates
created by combining NNs with particle swarm optimization
algorithms. The method is proﬁtable given small commission
costs, but does not exceed the S&P500 returns when realistic
commissions are introduced. Jaruszewicz and Mandziuk [29]
train ANNs using both technical analysis variables and inter
market data, to predict one day changes in the NIKKEI index.
They achieve good results using moving average convergence
divergence (MACD), Williams, and two averages, along with
related market data from the National Association of Securities
Dealers Automated Quotation System (NASDAQ) and DAX.
It has been a new tendency that combining the SC technolo
gies of NNs, fuzzy logic (FL), and genetic algorithms (GAs)
may signiﬁcantly improve an analysis [1], [2], [9], [11]–[13],
[19], [24], [25], [27], [31], [36], [39], [40], [51], [53], [55]–
[57], [61]. In generally, NNs are used for learning and curve
ﬁtting, FL is used to deal with imprecision and uncertainty, and
GAs are used for search and optimization [13], [28], [39], [62].
Zadeh [65] pointed out that merging these technologies allows
for the exploitation of a tolerance for imprecision, uncertainty,
and partial truth to achieve tractability, robustness, and low so
lution cost.
Wavelet analysis is a relatively new ﬁeld in signal process
ing [18]. Wavelets are mathematical functions that decompose
data into different frequency components, and then, study each
component with a resolution matched to its scale—a scale refers
to a time horizon [47]. Wavelet ﬁltering is particularly rele
vant to volatile and timevarying characteristics of realworld
time series and is not restrained by the assumption of station
arity [44]. The wavelet transform decomposes a process into
different scales, which makes it useful in differentiating season
alities, revealing structural breaks and volatility clusters, and
identifying local and global dynamic properties of a process
at these timescales [5], [23], [45]. Wavelet analysis has been
shown to be especially productive in analyzing, modeling, and
predicting the behavior of ﬁnancial instruments as diverse as
stocks and exchange rates [43], [46].
In this paper, a wavelet ﬁltering is used and that is because of
the property of wavelets for economic analysis in data decompo
sition by time scale. Economic and ﬁnancial systems, like many
other systems, contain variables that operate on a variety of time
scales simultaneously so that the relationships between variables
may well differ across time scales. A wavelet will decompose
the time series into a range of frequency scales [7], [19], [22].
The lower level of the decomposition can capture the long range
dependencies with only a few coefﬁcients, while the higher lev
els capture the usual shortterm dependencies. This research,
motivated by the effective preprocessing capability of wavelets
and the predictive power of fuzzy rule system, presents a hybrid
system by integrating the wavelet and a TSK fuzzy rule system
for stock price prediction.
III. M
ETHODOLOGY
In tradition, there are two major factors to be considered in
stock forecasting, and they are technical analysis and funda
mental analysis. Technical analysis concentrates on the study
of market action, and fundamental analysis concentrates on the
economic forces of supply and demand that cause price move
ments. In addition, as explained in [68], statistics, technical anal
ysis, fundamental analysis, and linear regression are all used to
attempt to predict the market’s direction. However, technical
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804 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 6, NOVEMBER 2008
indexes themselves alone often miss a lot of potential chances
in the stock movement before the appropriate trading signal is
generated. Thus, although technical analysis may yield insights
into the market, its highly subjective nature and inherent time
delay does not make it ideal for the fast, dynamic trading mar
kets. That is why a wavelet and TSK fuzzyrulebased model is
developed in this research.
Fundamental analysis must rely on the reasons of price move
ment, and this process is very complicated since there are so
many factors that may affect the price change such as political,
psychotically events, etc. Therefore, the basic assumption of
this research is that the price movement is closely related to the
variation of technical index as widely applied in ﬁnancial time
series researches. A set of technical indexes will be applied as
input factors and the output will be the stock price. To study
their relationship, a hybrid method that integrates a wavelet and
Takagi and Sugeno fuzzysystembased forecasting model is
developed and implemented in this research for Taiwan stock
price prediction. The main procedures of the hybrid system are
shown in Fig. 1 and inputs and outputs of each block are further
explained in the following sections.
The notation of variables applied in the following sections is
shown in Table I.
A. Data Preprocessing Using Wavelet Theory
The reason for applying wavelet theory as a data preprocess
ing method is because that as mentioned by Ramsey [46], the
process of representation in wavelet is able to deal with the non
stationarity involved with economic and ﬁnancial time series.
One of the beneﬁts of a wavelet approach is the ﬂexibility in
handling very irregular data series, as illustrated in [47]. Eco
nomic and ﬁnancial systems contain variables that operate on a
variety of time scale simultaneously so that the relationship be
tween variables may differ across time scale. The most important
property of wavelets for economic analysis is decomposition by
time scale.
In this research, Haar wavelet is applied as our major wavelet
transform tools. Haar wavelet is a wavelet evolved from con
tinuous wavelet transform. According to [3], wavelet not only
decompose the data in terms of times and frequency, but also can
reduce lots of processing times. For a time series of size N,the
wavelet decomposition used here can be determined in O(n)
time. In considering Haar wavelet and Coiﬂets wavelet [18],
Coiﬂets wavelet considers more aspects than Harr wavelet, es
pecially in combining compact support with various degree of
smoothness and numbers of vanishing moments [3], but Haar
wavelet still provides easily and quickly process time without
losing much in performance than other wavelet systems [48].
Haar wavelet has been widely applied in time series forecast
ing [6], [48].
Depending on normalization rules, there are two types of
Haar wavelets within a given function/family: father and mother
wavelets
Φ
a,b
=2
−1/2
Φ
t − 2
a
b
2
a
ψ
a,b
=2
−1/2
ψ
t − 2
a
b
2
a
Fig. 1. Framework of the hybrid model.
Father wavelets
Φ(t)dt =1
Mother wavelets
ψ(t)dt =0. (1)
Father wavelets are used for the “lowest frequency” smooth
components; those requiring wavelets with the widest support
and mother wavelets are used for the “higher frequency” detail
components. Father wavelets are used for the “trend compo
nents” and mother wavelets are used for all the deviations from
trend. While a sequence of mother wavelets is used to represent
a function, only one father wavelet is used.
A time series data, i.e., functionf(t), is an input to be repre
sented by a wavelet analysis, and it can be built up as a sequence
of projections onto father and mother wavelets indexed by both
{b}, b = {0, 1, 2,...} and by {s} =2
a
, {a =1, 2, 3,...}.In
actual data analysis using discretely sampled data, it is neces
sary to create a lattice over which the calculations will be made.
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CHANG AND FAN: HYBRID SYSTEM INTEGRATING A WAVELET AND TSK FUZZY RULES 805
TABLE I
N
OTATION OF VARIABLES APPLIED IN SECTION III
Mathematically, it is convenient to use a dyadic expansion as
illustrated in (1).
The coefﬁcients in the expansion are given by the projections:
s
A,b
=
f(t)Φ
A,b
(t)dt
d
a,b
=
f(t)ψ
a,b
(t)dt, and
a =1, 2,...,A (2)
where A is the maximum scale sustainable by the number of
data points. The representation of the signal f(t) can now be
given by:
f(t)=
b
s
A,b
Φ
A,b
(t)+
b
d
A,b
ψ
A,b
(t)
+
b
d
A−1,b
ψ
A−1,b
(t)+···+
b
d
1,b
ψ
1,b
(t). (3)
The approximation can be represented:
f(t)=S
A
(t)+D
A
(t)+D
A−1
(t)+···+ D
a
+ ···+ D
1
S
A
(t)=
b
S
A,b
φ
A,b
(t)
D
A
(t)=
b
D
A,b
ψ
A,b
(t). (4)
When n, the number of observations is divisible by 2
J
, and
then, the number of coefﬁcients of each type is given by the
following:
1) at the ﬁnest scale, 2
1
: n/2 coefﬁcients d
1,b
;
2) at the next scale, 2
2
: n/2
2
coefﬁcients d
2,b
;
3) at the coarsest scale, 2
A
: n/2
A
coefﬁcients d
A,b
; and
4) at the coarsest scale, 2
A
: n/2
A
coefﬁcients s
A,b
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806 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 6, NOVEMBER 2008
Fig. 2. Wavelet transforming process from f(t).
so that
n =
n
2
+
n
4
+ ···+
n
2
A−1
+
n
2
A
. (5)
As shown in Fig. 2, f (t) represents the original data, S
1
represents an approximation signal, and D
1
represents a detailed
signal. We can deﬁne the multiresolution decomposition of a
signal by specifying:
SA coarsest scale, and
S
A−1
= S
A
+ D
A
. (6)
In general
S
a−1
= S
a
+ D
a
(7)
where {S
A
,S
A−1
,...,S
1
} is a sequence of multiresolution ap
proximations of the function f(t), at ever increasing levels of
reﬁnement. The corresponding multiresolution decomposition
of f(t) is given by {S
A
,D
A
,D
A−1
,...,D
a
,...,D
1
}.
The sequence of terms S
A
,D
A
,D
A−1
,...,D
a
,...,D
1
rep
resents a set of orthogonal signal components that provide rep
resentations of the signal at resolutions 1 to A; each DA−k
provides the orthogonal increment to the representation of the
function f (t) at the scale, or resolution 2
J −k
. When the data
pattern is very rough, the wavelet process will be repeatedly
applied. In the preprocessing, the target is to minimize the mean
absolute percentage error (MAPE) between the signal before
and after transformation. In this way, the noise in the original
data can be removed.
B. TSKFuzzySystemBased Prediction
1) Input Selection Using Stepwise Regression Analysis: A
set of important technical factors, as shown in Table II, which
will affect the forecasting result, have been identiﬁed by [14].
These important input factors will be further selected through
stepwise regression analysis (SRA) model in this research.
There are totally six important indexes to be selected from and
they are X
1
= six day moving average; X
2
= six day bias
(BIAS); X
3
= six day relative strength index (RSI); X
4
= nine
day stochastic line (KD); X
5
= the moving average divergence
(MABIAS); and X
6
= the 13 days psychological line. The out
put is Y = stock price.
The SRA is applied to determine the set of independent vari
ables that most closely affect the dependent variable. This is
accomplished by the repetition of a variable selection. The step
bystep procedure of the SRA approach is explained in details
in the following.
Step 1) Calculate the correlation coefﬁcient (r) of each input
variable (X
1
X
2
···X
n
) and output data (Y ). Then, a
correlation matrix is derived.
Step 2) Rank each variable according to its square (r
2
) from
correlation matrix (suppose X
i
is the largest one in
the current stage), and check the linear regression
of this variable to the output data, i.e., derive a re
gression model as
ˆ
Y = f(X
i
). α value is applied to
consider the signiﬁcance of each input variable. Re
peat this process until all variables are tested. Finally,
select those statistically signiﬁcant variables for fur
ther veriﬁcation and assume that these variables are
(X
1
X
2
···X
k
).
Step 3) Calculate partial F value for those statistically sig
niﬁcant variables, as shown in (9), and choose the
largest correlation coefﬁcient among these input vari
ables (assume that it is X
j
). Then, derive another
regression model
ˆ
Y = f(X
5
,X
4
) again
F
∗
j
=
MSR(X
j
/X
i
)
MSE(X
j
/X
i
)
=
SSR(X
j
/X
i
)/(k − 2)
SSE(X
j
/X
i
)/(n − k)
,i∈ I
(8)
F
∗
j
=Max
1≤j ≤N,j/∈I
(F
j
). (9)
Step 4) Calculate the partial F value of the original data for
input variable X
j
. If the value is smaller than a user
deﬁned threshold, it is removed from the model be
cause X
j
is not statistically signiﬁcant for the output.
Step 5) Repeat steps 3 to 4. If every input number’s partial F
value is greater than the user deﬁned threshold, stop.
It means that every input value should have signif
icant inﬂuences on output value. According to [10]
and [13], we always set the threshold value as 4. If
the F value of a speciﬁc variable is greater than the
user deﬁned threshold, it is added to the model as
a signiﬁcant factor. When the F value of a speciﬁc
variable is smaller than a user deﬁned threshold, it
is removed from the model. The statistical software
Statistical Package for the Social Sciences (SPSS) for
Windows 10.0 was used for SRA in this research. The
ﬂow diagram of SRA is shown in Fig. 3.
2) TSK Fuzzy Rule Systems: The TSK fuzzy systems is se
lected as a universal function approximation for the stock pre
diction problems due to its ability to explain nonlinear relations
using a relatively low number of simple rules. The structure of
our TSK model is a multipleinput singleoutput fuzzy system
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CHANG AND FAN: HYBRID SYSTEM INTEGRATING A WAVELET AND TSK FUZZY RULES 807
Fig. 3. Flow diagram of SRA.
and its associated fuzzy inference method comprises a set of K
IFTHEN rules in the following form:
R
i
: If x
1
is A
i1
,x
2
is A
i2
...x
n
is A
in
,
then y
i
= β
i1
+ β
i1
x
1
+ ···+ β
in
x
n
where x
j
,j =1,...,n, are the inputs of the fuzzy system, n is
the number of inputs, A
ij
,i=1,...,N, are fuzzy subsets, N
is the number of fuzzy rules, y
i
is the output of ith rule, and
β
ji
are parameters for consequence rules. It is a ﬁrstorder TSK
fuzzy rule system, and in this paper, Gaussian fuzzy membership
functions are adopted
A
ij
(x
j
) = exp
−
(x
j
− a
ij
)
2
σ
2
ij
(10)
where a
ij
and σ
ij
are the mean and standard deviation of the
Gaussian functions. Given a crisp input pair (x
0
1
···x
0
n
); the
crisp output of the TSK model is described by
y =
N
i=1
w
i
y
i
N
i=1
w
i
(11)
where w
i
is the strength of rule i determined by
w
i
=
n
j=1
A
ij
(x
0
j
) (12)
and
y
i
= β
i0
+ β
i1
x
0
1
+ ···+ β
in
x
0
n
. (13)
The main task in TSK fuzzyrulebased prediction is to de
termine the parameters in the fuzzy membership functions and
in the rule consequences using a learning algorithm, given a set
of training data specifying the functional mapping between the
inputs and the output.
3) Data Clustering: The purpose of data clustering is to clus
ter the set of ﬁnancial time series data into different groups, and
data in each group will have a more homogeneous characteris
tic. However, it is very important to determine how many fuzzy
rules should be generated beforehand. If a standard rule struc
ture is used, rule explosion occurs when the number of inputs is
high. To resolve this problem, we divide the training data into
a number of clusters based on the output data (stock price) and
one fuzzy rule is generated for each cluster [32]. By doing this,
the number of fuzzy rules can be reduced effectively. Besides,
we can determine the fuzzy membership functions using the
mean and standard deviation of the data points that belongs to
each cluster.
The Kmeans clustering algorithm is employed for data clus
tering. Kmeans is a nonhierarchical clustering technique in
which the dataset is partitioned into K clusters. During the
clustering, the data points are randomly assigned to the clusters
to minimize the following squared error (SE):
SE =
K
i=1
p∈C
i
p − m
i

2
(14)
where p are data points in the cluster C
i
, m
i
is the center of
cluster C
i
, and K is the number of clusters.
Once the training data are clustered, we can calculate the
parameters of the membership functions for each cluster as
follows:
a
ij
=
1
s
i
s
i
i=1
x
j
σ
ij
=
1
s
i
− 1
(x
j
− a
ij
)
2
(15)
where S
i
is the number of data points in cluster i. In addition,
the output of the training data is also normalized using the mean
and standard deviation of the data in each cluster.
4) Optimization of the Parameters in Fuzzy Rules Using
Simulated Annealing: The purpose of applying the SA is to
ﬁnd a set of best values for the parameters within the fuzzy
rules. Traditionally, the parameter settings of TSK’s rules are
generated using the gradient method. The generalized gradient
algorithm searches for the solution in a multidimensional space
along the steepest ascent direction. However, such a search can
be extremely slow and ineffective if the equation has many
plateaus distributed throughout the landscape. Therefore, this
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808 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 6, NOVEMBER 2008
Fig. 4. Gradient method as compared with a simulated annealing approach.
method may not be able to derive an optimal solution and can
be trapped in a local optimum, as shown in Fig. 4. In such cases,
statistical search methods may offer better strategy in resolving
both problems [60], [66].
One of the most widely used statistical search method is the
SA [33], which uses the metropolis algorithm to decide whether
to accept or reject a conﬁguration that results in an increased
cost during its attempts in searching for the minimum cost.
The main characteristics of SA are its simplicity and the rapid
convergence. To properly adjust parameters’ weights of TSK
fuzzy rules, the SA approach is effective if the chosen energy,
or cost function, for the global system is appropriate. In this
study, the cost function is deﬁned as the MAPE for the set of
testing data, i.e., a series of stock index. The procedure of SA
is well known, as described in [33]. First, it is necessary to
generate random values of the parameters’ weights, and second,
to compute the associated cost of the system. This cost will
be minimized when the parameters’ weights achieve a global
minimum, the method thus allowing escape from local minima.
The detailed set up of the parameters for SA will be described
in the next section. Through a proper setup of the cooling sched
ule, ﬁnally SA can be applied to derive a set of nearoptimal
parameters for these TSK rules, as shown in Fig. 4.
5) Using KNearestNeighbor as a Sliding Window: The ba
sic idea of KNN [20] is to identify similar patterns of current
data trend from the historic data. We use KNN as a sliding
window to forecasting the data value for next day and use the
current k data as a time window to search within the historic
data to see if there are any similar patterns identiﬁed. Basically,
our approach is categorized as a onestep ahead prediction. The
selected data are preprocessed with the wavelet, and then, TSK
model is applied to generate a set of fuzzy rules for prediction
of stock price. In addition, the KNN sliding window is further
applied to reduce the forecasting errors. The set of historic data
is divided into training and testing set for cross validation. The
KNN is simpler than other SC approaches because there is no
model to train on the data series. Instead, the data series is be
ing searched for situations similar to the current, each time a
forecast needs to be made.
To describe the KNN process, several terms have to be deﬁned
ﬁrst. Assume the window size is L, which means there are L data
in each window to be considered. The ﬁnal data points of the
data series are the reference data, and the length of the reference
is the window size. To forecast the data series’ next data point,
the reference is compared to the ﬁrst group of data points in
the historical data series, called a candidate, and an error is
computed. Then, the reference is moved one data point forward
to the next candidate and another error is computed, and so on.
An error is calculated by subtracting the candidate value from
the reference value. All errors of the testing data are sorted and
stored in an array. Assume that the number of nearest neighbors
is H. Then, the smallest H errors corresponding to these H
candidates will be selected. Finally, the forecasted value will be
equal to the average of these k data points. Then, to forecast
the next data point, the process is repeated with the previously
forecasted data point appended to the end of the data series.
This process can be iteratively repeated until all n data points
are calculated.
Use KNN to calculate the new forecasted value.
Step 1) Use original data as a contrast data. Suppose to fore
cast number i data from index number t +1, i.e.,
number
ˆ
X
i,t+1
value.
Step 2) Use number t to t − L + 1 data for contrast base.
Using the sliding window method, one by one com
pare the data from 1 to t − L, and also calculate the
Euclidean distance from every interval D
(1)
j
, and ﬁnd
the corresponding forecasting value F
(1)
j
D
(1)
j
=
L
l=1
(X
i,t−L+ l
− X
i,l+j −1
)
2
F
(1)
j
= X
i,j+L
. (16)
In (14), j =1∼ t − L.
Step 3) Consider all D
(1)
j
and ﬁnd the kth smallest number.
It is KNN’s K option value.
Step 4) Use the weighted voting method to ﬁnd the last fore
casting value
ˆ
X
i,t+1
. The equation is
ˆ
X
i,t+1
=
H
k=1
F
k
/W
k
H
k=1
1/W
k
. (17)
W
k
means the kth smallest D
(1)
j
value, F
k
means
the F
(1)
j
value corresponding to the kth smallest D
(1)
j
value, H =1− k. And the parameter set of the slid
ing windows is (L, H). A simple example for KNN
forecasting with window size L = 3 and H = 2is
shown in Fig. 5.
C. Different Models to be Compared With
In this research, we use traditional backpropagation neural
networks (BPNs), the MRM, and a forecasting method by inte
grating GA with Wang and Mendal’s algorithm for fuzzy rule
generation (GAWM) [13], [58] to compare with our wavelet
TSK fuzzy rule forecasting system. These three compared mod
els will be brieﬂy introduced.
BPN [49] is a popular system that has been widely employed
in ﬁnancial forecasting. The most popular training method for
BPN is the supervised learning, i.e., learning by samples, which
will be selected in this research to train the system. After learning
(or training), the trained connection weights can be used for the
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CHANG AND FAN: HYBRID SYSTEM INTEGRATING A WAVELET AND TSK FUZZY RULES 809
TABLE I I
T
ECHNICAL INDICES USED AS INPUT VARIABLES
prediction of future occurrences. Technical indices selected by
SRA will be applied as the inputs to BPN and the output will
be the Taiwan stock index. The detailed parameter’s setting for
BPN will be described in Section IV.
Multiple regression analysis (MRA) [12] is one of the most
popular methods applied in business forecasting. MRA is em
ployed for testing hypothesis with regard to the relationship
between a dependent variable (Y ) and two or more independent
variables (Xs). It is easy to establish a model when there is a
linear relationship between the independent variable and depen
dent variable. On the other hand, it is very difﬁcult to establish an
accurate model within a nonlinear relation. In this research, the
output factor (Y ) is the stock price and the input factors include
six day moving average (X1), six day bias (X2), six day RSI
(X3), nine day stochastic line (X4), moving average divergence
(X5), and the 13 days psychological line (X6). The multiple
regression formula of this problem can be deﬁned as follows:
Y = a
1
X
1
+ a
2
X
2
+ a
3
X
3
+ a
4
X
4
+ a
5
X
5
+ a
6
X
6
+ b.
(18)
Among them, parameters a
1
, a
2
, a
3
, a
4
, a
5
, a
6
, and b are all
calculated via SPSS statistics software.
Wang and Mendel [58] developed a method to create a fuzzy
rule base, which is a combination of rules generated from nu
merical examples, i.e., historic stock data and linguistic rules
supplied by human experts [15]. The Wang and Mendel (WM)
method is evolved with GAs and the idea is similar to evolving
NN [60]. Essentially, a simple GA is used to determine the near
optimal number of fuzzy terms for each variable; as a result, the
objective function can be better improved by this evolution.
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810 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 6, NOVEMBER 2008
Fig. 5. KNN forecasting with window size L = 3andH = 2.
D. Performance Measures
There are many measures of prediction accuracy used to com
pare forecasting methods out of sample [10]. In considering
about the dimensionality of the data, in this research, there are
six input factors and one output value, and 494 training data.
The mean square error (MSE) of each model is very large after
training this data set. Since the purpose of the stock prediction is
to make proﬁt instead of just predict the future price accurately,
we use MAPE as a performance measure instead of MSE. The
equation of MAPE is listed later.
1) Mean Absolute Percentage Error: The accuracy of pre
dictions was measured with the following indicator, i.e., MAPE.
The average forecast error is measured as a percentage of his
torical results. The absolute value allows for the effect of adding
different signs. It is calculated as follows:
MAPE =
1
n
n
t=1
X
t
− F
t

X
t
× 100 (19)
where X
t
is the true value and F
t
is the predicted value at time
t. The MAPE is an average over n test sets.
IV. S
IMULATION RESULTS
The Taiwan Stock Exchange (TSE) began operations since
1962. At the end of January, 2005, Taiwan Stock Exchange Cor
poration (TSEC) had 699 listed companies with market capital
topping NT$13.7 trillion (US $396 billion). Most stock trading
goes to the listed IT companies and the trading value of TSE
stock market places it in the top ten of stock exchanges in the
world.
The data set applied for test in this research is the TSE in
dex, and it has been decomposed into three different sets: the
training data, test data, and validation data. The data for TSE
Fig. 6. Figures before and after a wavelet transformation for 492 stock price
data from Taiwan Stock Exchange index.
index are from July 18, 2003 to December 31, 2005, totally 614
records. During this period, the stock market has gone through a
rough up and down period owing to the national political issues.
Therefore, these data are very representative and suitable for
study and analysis. The ﬁrst 492 records will be training and
crossvalidation data and the rest of the data, i.e., 122 records
will be for outofsample test data. To avoid the interaction
among these factors, we will test each factor using SRA and
identify the factor that will affect the ﬁnal forecasted results
signiﬁcantly. The ﬁnal combination of the factors will be ﬁ
nalized after the analysis. The factors selected ﬁnally are MA6
and BIAS6; these two index and the output variables are TSE
index.
Before training the TSK fuzzy model, a wavelet transfor
mation has been applied to preprocess the data. According to
the MAPE, a threelevel wavelet preprocessing is thus applied.
Through this process, the noise in the original data can be re
moved. The result of a waveletbased decomposition process
is depicted in Fig. 6. According to the RSA method described
in Section IIIB, six input variables are ﬁnally selected as the
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CHANG AND FAN: HYBRID SYSTEM INTEGRATING A WAVELET AND TSK FUZZY RULES 811
Fig. 7. MAPEs of the hybrid model for different number of data clusters.
TABLE III
P
ARAMETER SETTINGS FOR SIMULATED ANNEALING
inputs to the TSK model to predict the stock price. They are six
day moving average (MA), six day bias (BIAS), six day RSI,
nine day stochastic line (KD), the moving average divergence
(MABIAS), and the 13 days psychological line (PSY).
In the kmeans clustering algorithm, the number of clusters
must be predeﬁned. It is a very interesting subject to be fur
ther investigated since there is no exact theorem explaining
the effect of number of clusters to the forecasting accuracy.
To check the sensitivity of the performance of our model on
the number of clusters, various numbers of data clusters are
investigated. In the experiment, the stock price data were clus
tered into two to eight clusters. The performance (MAPE) of
the algorithms with different number of clusters is shown in
Fig. 7.
As can be observed from the ﬁgure, MAPE will start to de
crease as the number of clusters increase. However, as the num
ber of clusters reaches a certain value, MAPE starts to increase.
Part of the reasons is because when the number of clusters is too
large, the number of the data in each cluster is too small. These
data in each cluster are not representative enough to generate a
model to forecast the future stock index. Therefore, in this re
search, the number of clusters will be three since it provides best
performance (the smallest MAPE). This number of clusters is
not deﬁnite and it has to be decided experimentally for different
application purposes.
The parameter setting in three different levels for the SA is
provided in Table III. Then, we use statistical software Minitab
R14 to run the Taguchi experiments and the results of different
factor levels are shown in Table IV. Table V lists the ﬁnal setting
for each factor in SA procedure. The factor response graph of
these experimental results is shown in Fig. 8.
The convergence diagram of the learning process of the
rule consequence parameters using the SA is shown in Fig. 9.
Finally, the MAPE of the forecasting model gradually de
creases to 3.8% after the temperature drops to a certain level.
It is justiﬁed that the proposed SA approach can ﬁnd near
optimal solutions for the set of parameters of consequence
rules.
We compared the proposed hybrid method combining the
wavelet and TSK fuzzy rules with three existing methods. To
justify the use of SA and KNN sliding window, a set of experi
mental results are listed in Table VI. In this table, rule number
is decided by (14) and (15), and the ﬁrst number means the
window size L; the second number means the number of best
neighborhood data H. According to a series of experiments
where L is setup as 2, 4, 6, and 8 and H is setup as 2, 3, 4, and
5, the best forecasting result in terms of the minimum MAPE
value from the hybrid model is in (2,5) with an average of
0.792.
The four different algorithms to be compared with are the
traditional backpropagation neural networks (BPNs), a stan
dard TSK, the MRM, and a forecasting method by integrating
GA with Wang and Mendal’s algorithm for fuzzy rule gener
ation (GAWM) [13], [58]. Tables VII and VIII are the best
parameter setting for BPN and GAWM using design of
experiments. Fig. 10 also reveals the experimental results that
GAWM converged after 100 generations. Table VIII shows the
MAPE value for all different methods.
As observed from Table IX, MRM has the largest MAPE
value and part of the reasons is because MRM cannot fully
explain the nonlinear relationship among the stock price and
the technical indexes. BPN also has a large error as compared
with other models and that is due to the tremendous noise
and complex dimensionality of stock price data. Besides, the
quantity of data itself and the input variables may also in
terfere with each other. In addition, BP learning algorithm is
subject to getting stuck in a local optimum, while the TSK
is less likely. Therefore, the result may not be that convinc
ing. In addition, BPN methods do not provide an insight into
the nature of the interactions between the technical indicators
and the stock market ﬂuctuations. As for GAWM, the fuzzy
rules generated from the training data are very large when
compared with TSK and these rules may interact with each
other.
From the experimental tests, we can observe that TSK fuzzy
system is more suitable in handling large amount of data.
The set of data is clustered according its mean and standard
deviation. After this clustering, these set of data will be de
composed into couples of subclusters. These subclusters have
more homogeneous characteristics within themselves, and each
subcluster will be transformed into a TSK fuzzy rule. There
fore, the fuzzy rules generated from TSK are quite small since
each cluster of data only generates one single rule. For our
experimental tests, the clusters have been preset into two to
eight; therefore, there are two to eight fuzzy rules within each
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812 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 6, NOVEMBER 2008
TABLE I V
T
AGUCHI EXPERIMENTAL RESULT OF DIFFERENT FACTOR LEVELS
TABLE V
B
EST PARAMETER SET FOR SA
Fig. 8. Factor response graph.
Fig. 9. SA training astringent diagram.
TABLE V I
R
ESULTS OF DIFFERENT NUMBER OF SLIDING WINDOWS (L) AND H NEAREST
NEIGHBORS APPLIED IN TSE INDEX FORECASTING (IN PERCENT)
TABLE V II
BPN P
ARAMETER SET
system. The best forecasting results will be in three clusters. In
addition, KNN has been further applied to reduce the fore
casting errors. That is why, TSK has a better forecasting
accuracy.
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CHANG AND FAN: HYBRID SYSTEM INTEGRATING A WAVELET AND TSK FUZZY RULES 813
TABLE VIII
P
ARAMETER SETTING FOR GAWM
Fig. 10 Convergence diagram of GAW& M model.
TABLE I X
MAPE P
ERFORMANCE FROM DIFFERENT METHODS
V. C ONCLUSION
This paper proposed a TSK fuzzy model for stock price pre
diction. To facilitate the prediction, the data are preprocessed
using the Haar wavelet. Then, SRA technique is employed to
select the most relevant factors for prediction. To avoid rule ex
plosion, the kmeans clustering algorithm is employed to group
the data into a number of clusters and one fuzzy rule is generated
for each cluster. As an additional beneﬁt, the fuzzy membership
function can be determined automatically using the mean and
variance of the data in each cluster. The parameters in the con
sequences of the TSK rules are optimized using the SA. A KNN
sliding window is applied to retrieve the similar patterns in the
historical data and further adjust the forecasted value from the
TSK model.
The proposed model is compared with the BNP, TSK, MRM,
and GAWM for stock price prediction. Simulation results show
that the TSK model with waveletbased preprocessing greatly
outperforms the other three models. To the best of knowledge,
the combination of the TSK fuzzy model with waveletbased
preprocessing is new for stock price forecasting. Due to its very
promising performance, we are going to apply the system for
realtime daily trading.
In the future, a different TSK fuzzy model, such as a nonlinear
model using NNs as a consequence, can be further applied in a
more complex time series problem. In addition, more advanced
pattern matching algorithm can be embedded in the system to
retrieve signiﬁcant patterns from the historic stock data for com
parison with the current trend of the data. As a result, intelligent
trading signals instead of stock price can be identiﬁed.
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CHANG AND FAN: HYBRID SYSTEM INTEGRATING A WAVELET AND TSK FUZZY RULES 815
PeiChann Chang received the M.S. and Ph.D. de
grees from Lehigh University, Bethlehem, PA, in
1985 and 1989.
He is currently a Professor at the Department
of Information Management, Yuan Ze University,
Taoyuan, Taiwan, R.O.C. His current research inter
ests include ﬁnancial time series forecasting, evolu
tionary computation, fuzzy neural applications, pro
duction scheduling, forecasting, casebased reason
ing, and applications of soft computing. He is the
author or coauthor of more than 60 papers published
in international journals. He is the Senior Editor of the Journal of Chinese In
stitute of Industrial Engineering (JCIIE).
ChinYuan Fan received the B.S. degree from the
ChineseCulture University, Taipei, Taiwan, R.O.C.,
2001, and the M.S. degree from DaYeh University,
Changhua, Taiwan, in 2003, both in management. He
is currently working toward the Ph.D. degree at the
Industrial Engineering and Management Department,
Yuan Ze University, Taoyuan, Taiwan.
His current research interests include applications
of soft computing, ﬁnancial time series forecasting,
multiobjective optimization problems, and multicri
teria decision making.
Authorized licensed use limited to: Hong. Downloaded on February 8, 2009 at 00:06 from IEEE Xplore. Restrictions apply.