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A fuzzy support vector regression model for business cycle predictions
Expert Systems with Applications 37 (2010) 5430–5435
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A fuzzy support vector regression model for business cycle predictions Kuo-Ping Lin a,*, Ping-Feng Pai b,c a
Department of Information Management, Lunghwa University of Science and Technology, Taoyuan 333, Taiwan Department of Information Management, National Chi Nan University, University Rd., Puli, Nantou 545, Taiwan c Department of International Business Studies, National Chi Nan University, University Rd., Puli, Nantou 545, Taiwan b
a r t i c l e
i n f o
Keywords: Business cycle Fuzzy set theory Support vector regression
a b s t r a c t Business cycle predictions face various sources of uncertainty and imprecision. The uncertainty is usually linguistically determined by the beliefs of decision makers. Thus, the fuzzy set theory is ideally suited to depict vague and uncertain features of business cycle predictions. Consequently, the estimation of fuzzy upper and lower bounds become an essential issue in predicting business cycles in an uncertain environment. The support vector regression (SVR) model is a novel forecasting approach that has been successfully used to solve time series problems. However, the SVR approach has not been widely applied in fuzzy forecasting problems. This study employs support vector regressions to calculate fuzzy upper and lower bounds; and presents a fuzzy support vector regression (FSVR) model for forecasting indices of business cycles. A numerical example of a business cycle prediction in Taiwan was used to demonstrate the forecasting performance of the FSVR model. The empirical results are satisfactory. Therefore, the FSVR model is an effective alternative in forecasting business cycles under uncertain circumstances. Crown Copyright 2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction Accuracy in forecasting business cycles is an important issue in economic study, and statistical methods have usually been employed to analyze them. Many investigations have been done in the analysis of business cycles (Banerji & Hiris, 2001; Layton, 1996, 1998; Seip & McNown, 2007; Wu & Tseng, 2002; Yang & Kim, 2005). However, business cycles are often determined by a panel of macroeconomic experts, and thus, it is difficult to predict the index of business cycles. The difficulty arises from assumptions made from the probability distributions and business cycle data, which are usually vague. The index of business cycles in Taiwan is composed of nine exogenous variables, and five lights are used to represent different economic activities. The five lights include some uncertain factors in predicting business cycles. Hence, the fuzzy set theory (Zadeh, 1965) is a proper approach to analyze Taiwan business cycles. Unlike most of traditional technologies SVR (Vapnik, Golowich, & Smola, 1996) implementing neural network models, SVR adopts a structural risk minimization principle, which seeks to minimize the upper bounds of the generalization error rather than minimize the training error. In recent years, SVR schemes have been extended to cope with forecasting problems, and have provided many promising results in customer demand (Levis & Papageorgiou,
* Corresponding author. E-mail addresses: [email protected] (K.-P. Lin), [email protected] (P.-F. Pai).
2005), finance (Huang, Nakamori, & Wang, 2005; Kim, 2003; Tay & Cao, 2002), intermittent demand (Hua & Zhang, 2006), tourism demand (Pai & Hong, 2005), air quality (Lu & Wang, 2005), wind speed (Mohandes, Halawani, Rehman, & Hussain, 2004), plant control systems (Xi, Poo, & Chou, 2007), rainfall (Hong & Pai, 2007), prices for the electricity market (Gaoa, Bompard, Napoli, & Cheng, 2007), and flood control (Yu, Chen, & Chang, 2006). Hong and Hwang (2003) proposed a support vector fuzzy regression machine model for modifying convex optimization problems of multivariate fuzzy linear regression models. Empirical results indicate that the developed model derives satisfying solutions efficiently. Jeng, Chuang, and Su (2003) developed a support vector interval regression network to efficiently handle interval output data. Yao and Yu (2006) developed a fuzzy regression based on asymmetric support vector machines, which overcome limitations of traditional nonlinear fuzzy regression, and can be effectively used for parameter estimation. Chuang (2008) presented an interval support vector regression network model, which can handle interval input and output data. Hao and Chiang (2008) developed a fuzzy regression analysis model based on support vector learning techniques, and suggested that the developed model can perform automatic and accurate control in fuzzy regression analysis tasks. In this study, a fuzzy support vector regression model is presented to forecast an index of business cycles. Support vector regression was used to calculate fuzzy upper and lower bounds, and then make predictions by fuzzy H-level set (H-cut). In addition, genetic algorithms (GA) were employed to select three parameters of SVR models. The remainder of this paper is organized as follows.
0957-4174/$ - see front matter Crown Copyright 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.02.071
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A brief introduction of the theory of SVR is given in Section 2. The fuzzy support vector regression model is derived in Section 3. A numerical example of business cycle predictions and empirical results are presented in Section 4. Some concluding remarks are offered in Section 5.
This constrained optimization problem can be solved using the following primal Lagrangian form:
2. Support vector regression models
! N N X X 1 ðni þ ni Þ bi ½w/ðxi Þ þ b Y i þ e þ ni kwk2 þ C 2 i¼1 i¼1
Min
N X
bi ½Y i w/ðxi Þ b þ e þ ni
i¼1
Presented by Vapnik (1995), the support vector machine was originally applied to pattern recognition problems. Then, the support vector machine model has been successfully extended for dealing with nonlinear regression problems (Vapnik et al., 1996). The support vector regression model is based on the idea of mapping the original data x nonlinearly into a higher dimensional feature space. The SVR approach is to approximate an unknown function by a training data set {(xi, Yi), i = 1, . . . , N}. The regression function can be formulated as follows:
F ¼ w/ðxi Þ þ b;
N 1 X 1 RðFÞ ¼ C Le ðY i ; F i Þ þ kwk2 ; N i¼1 2
ð2Þ
Le ðY i ; F i Þ ¼
if jY i F i j 6 e
0
jY i F i j e otherwise
ð3Þ
;
! N X 1 2 f ðw; n; n Þ ¼ kwk þ C ðni þ ni Þ 2 i¼1
Subjectiv e to w/ðxi Þ þ b Y i 6 e þ ni ;
i ¼ 1; 2; . . . ; N;
Y i w/ðxi Þ b 6 e þ ni ;
i ¼ 1; 2; . . . ; N;
ni ; ni P 0;
i¼1
Subjectiv e to
N N X N X 1X bi þ bi ðbi bi Þðbj bj ÞKðxi ; xj Þ 2 i¼1 i¼1 j¼1
N X ðbi bi Þ ¼ 0; i¼1
f 0 6 bi 6 C; i ¼ 1; 2; . . . ; N; 0 6 bi 6 C; i ¼ 1; 2; . . . ; N: ð6Þ
bi
The Lagrange multipliers in Eq. (6) satisfy the equality bi ¼ 0. The Lagrange multipliers, bi and bi , are determined, and an optimal weight vector of the regression hyperplane is written by Eq. (7). N X ðbi bi ÞKðx; xi Þ:
ð7Þ
ð4Þ
Thus, the regression function is given by:
Fðx; b; b Þ ¼
N X ðbi bi ÞKðx; xi Þ þ b:
ð8Þ
i¼1
Herein, K(xi, xj) denotes a Kernel function whose value equals the inner product of two vectors, xi and xj , in the feature space / (xi) and /(xj), meaning that K(xi, xj)= /(xi) /(xj). Any function that satisfies Mercer’s condition (Mercer, 1909) can act as the Kernel function. This work uses the Gaussian function.
3. A fuzzy support vector regression model Introduced by Zadeh (1965), fuzzy set theory has been applied to deal with uncertain problems in many fields. A brief of fuzzy set theory is depicted as follows. A membership function is defined for all elements a in the referential set U. A fuzzy set (A) can be defined by the membership functions lA(a), which can have values of [0, 1]. Thus, a fuzzy set (A) is said to be convex if all ordinary subsets of A are convex. A fuzzy set is normalized if $a 2 U, lA(a) = 1. For an interval, level of confidence, or called a-cut or a-level set at level a (0, 1], an ordinary subset of A can be defined and denoted as [A]a,
½Aa ¼ fa 2 U lA ðaÞ P ag;
i ¼ 1; 2; . . . ; N:
a 2 ð0; 1:
ð9Þ
Furthermore, a fuzzy number (FN) can be defined as a convex, a normalized fuzzy set on a real line, with an upper semi-continuous membership function and bounded support. Dubois and Prade (1980) have defined a general representation form for FNs, which can be called the L–R type FNs. Here, FNs are represented as follows:
Lε (Yi, Fi)
−ε
Y i ðbi bi Þ e
i¼1
where C and e are user-defined parameters. The parameter e is the difference between actual values and values calculated from the regression function. This difference can be viewed as a tube around the regression function. The points outside the tube are regarded as training errors. In Eq. (2), Le(Yi, Fi) is called an e-insensitive loss function, and can be illustrated as Fig. 1. The loss equals zero if the approximate value is within the etube. Additionally, the second item of Eq. (2), 12 kwk2 , is adopted to estimate the flatness of a function which can avoid overfitting. Therefore, C indicates a parameter determining the trade-off between the empirical risk and the model flatness. Two positive slack variables (ni and ni ), representing the distance from actual values to the corresponding boundary values of the e-tube, are then introduced. These two slack variables equal zero when the data points fall within the e-tube. Eq. (2) is then reformulated into the following constrained form:
Min
N X
Max
w ¼
where
ð5Þ
i¼1
Eq. (5) is minimized with respect to primal variables w, b, n, and n*, and is maximized with regard to non-negative Lagrangian multipliers ai, ai , bi, and bi . Finally, Karush–Kuhn–Tucker conditions are applied to Eq. (4), and the dual Lagrangian form given by Eq. (6).
ð1Þ
where /(xi) denotes the feature of the inputs, and w and b indicate coefficients. The coefficients (wi and b) are estimated by minimizing the following regularized risk function.
N X ðai ni þ ai ni Þ:
+ε
Fig. 1. The e-insensitive loss function.
Yi − Fi
Definition 1 (The bound form representation). A symmetrical FN can be written as A = (w c, w, w + c)LR, where w c and w + c, respectively, denote the lower and upper bounds, w the mode or center value, (w c, w + c) forms the support, and L, R, respectively, denote the left and right reference (or shape) functions of A. In the case L R, and A has the membership function:
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8 0; > > > < Lðða wÞ=cÞ; lA ðaÞ ¼ > Rððw aÞ=cÞ; > > : 0;
a < w c; w c 6 a 6 w; w 6 a 6 w þ c;
ð10Þ
a > w þ c:
Likewise, non-symmetrical FNs can be defined. In addition, for example, a symmetrical triangular FN may be denoted as A = (w c, w, w + c)T specially with the symmetrical triangular membership function:
8 0; > > > < ða wÞ=c; lA ðaÞ ¼ > ðw aÞ=c; > > : 0;
a < w c; w c 6 a 6 w; w 6 a 6 w þ c;
ð11Þ
where the Lagrange multipliers, bLi and bLi , and an optimal weight vector wLi of the lower bound regression hyperplane can be estimated by Eq. (6):
Y U ðxi Þ ¼ wUi þ bU ¼
where the Lagrange multipliers, bUi and bUi , and an optimal weight vector wUi of the upper bound regression hyperplane can be estimated by Eq. (6). Then the fuzzy prediction model uses Eqs. (13) e ðxi Þ, which degree of membership and (14) to estimate the output Y function approaches precisely the H-level set. That is:
" N X ðbLi bLi ÞKðx; xi Þ þ bL ; i¼1
The representation can thus be called the bound form representation. Another representation can be used, and is given below. Definition 2 (The spread form representation). A symmetric FN can be written as A = (w, c)L, with the membership function lA(a) = L((a w)/c) "a 2 R, where w and c, respectively, represent the mode or center value, and the spread around the mode of A. The reference function L(x) thus has the properties (i) L(x) = L(x), (ii) L(0) = 1, L(1) = 0, (iii) L is decreasing on [0, 1], and (iv) L is invertible on [0, 1]. Based on the fuzzy set theory, fuzzy regression (FR) analysis is a methodology giving rise to a possibility distribution for an imprecise or vague phenomenon, which can be expressed by yielding fuzzy parameters. The FR can represent the data accrual without losing the original meaning, and analyze the trends of both variability and mean in data. FR analysis was first introduced by Tanaka, Uejima, and Asai (1980), who represented the observational uncertainty or fuzziness of a system by the fuzzy parameters for its indefinite structure. The objective is to minimize the overall fuzziness of the estimated outputs with the condition that an H-level-set inclusion of all data observed holds. Increasing the H-factor may expand the confidence interval of the FR, and increase the probability that out-of-sample data may fall within the model description. It is comparable to increase the confidence in the statistical regression model by increasing the confidence interval. However, in fuzzy input data exist certain outliers, according to the linear programming estimation technique, the approach could obtain a huge but useless estimated fuzzy output. Therefore, a more precise explanation should be considered, which the H-level set of membership functions for representative input data should be closed estimated precisely H-level set of membership function (Wu & Tseng, 2002). A fuzzy input data Yi belongs to Triangular FNs (TFNs), the membership of Yi is distributed on the interval [yi ci, yi + ci] and can be defined a TFNs as (yi ci, yi, yi + ci), where YLi = yi ci is lower bound value, YUi = yi + ci is upper bound value, and YCi = yi is mode value of FNs. The YLi and YUi must approach precisely H-level set. Hence, given the significant level H, the FSVR model satisfies Eq. (12):
leY ðx Þ ðyi þ ci Þ H; leY ðx Þ ðyi ci Þ H; i
where
ð14Þ
i¼1
e ðxi ÞH ¼ ½Y
a > w þ c:
N X ðbUi bUi ÞKðx; xi Þ þ bU ;
# N X ðbUi bUi ÞKðx; xi Þ þ bU : i¼1
ð15Þ Then, we can obtain estimated central vale YC(xi) by a simple calculation as follows:
Y C ðxi Þ ¼
N X
ðððbLi bLi Þ þ ðbUi bUi ÞÞKðx; xi Þ þ bL þ bU Þ=2:
ð16Þ
i¼1
Moreover, based on YC(xi), and Eqs. (9)–(11), we can derive the lower bound approach values with H-level set as follows:
Y HL ðxi Þ ¼
N N X X ðbLi bLi ÞKðx; xi Þ þ bL H ðððbLi bLi Þ i¼1
i¼1
!
þ bUi
bUi ÞÞKðx; xi Þ
þ bL þ bU Þ=2 =ð1 HÞ:
ð17Þ
Fuzzy input data
Find the lower and upper bounds
Training data
Testing data
Use SVR to approach the lower and upper bounds Calculate mode of fuzzy variables and estimate the lower and upper bounds with H-cut of fuzzy variables
Use FSVR model to forecast the lower and upper bounds with H-cut of fuzzy variables
Estimate mode of fuzzy variables
Use GA to select parameters of FSVR Calculate testing errors Calculate training errors
Is the GAs stop criterion satisfied ?
NO
Yes
ð12Þ
i
Fig. 2. The integrated flowerchart of the prediction procedure.
leY ðx Þ ðyi þ ci Þ is a member of the upper bound, and i
leY ðx Þ ðyi ci Þ is a member of the lower bound. i
In the FSVR model, the lower and upper regression lines must first be constructed by the SVR method with the lower bound of {(xi, YLi), i = 1, 2, . . . , N} and the upper bound of {(xi, YUi), i = 1, 2, . . ., N}, respectively, which can be written as follows:
Y L ðxi Þ ¼ wLi þ bL ¼
N X ðbLi bLi ÞKðx; xi Þ þ bL ; i¼1
ð13Þ
Table 1 The index of business cycles in Taiwan. Lights
Intervals of business cycle
Meanings of business activities
Red Yellow–red Green Yellow–blue Blue
[38, 45] [32, 37] [23, 31] [17, 22] [9, 16]
Excellent Good Medium Retrieving Not good
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Actual values of business index
Fuzzy lower/upper bounds
Training data
Testing data
Fig. 3. The index of Taiwanese business cycles with fuzzy intervals.
The upper bound approach values can be calculated by a similar method with H-level set as follows:
Y HU ðxi Þ ¼
N N X X ðbUi bUi ÞKðx; xi Þ þ bU H ðððbLi bLi Þ i¼1
i¼1
!
þðbUi bUi ÞÞKðx; xi Þ þ bL þ bU Þ=2 =ð1 HÞ:
ð18Þ
were used to design the forecasting model, and the testing data were employed to measure the forecasting performance of the different forecasting models. In addition, the root mean square error (RMSE), mean absolute percentage error (MAPE), and fitness function (J) of GA were used to measure the forecasting accuracy of the two models. Eqs. (19)–(21) illustrate the expression of RMSE, MAPE, J, respectively.
(
Therefore, the estimation of the FSVR model is simpler than the linear programming method. Furthermore, the FSVR model uses Genetic Algorithm (GA; Holland, 1962, 1975) to select the three parameters of SVR. Fig. 2 shows the flowchart of the proposed FSVR model. 4. A numerical example and empirical results
)0:5 M 1 X ðY Ct Y C ðxt ÞÞ2 ; M t¼1 M 100 X Y Ct Y C ðxt Þ; MAPE ð%Þ ¼ M t¼1 Y Ct
RMSE ¼
ð19Þ ð20Þ
M X ððY Ct Y C ðxt ÞÞ2 þ ðY Lt Y HL ðxt ÞÞ2 þ ðY Ut Y HU ðxt ÞÞ2 Þ
J¼
!0:5 ;
t¼1
Actual values
Predicted fuzzy bounds
Actual bounds
1 /1 0/
/1 /9
/1
08 20
/8
08 20
/1 /7
08 20
/1 /6
08 20
/1 /5
08 20
/1 20
08
/4
/1 /3
08 20
/1
08
/2 08
/1 20
10 40 0.5 0.1 100 epochs Two chromosomes
08
Value
Population size Crossover probability Mutation probability Termination condition Elitism
20
Parameter
Predicted central values 40 35 30 25 20 15 10 5 0 /1
Table 2 Parameters of GA.
ð21Þ
20
To represent the vague characteristics of business cycles index in Taiwan, the Council of Economic Planning and Development uses five business lights to indicate the various degrees of business cycles. Table 1 shows the lights, intervals, and meanings of economic activities of the different degrees of business cycles from the Council of Economic Planning and Development. Fig. 3 shows the monthly Taiwanese business index with fuzzy intervals from January 1998 to October 2008. The monthly data were divided into two sets: the training set (January 1998 to December 2007) and the testing set (from January 2008 to October 2008). The training data
Fig. 4. Illustration of the actual values of business cycles and the predicted values of the FSVR model with H = 0.3.
Table 3 The predicted results of the FSVR models, with sensitive analysis on the H-level. Date
2008/1/1 2008/2/1 2008/3/1 2008/4/1 2008/5/1 2008/6/1 2008/7/1 2008/8/1 2008/9/1 2008/10/1
Actual value
Parameters of FSVR
H = 0.1
yL
yR
yC
C/e/r
YL
YR
H = 0.3 YL
YR
H = 0.5 YL
YR
H = 0.7 YL
YR
YC
23 23 23 23 17 17 9 17 9 9
31 31 31 31 22 22 16 22 16 16
29 27 26 27 22 20 16 18 12 12
62.55/0.39/0.99 65.84/0.31/0.99 52.77/0.39/0.99 35.84/0.22/0.88 64.68/0.61/1.03 52.77/0.39/0.99 57.77/0.39/0.99 50.55/0.14/1.02 63.42/0.32/1.00 61.39/0.33/0.95
22.16 16.41 22.37 31.51 21.94 22.16 22.34 22.42 8.93 8.98
31.05 21.97 31.04 37.06 30.83 31.05 31.58 16.86 16.71 16.02
20.89 15.62 21.14 30.71 20.67 20.89 21.02 16.07 7.82 7.97
32.32 22.76 32.28 37.86 32.10 32.32 32.90 23.21 17.82 17.03
18.61 14.19 18.91 39.28 18.39 18.61 18.64 14.64 5.82 6.16
34.61 24.19 34.51 29.28 34.39 34.61 35.27 24.64 19.82 18.84
13.27 10.86 13.71 25.95 13.05 13.27 13.10 11.31 1.15 1.93
39.94 27.52 39.71 42.62 39.72 39.94 40.81 27.97 24.49 23.07
26.61 19.19 26.71 34.28 26.39 26.61 26.96 19.64 12.82 12.50
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Table 4 The predicted results of Wu and Tseng’s method, with sensitive analysis on the H-level. Date
Wu and Tseng-FR line YC = x0 + x1x + x2x
2008/1/1 2008/2/1 2008/3/1 2008/4/1 2008/5/1 2008/6/1 2008/7/1 2008/8/1 2008/9/1 2008/10/1
2
YC = 20.0810 + 0.0596x 0.00009x2 YC = 19.5568 + 0.0759x 0.00029x2 YC = 18.9826 + 0.0935x 0.00030x2 YC = 18.3551 + 0.1123x 0.00042x2 YC = 18.3322 + 0.1120x 0.00041x2 YC = 18.0626 + 0.1252x 0.00053x2 YC = 17.7887 + 0.1379x 0.00064x2 YC = 17.2515 + 0.1633x 0.00086x2 YC = 16.9793 + 0.1746x 0.00096x2 YC = 16.4222 + 0.1990x 0.00120x2
H = 0.1
H = 0.3
H = 0.5
H = 0.7
YL
YR
YL
YR
YL
YR
YL
YR
YC
19.93 22.53 22.41 22.27 22.27 21.83 21.41 20.39 20.01 19.04
32.14 29.40 29.35 29.28 29.43 28.90 28.39 27.45 26.99 26.11
21.68 21.55 21.41 21.27 21.25 20.83 20.41 19.38 19.02 18.03
30.40 30.38 30.34 30.28 30.46 29.91 29.38 28.46 27.99 27.13
26.04 19.79 19.63 19.46 19.41 19.01 18.62 17.57 17.22 16.21
32.14 32.14 32.12 32.08 32.30 31.73 31.18 30.28 29.79 28.94
15.86 15.67 15.46 15.26 15.11 14.77 14.44 13.33 13.03 11.96
36.21 36.26 36.29 36.29 36.60 35.96 35.36 34.51 33.97 33.19
26.04 25.96 25.88 25.77 25.85 25.37 24.90 23.92 23.50 22.58
where M is the number of forecasting periods; Y Ct ; Y Lt and Y Ut are, respectively, the actual value, lower and upper bound at period t; and Y C ðxt Þ is the forecasting value at period t; Y HL ðxt Þ and Y HU ðxt Þ are, respectively, the forecasting lower and upper bound value at period t with H-level set. Table 2 indicates the related parameters of GA. Table 3 illustrates the actual values and experimental results of the FSVR model with different H-level from January 2008 to October 2008. In addition, Table 3 shows the optimal parameters of the proposed FSVR model. Fig. 4 makes a point-to-point comparison of actual values and predicted values of FSVR. Fig. 4 shows that the FSVR model can efficiently capture the trend of data. Table 4 shows experimental result obtained by Wu and Tseng’s method with different H-levels. In Table 4, it is observed that the value of intercept coefficient (x0) gradually decreases. Fig. 5 shows a point-to-point comparison of the actual values and predicted values of Wu and Tseng’s method with H = 0.3. Fig. 5 indicates that Wu and Tseng’s method can not precisely capture the trend of data. Hence, the proposed FSVR model has better performance than Wu and Tseng’s method in predicting business cycles under uncertain circume ðxi Þ, which is represented stances. This study utilizes the mode of Y by YC, to measure performance of the two models. As shown in Table 5, the FSVR has better performance than Wu and Tseng’s
method with YC. Moreover, the study has examined the sensitive analysis of H-levels. Both models have the best performance with H = 0.3. In addition, the FSVR model has lower testing errors than Wu and Tseng’s method with H = 0.3. 5. Conclusions Due to the recent global economic recession, analysis of business cycles is increasingly crucial. This study develops a FSVR model to exploit the unique strength of the fuzzy set theory and the SVR technique, in order to predict business cycles in Taiwan. Simulation results indicate that the FSVR model offers a promising alternative in business cycles in uncertain circumstances. The superior performance of the FSVR model can be ascribed to two causes. First, the SVR can efficiently capture trends of nonlinear data, and precisely estimate upper bounds and lower bounds of fuzzy numbers. Second, based on sensitive analysis of H-level, the FSVR model can provide creditable predictions for Taiwanese business cycle predictions. For future work, forecasting other types of uncertain time series data by the FSVR model is a challenging issue for study. Future studies can also consider using data preprocessing techniques to improve the forecasting accuracy of the FSVR model. Acknowledgments
Predicted central values (Wu and Tseng-FR) This research was conducted with the supports of National Science Council (NSC 97-2410-H-262 -006) and (NSC 96-2628-E-260001-MY3).
Predicted fuzzy bounds (Wu and Tseng-FR)
References
Fig. 5. Illustration of actual values of business cycles and predicted values of Wu and Tseng’s method with H = 0.3.
Table 5 Comparison of performance of Wu and Tseng’s method with FSVR. Forecasting models
Wu and Tseng’s method
FSVR
RMSE of YC MAPE (%) of YC Testing error (J) with H = 0.3
6.43 33.59 37.38
5.54 20.85 34.98
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K.-P. Lin, P.-F. Pai / Expert Systems with Applications 37 (2010) 5430–5435 Huang, W., Nakamori, Y., & Wang, S. Y. (2005). Forecasting stock market movement direction with support vector machine. Computer and Operations Research, 32, 2513–2522. Jeng, J.-T., Chuang, C.-C., & Su, S.-F. (2003). Support vector interval regression networks for interval regression analysis. Fuzzy Set and Systems, 138, 283–300. Kim, K. J. (2003). Financial time series forecasting using support vector machines. Neurocomputing, 55, 307–319. Layton, A. P. (1996). Dating and predicting phase changes in the US business cycle. International Journal of Forecasting, 12, 417–428. Layton, A. P. (1998). A further test of the influence of leading indicators on the probability of US business cycle phase shifts. International Journal of Forecasting, 14, 63–70. Levis, A. A., & Papageorgiou, L. G. (2005). Customer demand forecasting via support vector regression analysis. Chemical Engineering Research and Design, 83, 1009–1018. Lu, W. Z., & Wang, W. J. (2005). Potential assessment of the ‘‘support vector machine” method in forecasting ambient air pollutant trends. Chemosphere, 59, 693–701. Mercer, J. (1909). Function of positive and negative type and their connection with the theory of integral equations. Philosophical Transactions of the Royal Society, A209, 415–446. Mohandes, M. A., Halawani, T. O., Rehman, S., & Hussain, A. A. (2004). Support vector machines for wind speed prediction. Renewable Energy, 29, 939–947. Pai, P. F., & Hong, W. C. (2005). An improved neural network model in forecasting tourist arrivals. Annals of Tourism Research, 32, 1138–1141.
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
A fuzzy support vector regression model for business cycle predictions Kuo-Ping Lin a,*, Ping-Feng Pai b,c a
Department of Information Management, Lunghwa University of Science and Technology, Taoyuan 333, Taiwan Department of Information Management, National Chi Nan University, University Rd., Puli, Nantou 545, Taiwan c Department of International Business Studies, National Chi Nan University, University Rd., Puli, Nantou 545, Taiwan b
a r t i c l e
i n f o
Keywords: Business cycle Fuzzy set theory Support vector regression
a b s t r a c t Business cycle predictions face various sources of uncertainty and imprecision. The uncertainty is usually linguistically determined by the beliefs of decision makers. Thus, the fuzzy set theory is ideally suited to depict vague and uncertain features of business cycle predictions. Consequently, the estimation of fuzzy upper and lower bounds become an essential issue in predicting business cycles in an uncertain environment. The support vector regression (SVR) model is a novel forecasting approach that has been successfully used to solve time series problems. However, the SVR approach has not been widely applied in fuzzy forecasting problems. This study employs support vector regressions to calculate fuzzy upper and lower bounds; and presents a fuzzy support vector regression (FSVR) model for forecasting indices of business cycles. A numerical example of a business cycle prediction in Taiwan was used to demonstrate the forecasting performance of the FSVR model. The empirical results are satisfactory. Therefore, the FSVR model is an effective alternative in forecasting business cycles under uncertain circumstances. Crown Copyright 2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction Accuracy in forecasting business cycles is an important issue in economic study, and statistical methods have usually been employed to analyze them. Many investigations have been done in the analysis of business cycles (Banerji & Hiris, 2001; Layton, 1996, 1998; Seip & McNown, 2007; Wu & Tseng, 2002; Yang & Kim, 2005). However, business cycles are often determined by a panel of macroeconomic experts, and thus, it is difficult to predict the index of business cycles. The difficulty arises from assumptions made from the probability distributions and business cycle data, which are usually vague. The index of business cycles in Taiwan is composed of nine exogenous variables, and five lights are used to represent different economic activities. The five lights include some uncertain factors in predicting business cycles. Hence, the fuzzy set theory (Zadeh, 1965) is a proper approach to analyze Taiwan business cycles. Unlike most of traditional technologies SVR (Vapnik, Golowich, & Smola, 1996) implementing neural network models, SVR adopts a structural risk minimization principle, which seeks to minimize the upper bounds of the generalization error rather than minimize the training error. In recent years, SVR schemes have been extended to cope with forecasting problems, and have provided many promising results in customer demand (Levis & Papageorgiou,
* Corresponding author. E-mail addresses: [email protected] (K.-P. Lin), [email protected] (P.-F. Pai).
2005), finance (Huang, Nakamori, & Wang, 2005; Kim, 2003; Tay & Cao, 2002), intermittent demand (Hua & Zhang, 2006), tourism demand (Pai & Hong, 2005), air quality (Lu & Wang, 2005), wind speed (Mohandes, Halawani, Rehman, & Hussain, 2004), plant control systems (Xi, Poo, & Chou, 2007), rainfall (Hong & Pai, 2007), prices for the electricity market (Gaoa, Bompard, Napoli, & Cheng, 2007), and flood control (Yu, Chen, & Chang, 2006). Hong and Hwang (2003) proposed a support vector fuzzy regression machine model for modifying convex optimization problems of multivariate fuzzy linear regression models. Empirical results indicate that the developed model derives satisfying solutions efficiently. Jeng, Chuang, and Su (2003) developed a support vector interval regression network to efficiently handle interval output data. Yao and Yu (2006) developed a fuzzy regression based on asymmetric support vector machines, which overcome limitations of traditional nonlinear fuzzy regression, and can be effectively used for parameter estimation. Chuang (2008) presented an interval support vector regression network model, which can handle interval input and output data. Hao and Chiang (2008) developed a fuzzy regression analysis model based on support vector learning techniques, and suggested that the developed model can perform automatic and accurate control in fuzzy regression analysis tasks. In this study, a fuzzy support vector regression model is presented to forecast an index of business cycles. Support vector regression was used to calculate fuzzy upper and lower bounds, and then make predictions by fuzzy H-level set (H-cut). In addition, genetic algorithms (GA) were employed to select three parameters of SVR models. The remainder of this paper is organized as follows.
0957-4174/$ - see front matter Crown Copyright 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.02.071
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A brief introduction of the theory of SVR is given in Section 2. The fuzzy support vector regression model is derived in Section 3. A numerical example of business cycle predictions and empirical results are presented in Section 4. Some concluding remarks are offered in Section 5.
This constrained optimization problem can be solved using the following primal Lagrangian form:
2. Support vector regression models
! N N X X 1 ðni þ ni Þ bi ½w/ðxi Þ þ b Y i þ e þ ni kwk2 þ C 2 i¼1 i¼1
Min
N X
bi ½Y i w/ðxi Þ b þ e þ ni
i¼1
Presented by Vapnik (1995), the support vector machine was originally applied to pattern recognition problems. Then, the support vector machine model has been successfully extended for dealing with nonlinear regression problems (Vapnik et al., 1996). The support vector regression model is based on the idea of mapping the original data x nonlinearly into a higher dimensional feature space. The SVR approach is to approximate an unknown function by a training data set {(xi, Yi), i = 1, . . . , N}. The regression function can be formulated as follows:
F ¼ w/ðxi Þ þ b;
N 1 X 1 RðFÞ ¼ C Le ðY i ; F i Þ þ kwk2 ; N i¼1 2
ð2Þ
Le ðY i ; F i Þ ¼
if jY i F i j 6 e
0
jY i F i j e otherwise
ð3Þ
;
! N X 1 2 f ðw; n; n Þ ¼ kwk þ C ðni þ ni Þ 2 i¼1
Subjectiv e to w/ðxi Þ þ b Y i 6 e þ ni ;
i ¼ 1; 2; . . . ; N;
Y i w/ðxi Þ b 6 e þ ni ;
i ¼ 1; 2; . . . ; N;
ni ; ni P 0;
i¼1
Subjectiv e to
N N X N X 1X bi þ bi ðbi bi Þðbj bj ÞKðxi ; xj Þ 2 i¼1 i¼1 j¼1
N X ðbi bi Þ ¼ 0; i¼1
f 0 6 bi 6 C; i ¼ 1; 2; . . . ; N; 0 6 bi 6 C; i ¼ 1; 2; . . . ; N: ð6Þ
bi
The Lagrange multipliers in Eq. (6) satisfy the equality bi ¼ 0. The Lagrange multipliers, bi and bi , are determined, and an optimal weight vector of the regression hyperplane is written by Eq. (7). N X ðbi bi ÞKðx; xi Þ:
ð7Þ
ð4Þ
Thus, the regression function is given by:
Fðx; b; b Þ ¼
N X ðbi bi ÞKðx; xi Þ þ b:
ð8Þ
i¼1
Herein, K(xi, xj) denotes a Kernel function whose value equals the inner product of two vectors, xi and xj , in the feature space / (xi) and /(xj), meaning that K(xi, xj)= /(xi) /(xj). Any function that satisfies Mercer’s condition (Mercer, 1909) can act as the Kernel function. This work uses the Gaussian function.
3. A fuzzy support vector regression model Introduced by Zadeh (1965), fuzzy set theory has been applied to deal with uncertain problems in many fields. A brief of fuzzy set theory is depicted as follows. A membership function is defined for all elements a in the referential set U. A fuzzy set (A) can be defined by the membership functions lA(a), which can have values of [0, 1]. Thus, a fuzzy set (A) is said to be convex if all ordinary subsets of A are convex. A fuzzy set is normalized if $a 2 U, lA(a) = 1. For an interval, level of confidence, or called a-cut or a-level set at level a (0, 1], an ordinary subset of A can be defined and denoted as [A]a,
½Aa ¼ fa 2 U lA ðaÞ P ag;
i ¼ 1; 2; . . . ; N:
a 2 ð0; 1:
ð9Þ
Furthermore, a fuzzy number (FN) can be defined as a convex, a normalized fuzzy set on a real line, with an upper semi-continuous membership function and bounded support. Dubois and Prade (1980) have defined a general representation form for FNs, which can be called the L–R type FNs. Here, FNs are represented as follows:
Lε (Yi, Fi)
−ε
Y i ðbi bi Þ e
i¼1
where C and e are user-defined parameters. The parameter e is the difference between actual values and values calculated from the regression function. This difference can be viewed as a tube around the regression function. The points outside the tube are regarded as training errors. In Eq. (2), Le(Yi, Fi) is called an e-insensitive loss function, and can be illustrated as Fig. 1. The loss equals zero if the approximate value is within the etube. Additionally, the second item of Eq. (2), 12 kwk2 , is adopted to estimate the flatness of a function which can avoid overfitting. Therefore, C indicates a parameter determining the trade-off between the empirical risk and the model flatness. Two positive slack variables (ni and ni ), representing the distance from actual values to the corresponding boundary values of the e-tube, are then introduced. These two slack variables equal zero when the data points fall within the e-tube. Eq. (2) is then reformulated into the following constrained form:
Min
N X
Max
w ¼
where
ð5Þ
i¼1
Eq. (5) is minimized with respect to primal variables w, b, n, and n*, and is maximized with regard to non-negative Lagrangian multipliers ai, ai , bi, and bi . Finally, Karush–Kuhn–Tucker conditions are applied to Eq. (4), and the dual Lagrangian form given by Eq. (6).
ð1Þ
where /(xi) denotes the feature of the inputs, and w and b indicate coefficients. The coefficients (wi and b) are estimated by minimizing the following regularized risk function.
N X ðai ni þ ai ni Þ:
+ε
Fig. 1. The e-insensitive loss function.
Yi − Fi
Definition 1 (The bound form representation). A symmetrical FN can be written as A = (w c, w, w + c)LR, where w c and w + c, respectively, denote the lower and upper bounds, w the mode or center value, (w c, w + c) forms the support, and L, R, respectively, denote the left and right reference (or shape) functions of A. In the case L R, and A has the membership function:
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8 0; > > > < Lðða wÞ=cÞ; lA ðaÞ ¼ > Rððw aÞ=cÞ; > > : 0;
a < w c; w c 6 a 6 w; w 6 a 6 w þ c;
ð10Þ
a > w þ c:
Likewise, non-symmetrical FNs can be defined. In addition, for example, a symmetrical triangular FN may be denoted as A = (w c, w, w + c)T specially with the symmetrical triangular membership function:
8 0; > > > < ða wÞ=c; lA ðaÞ ¼ > ðw aÞ=c; > > : 0;
a < w c; w c 6 a 6 w; w 6 a 6 w þ c;
ð11Þ
where the Lagrange multipliers, bLi and bLi , and an optimal weight vector wLi of the lower bound regression hyperplane can be estimated by Eq. (6):
Y U ðxi Þ ¼ wUi þ bU ¼
where the Lagrange multipliers, bUi and bUi , and an optimal weight vector wUi of the upper bound regression hyperplane can be estimated by Eq. (6). Then the fuzzy prediction model uses Eqs. (13) e ðxi Þ, which degree of membership and (14) to estimate the output Y function approaches precisely the H-level set. That is:
" N X ðbLi bLi ÞKðx; xi Þ þ bL ; i¼1
The representation can thus be called the bound form representation. Another representation can be used, and is given below. Definition 2 (The spread form representation). A symmetric FN can be written as A = (w, c)L, with the membership function lA(a) = L((a w)/c) "a 2 R, where w and c, respectively, represent the mode or center value, and the spread around the mode of A. The reference function L(x) thus has the properties (i) L(x) = L(x), (ii) L(0) = 1, L(1) = 0, (iii) L is decreasing on [0, 1], and (iv) L is invertible on [0, 1]. Based on the fuzzy set theory, fuzzy regression (FR) analysis is a methodology giving rise to a possibility distribution for an imprecise or vague phenomenon, which can be expressed by yielding fuzzy parameters. The FR can represent the data accrual without losing the original meaning, and analyze the trends of both variability and mean in data. FR analysis was first introduced by Tanaka, Uejima, and Asai (1980), who represented the observational uncertainty or fuzziness of a system by the fuzzy parameters for its indefinite structure. The objective is to minimize the overall fuzziness of the estimated outputs with the condition that an H-level-set inclusion of all data observed holds. Increasing the H-factor may expand the confidence interval of the FR, and increase the probability that out-of-sample data may fall within the model description. It is comparable to increase the confidence in the statistical regression model by increasing the confidence interval. However, in fuzzy input data exist certain outliers, according to the linear programming estimation technique, the approach could obtain a huge but useless estimated fuzzy output. Therefore, a more precise explanation should be considered, which the H-level set of membership functions for representative input data should be closed estimated precisely H-level set of membership function (Wu & Tseng, 2002). A fuzzy input data Yi belongs to Triangular FNs (TFNs), the membership of Yi is distributed on the interval [yi ci, yi + ci] and can be defined a TFNs as (yi ci, yi, yi + ci), where YLi = yi ci is lower bound value, YUi = yi + ci is upper bound value, and YCi = yi is mode value of FNs. The YLi and YUi must approach precisely H-level set. Hence, given the significant level H, the FSVR model satisfies Eq. (12):
leY ðx Þ ðyi þ ci Þ H; leY ðx Þ ðyi ci Þ H; i
where
ð14Þ
i¼1
e ðxi ÞH ¼ ½Y
a > w þ c:
N X ðbUi bUi ÞKðx; xi Þ þ bU ;
# N X ðbUi bUi ÞKðx; xi Þ þ bU : i¼1
ð15Þ Then, we can obtain estimated central vale YC(xi) by a simple calculation as follows:
Y C ðxi Þ ¼
N X
ðððbLi bLi Þ þ ðbUi bUi ÞÞKðx; xi Þ þ bL þ bU Þ=2:
ð16Þ
i¼1
Moreover, based on YC(xi), and Eqs. (9)–(11), we can derive the lower bound approach values with H-level set as follows:
Y HL ðxi Þ ¼
N N X X ðbLi bLi ÞKðx; xi Þ þ bL H ðððbLi bLi Þ i¼1
i¼1
!
þ bUi
bUi ÞÞKðx; xi Þ
þ bL þ bU Þ=2 =ð1 HÞ:
ð17Þ
Fuzzy input data
Find the lower and upper bounds
Training data
Testing data
Use SVR to approach the lower and upper bounds Calculate mode of fuzzy variables and estimate the lower and upper bounds with H-cut of fuzzy variables
Use FSVR model to forecast the lower and upper bounds with H-cut of fuzzy variables
Estimate mode of fuzzy variables
Use GA to select parameters of FSVR Calculate testing errors Calculate training errors
Is the GAs stop criterion satisfied ?
NO
Yes
ð12Þ
i
Fig. 2. The integrated flowerchart of the prediction procedure.
leY ðx Þ ðyi þ ci Þ is a member of the upper bound, and i
leY ðx Þ ðyi ci Þ is a member of the lower bound. i
In the FSVR model, the lower and upper regression lines must first be constructed by the SVR method with the lower bound of {(xi, YLi), i = 1, 2, . . . , N} and the upper bound of {(xi, YUi), i = 1, 2, . . ., N}, respectively, which can be written as follows:
Y L ðxi Þ ¼ wLi þ bL ¼
N X ðbLi bLi ÞKðx; xi Þ þ bL ; i¼1
ð13Þ
Table 1 The index of business cycles in Taiwan. Lights
Intervals of business cycle
Meanings of business activities
Red Yellow–red Green Yellow–blue Blue
[38, 45] [32, 37] [23, 31] [17, 22] [9, 16]
Excellent Good Medium Retrieving Not good
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Actual values of business index
Fuzzy lower/upper bounds
Training data
Testing data
Fig. 3. The index of Taiwanese business cycles with fuzzy intervals.
The upper bound approach values can be calculated by a similar method with H-level set as follows:
Y HU ðxi Þ ¼
N N X X ðbUi bUi ÞKðx; xi Þ þ bU H ðððbLi bLi Þ i¼1
i¼1
!
þðbUi bUi ÞÞKðx; xi Þ þ bL þ bU Þ=2 =ð1 HÞ:
ð18Þ
were used to design the forecasting model, and the testing data were employed to measure the forecasting performance of the different forecasting models. In addition, the root mean square error (RMSE), mean absolute percentage error (MAPE), and fitness function (J) of GA were used to measure the forecasting accuracy of the two models. Eqs. (19)–(21) illustrate the expression of RMSE, MAPE, J, respectively.
(
Therefore, the estimation of the FSVR model is simpler than the linear programming method. Furthermore, the FSVR model uses Genetic Algorithm (GA; Holland, 1962, 1975) to select the three parameters of SVR. Fig. 2 shows the flowchart of the proposed FSVR model. 4. A numerical example and empirical results
)0:5 M 1 X ðY Ct Y C ðxt ÞÞ2 ; M t¼1 M 100 X Y Ct Y C ðxt Þ; MAPE ð%Þ ¼ M t¼1 Y Ct
RMSE ¼
ð19Þ ð20Þ
M X ððY Ct Y C ðxt ÞÞ2 þ ðY Lt Y HL ðxt ÞÞ2 þ ðY Ut Y HU ðxt ÞÞ2 Þ
J¼
!0:5 ;
t¼1
Actual values
Predicted fuzzy bounds
Actual bounds
1 /1 0/
/1 /9
/1
08 20
/8
08 20
/1 /7
08 20
/1 /6
08 20
/1 /5
08 20
/1 20
08
/4
/1 /3
08 20
/1
08
/2 08
/1 20
10 40 0.5 0.1 100 epochs Two chromosomes
08
Value
Population size Crossover probability Mutation probability Termination condition Elitism
20
Parameter
Predicted central values 40 35 30 25 20 15 10 5 0 /1
Table 2 Parameters of GA.
ð21Þ
20
To represent the vague characteristics of business cycles index in Taiwan, the Council of Economic Planning and Development uses five business lights to indicate the various degrees of business cycles. Table 1 shows the lights, intervals, and meanings of economic activities of the different degrees of business cycles from the Council of Economic Planning and Development. Fig. 3 shows the monthly Taiwanese business index with fuzzy intervals from January 1998 to October 2008. The monthly data were divided into two sets: the training set (January 1998 to December 2007) and the testing set (from January 2008 to October 2008). The training data
Fig. 4. Illustration of the actual values of business cycles and the predicted values of the FSVR model with H = 0.3.
Table 3 The predicted results of the FSVR models, with sensitive analysis on the H-level. Date
2008/1/1 2008/2/1 2008/3/1 2008/4/1 2008/5/1 2008/6/1 2008/7/1 2008/8/1 2008/9/1 2008/10/1
Actual value
Parameters of FSVR
H = 0.1
yL
yR
yC
C/e/r
YL
YR
H = 0.3 YL
YR
H = 0.5 YL
YR
H = 0.7 YL
YR
YC
23 23 23 23 17 17 9 17 9 9
31 31 31 31 22 22 16 22 16 16
29 27 26 27 22 20 16 18 12 12
62.55/0.39/0.99 65.84/0.31/0.99 52.77/0.39/0.99 35.84/0.22/0.88 64.68/0.61/1.03 52.77/0.39/0.99 57.77/0.39/0.99 50.55/0.14/1.02 63.42/0.32/1.00 61.39/0.33/0.95
22.16 16.41 22.37 31.51 21.94 22.16 22.34 22.42 8.93 8.98
31.05 21.97 31.04 37.06 30.83 31.05 31.58 16.86 16.71 16.02
20.89 15.62 21.14 30.71 20.67 20.89 21.02 16.07 7.82 7.97
32.32 22.76 32.28 37.86 32.10 32.32 32.90 23.21 17.82 17.03
18.61 14.19 18.91 39.28 18.39 18.61 18.64 14.64 5.82 6.16
34.61 24.19 34.51 29.28 34.39 34.61 35.27 24.64 19.82 18.84
13.27 10.86 13.71 25.95 13.05 13.27 13.10 11.31 1.15 1.93
39.94 27.52 39.71 42.62 39.72 39.94 40.81 27.97 24.49 23.07
26.61 19.19 26.71 34.28 26.39 26.61 26.96 19.64 12.82 12.50
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Table 4 The predicted results of Wu and Tseng’s method, with sensitive analysis on the H-level. Date
Wu and Tseng-FR line YC = x0 + x1x + x2x
2008/1/1 2008/2/1 2008/3/1 2008/4/1 2008/5/1 2008/6/1 2008/7/1 2008/8/1 2008/9/1 2008/10/1
2
YC = 20.0810 + 0.0596x 0.00009x2 YC = 19.5568 + 0.0759x 0.00029x2 YC = 18.9826 + 0.0935x 0.00030x2 YC = 18.3551 + 0.1123x 0.00042x2 YC = 18.3322 + 0.1120x 0.00041x2 YC = 18.0626 + 0.1252x 0.00053x2 YC = 17.7887 + 0.1379x 0.00064x2 YC = 17.2515 + 0.1633x 0.00086x2 YC = 16.9793 + 0.1746x 0.00096x2 YC = 16.4222 + 0.1990x 0.00120x2
H = 0.1
H = 0.3
H = 0.5
H = 0.7
YL
YR
YL
YR
YL
YR
YL
YR
YC
19.93 22.53 22.41 22.27 22.27 21.83 21.41 20.39 20.01 19.04
32.14 29.40 29.35 29.28 29.43 28.90 28.39 27.45 26.99 26.11
21.68 21.55 21.41 21.27 21.25 20.83 20.41 19.38 19.02 18.03
30.40 30.38 30.34 30.28 30.46 29.91 29.38 28.46 27.99 27.13
26.04 19.79 19.63 19.46 19.41 19.01 18.62 17.57 17.22 16.21
32.14 32.14 32.12 32.08 32.30 31.73 31.18 30.28 29.79 28.94
15.86 15.67 15.46 15.26 15.11 14.77 14.44 13.33 13.03 11.96
36.21 36.26 36.29 36.29 36.60 35.96 35.36 34.51 33.97 33.19
26.04 25.96 25.88 25.77 25.85 25.37 24.90 23.92 23.50 22.58
where M is the number of forecasting periods; Y Ct ; Y Lt and Y Ut are, respectively, the actual value, lower and upper bound at period t; and Y C ðxt Þ is the forecasting value at period t; Y HL ðxt Þ and Y HU ðxt Þ are, respectively, the forecasting lower and upper bound value at period t with H-level set. Table 2 indicates the related parameters of GA. Table 3 illustrates the actual values and experimental results of the FSVR model with different H-level from January 2008 to October 2008. In addition, Table 3 shows the optimal parameters of the proposed FSVR model. Fig. 4 makes a point-to-point comparison of actual values and predicted values of FSVR. Fig. 4 shows that the FSVR model can efficiently capture the trend of data. Table 4 shows experimental result obtained by Wu and Tseng’s method with different H-levels. In Table 4, it is observed that the value of intercept coefficient (x0) gradually decreases. Fig. 5 shows a point-to-point comparison of the actual values and predicted values of Wu and Tseng’s method with H = 0.3. Fig. 5 indicates that Wu and Tseng’s method can not precisely capture the trend of data. Hence, the proposed FSVR model has better performance than Wu and Tseng’s method in predicting business cycles under uncertain circume ðxi Þ, which is represented stances. This study utilizes the mode of Y by YC, to measure performance of the two models. As shown in Table 5, the FSVR has better performance than Wu and Tseng’s
method with YC. Moreover, the study has examined the sensitive analysis of H-levels. Both models have the best performance with H = 0.3. In addition, the FSVR model has lower testing errors than Wu and Tseng’s method with H = 0.3. 5. Conclusions Due to the recent global economic recession, analysis of business cycles is increasingly crucial. This study develops a FSVR model to exploit the unique strength of the fuzzy set theory and the SVR technique, in order to predict business cycles in Taiwan. Simulation results indicate that the FSVR model offers a promising alternative in business cycles in uncertain circumstances. The superior performance of the FSVR model can be ascribed to two causes. First, the SVR can efficiently capture trends of nonlinear data, and precisely estimate upper bounds and lower bounds of fuzzy numbers. Second, based on sensitive analysis of H-level, the FSVR model can provide creditable predictions for Taiwanese business cycle predictions. For future work, forecasting other types of uncertain time series data by the FSVR model is a challenging issue for study. Future studies can also consider using data preprocessing techniques to improve the forecasting accuracy of the FSVR model. Acknowledgments
Predicted central values (Wu and Tseng-FR) This research was conducted with the supports of National Science Council (NSC 97-2410-H-262 -006) and (NSC 96-2628-E-260001-MY3).
Predicted fuzzy bounds (Wu and Tseng-FR)
References
Fig. 5. Illustration of actual values of business cycles and predicted values of Wu and Tseng’s method with H = 0.3.
Table 5 Comparison of performance of Wu and Tseng’s method with FSVR. Forecasting models
Wu and Tseng’s method
FSVR
RMSE of YC MAPE (%) of YC Testing error (J) with H = 0.3
6.43 33.59 37.38
5.54 20.85 34.98
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