Empirical mode decomposition-based least squares support vector regression for foreign exchange rate forecasting

Empirical mode decomposition

–based least squares support vector regression for

foreign exchange rate forecasting

Chiun-Sin Lin

, Sheng-Hsiung Chiu

, Tzu-Yu Lin

⁎

Department of Business and Entrepreneurial Management, Kainan University, No.1 Kainan Road, Luzhu Shiang, Taoyuan 33857, Taiwan

b_{Department of Management Science, National Chiao Tung University, No. 1001, Ta-Hsueh Road, Hsinchu City 30010, Taiwan}

a b s t r a c t

a r t i c l e i n f o

Article history: Accepted 11 July 2012 Keywords:

Empirical mode decomposition Least-squares support vector regression Foreign exchange rate forecasting Intrinsic mode function

To address the nonlinear and non-stationary characteristics offinancial time series such as foreign exchange rates, this study proposes a hybrid forecasting model using empirical mode decomposition (EMD) and least squares support vector regression (LSSVR) for foreign exchange rate forecasting. EMD is used to decompose the dynamics of foreign exchange rate into several intrinsic mode function (IMF) components and one residual component. LSSVR is constructed to forecast these IMFs and residual value individually, and then all these fore-casted values are aggregated to produce thefinal forecasted value for foreign exchange rates. Empirical results show that the proposed EMD-LSSVR model outperforms the EMD-ARIMA (autoregressive integrated moving average) as well as the LSSVR and ARIMA models without time series decomposition.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Financial time series forecasting has come to play an important role in the world economy as a result of its ability to forecast economic

benefits and influence countries' economic development; it has

attracted increasing attention from academic researchers and business practitioners for its theoretical possibilities and practical applications (Hadavandi et al., 2010; Lu et al., 2009). Since the breakdown of

finan-cial market boundaries in order to enhance the efficiency of capital

funding, for example, the Bretton Woods system of monetary manage-ment was officially ended in the 1973; currencies traded internationally

has become crucial economic indices for international trade,financial

markets, the alignment of economic policy by governments, and

corpo-rate_{financial decision-making. However, it is widely known that}

finan-cial time series forecasting has shortcomings, including its inherent nonlinearity and non-stationarity (Huang et al., 2010; Lu et al., 2009). Therefore,_{financial time series forecasting is one of the most} challeng-ing tasks in thefinancial markets.

For modeling financial time series, autoregressive integrated

moving average (ARIMA) models have been popular and are widely chosen for academic research observing the behavior of foreign exchanges and stock markets, because of their statistical properties

and forecasting performance (Khashei et al., 2009). However, some

problems arise when forecasting financial time series with ARIMA

models, as follows. First is the characteristic linear limitation of ARIMA

models, in contrast to real-wordfinancial time series, which are often

nonlinear (Khashei et al., 2009; Zhang, 2001; Zhang et al., 1998) and are rarely pure linear combinations. Second is the robustness limitation

of ARIMA models—the ARIMA model selection procedure depends

greatly on the competence and experience of the researchers to yield desired results. Unfortunately, choice among competing models can be arbitrated by similar estimated correlation patterns and may fre-quently reach inappropriate forecasting results.

With the recent development of machine-learning algorithms, several methods have been utilized that work more effectively than the traditional linear model in time series forecasting problems. For example, the support vector machine (SVM) is a novel machine-learning approach. SVM's generalization capability in obtaining a

unique solution (Lu et al., 2009) and structural risk minimization

principle (SRM) in achieving high performance (Duan and Stanley,

2011) have drawn attention to SVM's research applications. Support

vector regression (SVR) is the regression model of SVM (Vapnik,

2000). It has been applied to investigate the forecasting ability of finan-cial time series.Lu et al. (2009)used SVR to construct a stock price

fore-casting model, andHuang et al. (2010)andNi and Yin (2009)both

implemented SVR in exchange rate forecasting models. However, the training phrase of SVR is a time-consuming process when there is a lot of data to deal with. Therefore, least squares support vector

regression (LSSVR), proposed by Suykens and Vandewalle (1999),

has been applied in much literature as an alternative (He et al., 2010; Khemchandani et al., 2009); it is a simplified version of traditional SVR that alters inequality constraints into equal conditions and employs a squared loss function, which is a differential setting relative to

tradi-tional SVR (Wang et al., 2011), to achieve higher calculation speed

and efficiency while retaining the advantage of the structural risk min-imization principle.

⁎ Corresponding author. Tel.: +886 921865025. E-mail address:blackmallows@gmail.com(T.-Y. Lin).

0264-9993/$– see front matter © 2012 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.econmod.2012.07.018

Contents lists available atSciVerse ScienceDirect

Economic Modelling

When we modelfinancial time series using LSSVR or ARIMA, we

must remember that these financial time series are inherently

nonlinear and non-stationary. If we ignore this problem, it will result in worse forecasting. The property offinancial time series and the divide and conquer principle (Yu et al., 2008) are important in constructing a financial time series forecasting model. Therefore, hybrid models are widely used to solve the limitations infinancial time series forecasting.

Empirical mode decomposition (EMD) is suitable for financial time

series in terms offinding fluctuation tendency, which simplifies the

forecasting task into several simple forecasting subtasks. EMD as a

time–frequency resolution approach offers a new way by which the

-stationary and nonlinear behavior of time series can be decomposed

into a series of valuable independent time resolutions (Tang et al.,

2012). It also can reveal the hidden patterns and trends of time series, which can effectively assist in designing forecasting models for various applications (An et al., 2012; Guo et al., 2012).Guo et al. (2012), for ex-ample, decomposed wind-speed series using EMD and then forecasted

them using a feed-forward neural network, whereas Chen et al.

(2012)proposed an EMD approach combined with an arti_{ficial neural} network for tourism-demand forecasting.

In this paper, EMD and LSSVR are used to present afinancial time

series forecasting model for foreign exchange rate, in which

consider-ation of the decomposedfinancial time series structure will increase

the accuracy and practicability of the proposed model in terms of over-coming the nonlinearity and non-stationarity limitations to the linear statistical model. The proposed approach is compared with the com-bination of EMD and ARIMA as well as with the existing LSSVR and ARIMA models, and it is shown that the proposed model can yield more accurate results. Threefinancial time series are used as illustrative examples, as follows: USD/NTD exchange rate, JPY/NTD exchange rate, and RMB/NTD exchange rate.

2. Methodology

2.1. Empirical mode decomposition

Empirical mode decomposition (EMD) is a nonlinear

signal-transformation method developed byHuang et al. (1998, 1999). It is

used to decompose a nonlinear and non-stationary time series into a sum of intrinsic mode function (IMF) components with individual

intrinsic time scale properties. According to Huang et al. (1998),

each IMF must satisfy the following two conditions. First, the number

of extreme values and zero-crossings either are equal or differ at the most by one; and second, the mean value of the envelope constructed by the local maxima and minima is zero at any point. The

detail-decomposition process of EMD is presented byHuang et al. (1998).

Suppose that a data time series can be decomposed according to the fol-lowing procedure.

(1) Identify all the local maxima and minima of x(t).

(2) Obtain the upper envelope xu(t) and the lower envelope xl(t)

of the x(t).

(3) Use the upper envelope xu(t) and the lower envelope xl(t) to

compute the first mean time series m1(t), that is, m1(t) =

[xu(t) + xl(t)]/2.

(4) Evaluate the difference between the original time series x(t) and the mean time series and get thefirst IMF h1(t), that is, h1(t)=

x(t)−m1(t). Moreover, we see whether h1(t) satisfies the two

con-ditions of an IMF property. If they are not satisfied, we repeat steps

1–3 of the decomposition procedure to eventually find the first

IMF.

(5) After we obtain thefirst IMF, a repeat of the above steps is

neces-sary tofind the second IMF, until we reach the final time series

r(t) as a residue component that becomes a monotonic function, which is suggested for stopping the decomposition procedure (Huang et al., 1999).

The original time series x(t) can be reconstructed by summing up

all the IMF components and one residue component as Eq.(1), as

follows.

x tð Þ ¼X

n

i¼1

hið Þ þ r tt ð Þ: ð1Þ

2.2. Least squares support vector regression

The support vector machine (SVM) developed byVapnik (1995,

2000) is based on the SRM principle. It aims to minimize the upper

bound of the generalization error, instead of the empirical error as in other neural-network methods such as back-propagation networks (BPN). SVM explores not only the problem of classification but also the regression application of forecasting.Vapnik et al. (1997)proposed sup-port vector regression (SVR) as an SVM regression estimation model, introducing the concept of theε-loss function.

Foreign Exchange Rate

Data

EMD

IMF1

IMF2

...

IMFn-1

IMFn

R

SVR1

SVR2

...

SVRn-1

SVR

SVRn+1

Prediction Results

Input

Output

put

SVR performs by nonlinearly mapping the input space into a high-dimensional feature space, and then runs the linear regression in the output space. This allows us to formulate the nonlinear relationship between input data and output data. The formulation of SVR basically is represented the following linear estimation function:

f xð Þ ¼ ω⋅ϕ xð Þ þ b;i ð2Þ

whereω denotes the weight vector, b is the bias, ϕ(xi) represents a

mapping function that aims to map the input vectors into a

high-dimensional feature space, andω⋅ϕ(xi) describes the dot production

in the feature space.

In SVR, the problem of nonlinear regression in the low-dimension input space is transformed into a linear regression problem in a high-dimension feature space. That is, the original optimization problem

involving a nonlinear regression is recast as a search for theflattest

function in the feature space, not in the input space. However, LSSVR

is the least squares version of SVR, andfinds the solution by solving a

set of linear equations instead of a quadratic programming problem (Iplikci, 2006). In LLSVR, the regression problem can be applied to the following optimization problem:

min1 2

‖

‖

where eirepresents the error from the training set and C is the penalty

parameter to be used to limit the minimization of estimation error and function smoothness.

In order to derive the optimization problem of Eq. (3), the

Lagrange function is formulated for Eq.(3)tofind out the solutions

toω and e; this can be written as follows.

L wð ; b; e; αÞ ¼1 2

‖

‖

whereαi= (α1,…,αl) are Lagrange multipliers, which can be expressed

as either positive or negative. Thefirst-order conditions for optimality are as follows. ∂L ∂ω¼ ω− Xl i¼1 αiϕ xð Þ ¼ 0i ð5Þ ∂L ∂b¼ Xl i¼1 αi¼ 0 ð6Þ

Fig. 2. The daily USD/NTD exchange rate form July 2005 to December 2009.

Fig. 3. The daily JPY/NTD exchange rate form July 2005 to December 2009.

Fig. 4. The daily RMB/NTD exchange rate form July 2005 to December 2009.

Table 1

Descriptive statistics of the exchange rare data. Currencies Numbers Mean Standard

deviation Max Min USD/NTD All sample 1130 32.537 0.895 35.174 30.010 Training 904 32.424 0.894 34.050 30.010 Testing 226 32.987 0.745 35.174 32.030 JPY/NTD All sample 1130 0.303 0.032 0.383 0.264 Training 904 0.291 0.023 0.383 0.264 Testing 226 0.351 0.010 0.375 0.329 RMB/NTD All sample 1130 4.403 0.313 5.138 3.829 Training 904 4.297 0.251 4.294 3.829 Testing 226 4.829 0.107 5.138 4.692 Table 2

Performance and their definition. Metrics Calculation MAPE MAPE¼1 n Xn i¼1 Ti Ai Ti     100% RMSE RMSE¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n Xn i¼1 Ti Ai ð Þ2 s MAD MAD¼1 n Xn i¼1 Ti Ai j j DS DS¼100 n Xn u¼1 di; where di¼ 1 Ti−Tt−1 ð Þ Aði−At−1Þ ≥ 0 0 otherwise   CP CP¼100 n1 Xn u¼1

di; where di¼ 1 A₀ð I−A_otherwiset−1Þ ≻ 0 and Tð i−Tt−1Þ Að i−At−1Þ ≥ 0



CD CD¼100_n

2

Xn u¼1

di; where di¼ 1 A₀ð I−A_otherwiset−1Þ ≺ 0 and Tð i−Tt−1Þ Að i−At−1Þ ≥ 0



Note that A and T represent the actual and forecasted value, respectively. n is total number of data points, n1is number of data points belong to up trend and n2is number of data

∂L ∂ei ¼ C·ei−αi¼ 0 ð7Þ ∂L ∂αi ¼ ω·ϕ xð Þ þ b þ ei i−yi¼ 0: ð8Þ

By solving the above linear system, the forecasting formulation of LSSVR can be represented in the following equation:

f xð Þ ¼X l i¼1 αiK xi; xj   þ b; ð9Þ

where K(xi,xj) is called the“kernel function” and must satisfy Mercer's

theorem (Vapnik, 1995). The value of the kernel equals the inner

product of two vectors, xi and xj, in the feature space ϕ(xi) and

ϕ(xj); that is, K(xi,xj) =ϕ(xi)⋅ϕ(xj).

The most widely used kernel function is the Gaussian radial basis function (RBF), defined as K x i; xj ¼ exp −

‖

‖

2

2σ2 !

, whereσ

de-notes the width of the RBF. Moreover, the RBF kernel is not only easier to implement than alternatives, but also capable of nonlinearly map-ping the training data into an infinite-dimensional space; thus, it deals suitably with nonlinear relationship problems. Thus, the Gaussian RBF kernel function is used in this work. LSSVR parameter selection is

most important, so that we can see that the established LLSVR model with Gaussian RBF kernel function goes well, because these parameters can signi_{ficantly affect generalized performance. Grid search (}He et al.,

2010), one of the most useful methods for parameter optimization,

is applied tofind the optimal parameters, C and σ, in LSSVR model

construction.

3. Proposed EMD-LSSVR model

Many studies have used least squares support vector regression in practice problems (Khemchandani et al., 2009; Lin et al., in press). In financial time series forecasting, however, the major problems are inher-ent nonlinearity and non-stationary properties affecting the robustness of

the LSSVR model significantly. For this reason, the proposed EMD-LSSVR

model is employed according to the principle of decomposition and ensemble (He et al., 2010;Wang et al., 2011). The procedure of the

pro-posed EMD-LSSVR structure is shown inFig. 1and generally consists

of the following three steps: (1) data decomposition, (2) forecasting-model construction, and (3) data reconstruction and validation.

(1) Suppose there is a foreign exchange rate time series x(t) that also

can be decomposed into n IMF components, hi(t), i = 1, 2,…,n,

and one residue component, r(t), through the EMD approach, as in Eq.(1).

(2) After the data decomposition, each obtained IMF component and residual component is further input to the LSSVR forecasting

model; consequently, the corresponding forecasted values for all IMF and residual components are acquired from the forecasting tool.

(3) The forecasted value of each IMF and residual component in the previous stage can be reconstructed as a sum of superposition of all components, which can be used as the_{final forecasting result,} and then compared with the original time series according to several criteria for measuring the performance capability of this proposed model.

4. Experimental results 4.1. Exchange rate dataset

To evaluate the performance of the proposed forecasting models using EMD with LSSVR methodologies, this study uses daily USD/NTD, JPY/NTD, and RMB/NTD exchange rates obtained from the Central Bank of Taiwan and Yahoo Finance. The whole daily data of exchange rates from July 1, 2005 to December 31, 2009 are used, for a total of

1130 data points, as illustrated inFigs. 2–4for each of the three rates respectively. These datasets are also divided into training group and

testing group in separate foreign exchange rate categories. The_first

904 data points (80% of the total dataset) are used as the training group and the remaining 226 data points are used as the testing

group.Table 1shows some basic summary statistics for total, training,

and testing data within the three foreign exchange rate datasets. 4.2. Performance criteria

FollowingLu et al. (2009)andTay and Cao (2001), the following

performance measures are used and evaluated respectively: applied

mean absolute percentage error (MAPE), root‐mean‐square error

(RMSE), mean absolute difference (MAD), directional symmetry (DS), correct uptrend (CP), and correct downtrend (CD) for

consider-ation. The definition of these criteria can be summarized inTable 2.

MAPE, RMSE, and MAD are measures of the deviation between the actual and forecasted value. They can be used to evaluate forecasting error. The smaller the values of the criteria, the closer the forecasted value to the actual value. DS provides the correctness of the forecasted direction of the exchange rate in terms of percentage, while CP and CD provide the correctness of the forecasted up trend and down trends of exchange rate, also in terms of percentage. DS, CP, and CD can be uti-lized to provide forecasting accuracy.

4.3. Forecasting results

In this section, the forecasting results of the EMD-LSSVR model are compared to those of other linear and nonlinear models. First is another hybrid forecasting model, one that integrates EMD with ARIMA. EMD is applied to decompose the foreign exchange rate time series, and gathered components that have a monotonic function, enhancing the forecasting ability of LSSVR and ARIMA. The others are the single LSSVR and ARIMA models without algorithms or treatments for fore-casting. That is, the single LSSVR and ARIMA model were directly ap-plied to forecast future exchange rates. The purpose of doing so is to

explore the problem offinancial time series forecasting based on linear

and nonlinear models, whether we can preprocess forecasting variables using the EMD approach or not, thus helping to further managerial applications.

The modeling steps of the proposed EMD-LSSVR are shown in

Section 3. Using the EMD approach in the data decomposition, the three foreign exchange rate time series can be decomposed into nine independent IMFs and one residue component, respectively, as

illus-trated in Figs. 5–7. These decomposition results may enhance the

model's forecasting ability in terms of the divide and conquer concept (Yu et al., 2008). Then, the decomposed forecasting variables, the inde-pendent IMFs, and residual components from the previous step, are used in LSSVR model construction. Parameter selection is essential for LSSVR model construction; we employ the Gaussian RBF as the kernel

function of LSSVR. The parameter combination (C andσ) was selected

by grid search, as suggested in He et al. (2010) andVan Gestel et al.

(2004). The optimal values of C and_{σ for each EMD-LSSVR forecasting} model are presented inTable 3. At the reconstruction step, we combine all forecasted values from the individual EMD-LSSVR models in order to compare them with the actual foreign exchange rate date, so as to vali-date the forecasting ability of the EMD-LSSVR model.

The same EMD-based methodology steps are also fed into ARIMA in order to build the hybrid linear foreign exchange rate forecasting model, namely, the EMD-ARIMA model, the results of which are com-pared with those of the EMD-LSSVR model. As well, the pure LSSVR and ARIMA models are applied for comparison. The optimal parameter combination selected for the single LSSVR model is listed inTable 4, and the performance evaluation of each forecasting model is based on the several performance criteria fromSection 4.2, as listed inTable 2. The performance measurements of the selected forecasting models are given inTable 5.

4.4. Comparison of forecasting results

In order to verify the forecasting capability of the proposed EMD-LSSVR model, the EMD-ARIMA, EMD-LSSVR and ARIMA models are employed for comparison, using three foreign exchange rate data sets: (1) the USD/NTD exchange rate data set, (2) the JPY/NTD exchange rate data set, and (3) the RMB/NTD exchange rate data set. MAPE, RMSE, MAD, DS, CP and CD, which are computed from the equations mentioned in

Table 2, are used as performance indicators to further survey the fore-casting performance of the proposed EMD-LSSVE model as compared to other linear and nonlinear models.

Take the USD/NTD exchange rate as an example-the forecasting results using EMD-LSSVR, EMD-ARIMA, LSSVR, and ARIMA are

comput-ed and listcomput-ed inTable 5, where can be seen that the MAPE, RMSE, and

MAD of the EMD-LSSVR model are, respectively, 0.21%, 0.0188, and 0.0135. These values are the smallest of all the forecasting models, that the deviation between actual and forecasted values in the EMD-LSSVR model is the smallest. Moreover, EMD-EMD-LSSVR also has higher

Table 3

The parameter selection of EMD-LSSVR forecasting model.

Kernel USD/NTD JPY/NTD RMB/NTD RBF Set C σ Set C σ Set C σ

1 0.5 0.00390625 1 8 0.00390625 1 0.5 0.015625 2 64 1 2 1 0.00390625 2 0.5 0.00390625 3 64 1 3 64 0.5 3 1 0.0078125 4 16 0.5 4 64 0.5 4 64 1 5 64 1 5 64 1 5 32 1 6 64 1 6 64 1 6 64 1 7 64 1 7 64 1 7 32 1 8 64 1 8 64 1 8 64 1 9 64 1 9 64 0.015625 9 32 0.25 10 64 1 10 64 1 10 64 1 Table 4

The parameter selection of single LSSVR forecasting model.

Kernel USD/NTD JPY/NTD RMB/NTD

RBF C σ C σ C σ

64 0.25 64 0.125 64 0.0312

Table 5

The exchange rate forecasting results using EMD-LSSVR, EMD-ARIMA, LSSVR and ARIMA models.

Models Indicators

MAPE (%) REMSE MAD DS (%) CP (%) CD (%) USD/NTD EMD-LSSVR 0.48 0.0109 0.0081 86.37 87.88 86.28 EMD-ARIMA 1.21 0.0173 0.0159 80.78 74.46 84.19 LSSVR 1.06 0.0262 0.0231 83.54 82.12 83.42 ARIMA 11.06 0.1039 0.1020 73.53 74.77 73.09 JPY/NTD EMD-LSSVR 0.57 0.0026 0.0020 86.37 87.63 82.74 EMD-ARIMA 2.29 0.0183 0.0150 83.89 82.54 86.17 LSSVR 2.94 0.0267 0.0212 81.77 78.55 82.22 ARIMA 7.19 0.0302 0.0400 73.67 76.17 70.75 RMB/NTD EMD-LSSVR 0.48 0.0109 0.0081 86.37 87.88 86.28 EMD-ARIMA 1.21 0.0173 0.0159 80.78 74.46 84.19 LSSVR 1.06 0.0262 0.0231 83.54 82.12 83.42 ARIMA 11.06 0.1039 0.1020 73.53 74.77 73.09

DS, CP, and CD ratios, 87.26%, 85.88%, and 89.38%, respectively. DS, CP, and CD provide a good measure of forecasting consistency of moving exchange rate trends. In sum, it can be concluded that EMD-LSSVR provides better forecasting accuracy and direction criteria for USD/ NTD exchange rate than EMD-ARMA, LSSVR, or ARIMA. In addition, the results of EMD-LSSVR are consistent with the principle of

decompo-sition and ensemble (He et al., 2011; Wang et al., 2010). Time series

decomposition may enhance forecasting ability. For example, in terms of DS indicators from the USD/NTD exchange rate forecasting, as

shown inTable 6, relative to the comparison models, the improvement

percentages of the proposed model are 8.02%, 5.01% and 17.51%, respectively.

The forecasting results and performance comparisons of the four forecasting models for JPY/NTD and RMB/NTD are also reported in

Tables 5 and 6. In addition, we see that the decomposition of time series in EMD can enhance the forecasting ability of nonlinear and linear models.

5. Conclusions

There has been increasing attention given tofinding an effective

model to address the problem offinancial time series forecasting in

terms of nonlinear and non-stationary characteristics. In this paper, an EMD-based LSSVR forecasting model is proposed. EMD is used to detect the moving trend offinancial time series data and improve the forecast-ing success of LSSVR. Through empirical comparison of several models of foreign exchange rate forecasting, the proposed EMD-LSSVR model outperforms EMD-ARIMA, LSSVR and ARIMA on several criteria. Thus, it can be concluded that the proposed EMD-LSSVR model may be an effective tool forfinancial time series forecasting.

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Table 6

Percentage improvement of forecasting performance of the proposed EMD-LSSVR model in comparison with other forecasting models.

Models Indicators

MAPE REMSE MAD DS CP CD USD/NTD EMD-ARIMA 82.79 62.85 66.67 8.02 15.71 3.56 LSSVR 91.03 79.76 80.93 5.01 4.01 7.00 ARIMA 98.23 81.01 81.38 17.51 13.06 26.91 JPY/NTD EMD-ARIMA 75.11 85.79 86.67 2.96 6.17 3.98 LSSVR 80.61 90.26 90.57 5.63 11.56 0.63 ARIMA 92.07 91.39 95.00 17.24 15.05 16.95 RMB/NTD EMD-ARIMA 60.33 36.99 49.06 6.92 18.02 2.48 LSSVR 54.72 58.40 64.94 3.39 7.01 3.43 ARIMA 95.66 89.51 92.06 14.46 17.53 18.05