Data characteristic analysis and model selection for container throughput forecasting within a decomposition-ensemble methodology

Transportation Research Part E 108 (2017) 160–178

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Data characteristic analysis and model selection for container throughput forecasting within a decomposition-ensemble methodology Gang Xie ⇑, Ning Zhang, Shouyang Wang Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

a r t i c l e

i n f o

Article history: Received 12 January 2017 Received in revised form 17 August 2017 Accepted 30 August 2017 Available online 1 November 2017 Keywords: Container throughput Data characteristic analysis Model selection Time series forecasting Decomposition-ensemble methodology

a b s t r a c t In this study, a novel decomposition-ensemble methodology is proposed for container throughput forecasting. Firstly, the sample data of container throughput at ports are decomposed into several components. Secondly, the time series of the various components are thoroughly investigated to accurately capture the data characteristics. Then, an individual forecasting model is selected for each component based on the data characteristic analysis (DCA). Finally, the forecasting results are combined as an aggregated output. An empirical analysis is implemented for illustration and verification purposes. Our results suggest that proposed hybrid models can achieve better performance than other methods. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction With the current development of economic globalization, ports have become more and more important in the operation of international trade activities (Notteboom, 2016; Yap and Lam, 2006). A port is characterized by a functional and spatial clustering of cargo transport, storage and transformation processes, all of which are linked to global supply chains (Twrdy and Batista, 2016). The handling of maritime cargo at specialized terminals remains a core function of ports (de Gooijer and Klein, 1989; Steenken et al., 2004; Coto-Millán et al., 2005). As many manufacturing industries have experienced rapid growth, particularly in emerging economies, many coastal cities in those economic centers have invested heavily in ports (Jeevan et al., 2015). The prediction of container throughput at a port helps port managers make not only strategic decisions on developing the port in terms of scale, general layout and district division, but also helps with tactical and operational decisions (mid-term and short term decisions) such as operations planning decisions within the port, the scheduling of port equipment, etc. Accurate predictions of container throughput mean that investments in port capacity and other transportation infrastructure can be made consistent with the needs generated by that traffic (Lam et al., 2004; Levine et al., 2009; Petering, 2009). If these throughput predictions are not accurate enough, policy bias will occur, which could cause huge financial losses. Therefore, developing an effective container throughput forecasting model has become a crucial task (Geng et al., 2015). Containerization is an important element of the logistics and security innovations that revolutionized freight handling in the 20th century. The pattern characteristics of container throughput time series include cyclicity, seasonality, mutability and randomicity. These traits are determined by the economic structure and market development of the port’s hinterland ⇑ Corresponding author. Tel. +86-10-82541368; fax: +86-10-62541823. E-mail addresses: [email protected] (G. Xie), [email protected] (N. Zhang), [email protected] (S. Wang). https://doi.org/10.1016/j.tre.2017.08.015 1366-5545/Ó 2017 Elsevier Ltd. All rights reserved.

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(Tian et al., 2013). Currently, the length of the available time series is from 10 to 30 years; the time granularity is one month or one year. In practice, there are distinct seasonal characteristics of container throughput at ports (Chou et al., 2003; Peng, 2006). For example, the Chinese New Year has a significant impact on container throughput at all Chinese ports (Liang and Chou, 2003; Chen and Chen, 2010). In addition, the time series of container throughput at ports may fluctuate due to specific events, including, for example, the financial crisis of 2008, large-scale manufacturing transfers, a dock workers’ strike, etc. Any or all of these (or similar) events may result in breakpoints within the time series. As a consequence, the intrinsic complexity and volatility of the global economy and trade is what causes container throughput series to appear nonlinear and non-stationary. Container throughput forecasting is also a very complex and dynamic process. In this study, we further analyze the data characteristic of complexity, which covers various nonlinear characteristics, e.g., chaoticity, fractality, irregularity and long-range memorability (Tang et al., 2014; Yu et al., 2015). To achieve better forecasting performance, conducting a data characteristics analysis (DCA) is a useful exercise to undertake before attempting to predict the container throughput at any given port. Some econometric models have previously been used for container throughput forecasting. Those are the error correction model (ECM) (Fung, 2002; Hui et al., 2004), the multivariate regression model (Chou et al., 2008; Veenstra and Haralambides, 2001), and the seasonal autoregressive integrated moving average (SARIMA) model (Schulze and Prinz, 2009), as well as other models, for example, exponential smoothing (Liu et al., 2010) and the Vector Auto-Regression (VAR) model (Tian et al., 2013). However, econometric models are built on linear assumptions. As such, they cannot capture the nonlinear patterns hidden in the original data, which in turn leads to poor forecasting performance, especially with regard to some time series with nonlinearity. For the description of nonlinear characteristics in the time series of container throughput at ports, artificial intelligence (AI) models have been used. These AI models include a back-propagation neural network (BPNN) model (Liu et al., 2010), a genetic programming (GP) model (Chen and Chen, 2010) and a transfer forecasting model guided by a discrete particle swarm optimization (TF-DPSO) (Xiao et al., 2014) algorithm. For better forecasting performance, many hybrid approaches have been developed. Combining projection pursuit regression (PPR) with a genetic programming (GP) algorithm, Huang et al. (2015) proposed a hybrid method to forecast the container throughput of Qingdao Port. To eliminate the influence of outliers, Huang extended a local outlier factor (LOF) as a means to detect outliers in the time series. Then, different dummy variables were constructed to capture the effect of outliers, based on domain knowledge. The results of Huang’s research show that the proposed method significantly outperforms artificial neural network (ANN), SARIMA, and PPR models. In addition, decomposition has been used to develop hybrid approaches for forecasting container throughput. Peng and Chu (2009) presented the classical decomposition (CD) model, which decomposes the time series of container throughput at ports into four separate components, namely trend, cyclical, seasonal and irregular factors. In the CD approach, the least square method is applied, in order to derive a linear regression equation for the estimate of the trend component. However, the CD model is also built on linear assumptions. As such, the CD model cannot capture the nonlinear patterns hidden in the original container throughput time series. Consequently, Xie et al. (2013) proposed several hybrid approaches based on the least squares support vector regression (LSSVR) model and preprocessed methods, including SARIMA, seasonal decomposition and classical decomposition. The empirical analysis presented in Xie et al. (2013) concluded that the proposed hybrid approaches can achieve better forecasting performance than individual approaches. For good forecasting performance, it is important to describe the seasonal nature and nonlinear characteristics of container throughput series, which can be realized efficiently by employing decomposition and the ‘‘divide and conquer” principle (Xie et al., 2013). However, in the above-named studies, DCA has not yet been implemented to the time series of the components before models are developed for the purpose of container throughput forecasting. Existing studies merely consider certain given data characteristics of time series. For example, Peng and Chu (2009) investigated the seasonality of container throughput at ports, and Xie et al. (2013) took the mutability of a time series into account, while other important data characteristics were ignored in the modeling process in these studies. Therefore, it is necessary to conduct DCA before a container throughput forecasting model is developed. As such, we propose a forecasting procedure within a decomposition-ensemble methodology. To the best of our knowledge, the application of DCA for container throughput forecasting has not yet been studied or published in any existing literature. In this study, we propose a novel DCA-based decomposition-ensemble methodology. With our methodology, a number of hybrid approaches are developed for container throughput forecasting. Firstly, the sample data of container throughput at ports are decomposed into several components. Secondly, the time series of the components are thoroughly investigated, in order to capture the data characteristics (including stationarity, seasonality, mutability and complexity). Then, a single model is selected for the prediction of each component based on the DCA results. In particular, when the time series of the components have the characteristics of being nonstationary or complexity, artificial intelligence (AI) models are accordingly developed. Finally, the forecasting results (in terms of the decomposed subtasks) are combined as an aggregated output. An empirical analysis is implemented for illustration and verification purposes. Finally, some related issues are discussed, and our conclusions are drawn and presented. The remainder of this paper is organized as follows: The proposed novel decomposition-ensemble methodology is described in detail in Section 2. Here, the overall process of the proposed methodology, decomposition methods, data characteristic analysis (DCA), and forecasting modeling are thoroughly discussed. Section 3 illustrates the problems related to the forecasting of container throughput by using an empirical analysis with experiments. Then, a number of related issues are discussed in Section 4. Section 5 presents our conclusions and suggests some directions for future investigation.

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2. Methodology formulation In this section, firstly, the overall process of the proposed approach is presented. Then, the methods used to perform the decomposition and DCA of time series are described, respectively. Finally, our forecasting modeling is introduced, and we present the procedure of DCA and explain the selection of corresponding models. 2.1. Overall process of the proposed methodology To enhance the degree of prediction accuracy of container throughput forecasting at ports, a novel decompositionensemble methodology is proposed. Our proposed approach actually improves the existing decomposition-ensemble techniques in the ‘‘divide and conquer” framework (Tang et al., 2012), by incorporating DCA and model selection. The overall process of the proposed methodology can be described in Fig. 1 as being comprised of the following four main steps: Step 1: Data decomposition. The original time series xt (t = 1, 2, . . ., T) of container throughput at ports is decomposed into n components via decomposition techniques. Step 2: Data characteristic analysis (DCA) of components. All decomposed components are thoroughly analyzed, in order to capture the data characteristics. Step 3: Individual prediction. On the basis of the DCA, one model is selected for the prediction of each component. Accordingly, we can then obtain the prediction results for the corresponding component. Step 4: Ensemble prediction. The predicted results for the components are combined as an aggregated output of container throughput forecasts at ports. 2.2. Decomposition methods In order to capture the seasonal characteristics of observations in different years, we use one of the most popular seasonal decomposition methods, namely X-12-ARIMA. This method is the Census Bureau’s latest seasonal adjustment program

Time series of container throughput at a port

Decomposition

Component n

Component 1

Component 2

...

Data characteristic analysis

Data characteristic analysis

...

Data characteristic analysis

Selected model

Selected model

...

Selected model

Prediction results of Component 1

Prediction results of Component 2

...

Prediction results of Component n

Aggregation

Container throughput forecasts at a port Fig. 1. Overall process of the proposed decomposition-ensemble methodology.

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(Findley et al., 1998). The X-12-ARIMA method decomposes time series xt of container throughput at a port into three components, namely the trend-cycle component tct , seasonal factor component sft and irregular component ir t , which can all be combined into the original data in multiplicative and additive forms, as follows:

xt ¼ tct  sft  ir t

ð1Þ

xt ¼ tct þ sft þ ir t

ð2Þ

In this study, for the purpose of comparison, in addition to the X-12-ARIMA method with multiplicative form, the X-12ARIMA method with additive form is also employed for the seasonal decomposition of container throughput at ports. 2.3. Data characteristic analysis (DCA) After the sample time series data of container throughput at a port is decomposed into several components, we thoroughly investigate the various components, in order to capture the relevant data characteristics, including stationarity, seasonality, mutability and complexity. 2.3.1. Stationarity Stationarity is tested via a unit root test. This test will discover whether or not any unit root exists in the data generating process (DGP). If no unit root exists in the DGP, the time series data can then be termed as stationarity, and the time series can converge with time. Otherwise, the time series is deemed to have the characteristic of non-stationarity. Generally, the data characteristic of stationarity is the premise for using the econometric model for time series. Currently, the most popular unit root test in time series data is the Augmented Dickey-Fuller (ADF) test, which can be implemented via Eviews software. In this study, if a time series with non-stationarity characteristics can be transformed into one with stationarity by means of difference, we can deal with this as stationarity after difference. 2.3.2. Seasonality Seasonality means that some repeated characteristics (the autocorrelation coefficient of the time series is obviously not 0) occur at a fixed time interval in the time series. Usually, for a time series with monthly or quarterly frequencies, the fixed time interval covers 12 months or four quarters, respectively. Therefore, an autocorrelation analysis can be used to test whether or not seasonality exists in a time series. Analysis results indicate that an obviously cyclical pattern with a time window of 12 months (one year) is hidden in all monthly frequent sample data. This occurs because the absolute values of autocorrelation coefficients with a lag of 12 months are much higher than those with other lag times. It is worth noticing that these cyclical patterns which last for a period of one year can also be seen as seasonal patterns. These are the special cyclical factors which are usually hidden in the time series of container throughput at ports. Seasonality can also be tested via Eviews software. 2.3.3. Mutability As for mutability testing, two effective breakpoint tests can be used to verify whether or not any given time series has been structurally changed by certain extreme events (Yu et al., 2015). Firstly, the iterative cumulative sum of squares (ICSS) test (Inclan and Tiao, 1994) is performed to find the suspect points. In the ICSS algorithm, the time series should have the stationary unconditional variance of r21 until a sudden change occurs at point k1 . The new unconditional variance r22 is stationary until the next breakpoint k2 , and this process repeats itself. Secondly, the Chow test (Chow, 1960) is employed for each potential breakpoint k 2 fk1 ; k2 ; . . .g via the F-test:

F¼

SSESSE1 SSE2 m1 þ1 SSE1 þSSE2 N 1 þN2 2m1 2

 Fðm1 þ 1; N1 þ N 2  2m1  2Þ

ð3Þ

where m1 is the number of the explanatory variables, N 1 and N 2 are the amounts of the observations in the data subsets before and after breakpoint k, SSE is the sum of the residual squares in modeling the whole time series data with N 1 þ N 2  m1  1 degrees of freedom, and SSE1 and SSE2 are those of the data subsets before and after breakpoint k with degrees of freedom N 1  m1  1 and N 2  m1  1, respectively. When the time series has the data characteristic of mutability, the dummy variables p1;t and p2;t , for the structural changes respectively in the level and slope at breakpoint k, can be described as follows (Yu et al., 2015):


p1;t ¼

0

t
1 tPk


p2;t ¼

0

t
tk t Pk

ð4Þ

2.3.4. Complexity Due to the inherent advantages of simplicity and robustness, permutation entropy (PE), as proposed by Bandt and Pompe (2002), is used for complexity testing. In particular, the PE method performs well, even for complex data with high-level

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noise (Yu et al., 2015). In our study, PE measures the complexity of data dynamics by mapping the time series into a symbolic sequence. Firstly, PE transforms the original time series xt (t = 1, 2, . . ., T) of container throughput at a port into the embedding vectors as follows:

X t ¼ fxt ; xtþs ; . . . ; xtþðm1Þs g

ð5Þ

where m is the embedding dimension (the permutation entropy of order), Then, vector X t is rearranged with an increasing value:

s is the time delay, and t =1, 2, . . ., T  ðm  1Þs.

X t ¼ fxtþðj1 1Þs ; xtþðj2 1Þs ; . . . ; xtþðjm 1Þs g

ð6Þ

where xtþðjk 1Þs is the jk th value and xtþðj1 1Þs 6 xtþðj2 1Þs 6 . . . 6 xtþðjm 1Þs (k = 1, 2, . . ., m). For a given embedding dimension m, there can be m! possible order patterns (i.e., permutations)

p ¼ fj1 ; j2 ; . . . ; jm g Let f ðpÞ denote the frequency of

pðpÞ ¼

ð7Þ

p. The relative frequency can then be designated as:

f ðpÞ T  ðm  1Þs

ð8Þ

Accordingly, the permutation entropy can be estimated by: m! X Hp ðmÞ ¼  pðpÞ ln½pðpÞ

ð9Þ

1

Finally, the permutation entropy is normalized into:

Hp ¼

Hp ðmÞ lnðm!Þ

ð10Þ

As a consequence, Hp 2 ½0; 1, and the larger value of Hp represents the higher level complexity of the target time series. In this study, a threshold value h is given. When the PE value is estimated to be higher than h, the time series has the data characteristic of complexity. 2.4. Forecasting modeling In existing literature, time series models for container throughput forecasting can be described as:

xtþh ¼ f ðX t Þ þ et

ð11Þ

where xt denotes the container throughput at time t, X t ¼ fxt1 ; xt2 ; . . . ; xtl g are the history values before period t with lag l, h is the prediction horizon, and et represents the prediction errors following IID (independent identical distribution). Two types of forecasting models are currently employed: econometric models and artificial intelligence (AI) models. As for model type determination factors, econometric models are sufficient when modeling data with characteristics of stationary and low complexity. On the other hand, AI tools are used to capture irregular nonlinear patterns in time series with complexity (Tang et al., 2014). In this study, the econometric models used include ARIMA and SARIMA. The AI tool used in our study is the LSSVR model. 2.4.1. Least squares support vector regression model Compared with traditional linear models, AI tools have been shown to be effective individual forecasting models, mainly because of the AI tools’ adaptive self-learning capability and flexible function design. The most popular AI tools are the least squares support vector regression (LSSVR) model and the artificial neural networks (ANN) model. As the ANN model often suffers local minima and over-fitting (Xie et al., 2013), in this study, we use the LSSVR model for the prediction of the decomposed components of container throughput at ports. The support vector machine (SVM) has played an important role in pattern recognition, machine learning and prediction. The SVM does this by minimizing an upper bound of the generalization error (Vapnik, 1995). Also, SVM can be applied to classification and regression, i.e. support vector classification (SVC) and support vector regression (SVR). Since SVR adopts the structural risk minimization (SRM) principle, SVR can alleviate the over-fitting and local minima issues. As such, the SVR solution is the more stable and globally optimum choice (Xie et al., 2013). Moreover, in order to reduce the computational complexity of SVM, Suykens and Vandewalle (1999) proposed a least squares support vector regression (LSSVR) model. This model solves a system of equations as opposed to a quadratic programming (QP) problem. As such, the LSSVR model provides significantly improved speed of calculations. The LSSVR first maps the original data xt of container throughput at a port into a high dimensional feature space via a nonlinear kernel function and then performs linear regression, as follows:

^xt ¼ wT uðxt Þ þ b

ð12Þ

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where uðxt Þ is the nonlinear mapping function. The weight w and bias b are estimated according to structural risk minimization, as follows:

! T X   Min wT w =2 þ c e2t =2

ð13Þ

t¼1

s:t: ^xt ¼ wT uðxt Þ þ b þ et ; ðt ¼ 1; 2; . . . ; TÞ where c is the penalty parameter, and et is the estimation error at time t. Here, c is used to control the minimization of estimation errors and ensure function smoothness. In order to solve the optimization problem, the Lagrange function is developed as

! T T X X    2 Lðw; b; et ; at Þ ¼ w w =2 þ c et =2  at wT uðxt Þ þ b þ et  ^xt ; 

T

t¼1

ð14Þ

t¼1

where a ¼ ða1 ; a2 ; . . . ; aT Þ is the Lagrange multiplier. Differentiating L with respect to variables w, b, et and at , we obtain T X @L at uðxt Þ; ¼0!w¼ @w t¼1

ð15Þ

T X @L ¼0! at ¼ 0; @b t¼1

ð16Þ

@L ¼ 0 ! at ¼ cet ; @et

ð17Þ

@L ¼ 0 ! wT uðxt Þ þ b þ et  ^xt ¼ 0 @ at

ð18Þ

After solving the above functions, we can obtain the solution to the problem in the following form:

^x ¼

T X

at Kðx; xt Þ þ b

ð19Þ

t¼1

In the solution, KðÞ is the kernel function. Here, the usual Gaussian RBF Kðx; xi Þ ¼ exp½jjx  xi jj=ð2r2 Þ with a width of employed.

r is

2.4.2. A novel hybrid approach As the time series of container throughput is usually complex, seasonal and nonlinear, we adopt the ‘‘divide and conquer” policy. We firstly decompose the time series into several components. In this study, X-12-ARIMA models with both multiplicative and additive forms are used as decomposition methods. Next, we analyze the data characteristics of the components, which are the trend-cycle component tct , seasonal factor component sft and irregular component ir t from the seasonal decomposition methods X-12-ARIMA with both multiplicative and additive forms. The data characteristics include those of stationarity, seasonality, mutability and complexity. Then, on the basis of a data characteristic analysis (DCA), a single model is selected for the prediction of each component. In particular, when the time series of the components have the characteristic of complexity, then LSSVR models are accordingly developed. Accordingly, the prediction results for the corresponding component can be obtained. Finally, the predicted results for the components are combined as an aggregated output. Based on X-12-ARIMA with both multiplicative and additive forms, with respect to DCA, novel hybrid approaches are developed. For the hybrid approaches based on X-12-ARIMA with both multiplicative and additive forms, the aggregated outputs of container throughput forecasts at a port are:

b t  ir bt b t  sf ^xt ¼ tc

ð20Þ

b t þ ir bt b t þ sf ^xt ¼ tc

ð21Þ

In the above section, we introduced how to test data characteristics, including stationary, seasonality, mutability and complexity. We then proposed the procedure of data characteristic analysis and model selection shown in Fig. 2, where the LSSVR models are selected for the prediction of the components with complexity. Moreover, in order to verify the effectiveness of the proposed approaches, the time series of container throughput at the Port of Singapore and the Port of Los Angeles are used as a testing target, as illustrated in the next section.

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Time series of a component

Stationarity test

Is it stationarity/ difference stationarity?

No

Yes Mutability test

No

Does it have mutability? Yes Dummy variables are used

Seasonality test

Yes

Does it have seasonality? No Complexity test

Does it have complexity?

Yes

No SARIMA model

ARIMA model

LSSVR model

Prediction results of a component Fig. 2. Procedure of data characteristic analysis and model selection for the prediction of a component.

3. Empirical analysis 3.1. Data description and experiment design The International Maritime Organization (IMO) has reported that more than 90% of global trade volume is transported by sea, highlighting the fact that maritime transport is the dominant mode of transport in global trade (Xie et al., 2017). A port is a logistic and industrial node which accommodates seagoing vessels (Gao et al., 2016). In this study, two main global ports are selected for empirical analysis. The Port of Singapore (PS), located on the southern coast of The Republic of Singapore, near to the southeast side of the Straits of Malacca, is the largest container port in Southeast Asia. The Port of Los Angeles (PLA) is located on the southwest coast of San Pedro Bay, California. This is the largest container port by volume in the United

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States and has held this distinction consistently since 2000. In 2016, container throughput at PS and PLA was 30.90 million TEU and 8.86 million TEU, respectively. This means that PS ranked No. 2 and PLA ranked No. 18 in terms of container throughput volume among global ports (Wind Info, 2017). The time series data employed in this study were obtained from the WIND Database (Wind Info, 2017). The sample data are the monthly data of container throughput at PS and PLA (covering the period from January 1995 to January 2017, with a total of 265 observations at each port) as shown in Fig. 3. In our experiments, training datasets are used to determine the unknown parameters of the pre-defined models. Testing datasets are used to evaluate the forecasting performance. For each out-of-sample observation, its previous data are used as 6

3

x 10

PS PLA

Container throughput: TEU

2.5

2

1.5

1

0.5

0

0

50

100

150

200

250

Time Fig. 3. Monthly container throughput at the Port of Singapore and the Port of Los Angeles.

Table 1 Data characteristics of the components from the X-12-ARIMA method with multiplicative form. Port

Component

Stationarity

Complexity

Seasonality

Mutability

PS

tct sft ir t

U U U

 U U

 U 

U  

PLA

tct sft ir t

U U U

U U U

 U 

U  

Table 2 Data characteristics of the components from the X-12-ARIMA method with additive form. Port

Component

Stationarity

Complexity

Seasonality

Mutability

PS

tct sft ir t

U U U

 U U

 U 

U  

PLA

tct sft ir t

U U U

U U U

 U 

U  

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Table 3 Forecasting performance of models with different ratios of training sample sizes. Relative ratio

Models

Testing data Port of Singapore

Port of Los Angeles

RMSE

MAE

MAPE

Dstat

RMSE

MAE

MAPE

Dstat

60%

SARMA LSSVR EMD-LSSVR CD-LSSVR X12-A-LSSVR X12-M-LSSVR X12-A-SAL X12-M-SAL

85917.6 139839.7 78523.0 80062.3 105733.2 103377.1 45773.7 45787.6

66617.4 107415.7 62932.6 64879.3 81156.8 79588.7 30877.2 30270.3

2.67 4.26 2.56 2.58 3.27 3.21 1.24 1.23

0.774 0.642 0.830 0.849 0.717 0.755 0.943 0.924

54151.9 67891.6 57246.0 60773.1 60769.4 56817.5 44233.4 41995.7

38171.5 51848.5 39206.6 42838.6 44490.8 43011.4 27200.9 25662.5

5.87 8.01 6.11 6.47 7.04 6.78 4.21 3.84

0.802 0.566 0.783 0.764 0.774 0.708 0.877 0.835

70%

SARMA LSSVR EMD-LSSVR CD-LSSVR X12-A-LSSVR X12-M-LSSVR X12-A-SAL X12-M-SAL

80784.3 139588.1 72658.8 79298.2 101159.5 102257.3 45928.0 45787.6

63587.7 106355.3 59389.6 64230.9 79492.2 79685.2 30463.0 30270.3

2.45 4.06 2.31 2.46 3.07 3.07 1.18 1.17

0.810 0.633 0.861 0.861 0.734 0.772 0.937 0.924

57290.8 70031.1 61850.7 66101.0 61448.9 59706.1 47753.2 41995.7

41066.0 53275.4 42007.9 46393.2 45049.6 45130.3 29124.6 25662.5

6.15 7.95 6.38 6.75 6.88 6.91 4.36 3.90

0.759 0.544 0.759 0.734 0.785 0.696 0.861 0.835

80%

SARMA LSSVR EMD-LSSVR CD-LSSVR X12-A-LSSVR X12-M-LSSVR X12-A-SAL X12-M-SAL

85223.1 131752.7 71493.1 83028.4 103673.1 103872.1 49698.6 50025.8

65149.4 97301.5 57496.6 66237.1 81853.0 81466.5 31924.0 32165.4

2.46 3.61 2.18 2.48 3.08 3.06 1.21 1.22

0.811 0.623 0.830 0.849 0.679 0.736 0.925 0.906

65966.1 75490.5 69418.5 76810.8 63398.0 61417.6 56362.9 49106.6

48461.9 57529.8 47598.6 55709.7 45963.5 46036.2 35783.5 30790.1

7.22 8.45 7.22 8.00 6.92 7.00 5.34 4.68

0.698 0.547 0.774 0.660 0.811 0.717 0.868 0.849

Fig. 4. Comparison of the real data and forecasts of decomposed components at PS in X12-M-SAL.

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training samples to set the forecasting model for making one-step-ahead forecasts, where the lag period is determined by analyzing the autocorrelation and partial correlation of the time series. Also, we use the method of empirical model decomposition (EMD). The EMD method was developed by Huang et al. (1998) for decomposing a nonlinear, nonstationary time series into several components, referred to as Intrinsic Mode Functions (IMFs). An IMF is defined by two criteria: (1) The number of zero crossings and extrema are equal or differ at most by one; (2) The IMF is symmetric with respect to local zero mean. An iterative ‘‘sifting process” is employed to extract the IMFs and a residual series from the data. If we denote cit (i = 1, 2,. . ., n), to be the resultant set of IMFs and the residual series is rt , the original time series of container throughput at a port can be expressed as follows:

xt ¼

n X cit þ r t

ð22Þ

i¼1

The EMD method combined with the LSSVR model as EMD-LSSVR has been widely used for prediction within the ‘‘divide and conquer” principle. Applications include foreign exchange rate forecasting (Lin et al., 2012) and air passenger forecasting (Xie et al., 2014). In addition to EMD-LSSVR, the combination of classical decomposition (CD) and LSSVR, i.e. CD-LSSVR, as proposed in Xie et al. (2013), is used for comparison. The measures RMSE, MAE and MAPE have been used in the existent main literature on container throughput forecasting (Peng and Chu, 2009), and this study is an extension of that literature. For the purpose of comparison with same measures, we choose RMSE, MAE and MAPE as forecasting accuracy measures as follows:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX 2 RMSE ¼ t ðxt  ^xt Þ =N

ð23Þ

t¼1

MAE ¼

N X jxt  ^xt j=N t¼1

Fig. 5. Comparison of the real data and forecasts of decomposed components at PLA in X12-M-SAL.

ð24Þ

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G. Xie et al. / Transportation Research Part E 108 (2017) 160–178 N X MAPE ¼ 100 j1  ^xt =xt j=N

ð25Þ

t¼1

where N is the size of the testing set, xt is the real value of container throughput at a port, and ^ xt is the corresponding forecast of container throughput at a port in the t th month. Apart from the level prediction accuracy, directional prediction accuracy is another important criterion for forecasting models. The performance when predicting movement direction can be measured by a directional statistic (Ruiz-Aguilar et al., 2014), as follows:

Dstat ¼

N X dt =N

ð26Þ

t¼1




^t  xt1 Þ P 0 1; ðxt  xt1 Þðx . 0; otherwise Note that RMSE, MAE and MAPE are measures of deviation between the real and predicted values. Therefore, the forecasting performance is improved when the values of these measures are smaller. In addition, Dstat provides the correctness of the predicted direction and can also be used to evaluate the prediction accuracy. The higher the value of Dstat, the better is the forecasting performance. For further performance comparisons from a statistical perspective, Diebold-Mariano (DM) statistic (Diebold and Mariano, 1995) is used to test the difference in the significance of forecasting performances between different models. In the DM test, the null hypothesis assumes that forecast accuracy across the different models is the same, and a unilateral test is used. Using mean square prediction error (MSPE) as the loss function, the DM statistic S can be thus defined as where dt ¼

g S ¼ qffiffiffiffiffiffiffiffiffiffiffiffi ^ V g =N

ð27Þ

Fig. 6. Comparison of the real data and forecasts of decomposed components at PS in X12-A-SAL.

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P 2 2 ^ g ¼ c þ 2P1 c (c ¼ covðg ; g Þ); ^ where g ¼ Nt¼1 g t =N (g t ¼ ðxt  ^ xte;t Þ  ðxt  ^ xre;t Þ ) and V xte;t and ^ xre;t represent the pret tl 0 l l¼1 l dicted values respectively by the tested method te and the reference method re at time t. Accordingly, DM statistic S and pvalue are helpful in testing the superiority (or inferiority) of the tested method te compared to its benchmark model re. In this study, X-12-ARIMA is implemented via the EVIEWS software package (Quantitative Micro Software Corporation), while LSSVR models are implemented via the MATLAB software package. Also, ARIMA, SARIMA and DM tests are implemented via the R software package.

3.2. Prediction results and analysis Within a decomposition-ensemble methodology, we firstly decompose the original time series of container throughput at the Port of Singapore (PS) and the Port of Los Angeles (PLA) via X-12-ARIMA with multiplicative and additive forms, respectively. Then, we further analyze the data characteristics of the components, as shown in Tables 1 and 2. In these tables, ‘‘ U ” indicates that the specified component has the relevant characteristics. Conversely, ‘‘  ” means that the component does not possess those characteristics. Here, the threshold value h is set to 0.45. From the DCA of the component time series in Tables 1 and 2, we find that all of the components have the characteristics of stationarity. Also, the components sft and ir t have the characteristic of complexity at both ports. However, the component tct at PLA has the characteristic of complexity, while the component tct at PS does not. In particular, only the components sft have the characteristic of seasonality at both ports. Also, only the components tct have the characteristic of mutability at both ports. According to Fig. 2, from the results shown in Tables 1 and 2, SARIMA, ARIMA and LSSVR models are selected for one-stepahead forecasts of sft , tct and ir t at PS. The SARIMA, LSSVR and LSSVR models are selected for one-step-ahead forecasts of sft , tct and ir t at PLA. Then, the hybrid approaches X12-M-SAL and X12-A-SAL are developed. Also, for comparison purposes, other models (including single models and hybrid models) are developed for container throughput forecasting. Here, the single models are SARIMA and LSSVR, while other hybrid models are CD-LSSVR, EMD-LSSVR, X12-A-LSSVR and X12-M-LSSVR, where all decomposed components are predicted by LSSVR models.

Fig. 7. Comparison of the real data and forecasts of decomposed components at PLA in X12-A-SAL.

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In the LSSVR model, the values of c and r2 parameters are first determined via a 5-fold cross-validation grid search method, in the range of [0.01, 10000]. Those values are then adjusted using the trial-and-error approach to produce the smallest error in the training set (Tay and Cao, 2001). Apart from the single LSSVR, another single forecasting model SARIMA is used. In the SARIMA model, the best model for each training sample is determined through the minimization of Schwarz Criterion (SC). In our experiments, different ratios (60%, 70% and 80%) of training dataset to sample sizes are used. Then, the forecasting performances of the different models are presented in Table 3. In this table, bold font indicates the best forecasting performance among the methods. Obviously, the proposed DCA-based X12-A-SAL and X12-M-SAL methods can both achieve the best forecasting performance, according to forecasting accuracy measures. Obviously, with different ratios of training sample sizes, the proposed X12-M-SAL and X12-A-SAL methods can achieve the best performance in all measures. However, the forecasting performance for PLA is different to that of PS. The reason for this discrepancy is that the data characteristics of PLA are more complex than those of PS. For example, comparing the trend-cycle components (at the two ports) derived from X-12-ARIMA with both multiplicative and additive forms, we find that the PLA trend-cycle component has the data characteristic of complexity but the PS trend-cycle component does not. In particular, when the relative ratio of the training sample size is 60%, the comparison of forecast values of the testing dataset and the real values of the component series are shown in Figs. 5–9; specifically, X12-M-SAL for PS in Fig. 4, X12-MSAL for PLA in Fig. 5, X12-A-SAL for PS in Fig. 6, X12-A-SAL for PLA in Fig. 7, EMD-LSSVR for PS in Fig. 8 and EMD-LSSVR for PLA in Fig. 9. In order to further compare the predictive accuracy of the different forecasting models, the DM statistic is used to test the statistical significance of the forecasting performances of our different models. Tables 4 and 5 show the comparative forecasting performances of the different models through DM tests at PS and PLA, respectively, when the relative ratio of the training sample size is 60%. Here, the results listed in the tables are DM test values; p values are in brackets. Moreover, Tables 6 and 7 show the comparative forecasting performances of different models via DM tests at PS and PLA, respectively, when the relative ratio of the training sample size is set at 70%. Also, Tables 8 and 9 show the comparative forecasting performances of different models by means of DM tests at PS and PLA, respectively, when the relative ratio of the training sample size is set to 80%.

Fig. 8. Comparison of the real data and forecasts of decomposed components at PS in EMD-LSSVR.

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After illustrating the proposed approaches by means of our experiments, we further analyze a number of related issues in the following section. 4. Discussion This section presents an in-depth discussion on the use of data characteristic analysis (DCA) to conduct container throughput forecasting within the decomposition-ensemble methodology. The forecasting performance of the proposed hybrid approaches is firstly analyzed on the basis of the experimental results presented in Section 3. Then, deep insights are given with regard to container throughput forecasting.

Fig. 9. Comparison of the real data and forecasts of decomposed components at PLA in EMD-LSSVR.

Table 4 DM test results for time series at PS when the relative ratio of the training sample size is 60% Test model

X12-M-SAL X12-A-SAL X12-MLSSVR X12-ALSSVR CD-LSSVR EMD-LSSVR LSSVR

Reference model X12-A-SAL

X12-MLSSVR

X12-A-LSSVR

CD-LSSVR

EMD-LSSVR

LSSVR

SARIMA

0.1389 (0.5551)

5.0735 (1)

5.1988 (1)

5.2404 (1)

4.5097 (1)

6.2895 (1)

4.4219 (1)

5.1633 (1)

5.3077 (1) 0.66951 (0.7477)

5.3667 (1) 2.3349 (0.01072)

4.5148 (1) 3.0935 (0.001267) 3.1334 (0.00112)

6.285 (1) 3.1642 (0.999) 2.957 (0.9981) 4.7927 (1)

4.4917 (1) 1.6699 (0.04896) 1.8391 (0.03436) 1.4371 (0.9232) 1.0207 (0.8451) 4.0453 (5e05)

2.5209 (0.006604)

0.24329 (0.4041)

4.8828 (1)

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Table 5 DM test results for time series at PLA when the relative ratio of the training sample size is 60% Test model

X12-M-SAL

Reference model X12-A-SAL

X12-M-LSSVR

X12-A-LSSVR

CD-LSSVR

EMD-LSSVR

LSSVR

SARIMA

1.5784 (0.9413)

3.2142 (0.9991) 2.1264 (0.9821)

3.637 (0.9998) 3.1585 (0.999) 1.0675 (0.8559)

3.1985 (0.9991) 2.5703 (0.9942) 0.5816 (0.719)

1.9476 (0.9729) 1.428 (0.9219)

4.8684 (1)

3.2032 (0.9991)

4.0432 (0.9999) 2.3456 (0.9896) 1.5001 (0.9317) 1.4093 (0.9192) 1.3375 (0.908)

2.0992 (0.9809)

X12-A-SAL X12-MLSSVR X12-ALSSVR CD-LSSVR

0.052696 (0.521) 0.41874 (0.3381)

0.00057 (0.5002)

0.3768 (0.3535)

EMD-LSSVR LSSVR

0.45569 (0.3248) 1.1165 (0.1334) 1.3618 (0.08808) 0.3555 (0.3615) 3.5679 (0.0002719)

Table 6 DM test results for time series at PS when the relative ratio of the training sample size is 70% Test model

X12-M-SAL

Reference model X12-A-SAL

X12-MLSSVR

X12-A-LSSVR

CD-LSSVR

EMD-LSSVR

LSSVR

SARIMA

0.10235 (0.5406)

4.2948 (1)

4.388 (1)

4.3638 (1)

3.5908 (0.9997)

5.1108 (1)

3.8948 (0.9999)

4.3524 (1)

4.4637 (1) 0.40752 (0.3424)

4.4743 (1) 1.9788 (0.02568) 1.9444 (0.02773)

3.6235 (0.9997) 3.143 (0.001183)

5.1424 (1) 2.7627 (0.9964) 2.8791 (0.9974) 3.9806 (0.9999) 4.3097 (1)

3.9467 (0.9999) 1.7945 (0.03831)

X12-A-SAL X12-MLSSVR X12-ALSSVR CD-LSSVR

3.1829 (0.001047) 0.85539 (0.1975)

EMD-LSSVR LSSVR

1.7258 (0.04417) 0.41217 (0.6593) 0.90267 (0.8153) 3.6527 (0.000234)

Table 7 DM test results for time series at PLA when the relative ratio of the training sample size is 70% Test model

X12-M-SAL X12-A-SAL

Reference model X12-A-SAL

X12-M-LSSVR

X12-A-LSSVR

CD-LSSVR

EMD-LSSVR

LSSVR

SARIMA

1.6004 (0.9432)

2.671 (0.9954) 1.6272 (0.9461)

2.9052 (0.9976) 2.333 (0.9889)

2.9757 (0.9981) 2.3288 (0.9888) 0.75532 (0.7738) 0.58976 (0.7215)

1.766 (0.9593)

3.9983 (0.9999) 3.1557 (0.9989) 1.7544 (0.9584) 1.5439 (0.9367) 0.63902 (0.7377) 0.8295 (0.7953)

2.7247 (0.996)

X12-MLSSVR X12-ALSSVR CD-LSSVR EMD-LSSVR LSSVR

0.44791 (0.6723)

1.2486 (0.8922) 0.21063 (0.5831) 0.03887 (0.5155) 0.36814 (0.3569)

1.632 (0.9466) 0.32865 (0.3717) 0.58866 (0.2789) 1.4885 (0.07032) 0.41916 (0.3381) 2.8335 (0.002929)

4.1. Performance comparison and analysis Experimental forecasts of container throughput at the Port of Singapore (PS) and the Port of Los Angeles (PLA) were made by using the experiment design and methodologies mentioned above. Forecasting performances were then evaluated by the four main measurement criteria and the DM test.

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G. Xie et al. / Transportation Research Part E 108 (2017) 160–178 Table 8 DM test results for time series at PS when the relative ratio of the training sample size is 80% Test model

X12-M-SAL

Reference model X12-A-SAL

X12-M-LSSVR

X12-A-LSSVR

CD-LSSVR

EMD-LSSVR

LSSVR

SARIMA

0.18536 (0.4268)

3.2484 (0.999)

3.3438 (0.9992) 3.4318 (0.9994) 0.055809 (0.4779)

3.2053 (0.9988) 3.3074 (0.9991) 1.3836 (0.0862)

2.2177 (0.9845) 2.2872 (0.9869) 2.7004 (0.004663) 2.7931 (0.003645) 1.1103 (0.136)

3.5383 (0.9996) 3.5683 (0.9996) 1.6508 (0.9476) 1.7014 (0.9526) 2.5728 (0.9935) 2.9978 (0.9979)

2.9018 (0.9973)

3.3179 (0.9992)

X12-A-SAL X12-MLSSVR X12-ALSSVR CD-LSSVR

1.4082 (0.08252)

EMD-LSSVR LSSVR

2.9639 (0.9977) 1.1946 (0.1188) 1.1839 (0.1209) 0.46453 (0.6779) 1.1058 (0.863) 2.2859 (0.01318)

Table 9 DM test results for time series at PLA when the relative ratio of the training sample size is 80% Test model

X12-M-SAL

Reference model X12-A-SAL

X12-M-LSSVR

X12-A-LSSVR

CD-LSSVR

EMD-LSSVR

LSSVR

SARIMA

1.5991 (0.9421)

1.6751 (0.95)

1.8546 (0.9653) 1.0945 (0.8606) 0.37235 (0.6444)

2.7139 (0.9955) 2.053 (0.9774) 1.4689 (0.9261) 1.3717 (0.912)

1.3991 (0.9161) 0.89775 (0.8133) 0.60158 (0.725) 0.44352 (0.6704) 0.49524 (0.3113)

3.0411 (0.9982) 2.1766 (0.983)

2.3695 (0.9892) 1.291 (0.8988)

1.7653 (0.9583) 1.5932 (0.9414) 0.17205 (0.432)

0.51689 (0.6963) 0.30255 (0.6183) 1.4344 (0.07873) 0.24396 (0.4041)

X12-A-SAL X12-MLSSVR X12-ALSSVR CD-LSSVR EMD-LSSVR LSSVR

0.58552 (0.7196)

0.46708 (0.6788)

1.8856 (0.03247)

In Table 3, with respect to the criteria of measuring prediction accuracy, the X12-M-SAL and X12-A-SAL methods can achieve better performance than other methods in different relative ratios of training sample size. Here, other methods are LSSVR models (Xie et al., 2013), EMD-LSSVR (Lin et al., 2012), CD-LSSVR (Xie et al., 2013), X12-A-LSSVR and X12-MLSSVR. Currently, LSSVR models and CD-LSSVR are the methods used for container throughput forecasting, while EMDLSSVR is a popular decomposition ensemble method. Also, we propose X12-A-LSSVR and X12-M-LSSVR methods, which are compared with DCA-based X12-A-SAL and X12-M-SAL. As shown in Fig. 10, it is easy to conclude that the proposed DCA-based approaches X12-M-SAL and X12-A-SAL account for all of the best forecasts of container throughput in both ports, relative to the other models listed in this study. Furthermore, the DM test confirms the above conclusions. In Tables 4 and 5, when the relative ratio of the training sample size is set to 60%, X12-M-SAL and X12-A-SAL outperform their respective counterparts (i.e. X12-M-LSSVR, X12-A-LSSVR, CDLSSVR, EMD-LSSVR, LSSVR and SARIMA), under the confidence level of 100% at PS and 92% at PLA. In Tables 6 and 7, when the relative ratio of the training sample size is 70%, X12-M-SAL and X12-A-SAL outperform their respective counterparts under the confidence level of 99% at PS and 94% at PLA, except for EMD-LSSVR, with a p value of 0.8922 at Table 7. In Tables 8 and 9, when the relative ratio of the training sample size is 80%, X12-M-SAL and X12-A-SAL outperform their respective counterparts under the confidence level of 98% at PS and 90% at PLA, except for X12-M-LSSVR with a p value of 0.7196, X12-A-LSSVR with a p value of 0.8606, EMD-LSSVR with a p value of 0.8133 and SARIMA with a p value of 0.8988 at Table 9. 4.2. Deep insights From the performance comparisons and analyses detailed above, deep insights regarding container throughput forecasting can be derived as follows: Firstly, DCA is an important component of container throughput forecasting at ports. The DCA-based models used in this study performed better than the corresponding hybrid models without DCA. That is, the DCA-based hybrid approaches X12M-SAL and X12-A-SAL outperformed their counterparts X12-A-LSSVR and X12-M-LSSVR. The reason for this finding is that the seasonal factors can be more efficiently predicted by the SARIMA model in X12-M-SAL and X12-A-SAL than by the LSSVR

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Fig. 10. Number of getting best forecasting performance of models.

model in X12-A-LSSVR and X12-M-LSSVR. This is due to the relatively simple data characteristics of the seasonal factor components of container throughput at ports. In particular, as the trend-cycle component of container throughput at PS does not have the characteristic of complexity, the ARIMA model rather than the LSSVR model is used for the prediction. This result shows that the decomposition-ensemble methodology enhances the level of prediction accuracy for container throughput forecasting at ports. Secondly, it is important to deal with seasonality to achieve better performance of container throughput forecasting. In different ratios of training sample sizes, the SARIMA model can achieve better performance than the LSSVR model. The reason for this finding is that the time series of container throughput at the two ports have distinct seasonality. The SARIMA model can deal with the seasonality, but the LSSVR model cannot. Similarly, both X12-M-SAL and X12-A-SAL outperform the EMD-LSSVR method. This is because seasonal factors are predicted well by both X12-M-SAL and X12-A-SAL but they are not dealt with by EMD-LSSVR. Finally, a single forecasting model should be selected with respect to DCA of the decomposed component time series of container throughput at ports. Both X12-M-SAL and X12-A-SAL are better than CD-LSSVR, which exhibited the best forecasting performance in Xie et al. (2013). The reason for this result is that the ARIMA models are selected for the prediction of trend component trt and the SARIMA models are selected for the prediction of seasonal factor sft in X12-SAL based on DCA, rather than only the LSSVR model for trend components in CD-LSSVR. As a consequence, seasonal and irregular factors are not captured well by CD-LSSVR. Actually, our proposed approach improves the existing decomposition-ensemble techniques in the ‘‘divide and conquer” framework in which DCA is implemented before a forecasting model is selected for each decomposed component of container throughput time series at ports. 5. Conclusions and future work In this study, we propose a procedure of data characteristic analysis (DCA) and model selection within a decompositionensemble methodology, which includes data decomposition, DCA of components, individual prediction and ensemble prediction. Based on the time series of container throughput at the Port of Singapore and the Port of Los Angeles, the proposed approaches are illustrated and empirically compared with other benchmark methods. Finally, some related issues are discussed, and our conclusions are drawn. Specifically, we chose the two time series of container throughput at the Port of Singapore (PS) and the Port of Los Angeles (PLA). In fact, the two time series represent different growth trends of container throughput in two regions: (1) Asia; and (2)

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America. In recent years, the container throughput at most American ports has experienced slow growth or has even declined. For example, the container throughput at PLA has been in a stable state since 2003. On the contrary, the Asian ports are still achieving rapid growth, including PS and Chinese ports. The contribution of this study is in recommending that DCA and corresponding model selection be first used for container throughput forecasting at ports. We also develop new hybrid approaches for comparison with benchmark methods. Our investigation suggests that DCA and decomposition-ensemble techniques are effective tools to be used for container throughput forecasting. It is important to select models with respect to DCA for better forecasting performance. The DCAbased hybrid approaches can achieve better performance than other models. We expect that future researchers would benefit from concentrating on other methods which could be used for container throughput forecasting. 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