International tourism demand forecasting with machine learning models: The power of the number of lagged inputs

Abstract

This study explores how to select the optimal number of lagged inputs (NLIs) in international tourism demand forecasting. With international tourist arrivals at 10 European countries, the performances of eight machine learning models are evaluated using different NLIs. The results show that: (1) as NLIs increases, the error of most machine learning models first decreases rapidly and then tends to be stable (or fluctuates around a certain value) when NLIs reaches a certain cutoff point. The cutoff point is related to 12 and its multiples. This trend is not affected by the size of the test set; (2) for nonlinear and ensemble models, it is better to select one cycle of the data as the NLIs, while for linear models, multiple cycles are a better choice; (3) significantly different prediction results are obtained by different categories of models when the optimal NLIs are used.

Keywords

experimental study machine learning models number of lagged inputs tourism demand forecasting

Introduction

Due to the increasing importance of international tourism to the world economy, both the public and the private sectors have channeled a lot of resources and investment to this industry (Jiao and Chen, 2019; Peng et al., 2014). Accordingly, international tourism demand forecasting is of great interest to both tourism researchers and practitioners (Crouch, 1994; Gunter and Önder, 2015; Jorge-González et al., 2019; Kim et al., 2018; Song and Witt, 2006). Accurate forecasts are of enormous value to government, organizations, and tourism practitioners and support good decision-making (Bi et al., 2019; Palmer et al., 2006; Song and Li, 2008). For example, accurate demand forecasts can help government agencies design infrastructure for tourism, such as transportation systems and accommodation, can help organizations design marketing and other materials, such as tour brochures (Li et al., 2018), and can help practitioners make reasonable operational decisions, such as scheduling and staffing (Bi et al., 2020a). Therefore, to support international tourism-related decision-making, it is necessary to study international tourism demand forecasting.

Many models can be used to forecast international tourism demand (Song et al., 2019). These can be mainly divided into three categories, that is, time series models, econometric models, and machine learning models (Bi et al., 2020b; Song and Li, 2008; Song et al., 2019). A detailed introduction to these three categories is given in “Tourism demand forecasting models” section. Among the three categories, the machine learning models are relatively new and have shown excellent predictive ability. Therefore, this category of model has been widely studied and applied in tourism demand forecasting (Hassani et al., 2015, 2017; Li et al., 2018; Silva et al., 2019). For machine learning models to forecast tourism demand, it is first necessary to convert the tourism volume data in the form of time series into an input–output pair sequence. In this process, an important step is to determine the number of lagged inputs (NLIs), that is, the number of the time series’ lagged terms as inputs. Although selecting and determining the optimal NLIs values is very time-consuming, and the NLIs value will significantly affect the accuracy of the forecasts (Peng et al., 2014), there is no published research on how to determine the optimal NLIs and, consequently, there is a lack of guidance on how to reasonably and quickly select the NLIs in international tourism demand forecasting based on machine learning models.

To fill this research gap, this study explores how to select the NLIs in international tourism demand forecasting based on machine learning models. Specifically, we explore the following four questions.

Studies have shown that the NLIs can significantly affect the accuracy of international tourism demand forecasting (Peng et al., 2014). However, it is still unknown how the prediction errors change as NLIs increases. Therefore, the first research question is:

Q1: As NLIs increases, how do the forecasting errors of machine learning models change in international tourism demand forecasting?

In practice, the number of samples in the test set may not be the same. However, whether the optimal NLIs for a particular machine learning model will be affected by the size of the test set is unclear. In other words, for a given time series data set of tourist volume and a given machine learning model, it is still unknown whether the optimal NLIs will change with the number of samples. Therefore, the second research question is:

Q2: Are the optimal NLIs values of different machine learning models affected by the size of test set?

As a type of time series data, an important characteristic of international tourist volume data is seasonality. Intuitively, the optimal NLIs may be related to the seasonality of international tourist volume data. However, there is still no direct evidence showing the relationship between optimal NLIs and the seasonality of tourism time series. Whether the optimal NLIs of different models can be directly determined through the characteristic of international tourist volume data is still unclear. Consequently, there is no consensus about the selection of the optimal NLIs. That is why different NLIs are used in different studies for monthly international tourism demand forecasting (Álvarez-Díaz and Rosselló-Nadal, 2010; Burger et al., 2001; Hassani et al., 2017; Hong et al., 2011). Therefore, the third research question is:

Q3: Is there a relationship between optimal NLIs and the seasonality of international tourist volume data? If so, how can the optimal NLIs be selected quickly and reasonably when using different machine learning models to forecast international tourism demand?

Comparing the performance of different models in tourism demand forecasting is very helpful for developing new models. Although several studies have attempted to address this issue, their conclusions are not based on the optimal NLIs for different machine learning models; indeed, some of the machine learning models do not use the optimal NLIs in these studies. As a result, the generalizability and reliability of the conclusions obtained in these studies may be limited. Moreover, it is not known whether there is a significant difference in the predicted results of different machine learning models when each of those models uses its optimal NLI. Therefore, the fourth research question is:

Q4: Is there a significant difference among the predicted results of different machine learning models when optimal NLIs values are used?

To answer the above research questions, a comprehensive and systematic experimental analysis is carried out in this study. A framework for tourism demand forecasting based on machine learning models is first proposed, which includes two parts: data conversion and model training. Then, the experimental design is given. On this basis, 19,200 experiments were conducted on 10 data sets and 8 machine learning models to explore the relationship between the NLIs and the forecasting results. Answering the above questions will provide scientific guidance and help for practitioners to determine the optimal NLIs when using machine learning models to forecast international tourism demand and provide valuable information that can be used to make policy and business decisions more effectively.

Literature review

Tourism demand forecasting models

A variety of tourism demand forecasting models have been developed. These models can largely be classified into three categories, according to their associated theory and technology: time series analysis models, econometric models, and machine learning models (Pan and Yang, 2017; Song and Li, 2008). A brief literature review concerning these is presented below.

(1) Time series analysis models

Time series analysis models are the traditional models for noncausal time series tourism forecasting. In this category, the autoregressive moving average (ARIMA) and its improved versions (e.g. seasonal ARIMA) are the most commonly used models and have good performance (Athanasopoulos et al., 2011; Cho, 2001; Du Preez and Witt, 2003; Goh and Law, 2002; Lim and McAleer, 2002; Shahrabi et al., 2013). For example, Cho (2001) applied three time series forecasting models, namely exponential smoothing, ARIMA, and improved ARIMA, to predict travel demand for Hong Kong. ARIMA and improved ARIMA outperformed exponential smoothing and are more suitable to forecast fluctuating travel demand. Goh and Law (2002) compared the performance of 10 time series forecasting models based on data for tourist arrivals at Hong Kong and found that improved versions of ARIMA (i.e. seasonal ARIMA and multivariate ARIMA) had the highest prediction accuracy. Other time series models used in tourism forecasting have included state space models (Athanasopoulos and Hyndman, 2008; Beneki et al., 2012), generalized autoregressive conditional heteroskedastic models (Chan et al., 2005; Liang, 2014), and exponential smoothing (Fildes et al., 2011).

(2) Econometric models

Compared with time series models, econometric models are more suitable for causal time series tourism forecasting since they can analyze the causal relationship between tourism demand and factors that affect it. Those commonly used for tourism demand forecasting include error correction models (Shen et al., 2009; Wong et al., 2007), the vector autoregressive model (Assaf et al., 2018; Cao et al., 2017; Shan and Wilson, 2001; Song and Witt, 2006), the autoregressive distributed lag model (Song et al., 2003a, 2003b), the almost ideal demand system (De Mello and Fortuna, 2005; Li et al., 2006), and the structural equation model (Turner and Witt, 2001).

(3) Machine learning models

In recent years, machine learning models have emerged in the field of tourism forecasting, such as artificial neural networks (NNs) (Burger et al., 2001; Kon and Turner, 2005), support vector machines (Pai et al., 2014; Sencheong and Turner, 2005), and rough set approaches (Au and Law, 2000; Law and Au, 2000). The main advantage of machine learning models is that they do not require any assumptions regarding the data (e.g. distribution and probability); additionally, their adaptability and nonlinearity make them especially suitable for nonlinear prediction (Li et al., 2018; Song and Li, 2008). Machine learning models have therefore been increasingly applied in the field of tourism demand forecasting (Li et al., 2018), and they are the focus of this study.

NLIs in tourism demand forecasting

To use machine learning models to forecast tourism demand requires the time series data to be converted into a form that can be used in those models, that is, an input–output pair sequence. In this process, an important step is to determine the NLIs. The NLIs values used in previous studies with respect to different machine learning models and data sets are summarized in Table 1.

Table 1.

The NLIs used in previous studies of the use of machine learning models to forecast tourism demand.

Authors	Models	Data	NLIs
Burger et al. (2001)	Multiple regression, neural networks	US visitor arrivals to Durban, South Africa (1992–1998)	12
Chen and Wang (2007)	Support vector regression	Quarterly tourism arrivals in China (1985–2001)	4
Álvarez-Díaz and Rosselló-Nadal (2010)	Autoregressive neural network	Tourist arrivals in the Balearic Islands by air from the United Kingdom (December 1981–December 2006)	13
Hong et al. (2011)	Support vector regression	Total number of tourist arrivals in Barbados (1956–2003)	25
Chen et al. (2012)	Neural networks	The international tourist arrivals to Taiwan from the major markets of Japan, Hong Kong, and Macao (January 1971–August 2009)	8
Teixeira and Fernandes (2012)	Neural networks	Number of guest nights in the hotels in Northern Portugal (January 1987–December 2010)	12
Hassani et al. (2015)	Neural networks	Total monthly US tourist arrivals (January 1996–November 2012)	2
Hassani et al. (2017)	Neural networks	International tourist arrivals in 10 European countries (January 2000–December 2013)	2
Lin et al. (2018)	Neural network	Daily tourist arrivals of Jiuzhaigou in Sichuan, China (January 1–December 31, 2010)	1
Li and Cao (2018)	Neural networks	Daily tourist visitors to the Small Wild Goose Pagoda in Xi’an, China (January 2013–December 2015)	7

As shown in Table 1, different NLIs are used in different studies. It should be noted that selecting and determining the NLIs can be very time-consuming, and most studies do not explain why a particular NLI was chosen. Nevertheless, these studies have shown that NLIs can significantly affect the accuracy of forecasts (Peng et al., 2014). However, no research has been reported on how best to determine which NLIs to use. Therefore, a comprehensive and systematic experimental study exploring the relationship between NLIs and the accuracy of forecasts is necessary.

Framework for tourism demand forecasting with machine learning and related models

A framework for tourism demand forecasting with machine learning

The proposed framework for tourism demand forecasting based on machine learning is shown in Figure 1. It has two parts: data conversion and model training. Detailed descriptions of these two parts are given below.

Figure 1.

Framework for tourism demand forecasting based on machine learning.

(1) Data conversion

Tourist volume is a type of time series data, which cannot be directly used by machine learning models. Therefore, it is first necessary to convert the tourist volume data in the form of time series into the data form that can be used by machine learning models, namely input–output pair sequences.

Let $X = (x_{1}, x_{2}, x_{3}, x_{4}, ..., x_{T - 3}, x_{T - 2}, x_{T - 1}, x_{T})$ denote a time series of tourist volume data with T observation points, where x_t denotes the tourist volume of the tth observation point, t = 1, 2,…, T. Let n denote the number of lagged inputs, that is, using the tourist volume of n previous observation points to forecast the tourist volume of the following observation points, $n \in \{1, 2, ..., T - 1\}$ . Based on the above definitions, the time series of tourist volume data $X = (x_{1}, x_{2}, x_{3}, x_{4}, ..., x_{T - 3}, x_{T - 2}, x_{T - 1}, x_{T})$ can be converted into the form required by machine learning models, that is

\begin{array}{l} v_{1} v_{2} ... v_{n} y \\ [\begin{matrix} x_{1} & x_{2} & ... & x_{n} \\ x_{2} & x_{3} & ... & x_{n + 1} \\ ... & ... & ... & ... \\ x_{T - n} & x_{T - n + 1} & x_{T - 1} \end{matrix}] \to [\begin{matrix} x_{n + 1} \\ x_{n + 2} \\ ... \\ x_{T} \end{matrix}] \end{array}

where the matrix on the left-hand side is the input of machine learning models ( $v_{1}, v_{2}, ..., v_{n}$ denote n independent variables) and that on the right-hand side is the output of machine learning models (y denotes the dependent variable).

To illustrate the above conversion process more clearly, an example is given here. Table 2 presents a time series of tourist volume data with 10 observation points. In the case of n = 3, the data in Table 2 can be converted into the data shown in Table 3.

Table 2.

An example of a time series of tourist volume data with 10 observation points.

Time	1	2	3	4	5	6	7	8	9	10
Tourist volume	12,059	12,804	9745	9233	10,184	13,172	17,998	18,741	13,860	8691

Table 3.

The converted data corresponding to the data in Table 2 in the case of n = 3.

v ₁	v ₂	v ₃	y
12,059	12,804	9745	9233
12,804	9745	9233	10,184
9745	9233	10,184	13,172
9233	10,184	13,172	17,998
10,184	13,172	17,998	18,741
13,172	17,998	18,741	13,860
17,998	18,741	13,860	8691

(2) Model training

After the data have been converted, machine learning models can be trained, and a tourist volume predictor can be obtained. Here, the “tourist volume predictor” refers to the machine learning model with actual prediction ability, which is trained on the tourist volume data. Based on the obtained predictor, future tourist volumes can be forecast.

Machine learning models for tourism demand forecasting

At present, machine learning models that can be used to forecast tourism demand fall mainly into three categories: linear machine learning models, nonlinear machine learning models, and ensemble machine learning models. The most commonly used linear machine learning models are linear regression (LR) and ridge regression (RR); the most commonly used nonlinear machine learning models include the decision tree regression (DTR) model, the K nearest neighbor regression (KNNR) model, the support vector regression (SVR), and NNs; the most commonly used ensemble machine learning models are bootstrap aggregating (BA) and adaptive boosting (AB). To analyze the effect of NLIs values on forecasting results more comprehensively, all eight of these models are tested in this study. A brief introduction to the eight models is given below.

(1) Linear regression

LR is used to analyze the relationship between a dependent variable and one or more independent variables. If there is only one independent variable, then the model is called simple LR; if there are two or more independent variables, then the model is called multiple LR. Let $v_{1}, v_{2}, ..., v_{n}$ denote n independent variables and y denote the dependent variable. Then, the LR model can be constructed as follows

y = α_{0} + α_{1} v_{1} + α_{2} v_{2} + ... + α_{n} v_{n} + ε

where $α_{0}$ is a constant, $α_{1}, α_{2}, ..., α_{n}$ are regression coefficients, and $ε$ is an error term.

To determine the parameters in equation (1), the least squares method can be used, where the loss function and objective function are given in equations (2) and (3), respectively

J (α) = \frac{1}{2 M} \sum_{m = 1}^{M} {(y^{(m)} - α^{T} v^{(m)})}^{2}

Min J (α_{0}, α_{1}, ..., α_{n})

where M is the number of samples, $y^{(m)}$ and $α^{T} v^{(m)}$ denote the dependent variable corresponding to the $m th$ sample and the prediction result obtained by the sample, respectively, $m = 1, 2, ..., M$ .

(2) Ridge regression

RR, also known as Tikhonov regularization and weight decay, is a biased estimation regression method specially used for analysis of collinear data. In essence, it is an improved least squares method. By abandoning the unbiasedness of the least squares method, more reasonable regression coefficients can be obtained at the cost of losing part of the information and a reduction in accuracy. RR is more suitable for ill-posed problems than the least squares method.

The function form of RR and LR is the same, but their loss function is different when estimating the parameters. The main difference is that a regularization term is introduced into the loss function of RR to avoid the possible problems of LR when estimating the parameters, such as over-fitting. The loss function of RR is shown in equation (4)

J (α) = \frac{1}{2 M} \sum_{m = 1}^{M} {(y^{(m)} - α^{T} v^{(m)})}^{2} + λ \sum_{j = 1}^{N} α_{j}^{2}

where $λ$ is a parameter of the regularization term.

(3) Decision tree regression

DTR is a type of nonparametric supervised learning model. The goal of DTR is to create a model for predicting the value of target variables by the decision rules inferred from data features. Several algorithms can be used to construct the DTR, such as M5P, and classification and regression tree (CART). In these algorithms, CART, proposed by Breiman et al. (1984), is regarded as one of the most powerful nonparametric algorithms for generalization operations. Compared with other parametric algorithms, CART is less affected by abnormal data. CART generates an optimized binary tree by calculating the Gini index of each node and using binary recursive partitioning to achieve regression. The Gini index at node t can be calculated by

G (t) = 1 - \sum_{k = 1}^{K} p^{2} (k| t)

where K is the number of output categories and $p (k| t)$ is the probability of output category k in node t.

The difference loss of node t under branch condition $ξ$ is as follows

Δ G (ξ, t) = G (t) - \frac{|S_{R}|}{|S_{R}| + |S_{L}|} G (t_{R}) - \frac{|S_{L}|}{|S_{R}| + |S_{L}|} G (t_{L})

where $|S_{L}|$ and $|S_{R}|$ are the number of samples on the left and right branches of CART, respectively.

The conditions for choosing branches are computed as

ξ_{max} = arg max_{ξ \in ξ (t)} Δ G (ξ, t)

(4) K nearest neighbor regression

K nearest neighbor (KNN) model is one of the simplest and most commonly used algorithms in the field of data mining. The KNN model can be used not only for classification but also for regression. In this study, the KNN used for regression (i.e. KNNR) is employed. The basic idea of KNNR is to predict the value of a sample to be predicted based on its KNNs. The process has four steps: (a) calculate the Euclidean distances between the sample to be predicted and other samples; (b) rank the samples from small to large based on the obtained Euclidean distances; (c) determine the optimal value of K; and (d) determine the predicted values by calculating the weighted average values of the K nearest samples.

(5) Support vector regression

SVR is a type of support vector machine for regression (Drucker et al., 1997). Its core idea is to find a hyperplane satisfying the condition that all the data in the training set are as close as possible to the hyperplane. The process of finding the hyperplane can be transformed into solving the following optimization problems

min \frac{1}{2} {‖ω‖}^{2}

s.t. \{\begin{matrix} y^{(m)} - ω v^{(m)} - b \leq ε \\ ω v^{(m)} + b - y^{(m)} \leq ε \end{matrix}, m = 1, 2, ..., M

where $v^{(m)}$ denotes the $m th$ sample, $y^{(m)}$ is the target value of $v^{(m)}$ , and $ε$ is a threshold indicating the range of the difference between the predicted value and the actual value.

(6) Neural network

NN is an algorithmic mathematical model which imitates the behavioral characteristics of animal NNs and carries out distributed parallel information processing. NN mainly achieves the purpose of processing information by adjusting the interconnection between a large number of internal nodes. Let $v_{1}, v_{2}, ..., v_{n}$ denote n independent variables and y denote the dependent variable. Then, according to the principle of NN, the following mathematical models can be constructed

y = f (\sum_{k = 1}^{n} w_{k} v_{k} + θ)

where w_k is the weight of input v_k, $θ$ indicates the bias term, and f is an activation function of NN. The training of NN is the process of determining w_k and $θ$ based on $v_{1}, v_{2}, ..., v_{n}$ and y.

(7) Adaptive boosting

AB is a kind of ensemble learning algorithm. AB can be used for both classification and regression. For regression problems, its core idea is to train different predictors (weak predictors) based on the same training set using different sample weights and then aggregate these weak predictors to form a stronger final predictor (strong predictor). The training process of AB has six steps: (a) initialize distribution weights of the training samples; (b) train weak predictors based on training samples and their corresponding weights; (c) calculate the prediction error of the trained weak predictor; (d) calculate the weight of the trained weak predictor and update the weight of each training sample according to the obtained prediction error; (e) repeat the above process T times, and T weak predictors and their corresponding weights can be obtained; and (f) integrate the obtained T weak predictors and their corresponding weights, and the final strong predictor can be obtained.

(8) Bootstrap aggregating

BA, also called bagging, is another kind of ensemble learning algorithm, which aims to improve the stability and accuracy of machine learning algorithms for classification and regression. For regression tasks, its core idea is to train different predictors (weak predictors) using different training data sampled from the training set uniformly and with replacement, and then the final prediction results can be obtained by calculating the average value of the outputs obtained by the weak predictors. The training process of BA has four steps: (a) generate a bootstrap sample from the training set; (b) train a weak predictor using the bootstrap sample; (c) repeat the above process T times, and T weak predictors can be obtained; (d) calculate the average value of the outputs obtained by the T weak predictors, and the final prediction results can be obtained.

Experimental design

Experimental data sets

The experimental data sets used in this study are monthly numbers of international tourist arrivals at 10 European countries: Austria, Belgium, Finland, Germany, Greece, Italy, Luxembourg, the Netherlands, Portugal, and Sweden. The data used in the experiment are obtained from the Eurostat database (https://ec.europa.eu/eurostat/data/database). The period spans from January 1995 to April 2018, giving 280 observation points for each country.

The experimental data are shown in Figure 2. In Figure 2, the horizontal axis and vertical axis of each sub-figure represent time and number of international tourist arrivals, respectively. As can be seen from Figure 2, the international tourist arrivals in these 10 countries show a distinct periodicity, with a 12-month cycle. Since the most suitable tourist months in different countries are not the same, the months when international tourist arrivals reach an extreme value are not the same. In addition, international tourist arrivals in these 10 countries show an overall upward trend over study period.

Figure 2.

The experimental data sets.

Experimental procedure

The experimental procedure is shown in Figure 3. It consists of three stages: (1) data conversion, (2) machine learning model training, and (3) comparison and evaluation of experimental results. Detailed descriptions of these three stages are given below.

Figure 3.

The experimental procedure.

(1) Data conversion

As described above, the machine learning models require the tourist volume data in the form of time series to be converted into input–output pair sequences. Specifically, the international tourist arrivals of each of the 10 countries with respect to the 280 months mentioned above are transformed into the data form with different NLIs required by the machine learning models. In this experiment, the range of NLIs is set to 1–60. Therefore, for each country, 60 sub-data sets with different NLIs can be obtained. In this stage, a total of 60 × 10 = 600 sub-data sets were obtained.

(2) Machine learning model training

To train tourist demand predictors and evaluate their performance, it is necessary to divide each sub-data set into a training set and a test set. To verify whether the optimal NLIs value is affected by the size of the test set, each sub-data set is divided in four ways, to give test sets of 3, 12, 24, and 36 samples. Therefore, for a sub-data set, four training sets and four test sets are obtained. Since there are 600 sub-data sets, 600 × 4 = 2400 training sets and 600 × 4 = 2400 test sets are obtained. Based on the obtained training sets, the above eight machine learning models are trained, and the corresponding predictors are obtained. On this basis, the corresponding prediction result of each predictor on each test set is obtained. Since there are eight machine learning models and 2400 test sets, 2400 × 8 = 19,200 prediction results are obtained in this stage.

(3) Comparison and evaluation of experimental results

To compare the performance of different models with different NLIs values, the root-mean-square error (RMSE) is adopted in this study because RMSE is the most commonly used indicator to measure the performance of models in tourism forecasting (Hassani et al., 2017). The RMSE can be calculated by equation (11)

RMSE = \sqrt{\frac{1}{N} \sum_{n = 1}^{N} {(y_{n} - {y^{'}}_{n})}^{2}}

where y_n and ${y^{'}}_{n}$ , respectively, denote the actual tourist volume and the forecast tourist volume and N denotes the number of observations. Obviously, the smaller the value of RMSE, the better is the performance of the model.

To verify whether there is a significant difference between the prediction results with respect to different NLIs or models, the Wilcoxon test is used in this study. The Wilcoxon test is a nonparametric statistical method to test hypotheses and is used to compare two related samples to assess whether their population mean ranks are significantly different. The method is briefly described as follows.

Let $α_{1}, α_{2}, ..., α_{L}$ and $β_{1}, β_{2}, ..., β_{L}$ denote two sets of observations, which may be the RMSE obtained by different models on different data sets or the RMSE of a model on different data sets with respect to different NLIs values. Let d_l denote the difference between $α_{l}$ and $β_{l}$ , that is

d_{l} = α_{l} - β_{l}, l = 1, 2, ..., L

Let $|d_{(1)}|, |d_{(2)}|, ..., |d_{(L)}|$ denote the ranking of the absolute values of $d_{1}, d_{2}, ..., d_{L}$ from the smallest to the largest. Let r_l denote the ranking position of d_l in $|d_{(1)}|, |d_{(2)}|, ..., |d_{(L)}|$ , $l = 1, 2, ..., L$ . Let I_l denote an indication variable regard to d_l, and the value of I_l can be determined by equation (13)

I_{l} = \{\begin{matrix} 1, & if d_{l} > 0 \\ 0.5, & if d_{l} = 0 \\ 0, & if d_{l} < 0 \end{matrix}, l = 1, 2, ..., L

Let $R_{}^{+}$ denote a score indicating that $α_{1}, α_{2}, ..., α_{L}$ is better than $β_{1}, β_{2}, ..., β_{L}$ , and let $R_{}^{-}$ denote a score indicating that $α_{1}, α_{2}, ..., α_{L}$ is worse than $β_{1}, β_{2}, ..., β_{L}$ . According to Wilcoxon (1945), $R_{l}^{+}$ and $R_{l}^{-}$ can be calculated by equation (14) and equation (15), respectively

R^{+} = \sum_{l = 1}^{L} r_{l} I_{l}

R^{-} = \sum_{l = 1}^{L} r_{l} (1 - I_{l})

Let $T = min \{R^{+}, R^{-}\}$ be the value for verifying the significance of the difference between $α_{1}, α_{2}, ..., α_{L}$ and $β_{1}, β_{2}, ..., β_{L}$ . According to T and L, the significance of the difference between $α_{1}, α_{2}, ..., α_{L}$ and $β_{1}, β_{2}, ..., β_{L}$ can be verified by looking up the Wilcoxon rank sum table.

Experimental setup

The experimental study is performed on a PC with a 64 GB RAM and a 3.10 GHz Intel i7-7920HQ 8-Core CPU, using the Windows 7 operating system. In the experimental study, Anaconda (https://www.anaconda.com/) is used, which is an open-source distribution of the Python language for data science and machine learning applications. The main parameters for the above eight models and their settings in the experiment are shown in Table 4.

Table 4.

The main parameters for the eight models and their settings in the experiment.

Model	The main parameter settings
LR	—
RR	Regularization strength = 1, whether to calculate the intercept = “True,” and precision of the solution = 0.001, solver = “auto.”
DTR	The minimum number of samples required to split an internal node = 2, and the minimum number of samples required to be at a leaf node = 1.
KNN	Number of neighbors to use = 5, leaf size = 30, and power parameter = 2.
AB	Base estimator = “DTR,” the maximum number of estimators = 50, learning rate = 1.0, and the loss function = “linear.”
BA	Base estimator = “DTR,” the number of base estimators = 10, and whether samples are drawn with replacement = “True.”
SVR	Kernel type = “radial basis function,” gamma = “auto deprecated,” tolerance parameter = 0.001, and penalty parameter C = 1
NN	Batch size = 16, epochs = 500, activation = “sigmoid,” optimizer = “adam,” and loss = “mean square error.”

Note: LR: linear regression; RR: ridge regression; DTR: decision tree regression; KNN: K nearest neighbor; AB: adaptive boosting; BA: bootstrap aggregating; SVR: support vector regression; NN: neural network.

Experimental results and analysis

How the prediction errors of different models vary as NLI increases

The RMSE curves of different models with respect to different sub-data sets as NLI increases are shown in Figures 4 to 13. To explain the meanings of Figures 4 to 13 more clearly, we take Figure 4 as an example. Figure 4 shows the RMSE curves of different models for international tourist arrivals in Austria, where parts (a) to (d) represent the RMSE curves on the sub-data sets with 3, 12, 24, and 36 test samples, respectively. The horizontal axis represents the NLIs values, from 1 to 60, and the vertical axis represents the RMSE.

Figure 4.

The RMSE curves of different models for international tourist arrivals in Austria.

Figure 5.

The RMSE curves of different models for international tourist arrivals in Belgium.

Figure 6.

The RMSE curves of different models for international tourist arrivals in Finland.

Figure 7.

The RMSE curves of different models for international tourist arrivals in Germany.

Figure 8.

The RMSE curves of different models for international tourist arrivals in Greece.

Figure 9.

The RMSE curves of different models for international tourist arrivals in Italy.

Figure 10.

The RMSE curves of different models for international tourist arrivals in Luxembourg.

Figure 11.

The RMSE curves of different models for international tourist arrivals in the Netherlands.

Figure 12.

The RMSE curves of different models for international tourist arrivals in Portugal.

Figure 13.

The RMSE curves of different models for international tourist arrivals in Sweden.

As can be seen from Figures 4 to 13, the RMSE of different models is affected by the NLIs value. Specifically, as NLIs increases, most of the RMSE obtained by the eight machine learning models first decrease and then remain unchanged. This trend has nothing to do with the size of the test set. Therefore, for research questions Q1 and Q2, we can draw the following conclusions:

A1: As NLIs increases, the RMSE of most machine learning models first decreases rapidly and then tends to be stable (or fluctuates around a certain value) when NLIs reaches a certain cutoff point. The cutoff point is usually related to 12 and its multiples.

A2: The optimal NLIs values for different machine learning models are not affected by the size of the test set.

Exploring the relationship between NLIs and tourist volume data

As can be seen from Figures 4 to 13, the optimal NLIs values, that is, those corresponding to the minimum RMSE obtained by different models, are related to the cycle of the data (i.e. 12 months). In other words, the optimal NLIs values for different models are 12, 24, 36, 48, or 60. However, some models, such as NNs, have large fluctuations in the prediction results with some sub-data sets. To determine the optimal NLIs more intuitively and accurately, the standard deviation of each set of 12 consecutive prediction results is calculated by equation (16)

σ_{j} = \sqrt{\frac{{(y_{j} - \frac{1}{12} (\sum_{k = j}^{j + 11} y_{k}))}^{2}}{12}}, j = 1, 2, ..., 49

where $σ_{j}$ represents the standard deviation of the predicted results obtained by predictors based on NLIs values from NLIs = j to NLIs = j + 11 and y_j represents the RMSE of the prediction result obtained by the predictor with respect to NLIs = j. Obviously, the smaller $σ_{j}$ is, the smaller the difference among the 12 consecutive predictions is, that is, the more stable the results are.

According to equation (16), the standard deviation of the predicted results obtained by the eight models with different NLIs on the 10 data sets can be determined. To comprehensively consider the case of test sets with different numbers of samples, the average value of RMSE of each model on each data set with different sample sizes for the test sets is calculated. The results are shown in Table 5. To clarify the meaning of the data in Table 5, we take the “154,034” in the upper left corner of the table as an example. The “154,034” denotes the average value of RMSE obtained by the LR model on the data set “Austria” with different numbers of samples in the test set in the case of NLIs = 12. In other words, “154,034” is the average value of the RMSE obtained by LR with respect to NLIs = 12 corresponding to the RMSE values in Figure 4, where parts (a) to (d) show the results with different numbers of samples in the test set. It can be seen from Table 5 that the optimal NLIs value can differ both across models and across the different national data sets. For example, LR achieves the best prediction result with NLIs = 60 with the data set “Austria,” while LR achieves the best prediction result with NLIs = 24 with the data set “Finland.” To verify whether there are significant differences among the results obtained by each model with different NLIs values, the Wilcoxon test is used. The results are shown in Table 6. For DTR, KNNR, AB, BA, SVR, and NN, in most cases there is no significant difference between the results obtained by these models with one cycle (i.e. 12 months) as the NLIs or multiple cycles (24, 36, 48, and 60 months) as the NLIs. Since the complexity of these models increases as NLIs increases, but the prediction results are generally not significantly improved, it is better to select one cycle as the NLIs in tourism demand forecasting.

Table 5.

The average value of RMSE of each model on each data set with different numbers of samples in the test sets.

	NLIs	LR	RR	DTR	KNNR	AB	BA	SVR	NN	Average
Austria	12	154,034	195,018	275,928	334,814	242,333	263,679	202,378	205,270	234,182
	24	128,880	154,068	285,643	341,135	214,826	238,026	213,216	175,250	218,880
	36	138,018	152,299	197,470	334,988	214,020	190,387	212,553	185,844	203,197
	48	127,602	146,839	233,549	334,988	218,328	214,444	214,408	181,025	208,898
	60	119,543	139,643	206,435	334,988	211,853	200,590	217,619	196,513	203,398
Belgium	12	71,888	70,184	81,898	78,308	69,426	63,174	64,261	72,936	71,509
	24	71,637	64,080	83,821	79,887	72,474	71,809	65,121	74,136	72,870
	36	63,606	55,096	78,920	76,242	64,786	63,977	65,530	67,001	66,895
	48	63,288	53,381	69,227	75,524	62,816	62,831	59,236	69,038	64,418
	60	59,539	49,911	67,335	75,234	62,961	51,926	54,131	70,003	61,380
Finland	12	21,329	23,853	30,608	39,375	29,755	24,260	32,552	25,283	28,377
	24	14,502	21,531	28,946	40,754	31,886	26,782	34,360	20,539	27,412
	36	15,311	20,257	34,481	41,463	35,294	29,485	35,172	20,139	28,950
	48	15,481	18,671	31,296	40,324	32,921	27,608	34,401	18,846	27,444
	60	16,599	18,054	31,168	40,040	29,646	28,343	34,738	19,443	27,254
Germany	12	84,331	143,155	160,447	317,074	150,184	127,884	232,538	186,572	175,273
	24	78,631	122,137	138,968	293,981	123,762	154,261	243,865	170,312	165,740
	36	63,046	110,531	115,675	293,268	135,096	142,777	223,652	148,874	154,115
	48	64,179	105,136	140,060	283,341	124,196	142,553	223,212	136,215	152,361
	60	69,130	104,170	162,711	268,175	130,744	138,972	231,981	140,986	155,859
Greece	12	116,590	167,520	213,684	336,535	195,212	176,462	263,566	218,524	211,012
	24	78,492	114,421	243,275	341,757	205,717	190,150	246,543	214,942	204,412
	36	64,518	102,941	221,750	343,521	180,608	188,523	238,015	225,683	195,695
	48	75,523	99,043	222,502	341,662	195,663	195,513	237,858	240,118	200,985
	60	83,705	98,949	248,570	339,065	182,607	187,910	237,088	248,659	203,319
Italy	12	319,521	398,590	588,603	762,868	505,241	526,696	532,025	548,366	522,739
	24	265,360	321,075	657,097	767,370	446,243	524,251	566,315	471,530	502,405
	36	211,137	301,290	589,587	764,551	440,358	487,498	627,915	488,901	488,905
	48	219,242	300,786	599,192	760,061	461,313	492,616	619,620	489,790	492,828
	60	201,238	290,076	525,880	760,061	458,608	418,327	628,889	531,235	476,789
Luxembourg	12	4044	3620	7852	5413	5912	5599	4354	6712	5439
	24	3254	3558	8333	5936	6985	6631	4717	7158	5822
	36	3152	3500	9113	6591	6949	7278	5430	9095	6389
	48	3564	3757	8637	7777	6837	5838	6948	10,132	6686
	60	4177	4019	8665	7898	6308	6782	7259	10,421	6941
The Netherlands	12	89,199	91,075	157,138	190,819	116,612	118,445	133,409	102,716	124,927
	24	87,341	86,495	126,977	188,541	128,551	128,073	141,636	98,545	123,270
	36	86,605	78,746	110,748	181,625	119,199	102,944	139,583	101,249	115,087
	48	85,847	75,054	102,637	190,177	115,813	106,812	141,158	100,687	114,773
	60	85,158	75,782	98,605	191,360	119,548	100,203	147,576	113,249	116,435
Portugal	12	136,216	130,425	186,985	233,453	179,240	183,103	197,446	146,696	174,195
	24	92,827	130,153	186,350	238,303	177,585	184,823	198,371	166,923	171,917
	36	73,889	125,814	174,693	236,556	171,193	170,220	202,275	158,553	164,149
	48	71,912	132,650	180,427	235,778	180,446	179,848	211,991	170,585	170,455
	60	73,974	133,438	193,194	236,127	189,989	175,607	220,459	163,037	173,228
Sweden	12	36,438	53,429	105,528	122,553	82,605	70,227	74,260	110,388	81,929
	24	34,953	52,848	94,999	125,686	68,015	65,324	79,974	108,149	78,743
	36	35,366	57,388	90,020	124,944	70,340	65,219	85,366	118,443	80,886
	48	38,068	60,847	99,963	122,583	70,290	64,669	91,550	134,408	85,297
	60	42,450	69,330	85,053	121,668	63,780	70,508	90,986	159,506	87,910
Average		83,287	106,693	172,013	240,303	148,301	149,397	181,550	157,173	154,840

Note: LR: linear regression; RR: ridge regression; DTR: decision tree regression; KNNR: K nearest neighbor regression; AB: adaptive boosting; BA: bootstrap aggregating; SVR: support vector regression; NN: neural network; NLIs: number of lagged inputs; RMSE: root-mean-square error.

Table 6.

Wilcoxon test results for the differences in the RMSE obtained by different models with different NLIs.

Comparison	LR	RR	DTR	KNNR	AB	BA	SVR	NN
12 versus 24	Rejected**	Rejected**	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted
12 versus 36	Rejected**	Rejected*	Accepted	Accepted	Rejected*	Accepted	Accepted	Accepted
12 versus 48	Rejected**	Rejected*	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted
12 versus 60	Rejected*	Rejected*	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted
24 versus 36	Accepted	Rejected*	Rejected*	Accepted	Accepted	Rejected*	Accepted	Accepted
24 versus 48	Accepted	Rejected*	Accepted	Rejected*	Accepted	Rejected*	Accepted	Accepted
24 versus 60	Accepted	Accepted	Accepted	Rejected*	Accepted	Rejected*	Accepted	Accepted
36 versus 48	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted
36 versus 60	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted	Rejected*
48 versus 60	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted	Accepted	Rejected*

* At the significance level of 0.05.

** At the significance level of 0.01.

It should be noted that for LR, there is a significant difference between the results from using one cycle (i.e. 12 months) as the NLIs and those from using multiple cycles (24, 36, 48, and 60 months) as the NLIs. However, there is no significant difference among the results from using two cycles (i.e. 24 months), three cycles (i.e. 36 months), four cycles (i.e. 48 months), and five cycles (i.e. 60 months) as the NLIs. Since the complexity of the model increases as NLIs increases, but the prediction results are not significantly improved, it is better for LR to select two cycles as the NLIs in tourism demand forecasting.

For RR, there is a significant difference between the results from using one cycle as the NLIs and using multiple cycles as the NLIs. In addition, there is a significant difference between using two cycles as the NLIs and using three cycles (i.e. 36 months) or four cycles (i.e. 48 months) as the NLIs. However, there is no significant difference between the results from using three cycles (i.e. 36 months) and using four cycles (i.e. 48 months). Therefore, it is better for RR to select three cycles as the NLIs in tourism demand forecasting. Therefore, for research question Q3, we can draw the following conclusions:

A3: There is a certain relationship between the optimal NLIs and the seasonality of international tourist volume data. Specifically, for the nonlinear machine learning models (DTR, KNNR, SVR, and NN) and ensemble machine learning models (AB and BA), it is better to select one cycle as the NLIs in tourism demand forecasting. For the linear machine learning models (LR and RR), using one cycle as the NLIs can obtain acceptable prediction results, but better prediction results may be obtained using multiple cycles. It is a better choice for LR and RR to select two cycles and three cycles as the NLIs in tourism demand forecasting, respectively.

Verifying whether there are significant differences among the predicted results from different models

The last line of Table 5 shows the average RMSE of the eight models with the optimal NLIs values on the 10 data sets. The ranking of the eight models in tourism demand forecasting according to this average is $LR ≻ RR ≻ AB ≻ BA ≻ NN ≻ DTR ≻ SVR ≻ KNNR$ . To further verify whether there are significant differences among the results obtained by different models, the Wilcoxon test is used again (in accordance with the process set out in the “Experimental procedure” section), and the results are shown in Table 7. Significant differences are not found for “SVR versus DTR,” “AB versus BA,” “AB versus NN,” and “BA versus NN.” However, significant differences are found for all other pairs of models. Surprisingly, the two simplest linear models (LR and RR) perform significantly better than the ensemble machine learning models and nonlinear machine learning models. This may be for the following reasons: (1) in the experiment, tourist arrivals in 10 European countries are selected as the data sets, and the time dependence of the selected data is linear, that is, the relationship between the tourist volume of n previous observation points (independent variables $v_{1}, v_{2}, ..., v_{n}$ ) and the tourist volume of the following observation points (dependent variable y) is linear; and (2) the time span of each data set is from January 1995 to April 2018, with 280 observation points. In addition, other exogenous variables were not considered in the experiment. Therefore, the data used can be classified as “small data.” The ensemble machine learning models and nonlinear machine learning models cannot be fully trained with “small data,” which means that their advantages cannot be fully realized. Therefore, when new tourism demand forecasting models are proposed or when comparing the performance of models in tourism demand forecasting, linear models, such as LR and RR, should be selected as benchmarks; this point has been overlooked in most studies on tourism demand forecasting.

Table 7.

Wilcoxon test results for the differences in the RMSE obtained by different models.

Comparison	R ⁺	R ⁻	Hypothesis	p Value
LR versus SVR	1241	34	Rejected	0.000
LR versus DTR	1275	0	Rejected	0.000
LR versus KNNR	1275	0	Rejected	0.000
LR versus AB	1268	7	Rejected	0.000
LR versus BA	1251	24	Rejected	0.000
LR versus NN	1275	0	Rejected	0.000
LR versus RR	1116	159	Rejected	0.000
SVR versus DTR	480	795	Accepted	0.128
SVR versus KNNR	1275	0	Rejected	0.000
SVR versus AB	177	1098	Rejected	0.000
SVR versus BA	115	1160	Rejected	0.000
SVR versus NN	278	997	Rejected	0.001
SVR versus RR	9	1266	Rejected	0.000
DTR versus KNNR	1239	36	Rejected	0.000
DTR versus AB	180	1095	Rejected	0.000
DTR versus BA	93	1182	Rejected	0.000
DTR versus NN	390	885	Rejected	0.017
DTR versus RR	0	1275	Rejected	0.000
KNNR versus AB	7	1268	Rejected	0.000
KNNR versus BA	6	1269	Rejected	0.000
KNNR versus NN	47	1228	Rejected	0.000
KNNR versus RR	0	1275	Rejected	0.000
AB versus BA	558	717	Accepted	0.443
AB versus NN	779	496	Accepted	0.172
AB versus RR	9	1266	Rejected	0.000
BA versus NN	749	526	Accepted	0.172
BA versus RR	32	1243	Rejected	0.000
NN versus RR	4	1271	Rejected	0.000

Based on the above analysis, for the research question Q4, we can draw the following conclusions:

A4: In most cases, there are significant differences among the prediction results obtained by different categories of models, that is, linear machine learning models, nonlinear machine learning models, and ensemble machine learning models, when optimal NLIs values are used. Surprisingly, the two simplest linear models (LR and RR) perform significantly better than the nonlinear machine learning models (DTR, KNNR, SVR, and NN) and ensemble machine learning models (AB and BA) in tourism demand forecasting.

Additionally, to better evaluate the results of different models, it is necessary to compare the performances of these machine learning models with the baseline models (i.e. ARIMA and the Naïve model) commonly used in tourism demand forecasting. For this, the average RMSE of each machine learning model with the optimal NLIs on each data set is respectively calculated, as shown in columns 2 to 9 of Table 8. Meanwhile, the average RMSEs of ARIMA and the Naïve model on each data set are also respectively calculated, as shown in the last two columns of Table 8. The results in Table 8 show that when each of those machine learning models uses its optimal NLIs, they significantly outperform the baseline models (i.e. ARIMA and the Naïve model) on these data sets. However, if the optimal NLIs is not used, the performance of these models may not be as good as the ARIMA in some cases. This provides evidence that the NLIs value significantly affects the accuracy of machine learning models in tourism demand forecasting.

Table 8.

The average RMSEs of different machine learning models and baseline models on each data set.

Country	LR	RR	DTR	KNNR	AB	BA	SVR	NN	ARIMA	Naïve
Austria	119,543	139,643	197,470	334,988	211,853	190,387	202,378	175,250	490,997	666,463
Belgium	59,539	49,911	67,335	75,524	62,816	51,926	54,131	67,001	65,255	104,157
Finland	14,502	18,054	28,946	39,375	29,755	24,260	32,552	18,846	45,313	64,952
Germany	63,046	104,170	115,675	268,175	123,762	127,884	223,212	136,215	345,968	460,716
Greece	64,518	98,949	213,684	336,535	180,608	176,462	237,088	214,942	300,284	591,861
Italy	201,238	290,076	525,880	760,061	440,358	418,327	532,025	471,530	727,655	1,317,063
Luxembourg	3152	3500	7852	5413	5912	5599	4354	6712	9623	14,939
The Netherlands	85,158	75,054	98,605	181,625	115,813	100,203	133,409	98,545	202,207	276,517
Portugal	71,912	125,814	174,693	233,453	171,193	170,220	197,446	146,696	199,059	244,705
Sweden	34,953	52,848	85,053	121,668	63,780	64,669	74,260	108,149	140,043	231,181
Average	71,756	95,802	151,519	235,682	140,585	132,994	169,086	144,389	252,640	397,255

Note: LR: linear regression; RR: ridge regression; DTR: decision tree regression; KNNR: K nearest neighbor regression; AB: adaptive boosting; BA: bootstrap aggregating; SVR: support vector regression; NN: neural network; RMSE: root-mean-square error; ARIMA: autoregressive moving average.

Conclusions

This study experimentally explores how to select optimal NLIs values in international tourism demand forecasting based on machine learning models. A framework for tourism demand forecasting based on machine learning models is proposed, which has two parts: data conversion and model training. Based on the proposed framework, an experimental design is presented in which the international tourist arrivals in 10 European countries (Austria, Belgium, Finland, Germany, Greece, Italy, Luxembourg, the Netherlands, Portugal, and Sweden) from January 1995 to April 2018 are selected as the data sets; the performance of eight commonly used machine learning models (LR, RR, DTR, KNNR, AB, BA, SVR, and NN) is evaluated using NLIs values ranging from 1 to 60. The Wilcoxon test is used to verify whether there are significant differences between the prediction results obtained using different NLIs values and different models. Based on 19,200 experiments, the following conclusions can be drawn:

As NLIs increases, the RMSE of most machine learning models first decreases rapidly and then tends to be stable (or fluctuates around a certain value) when NLIs reaches a certain cutoff point. The cutoff point is usually related to 12 and its multiples.

The optimal NLIs value of different machine learning models is not affected by the size of test set.

For the nonlinear machine learning models (i.e. DTR, KNNR, SVR, and NN) and ensemble machine learning models (i.e. AB and BA), it is better to select one cycle as the NLIs in international tourism demand forecasting. For the linear machine learning models (i.e. LR and RR), using one cycle as the NLIs can obtain acceptable prediction results, but better prediction results may be obtained using multiple cycles. Specifically, it is better for LR and RR to select two cycles and three cycles as the NLIs in international tourism demand forecasting, respectively. These results provide evidence that there is a certain relationship between the optimal NLIs and the seasonality of tourism time series, and the optimal NLIs can be determined according to the seasonality of tourism time series. It should be noted that one cycle is not always the optimal NLIs for different models, especially for linear machine learning models. These results are helpful to the formation of the theory of selecting the optimal NLIs in tourism demand forecasting with machine learning models.

In most cases, there are significant differences among the prediction results obtained by different categories of models, that is, linear machine learning models, nonlinear machine learning models, and ensemble machine learning models, when the optimal NLIs is used. Surprisingly, the two simplest linear models (LR and RR) perform significantly better than the nonlinear machine learning models (DTR, KNNR, SVR, and NN) and ensemble machine learning models (AB and BA) in international tourism demand forecasting.

The above conclusions can provide scientific guidance and help for practitioners to determine the optimal NLIs quickly and reasonably when using machine learning models to forecast international tourism demand and provide valuable information for making more effective policy and business decisions.

The study does have some limitations, which may serve as avenues for future research. First, the conclusions are based on the international tourist arrivals at 10 European countries, and further research is needed on the validity of these conclusions for other countries and regions. In addition, this study only focuses on the univariate forecasting problem in the field of tourism demand forecasting, thus how to determine the NLIs in multivariate forecasting is a promising future research direction.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partly supported by the Humanities and Social Science Fund of Ministry of Education of China under grant number 20YJC630002, the China Postdoctoral Science Foundation under grant numbers 2020T130318 and 2019M661000, the National Natural Science Foundation of China under grant numbers 71971124 and 71932005, and the Fundamental Research Funds for the Central Universities, NKU under grant number 63202074.

ORCID iD

Jian-Wu Bi

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