Forecast combination with multivariate grey prediction for tourism demand forecasting

Abstract

Empirical evidence has shown that forecast combination can improve the prediction accuracy of tourism demand forecasting. This paper aimed to develop a more accurate grey forecast combination method (GFCM) with multivariate grey prediction models In light of the practical applicability of grey prediction, which is not required to apply any statistical test to examine data series this research features the use of multivariate grey models through the genetic algorithm to synthesize forecasts from univariate grey prediction models commonly used in tourism forecasting into composite forecasts Empirical results showed that the proposed GFCM significantly outperformed the other combination methods considered. The results also suggested that the risk of forecast failures caused by selecting an inappropriate single model for tourism demand forecasting can be reduced by using the GFCM.

Keywords

Tourism demand forecast combination tourist arrivals grey system genetic algorithm

1. Introduction

With the growth of the tourism industry worldwide, forecasting tourism demand has received increasing attention from the industry because it is important for the private sector to formulate apt pricing and revenue management strategies [1]. Tourism demand forecasting can help authorities with investments in tourism infrastructure such as airports and highways. The tourism policies formulated by the relevant authorities significantly influence the intention of investment by the private sector [2]. Therefore, accurate forecasts of tourism demand are of great importance to both tourism policymakers and industry practitioners [1,3].

Due to the importance of tourism demand forecasting, there is an extensive range of forecasting methods available in the field of tourism. Time-series models, causal econometric approaches, and artificial intelligence-based methods prevail in tourism forecasting [2,4]. Related studies include vector autoregressive models [5,6] regression analysis [30], autoregressive integrated moving average models (ARIMA) [7,8,9], autoregressive models [10,11], support vector regression (SVR) [12], and neural networks (NNs) [13,14,15]. Time-series models and econometric approaches require the available data to follow specific statistical properties such as stationarity, to obtain reasonable performance [16,17]. Nevertheless, the requirement for statistical properties in statistics and economics is unrealistic with respect to the real world [18,19].

It is acknowledged that no single forecasting model has been found to be superior to all others in all situations [3,4,20]. Even though forecasting uncertainty suggests that the final selected model will not always be the best in future cases, combination forecasting is a feasible approach to considerably reduce such a risk by synthesizing forecasts generated from different models [2,20]. Empirical evidence has shown that combining forecasting can improve accuracy in tourism forecasting as well [3,4,21,22]. Linear combination methods such as those proposed by Gunter and Önder [23], Shen et al. [22,24]. Song et al. [20], Sun et al. [3], and Wu et al. [25], dominate tourism forecast combination, although. Cang [21] demonstrated the superiority of nonlinear combinations, including the support vector regression (SVR), the radial-basis function network (RBFN), and multi-layer perceptron (MLP), over linear combination methods. Altogether, the persistent development of more accurate forecast combination methods is of great importance to tourism practitioners [2,4]. It should be noted that combination forecasting is different from the decomposition ensemble approaches. A decomposition ensemble approach usually divides the original series into different modalities through decomposition methods, then a combination method is applied to obtain combined forecasts.

Interestingly, grey prediction receives relatively less attention in tourism forecasting [4], although it does not require the collected samples to satisfy any statistical property such that the inherent defects in statistical time series models can be avoided [26,27,28,29]. Little attention has been paid to exploring how grey prediction models can be used in the context of tourism forecasting, with some exceptions such as passenger demand with interval models [31] tourism demand [15,32,33,34,35,36,37,38,39,40] tourism flows [41], and tourist income [42]. Overall, much attention should be paid to the application of grey prediction to tourism demand.

In light of the advantages of grey prediction, the purpose of this research was to develop a grey forecast combination method (GFCM) by using univariate grey models commonly used in the tourism context to generate single-model forecasts. A multivariate grey model can then be applied to synthesize these single-model forecasts into a composite forecast Among multivariate grey models, GM(1, $N$ ), consisting of a system’s characteristic sequence and $N -$ 1 relevant factor sequences, is considered here because GM(1, $N$ ) is the most frequently used multivariate grey model and facilitates explanations of causality [28,43]. Furthermore, GM(1, $N$ ) and its variants have been widely applied to different real-world forecasting problems, including energy consumption [44,45], economic indices of high-tech industries [46], grain production [47], anaerobic digestion [48], traffic-related emissions [49], carbon dioxide emissions [50,51], bankruptcy prediction [52,54], pattern classification [55], forecast combination [33], and electronic waste [56].

The remainder of this paper is structured as follows Section 2 focuses on the proposed GFCM by briefly introducing four candidate models and GM(1, $N$ ), Section 3 introduces commonly used combination methods, Section 4 presents the results of evaluation which verified the forecasting accuracy of the proposed method, and Section 5 provides the conclusions.

2. Grey forecast combination method

Four univariate grey models, commonly used in the tourism context, are considered in this study. Individual single-model forecasts obtained from candidate models are synthesized by GM(1, $N$ ) to a combined forecast.

2.1 Candidate models

2.1.1 First-order grey model with one variable (GM(1,1))

Given an original sequence $x_{1}^{(0)} = (x_{1}^{(0)} (1)$ , $x_{1}^{(0)} (2)$ , $\dots$ , $x_{1}^{(0)} (n))$ , where $x_{1}^{(0)} (j)$ ( $j =$ 1, 2, $\dots$ , $n$ ) is taken at successive equally spaced points in time, GM(1,1) generates a sequence $x_{2}^{(1)} = (x_{2}^{(1)} (1)$ , $x_{2}^{(1)} (2)$ , $\dots$ , $x_{2}^{(1)} (n))$ through the first-order accumulative generation operation (1-AGO) as

\begin{aligned} x_{2}^{(1)} (k) = \sum_{j = 1}^{k} x_{1}^{(0)} (j), k = 1, 2, \dots, n \end{aligned}

(1)

The following equation can serve as the mathematical form of the GM(1,1):

\begin{aligned} \frac{d x_{2}^{(1)}}{d t} + a x_{2}^{(1)} = b \end{aligned}

(2)

The corresponding time response function is expressed as

\begin{aligned} {\hat{x}}_{2}^{(1)} (k) = (x_{1}^{(0)} (1) - \frac{b}{a}) e^{- a (k - 1)} + \frac{b}{a} \end{aligned}

(3)

where

a

and

b

can be estimated by ordinary least squares (OLS). Using the first-order inverse AGO (1-IAGO), the forecast at time point

k

(

k =

2, 3,

\dots

n

), which is equal to

{\hat{x}}_{2}^{(1)} (k) - {\hat{x}}_{2}^{(1)} (k - 1)

, is formulated as

\begin{aligned} {\hat{x}}_{2}^{(0)} (k) = (1 - e^{a}) (x_{1}^{(0)} (1) - \frac{b}{a}) e^{- a (k - 1)} \end{aligned}

(4)

2.1.2 Nonlinear grey Bernoulli model (NGBM(1,1))

The Bernoulli equation is used as the mathematical form of the NGBM(1,1) [57]:

\begin{aligned} \frac{d x_{3}^{(1)}}{d t} + a x_{3}^{(1)} = b (x_{3}^{(1)})^{η} \end{aligned}

(5)

where

η \neq

1 and

x_{3}^{(1)}

is a sequence by performing 1-AGO on

x_{1}^{(0)}

. The corresponding time response function is then written as

\begin{aligned} {\hat{x}}_{3}^{(1)} (k) = {[((x_{1}^{(0)} (1))^{1 - η} - \frac{b}{a}) e^{- a (k - 1) (1 - η)} + \frac{b}{a}]}^{1 / (1 - η)} \end{aligned}

(6)

Here, ${\hat{x}}_{3}^{(0)} (k)$ can be derived by 1-IAGO as GM(1,1).

2.1.3 Fractional GM(1,1) (FGM(1,1))

A sequence $x_{4}^{(p)} = (x_{4}^{(p)} (1)$ , $x_{4}^{(p)} (2)$ , $\dots$ , $x_{4}^{(p)} (n))$ , can be derived for FGM(1,1) by performing the AGO with order $p$ (0 $< p < 1$ ) on $x_{1}^{(0)}$ as follows [58]:

\begin{aligned} x_{4}^{(p)} (k) = \sum_{i = 1}^{k} (\begin{matrix} k - i + p - 1 \\ k - i \end{matrix}) x_{1}^{(0)} (i), k = 1, 2, \dots, n \end{aligned}

(7)

in which

\begin{aligned} (\begin{matrix} k - i + p - 1 \\ k - i \end{matrix}) = \frac{(k - i + p - 1) (k - i + p - 2) \dots (p + 1) p}{(k - i)!} \end{aligned}

(8)

The following differential equation with order $p$ can serve as the mathematical form of FGM(1,1):

\begin{aligned} \frac{d x_{4}^{(p)}}{d t} a^{(p)} x_{4}^{(p)} = b^{(p)} \end{aligned}

(9)

where the corresponding time response function is expressed as

\begin{aligned} {\hat{x}}_{4}^{(p)} (k) = (x_{1}^{(0)} (1) - \frac{b^{(p)}}{a^{(p)}}) e^{- a^{(p)} (k - 1)} + \frac{b^{(p)}}{a^{(p)}} \end{aligned}

(10)

where

{\hat{x}}_{4}^{(0)} (k)

can be obtained by performing AGO with order (

1 - p

) on (

{\hat{x}}_{4}^{(p)} (1)

{\hat{x}}_{4}^{(p)} (2)

\dots

{\hat{x}}_{4}^{(p)} (n)

) first and 1-IAGO on the resulting (

{\hat{x}}_{4}^{(1)} (1)

{\hat{x}}_{4}^{(1)} (2)

\dots

{\hat{x}}_{4}^{(1)} (n)

) next.

2.1.4 Fractional NGBM(1,1) (FNGBM(1,1))

The Bernoulli equation with order $p$ serves as the mathematical form FNGBM(1,1) and is expressed by the following:

\begin{aligned} \frac{d x_{5}^{(p)}}{d t} + a^{(p)} x_{5}^{(p)} = b^{(p)} (x_{5}^{(p)})^{η} \end{aligned}

(11)

where

η \neq

1 and

x_{5}^{(p)}

is a sequence by performing the AGO with order

p

x_{1}^{(0)}

. The formulation of the corresponding time response function is

\begin{aligned} {\hat{x}}_{5}^{(p)} (k) = {[((x_{1}^{(0)} (1))^{1 - η} - \frac{b^{(p)}}{a^{(p)}}) e^{- a (k - 1) (1 - η)} + \frac{b^{(p)}}{a^{(p)}}]}^{1 / (1 - η)} \end{aligned}

(12)

Here, ${\hat{x}}_{5}^{(0)} (k)$ can be obtained in a similar manner as FGM(1,1).

2.1.5 Rolling mechanism

The mean absolute percentage error (MAPE), which is the most commonly used measure of forecasting performance in tourism forecasting, is used to assess the forecasting performance as follows:

\begin{aligned} MAPE = \frac{1}{n} \sum_{k = 1}^{n} \frac{| {\hat{x}}_{1}^{(0)} (k) - x_{1}^{(0)} (k) |}{x_{1}^{(0)} (k)} \times 100 % \end{aligned}

(13)

where

{\hat{x}}_{1}^{(0)} (k)

is a combined forecast at time point

k

We used quarterly data of 2008Q1-2019Q4 with respect to the Taiwan inbound tourism demand from Japan, China, Hong Kong/Macau, the US, South Korea, and Southeast Asia to verify the forecasting accuracy of different combination methods. Over 15 forecasts in the period of 2010Q3-2014Q1, a rolling mechanism is employed to find the optimal series length to construct a single grey model. We take one-quarter-ahead forecasting as an example. For four-point rolling, for instance, a forecast at 2010Q3 is generated by constructing a single model using the period of 2009Q3-2010Q2. Then the period of 2009Q4-2010Q3 is used to construct a single model to generate a forecast at 2010Q4. This process continues until a forecast at 2014Q1 is computed. As a result, the average MAPE is computed over 15 forecasts for four-point rolling. The optimal length, say $α$ (4 $⩽ α ⩽$ 10), is determined in case $v$ -point rolling has the minimum average MAPE among all considered rolls [38].

2.2 Combination of forecasts

GM(1, $N$ ) is a linear model with $N$ variables [49], including a dependent variable denoting tourism demand, $x_{1}$ , and $N -$ 1 explanatory variables, $v_{1}$ , $v_{2}$ , $\dots$ , $v_{N}$ . Here, $N$ is equal to $z$ in the case of the $z$ -model combination ( $z =$ 2, 3, 4). Let a sequence $v_{i}^{(0)} = (v_{i}^{(0)} (1)$ , $v_{i}^{(0)} (2)$ , $\dots$ , $v_{i}^{(0)} (n))$ associate with a single model $i (1 ⩽ i ⩽ z)$ for the $z$ -model combination. For instance, for the combination of forecasts generated by GM(1,1), NGBM(1,1), and NGBM(1,1), when $z =$ 3, $v_{1}^{(0)}$ , $v_{2}^{(0)}$ , and $v_{3}^{(0)}$ are identical to ${\hat{x}}_{2}^{(0)}$ , ${\hat{x}}_{3}^{(0)}$ , and ${\hat{x}}_{5}^{(0)}$ , respectively. A new sequence $v_{i}^{(1)} = (v_{i}^{(1)} (1)$ , $v_{i}^{(1)} (2)$ , $\dots$ , $v_{i}^{(1)} (n))$ can be generated by the 1-AGO as

\begin{aligned} v_{i}^{(1)} (k) = \sum_{j = 1}^{k} v_{i}^{(0)} (j), k = 1, 2, \dots, n \end{aligned}

(14)

and $v_{1}^{(1)}$ can be approximated by a first-order differential equation,

\begin{aligned} \frac{d v_{1}^{(1)}}{d t} + a v_{1}^{(1)} = \sum_{i = 1}^{z} b_{i} v_{i}^{(1)} \end{aligned}

(15)

where

a

and

b_{i}

are the development and the driving coefficients, respectively.

With initial condition ${\hat{x}}_{1}^{(1)} (1) = x_{1}^{(0)}$ (1), the corresponding time response expression of ${\hat{x}}_{1}^{(1)}$ can be expressed by the following:

\begin{aligned} {\hat{x}}_{1}^{(1)} (k + 1) = (x_{1}^{(0)} (1) - \frac{1}{a} \sum_{i = 1}^{z} b_{i} v_{i}^{(1)} (k + 1)) e^{- a k} + \frac{1}{a} \sum_{i = 1}^{z} b_{i} v_{i}^{(1)} (k + 1) \end{aligned}

(16)

Finally, with 1-IAGO, the combined forecast of ${\hat{x}}_{1}^{(0)} (k)$ at time point $k$ is equal to ${\hat{x}}_{1}^{(1)} (k) - {\hat{x}}_{1}^{(1)} (k - 1)$ . A two-phase methodological framework is illustrated in Fig. 1. Two-, three-, and four-model combinations can be considered to generate combined forecasts. It turns out that there are 11 ( $C_{2}^{4} + C_{3}^{4} + C_{4}^{4}$ , $C_{3}^{4}$ ) different combination sets. As suggested by Hsu [59], without the use of OLS. this study uses the genetic algorithm toolbox in MATLAB R2017 to automatically derive $a$ , $b_{1}$ , $\dots$ , and $b_{z}$ by minimizing the MAPE.

Figure 1.

Methodological framework for the study.

3. Combination forecasting

3.1 Linear combination methods

Simple average (SA), variance – covariance (VACO), the discounted mean-squared forecasting error (DMSFE), and quadratic programming (QP) are adopted to combine individual forecasts generated by the candidate models. Let ${\hat{e}}_{k}$ denote a combined forecast generated at time point $k (1 ⩽ k ⩽ n)$ . SA generates it as follows:

\begin{aligned} {\hat{e}}_{k} = \sum_{i = 1}^{z} \frac{v_{i}^{(0)} (k)}{z} \end{aligned}

(17)

VACO generates ${\hat{e}}_{k}$ as

\begin{aligned} {\hat{e}}_{k} = \sum_{i = 1}^{z} \frac{{[\sum_{t = 1}^{n} (v_{i}^{(0)} (t) - x_{1}^{(0)} (t))^{2}]}^{- 1}}{\sum_{s = 1}^{z} {[\sum_{t = 1}^{n} (v_{s}^{(0)} (t) - x_{1}^{(0)} (t))^{2}]}^{- 1}} v_{i}^{(0)} (k) \end{aligned}

(18)

and ${\hat{e}}_{k}$ generated by the DMSFE takes the following form:

\begin{aligned} {\hat{e}}_{k} = \sum_{i = 1}^{z} \frac{{[\sum_{t = 1}^{n} β^{n - t + 1} (v_{i}^{(0)} (t) - x_{1}^{(0)} (t))^{2}]}^{- 1}}{\sum_{s = 1}^{z} {[\sum_{t = 1}^{n} β^{n - t + 1} (v_{s}^{(0)} (t) - x_{1}^{(0)} (t))^{2}]}^{- 1}} v_{i}^{(0)} (k) \end{aligned}

(19)

where

β

is a discounting factor (0

< β ⩽ 1

) that assigns a greater weight to more recent data. For QP, the optimal weights are determined as follows [60]:

\begin{aligned} Minimize \sum_{t = 1}^{n} {(\sum_{i = 1}^{z} w_{i} v_{i}^{(0)} (t) - x_{1}^{(0)} (t))}^{2} \\ subject to \sum_{i = 1}^{z} w_{i} = 1 and w_{i} ⩾ 0 \end{aligned}

(20)

3.2 Nonlinear combination methods

Commonly used nonlinear methods in the tourism literature include MLP, SVR, and RBFN. For each such method, forecasts from candidate models serve as the inputs. One hidden layer is sufficient for approximating any continuous function for NNs [61], and the number of nodes in the hidden layer should be twice the number of input nodes, as suggested by Heaton [62]. Python with the “scikit-learn” package was used to construct the MLP and SVR for a given period of weight estimation, and the “PyTorch” package was used to set-up the RBFN. A grid search was performed to determine the appropriate hyperparameters, including the learning rate of MLP and the kernel function for SVR. For simplicity, the details of MLP, SVR, and RBFN are omitted here, but they can be found in [21].

As for the fuzzy integral, the Choquet fuzzy integral along with a $λ$ -fuzzy measure $μ$ is often applied to combine single-model forecasts nonlinearly [63,64], where $μ$ : $P (X) \to$ [0, 1] and $P (X)$ denote the power set of $V = {v_{1}, v_{2}, \dots, v_{z}}$ . Let $f (v_{s}) = v_{s}^{(0)} (k)$ with respect to a $z$ -model combination at time point $k$ . The elements in $V$ are renumbered such that $f (v_{z}) ⩽ f (v_{z - 1}) ⩽ \dots ⩽ f (v_{1})$ . $μ (E_{s})$ is formulated as

\begin{aligned} μ (E_{s}) = \frac{1}{λ} [\prod_{i = 1.. s} (1 + λ μ_{i}) - 1] \end{aligned}

(21)

where

E_{s} = {v_{1}, v_{2}, \dots, v_{s}}

⩽ s ⩽ z

) and

μ_{i} = μ ({x_{i}}) (0 ⩽ μ_{i} ⩽ 1)

. A non-additive forecast combination method (NA-FCM) [34] applies the Choquet integral to combine forecasts at time point

k

\begin{aligned} {\hat{e}}_{k} = \sum_{s = 1}^{z} f (v_{s}) [μ (E_{s}) - μ (E_{s - 1})] \end{aligned}

(22)

where and

μ (E) =

0. The elements in

X

are renumbered such that

f (x_{z}) ⩽ f (x_{z - 1}) ⩽ \dots ⩽ f (x_{1})

, where

f (x_{1})

and

f (x_{z})

equal

max_{k = 1.. z} f (x_{k})

and

min_{k = 1.. z} f (x_{k})

, respectively. Here,

μ_{i}

can be determined to obtain an optimized value of

{\hat{e}}_{k}

by maximizing the forecasting performance measured by the MAPE. For each nonlinear model, an average MAPE was obtained over 10 independent trials.

4. Empirical study

4.1 Data description

This study collected quarterly data from the Tourism Statistics Database of the Taiwan Tourism Administration (https://stat.taiwan.net.tw/inboundSearch). We generated one-quarter-ahead and three-quarter-ahead forecasts based on the periods of 2008Q1-2010Q2 and 2008Q1-2009Q4, respectively. Ultimately, we generated 38 one-quarter-ahead and 38 three-quarter-ahead forecasts for the period of 2010Q3-2019Q4. The latest 23 forecasts derived by the individual grey models were employed for the accuracy comparison, whereas the other forecasts were employed to measure the weights. We then removed the seasonality from the series by seasonal adjustments. We omit the details of this process here, but they can be found in [65]. To reduce the impact of outliers, natural logarithms were used as well. The resulting data sequences were then applied to construct individual single models.

4.2 Results

Tables 1 and 2 show that no single forecasting model outperformed all others in all situations for three model combinations, all source markets, and both forecasting horizons. In three-quarter-ahead forecasting among the single models, for example, GM(1,1) yielded the best average result (2.552) in the case of Japan, but NGBM(1,1) delivered the best performance (7.707) in the case of China. Tables 1 and 2 illustrate that, except for MLP and RBFN, three-model combinations outperformed two-model combinations in terms of average accuracy, while four-model combinations were more accurate than three-model combinations. This finding is consistent with the results reported by Song et al. [20], who found that forecasting accuracy increased with the number of single models considered in a combination set.

Table 1
Forecasting accuracy of one-quarter-ahead single-model and combined forecasts.

Single model Japan HK/Macau Korea China USA Southeast Asia Average

GM(1,1) 1.100 1.853 3.730 4.225 1.269 1.622 2.300

NGBM(1,1) 1.327 2.232 4.137 4.721 1.427 1.650 2.582

FGM(1,1) 1.418 2.062 4.243 4.073 1.517 1.798 2.518

FNGBM(1,1) 1.198 2.800 3.371 3.956 1.363 1.896 2.431

Average 1.261 2.237 3.870 4.244 1.394 1.742

Linear combination

SA

Two-model 1.203 2.127 3.674 4.088 1.248 1.657 2.333

Three-model 1.185 2.102 3.637 4.076 1.205 1.634 2.307

Four-model 1.184 2.072 3.615 4.076 1.180 1.628 2.293

VACO

Two-model 1.195 2.070 3.611 4.066 1.241 1.659 2.307

Three-model 1.176 2.024 3.467 4.025 1.182 1.636 2.252

Four-model 1.165 1.997 3.423 3.999 1.152 1.630 2.228

DMSFE

Two-model 1.184 2.010 3.641 4.124 1.236 1.695 2.315

Three-model 1.165 1.980 3.597 4.093 1.171 1.691 2.283

Four-model 1.162 1.946 3.571 4.091 1.144 1.699 2.269

QP

Two-model 1.188 2.011 3.633 4.136 1.304 1.656 2.321

Three-model 1.173 1.943 3.407 4.100 1.203 1.631 2.243

Four-model 1.179 1.839 3.151 4.056 1.121 1.637 2.164

GFCM

Two-model 1.170 1.924 3.566 3.985 1.242 1.626 2.252

Three-model 1.103 1.859 3.396 3.834 1.146 1.598 2.156

Four-model 1.146 1.544 3.423 3.928 1.127 1.596 2.173

Nonlinear combination

MLP

Two-model 3.138 3.801 5.099 4.282 9.861 3.242 4.904

Three-model 3.664 2.486 8.050 4.092 10.681 1.819 5.132

Four-model 4.502 3.020 12.727 4.127 9.569 1.777 5.954

RBFN

Two-model 9.672 7.693 12.819 6.633 8.807 13.024 9.775

Three-model 9.927 9.746 6.795 10.022 4.904 10.259 8.609

Four-model 8.829 17.910 3.893 10.245 2.924 8.273 8.679

SVR

Two-model 2.988 2.979 4.750 3.940 9.795 2.258 4.452

Three-model 3.182 3.932 8.426 3.850 9.795 2.161 5.224

Four-model 3.192 4.054 5.676 3.798 9.796 1.606 4.687

NA-FCM

Two-model 1.185 1.950 3.838 4.051 1.320 1.609 2.326

Three-model 1.152 1.879 3.609 3.942 1.198 1.562 2.224

Four-model 1.166 1.781 3.308 4.058 1.105 1.587 2.168

Single model	Japan	HK/Macau	Korea	China	USA	Southeast Asia	Average
GM(1,1)	1.100	1.853	3.730	4.225	1.269	1.622	2.300
NGBM(1,1)	1.327	2.232	4.137	4.721	1.427	1.650	2.582
FGM(1,1)	1.418	2.062	4.243	4.073	1.517	1.798	2.518
FNGBM(1,1)	1.198	2.800	3.371	3.956	1.363	1.896	2.431
Average	1.261	2.237	3.870	4.244	1.394	1.742
Linear combination
SA
Two-model	1.203	2.127	3.674	4.088	1.248	1.657	2.333
Three-model	1.185	2.102	3.637	4.076	1.205	1.634	2.307
Four-model	1.184	2.072	3.615	4.076	1.180	1.628	2.293
VACO
Two-model	1.195	2.070	3.611	4.066	1.241	1.659	2.307
Three-model	1.176	2.024	3.467	4.025	1.182	1.636	2.252
Four-model	1.165	1.997	3.423	3.999	1.152	1.630	2.228
DMSFE
Two-model	1.184	2.010	3.641	4.124	1.236	1.695	2.315
Three-model	1.165	1.980	3.597	4.093	1.171	1.691	2.283
Four-model	1.162	1.946	3.571	4.091	1.144	1.699	2.269
QP
Two-model	1.188	2.011	3.633	4.136	1.304	1.656	2.321
Three-model	1.173	1.943	3.407	4.100	1.203	1.631	2.243
Four-model	1.179	1.839	3.151	4.056	1.121	1.637	2.164
GFCM
Two-model	1.170	1.924	3.566	3.985	1.242	1.626	2.252
Three-model	1.103	1.859	3.396	3.834	1.146	1.598	2.156
Four-model	1.146	1.544	3.423	3.928	1.127	1.596	2.173
Nonlinear combination
MLP
Two-model	3.138	3.801	5.099	4.282	9.861	3.242	4.904
Three-model	3.664	2.486	8.050	4.092	10.681	1.819	5.132
Four-model	4.502	3.020	12.727	4.127	9.569	1.777	5.954
RBFN
Two-model	9.672	7.693	12.819	6.633	8.807	13.024	9.775
Three-model	9.927	9.746	6.795	10.022	4.904	10.259	8.609
Four-model	8.829	17.910	3.893	10.245	2.924	8.273	8.679
SVR
Two-model	2.988	2.979	4.750	3.940	9.795	2.258	4.452
Three-model	3.182	3.932	8.426	3.850	9.795	2.161	5.224
Four-model	3.192	4.054	5.676	3.798	9.796	1.606	4.687
NA-FCM
Two-model	1.185	1.950	3.838	4.051	1.320	1.609	2.326
Three-model	1.152	1.879	3.609	3.942	1.198	1.562	2.224
Four-model	1.166	1.781	3.308	4.058	1.105	1.587	2.168

Table 2

Forecasting accuracy of three-quarter-ahead single-model and combined forecasts.

Single model	Japan	HK/Macau	Korea	China	USA	Southeast Asia	Average
GM(1,1)	2.552	2.336	5.905	7.777	1.937	4.165	4.112
NGBM(1,1)	3.520	3.227	8.073	7.707	2.753	3.661	4.823
FGM(1,1)	5.926	3.009	9.192	12.486	3.586	69.408	17.268
FNGBM(1,1)	3.728	4.134	9.813	8.866	3.276	4.012	5.638
Average	3.931	3.176	8.246	9.209	2.888	20.311
Linear combination
SA
Two-model	3.823	3.113	7.227	9.077	2.734	19.697	7.612
Three-model	3.761	3.089	6.695	9.032	2.681	19.506	7.461
Four-model	3.722	3.065	6.542	9.032	2.656	19.431	7.408
VACO
Two-model	3.564	2.888	7.191	9.140	2.634	17.433	7.142
Three-model	3.438	2.741	6.589	9.105	2.457	16.549	6.813
Four-model	3.362	2.663	6.313	9.107	2.307	16.165	6.653
DMSFE
Two-model	3.522	2.832	6.993	8.941	2.637	15.368	6.716
Three-model	3.405	2.591	6.584	8.570	2.444	13.087	6.114
Four-model	3.332	2.391	6.319	8.309	2.290	10.143	5.464
QP
Two-model	3.150	2.812	6.867	9.779	2.593	16.285	6.914
Three-model	2.824	2.559	6.504	10.143	2.335	16.380	6.791
Four-model	2.616	2.336	6.073	9.977	2.045	18.148	6.866
GFCM
Two-model	2.965	2.716	6.648	8.718	2.468	7.853	5.228
Three-model	2.700	2.514	6.050	8.491	2.228	4.320	4.384
Four-model	2.493	2.130	5.712	8.022	1.933	3.542	3.972
Nonlinear combination
MLP
Two-model	5.659	6.669	19.399	7.616	11.179	17.418	11.323
Three-model	5.977	4.941	19.271	7.894	11.297	20.957	11.723
Four-model	5.769	5.292	11.728	9.399	10.183	23.419	10.965
RBFN
Two-model	9.328	8.585	52.844	7.254	10.272	19.035	17.886
Three-model	9.141	8.370	48.178	7.196	8.888	18.861	16.772
Four-model	7.861	8.318	38.560	8.658	9.249	16.031	14.780
SVR
Two-model	7.669	6.109	33.262	7.063	10.160	12.531	12.799
Three-model	7.432	4.870	33.182	7.903	9.951	12.060	12.566
Four-model	8.067	3.676	32.954	7.098	10.106	12.287	12.365
NA-FCM
Two-model	3.078	2.780	7.145	8.322	2.496	11.885	5.951
Three-model	2.777	2.569	6.268	8.211	2.282	7.921	5.005
Four-model	2.607	2.324	5.758	7.973	2.041	10.044	5.125

As for each model combination, Tables 1 and 2 show that the combined forecasts of the proposed GFCM outperform single-model forecasts when the average accuracy is considered for all source markets and on both forecasting horizons. Table 3 shows that the worst forecasts of the GFCM were superior to the corresponding worst single-model forecasts from two- to four-model combinations for all source markets and on both forecasting horizons. The combined forecasts of the GFCM outperform not only the worst single-model forecasts but also the average forecasting accuracy of single models. These results confirm that the use of the GFCM can help decision-makers reduce the risk of uncertainty of model selection that one single model performs well in one time period but declines in other time periods. This has been noted for the advantage of combination forecasting in previous studies such as Hu and Wu [33], Li et al. [1], Shen et al. [22], and Sun et al. [3].

Table 3

Worst single-model forecasts and worst combined forecasts of GFCM.

One-quarter-ahead forecast	Japan	HK/Macau	Korea	China	USA	Southeast Asia
Single-model forecasts	1.418	2.800	4.243	4.721	1.517	1.896
GFCM
Two-model	1.341	2.187	4.059	4.143	1.359	1.745
Three-model	1.319	2.167	3.838	4.072	1.292	1.645
Four-model	1.146	1.544	3.423	3.928	1.127	1.596
Three-quarter-ahead forecast	Japan	HK/Macau	Korea	China	USA	Southeast Asia
Single-model forecasts	5.926	4.134	9.813	12.486	3.586	69.408
GFCM
Two-model	3.755	3.204	8.498	11.649	3.292	25.937
Three-model	3.250	3.072	7.606	9.993	3.054	5.125
Four-model	2.493	2.130	5.712	8.022	1.933	3.542

Table 4

Results of the Friedman test with a post-hoc test for GFCM and nonlinear combination methods.

	Average rank
Combination	MLP	RBFN	SVR	NA-FCM	GFCM	Friedman statistic	Matched comparison	Nemenyi test
Two-model	3.917	4.5	2.917	2.167	1.5	16.789^***	MLP vs. GFCM	2.417^***
							RBFN vs. GFCM	3^***
							SVR vs. GFCM	1.417
							NA-FCM vs. GFCM	0.667
Three-model	3.75	4.083	3.583	2.083	1.5	11.759^***	MLP vs. GFCM	2.25^***
							RBFN vs. GFCM	2.583^***
							SVR vs. GFCM	2.083^**
							NA-FCM vs. GFCM	0.583
Four-model	4.083	4.25	3.333	1.833	1.5	4.936^***	MLP vs. GFCM	2.583^***
							RBFN vs. GFCM	2.75^***
							SVR vs. GFCM	1.833^**
							NA-FCM vs. GFCM	0.333

Note: *** in ‘Friedman statistic’ column denotes the statistical significance of Friedman statistic at a 1% significance level. ***, **, and * in ‘Nemenyi test’ column denote significant differences between the average ranks of two methods in ‘Matched comparison’ column at 1%, 5%, and 10% significance levels, respectively.

Additionally, the validation on the forecasting results requires the multiple comparisons tests. Demšar [66] pointed that the nonparametric Friedman test is suitable for comparing multiple classifiers. In this study, the Friedman test has been used to validate whether the forecasting results of GFCM and the other combination methods considered here have a significant difference or not. For each model combination, Table 4 shows that the null hypothesis – which states that the average ranks of the GFCM and the nonlinear and linear combination methods are equal respectively – was rejected because the test results of the Friedman statistic indicates that the forecasting results of these methods have a significant difference at a 1% significance level. Table 5 likewise shows a significant difference. The post hoc Nemenyi test was conducted to detect differences between the two methods by comparing differences in their average ranks and critical differences between them. The “Matched comparison” columns in Tables 4 and 5 show pairwise comparisons between the GFCM with nonlinear and linear combination methods, respectively. The two methods delivered significantly different performance at significance levels of 1%, 5%, and 10% if their corresponding averages differed by critical differences of at least 2.101, 1.761, and 1.587, respectively. The results in the “Nemenyi test” column show that the GFCM significantly outperformed the MLP and RBFN across all model combinations. Although the proposed GFCM was not significantly superior to the NA-FCM with the Choquet integral for all model combinations, its average rank was less than that of the NA-FCM for each model combination. Table 5 shows that the GFCM significantly outperformed VACO, DMSFE, and SA across all model combinations. Because the results in the “Nemenyi test” column indicate that the GFCM was significantly superior to QP on two- and three-model combinations, and the difference between them was just below the critical difference of 1.587 for four-model combinations, we can conclude that the GFCM outperformed QP. Overall, the proposed GFCM showed superior forecasting accuracy compared to the linear and nonlinear methods considered here.

5. Concluding remarks

Because of the importance of accurate predictions of tourism demand for tourism practitioners, the methodological contribution of this study is to produce more accurate results for tourism demand forecasting through the development of the two-phase GFCM. In the first phase, four commonly used univariate grey models served as candidate models to produce individual single-model forecasts. In contrast with time series models and econometric approaches, grey prediction models have the characteristic of being free from the requirement for any statistical properties imposed on the collected data series. In the second phase, GM(1, $N$ ) was used to synthesize individual forecasts to generate combined forecasts through genetic algorithms. This study is the first to generate more accurate combined forecasts of tourism demand using the multivariate grey prediction model. The empirical results were encouraging because the proposed GFCM outperformed a range of combination methods considered.

Table 5

Results of the Friedman test with a post-hoc test for GFCM and linear combination methods.

	Average rank
Combination	VACO	DMSFE	QP	SA	GFCM	Friedman statistic	Matched comparison	Nemenyi test
Two-model	3.375	3.208	2.917	4.25	1.167	9.426^***	VACO vs. GFCM	2.208^***
							DMSFE vs. GFCM	2.042^**
							QP vs. GFCM	1.75^*
							SA vs. GFCM	3.083^***
Three-model	3.333	3.5	2.75	4.417	1	19.817^***	VACO vs. GFCM	2.333^**
							DMSFE vs. GFCM	2.5^***
							QP vs. GFCM	1.75^*
							SA vs. GFCM	3.417^***
Four-model	3.042	3.583	2.75	4.417	1.208	14.123^***	VACO vs. GFCM	1.833^**
							DMSFE vs. GFCM	2.375^***
							QP vs. GFCM	1.542
							SA vs. GFCM	3.208^***

The combined forecasts generated by the GFCM outperformed not only the worst single-model forecasts but also the average forecasting accuracy of single models for each source market over different forecasting horizons, which suggests that the GFCM helps avoid the risk of forecasting failures that arise from the selection of an inappropriate single model from GM(1,1), NGBM(1,1), FGM(1,1), and FNGBM(1,1). As no single grey model can outperform the others in all situations, the GFCM is a superior alternative among the combination methods considered because it can generate more accurate combined forecasts of tourism demand. In the practice of tourism demand forecasting, the GFCM can help tourism practitioners predict the trend of a source market so as to formulate more appropriate pricing and revenue strategies.

Interestingly, Tables 1 and 2 show that the linear combination methods outperformed the nonlinear methods considered in terms of the average accuracy for each model combination. These results contradict the findings by Cang [21], possibly because the forecasting performance of artificial intelligence methods is sensitive to hyperparameters such as the number of samples and network architecture [7]. Additionally, although GM(1, $N$ ) is a linear model, the proposed GFCM outperformed the nonlinear combination methods considered as well.

Additionally, the outbreak of COVID-19 pandemic has devastated international tourism since 2020. COVID-19 has presented a challenge for the development of accurate methods of forecasting tourism demand [67]. Against this backdrop, combination forecasting can play an important role for tourism demand forecasting. It would be better to re-construct candidate models for forecast combinations whenever structural changes in data series such as COVID-19 are detected [3]. Since tourism demand continues to remain complex and variable, it may not be possible to collect large samples or adhere to statistical properties in the early period when re-constructing candidate models for the late period. This means that it may be better for forecast combinations with time-varying weights to use grey prediction models instead of time-series models and econometric approaches. This highlights the practicability of the GFCM by providing forecasts to help the tourism sector formulate tourism-related strategies during the recovery from COVID-19. Our future research will explore the use of a time-varying weighting strategy for the GFCM instead of using constant weights for forecasting tourism demand.

Footnotes

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments. This research is supported by the National Science and Technology Council, Taiwan under grant NSTC 112-2410-H-033-017.

Compliance with ethical standards

The authors declare that they have no conflict of interest. This article does not contain any studies with human participants performed by the authors.

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