Application and Accuracy Improvement of Big Data Analytics in Market Demand Forecasting in Tourism Economy

The K-means algorithm is a classical clustering algorithm that minimizes the similarity measure between points within each cluster and maximizes the similarity measure between clusters. The steps of the K-means algorithm are as follows: first, the number of clusters K is set artificially by the writer or user of the algorithm, and then the initial K center-of-masses locations are randomly generated, and the distance from each point to each center-of-mass is calculated by choosing the appropriate distance measure and traversing the clusters to the nearest center-of-mass. Each point is assigned to the center of mass closest to its distance according to the distance minimization rule, so that each point is assigned to a cluster. However, since the initial K center-of-mass positions may not be optimal, the centers of each cluster need to be constantly updated for optimization purposes. The method of updating the center of mass position is to find the mean value of each point in the cluster, and the position where the mean value is located is the new cluster center of mass position, and after continuously finding the mean value to continuously update the cluster position, until the center of mass of the cluster does not change the magnitude of the change reaches the optimization criterion, the algorithm is over, and the final clustering result is the K clusters formed by K clusters with the center of mass that no longer changes [29].

For the K-mean clustering algorithm, the distance metric can be chosen in various forms, such as the Euclidean distance, Manhattan distance, Minkowski distance, and so on. Which distance method is used in practical applications should be selected according to the characteristics of the actual application. Since this paper is for time series to cluster, distance metrics such as Euclidean distance are not very suitable, so Dynamic Time Warping (DTW) is used for the measurement of time series similarity.

Dynamic Time Warping (DTW) is an algorithm that uses the idea of dynamic programming to sequentially match the similarity points between two time series, and uses the sum of the distances of all the similarity points to determine the similarity of the two time series, and the smaller the total distance is, the higher the similarity is [30].

Suppose there are two time series T and R, T and R of lengths n and m respectively(T = t₁, t₂, …, t_i, …, t_n; R = r₁, r₂, …, r_j, …, r_m). Construct a distance matrix D of n × m. Matrix element D(i, j) represents the distance between t_i and r_j. The regularization path is represented in the form of W = w₁, w₂, …, w_k, Max(n, m) ≤ k ≤ n + m − 1, w_k in the form of (i, j), where i, j is the sequence number of the similarity point of the two time series. DTW requires that the first point of the two time series is aligned, i.e., w₁ is represented as (1, 1), and w_k as (n, m). In addition, the regularization path has to be monotonically increasing, which ensures that every sequence point of the two sequences can be taken and that the regularization paths do not fold and cross. Monotonically increasing of the regularized path means: (1)wk=(i,j),wk+1=(i′,j′),i≤i′≤i+1,j≤j′≤j+1$${w_k} = (i,j),{w_{k + 1}} = (i',j'),i \le i' \le i + 1,j \le j' \le j + 1$$

Construct the regularized distance matrix WD to record the cumulative distance by the idea of dynamic programming, matrix element WD(i, j) is denoted as: (2)WD(i,j)=D(i,j)+min[WD(i−1,j),WD(i−1,j−1),WD(i,j−1)]$$WD(i,j) = D(i,j) + \min \left[ {WD(i - 1,j),WD(i - 1,j - 1),WD(i,j - 1)} \right]$$

Finally, the points of the regularized path can be found by backtracking on the regularized distance matrix WD w_k. WD(n, m) is the regularized distance of the two time series for similarity comparison, and the smaller the distance of the regularized path, the higher the similarity.

In this paper, we construct the following statistics to measure similarity between individuals. 1)

The full-time “absolute horizontal” distance between Individual i and Individual j, abbreviated as d_ij = (AQED): (3)dij(AQED)=∑t=1T(xit−xjt)2$${d_{ij}}(AQED) = \sqrt {\sum\limits_{t = 1}^T {{{({x_{it}} - {x_{jt}})}^2}} }$$

The composite distance is d_ij(CED) a weighted combination of the degree of absolute horizontal variation and the degree of dynamic variation. Weights α and β are calculated using the following idea: for all data, make the difference between all absolute level distances the same or similar to the difference between dynamic change distances. Specifically, the standard deviation of d_ij(AQED) and d_ij(ISED) is used as a reference indicator, i.e., a set of α and β is determined so that: (6)α⋅SD[dij(AQED)]=β⋅SD[dij(ISED)]i≠j∈{1,2,...,N}$$\alpha \cdot SD[{d_{ij}}(AQED)] = \:\beta \cdot SD[{d_{ij}}(ISED)]\quad i \ne j \in \{ 1,2,...,N\}$$ (7)α+β=1$$\alpha + \beta = 1$$

Standard deviation of the distances of changes SDl[d_ij(AQED)r] and SDl[d_ij(ISED)r]. Calculations are obtained: (8)α=sD[dij(ISED)]sD[dij(AQED)]+sD[dij(ISED)]$$\alpha = \frac{{sD[{d_{ij}}(ISED)]}}{{sD[{d_{ij}}(AQED)] + sD[{d_{ij}}(ISED)]}}$$ (9)β=sD[dij(AQED)]sD[dij(AQED)]+sD[dij(ISED)]$$\beta = \frac{{sD[{d_{ij}}(AQED)]}}{{sD[{d_{ij}}(AQED)] + sD[{d_{ij}}(ISED)]}}$$

The systematic clustering method is one of the most commonly used cluster analysis methods, and its clustering process depends on the definition of the distance between individuals and the distance between classes. The steps of the systematic clustering method are as follows: first, each sample of the clustering is treated as a class, then the similarity statistic between classes is determined, the closest two classes or a number of classes are merged into a new class, and then the aggregated subclasses are merged again according to their interclass distances, and the given data samples are disaggregated layer by layer until all the samples are merged into a single class. Using this type of clustering method a clustering tree consisting of data samples can be obtained with the characteristic of stopping the clustering division at any time [31]. The main interclass distances commonly used in practice are: the shortest distance method, the longest distance method, the middle distance method, the center of gravity method, the class average method, the variable class average method, and the Ward method. In this paper, Ward’s method is used as a measure of inter-class distance.

For time series {X_t, t ∈ T}, time series {x_t} is said to be a wide smooth time series if the following three conditions are satisfied simultaneously: 1)

∀t ∈ T, with EXt2<∞$$EX_t^2 < \infty$$.

Smoothness in time series analysis generally refers to wide smoothness, which is also known as weak smoothness or second-order smoothness. A series that does not satisfy the smoothness condition is called a non-smooth series. The smoothness test is the basis and premise of time series modeling, and its test methods are mainly the following two. 1)

Graph test method

Time series chart refers to the construction of the time as the horizontal axis, the sequence of values for the vertical axis of the plane of the two-dimensional coordinate chart, which can intuitively reflect the basic distribution characteristics of the time series. When the time series of a time series is always around a constant value of random fluctuations, and its fluctuation range has more obvious boundaries, then the time series is usually a smooth time series. On the contrary, if the time series plot shows more obvious periodic characteristics or trend characteristics, then the series is usually non-semi-stable. To be on the safe side, autocorrelation plots should be used to further assist in identification after observing the time series plot.

An autocorrelation plot is a planar two-dimensional coordinate hanging line plot in which the horizontal axis represents the autocorrelation coefficient and the vertical axis represents the number of delay periods, while the magnitude of the autocorrelation coefficient is represented by the hanging line. Usually, a smooth time series has short-term correlation, which means that the autocorrelation coefficient ρ^k$${\hat \rho _k}$$ of a smooth series will decay to zero in a relatively short period of time as the number of delay periods k increases, while the autocorrelation coefficient ρ^k$${\hat \rho _k}$$ of a non-smooth time series will decay to zero relatively slowly.

Its characteristic equation is: (11)λp−ϕ1λp−1−ϕ2λp−2−⋯−ϕp=0$${\lambda ^p} - {\phi _1}{\lambda ^{p - 1}} - {\phi _2}{\lambda ^{p - 2}} - \cdots - {\phi _p} = 0$$

Time series {x_t} is smooth if all the characteristic roots in Eq. (11) are within the unit circle, i.e. |λ_i| < 1, where i = 1, 2, 3, ⋯, p. If one unit root exists, then time series {x_t} is not smooth and the sum of autoregressive coefficients is exactly equal to 1. It may be useful to set λ_i = 1, and then 1 − ϕ₁ − ϕ₂ − ⋯ − ϕ_p = 0 can be deduced from Eq. (11), which in turn can be deduced: (12)ϕ1+ϕ2+⋯+ϕp=1$${\phi _1} + {\phi _2} + \cdots + {\phi _p} = 1$$

Assuming Order ρ = ϕ₁ + ϕ₂ + ⋯ + ϕ_p − 1, the hypothesis conditions for the AR(p)-process ADF unit root test can be determined as: (13)H0:ρ=0(Sequence{xt}is non - stationary) H1:ρ<0(Sequence{xt}is stationary)$$\begin{array}{l} {H_0}:\rho = 0\:({\text{Sequence}}\{ {x_t}\} {\text{is non - stationary)}} \\ {H_1}:\rho < 0\:({\text{Sequence}}\{ {x_t}\} {\text{is stationary)}} \\ \end{array}$$

Its test statistic is: (14)τ=ρ^/S(ρ^)$$\tau = \hat \rho /S(\hat \rho )$$

where S(ρ^)$$S(\hat \rho )$$ is the sample standard deviation of parameter ρ. The value of the calculated test statistic τ is compared with the table of critical values and if the original hypothesis is rejected, the series is smooth.

In time series analysis, the first step in analyzing time series observations, regardless of the method of analysis, is to take effective means to extract the deterministic information embedded in the time series observations.

From Cramer’s decomposition theorem, it can be seen that all time series can be decomposed into two parts, i.e., a deterministic trend determined by a polynomial and a smooth zero-mean error: (15)xt=μt+εt=∑j=0dβjtj+ψ(B)at$${x_t} = {\mu _t} + {\varepsilon _t} = \sum\limits_{j = 0}^d {{\beta _j}} {t^j} + \psi (B){a_t}$$

where {a_t} is a white noise sequence with zero mean, B is a delay operator, β₁, β₂, …, β_d is a constant coefficient, and d < ∞. Thus, {ε_t: ε_t = ψ(B)a_t} in Eq. (15) denotes the stochastic influence on the time series {x_t}, while the mean sequence {∑j=0dβjtj}$$\left\{ {\sum\limits_j = {0^d}{\beta _j}{t^j}} \right\}$$ reflects the deterministic influence in the time series {x_i}.

Performing the dst order difference operation on the discrete series is equivalent to performing the dnd order derivation on the continuous variable. From Eq. (15), the dth order difference operation on the time series {x_t} can fully extract the deterministic information in {x_t}. Expanding any dth order difference using the lag operator B yields Eq. (16): (16)∇dxt=(1−B)dxt=∑i=0d(−1)iCdixt−i$${\nabla ^d}{x_t} = {(1 - B)^d}{x_t} = \sum\limits_{i = 0}^d {{{( - 1)}^i}} C_d^i{x_{t - i}}$$

It can be obtained after conversion: (17)xt=∑i=1d(−1)i+1Cdixt−i+∇dxt$${x_t} = \sum\limits_{i = 1}^d {{{( - 1)}^{i + 1}}} C_d^i{x_{t - i}} + {\nabla ^d}{x_t}$$

From equation (17), it can be seen that the essence of dst order differencing is an dnd order autoregressive process. The historical data {xt−d}$$\left\{ {{x_{t - d}}} \right\}$$ from the delayed d period is used as the independent variable to explain the movement of the time series values {xt}$$\left\{ {{x_t}} \right\}$$ in the current period, i.e.: the deterministic information is extracted using autoregression. 1)

pth order differencing

pth order differencing is the operation of performing another 1st order differencing operation on the time series after p − 1th order differencing. It is denoted as: (18)∇pxt=∇p−1xt−∇p−1xt−1$${\nabla ^p}{x_t} = {\nabla ^{p - 1}}{x_t} - {\nabla ^{p - 1}}{x_{t - 1}}$$

For example, 1st order differencing is the operation of subtraction between two time series values separated by one period, i.e.: ∇x_t = x_t − x_t−1, where ∇x_t denotes the 1st order differencing of x_t. 2nd order differencing is the process of performing one 1st order differencing operation on a time series followed by another 1st order differencing operation, i.e.: ∇²x_t = ∇x_t − ∇x_t−1, where ∇²x_t denotes the 2nd order differencing of x_t. 2)

k-step differencing

The k-step differencing operation is a subtraction operation between two time series values that are separated by k periods. It is denoted as: (19)∇kxt=xt−xt−k$${\nabla _k}{x_t} = {x_t} - {x_{t - k}}$$

where ∇_kx_t denotes the k-step difference of x_t.

1)

ARIMA model

Difference operation has a strong ability to extract deterministic information, most of the non-smooth time series after difference operation will show the nature of smooth time series. If the series {xt}$$\left\{ {{x_t}} \right\}$$ is a non-stationary time series, but after the difference operation shows the nature of a stable series and through the series smoothness test, then the non-stationary series {xt}$$\left\{ {{x_t}} \right\}$$ is called the difference of a stable series, you can establish a summed autoregressive moving average model (ARIMA). The essence of the ARIMA(p, d, q) model is a combination of the difference operation and the autoregressive sliding average ARMA(p, q) model [32].

The ARIMA model can be abbreviated as: (20)ϕ(B)∇dxt=ϑ(B)εt$$\phi (B){\nabla ^d}{x_t} = \vartheta (B){\varepsilon _t}$$

where ϕ(B) is the autoregressive coefficient polynomial, i.e., ϕ(B) = 1 − ϕ_lB − ϕ₂B² − ⋯ − ϕ_pB^pϑ(B) is the shifted smoothing coefficient polynomial, i.e., ϑ(B) = 1 − θ₁B − θ₂B² − ⋯ − θ_qB^q. {ε_t} is the white noise series with zero mean, d denotes the number of differences, ∇ denotes the difference operator, and ∇ = 1 − B, ∇x_t = x_t − x_t−1. 2)

Seasonal product model (SARIMA)

Seasonal product model is the original non-stationary time series for differential transformation and then seasonal differential and through the smoothness test after the establishment of the model. The fitted model is actually the product of ARMA(p, q) and ARMA(P, Q) due to the multiplicative relationship between seasonal effects and short-term correlations. Combining the dth-order trend differencing and the Dth-order seasonal differencing with period S as the step size, the complete seasonal product model is obtained: (21)∇d∇SDxt=(ϑ(B)ϑS(B))/ϕ(B)ϕS(B)*εt$${\nabla ^d}\nabla _S^D{x_t} = (\vartheta (B){\vartheta _S}(B))/\phi (B){\phi _S}(B)*{\varepsilon _t}$$

In the formula: (22)ϑ(B)=1−θiB−⋯−θqBq,ϕ(B)=1−ϕlB−⋯−ϕpBp$$\vartheta (B) = 1 - {\theta _{\text{i}}}B - \cdots - {\theta _q}{B^q},\phi (B) = 1 - {\phi _{\text{l}}}B - \cdots - {\phi _{\text{p}}}{B^p}$$ (23)ϑs(B)=1−θ1Bs−⋯−θQBQS,ϕs(B)=1−ϕ1BS−⋯−ϕPBPS$${\vartheta _s}(B) = 1 - {\theta _1}{B^s} - \cdots - {\theta _Q}{B^{QS}},{\phi _s}(B) = 1 - {\phi _1}{B^S} - \cdots - {\phi _P}{B^{PS}}$$

The seasonal product model can be simply notated as ARIMA(p, d, q)(P, D, Q)_S where p is the autoregressive term. d is the number of times the original series becomes a smooth time series of differences. q is the moving average term. P is the seasonal autoregressive order. D is the seasonal difference order. Q is the seasonal sliding average order.

The data selected for this paper are the monthly data of Xiamen tourist trips from January 2014 to May 2019 (data source: data published by Xiamen Tourism Network). The original data series of tourism attendance is shown in Figure 3.

Figure 3.

The original data series of the tourist

Analyzing the time series characteristics of the data, the data is obviously non-stationary and seasonal, accompanied by certain cyclical fluctuations, and there are obvious peak seasons in a year, with peaks around April to May, July to August, and October.

In order to eliminate the trend and at the same time reduce the fluctuation of the sequence, the logarithm of the original sequence is taken and the sequence is named ly, whose time series is plotted in Fig. 4, and it is found that the sequence is still not smooth.

Figure 4.

Sequence diagram of the logarithm of the tourist

The first order difference is done on the sequence ly, the sequence is named ily, and its autocorrelation and partial autocorrelation analysis is shown in Fig. 5. From the figure, it can be seen that the trend of the series is basically eliminated, but when k=12, the sample autocorrelation coefficient and partial autocorrelation coefficient of the series are significantly not 0, which indicates the existence of seasonality.

Figure 5.

Self-correlation and partial correlation data of time series ily

Seasonal differencing is done on the sequence sily to obtain the new sequence sily, and the autocorrelation and partial autocorrelation analysis of the sequence sily are plotted, as shown in Fig. 6. The sample autocorrelation coefficients and partial autocorrelation coefficients of the sequence sily quickly fall into the random interval, so the sequence trend has been basically eliminated.

Figure 6.

The self-correlation and partial correlation data of sily of the seasonal difference

The unit root test is a formal method for testing the smoothness of a time series, and to further test whether the sequence sily is smooth or not, the ADF unit root test for the sequence sily is performed. Table 2 shows the results of unit root test for the sequence sily. The value of the t-statistic of the test is -7.86162, which is smaller than any critical value with a significance level of 1%, so the original hypothesis is rejected and it is concluded that there is no unit root in the sequence, and therefore the sequence sily is smooth.

In the “identification” stage of the model, we find that after the first-order logarithmic period-by-period differencing, the period of the sequence is basically eliminated, so d = 1, and after the first-order seasonal differencing, the seasonality is basically eliminated, so D = 1. Therefore, we initially choose the product-seasonal model (p, d, q) × (P, D, Q) = (p, 1, q) × (1, 1, 1). In addition, we observe the biased autocorrelation diagram of the sequence SILY, and the combinations of p = 2 and 1 are more appropriate. The autocorrelation diagram shows that q = 1 or 0 is more appropriate. Taken together, the available (p, q) combinations are (2, 0), (2, 1), (1, 0), and (1, 1).

The Akaike’s Criterion of Minimum Information (AIC) from the Best Criterion Function Order Fixing method was used to determine the order of the models. The relevant test results of the four selected models were summarized and Table 3 shows the comparison of the test results. After calculation, all four models satisfy the smooth condition and reversible condition of ARMA process, and the model setting is reasonable. In addition, the concomitant probability of the white noise test of the residual series shows that the residuals of each model satisfy the independence assumption and the models fit well. Comparing the test results of Table 3: Individual Models compared to each model, the first model (2, 1) has the largest adjusted R-squared value (0.83085), the smallest AIC and MAPE, and a smaller SC value. Thus the selection of the first model i.e. ARMA (2, 1, 1) (1, 1, 1) model is appropriate.

Based on the identification and selection of the model above, we choose ARMA (2, 1, 1) (1, 1, 1)1 as our best prediction model, and estimate the parameters of the model and the correlation test results of the model are shown in Table 4. The results show that the parameter estimates of the model ARMA (2, 1, 1) (1, 1, 1) are statistically significant. The prediction model is: (1 − 0.2046B¹²)(1 − 0.51131B − 0.19991B²) × (1 − B)(1 − B)¹²ln (IP_t) = (1 + 0.88047B¹²)ε_t.

The above model is used to forecast the tourist trips in Xiamen. Among them, the prediction results from 2014 to 2019 are shown in Figure 7. It can be seen that the predicted value is basically consistent with the actual value, and the results show that the model in this paper has a good fitting effect.

Figure 7.

Prediction of the number of tourists in Xiamen city

Using the above model, the predicted value of tourist arrivals to Xiamen for June-December 2019 is then given, as shown in Table 5. Comparing with the actual data, the situation is also basically consistent. Therefore, the model in this paper has obvious reference value for predicting and analyzing the tourism reception of tourist destinations.