Probabilistic En Route Sector Traffic Demand Prediction Based on Quantile Regression Neural Network and Kernel Density Estimation

Abstract

With the development of civil aviation in China, airspace congestion has become more and more serious and has gradually spread from airport terminal areas to en route networks. Traditionally, most prediction methods that obtain traffic flow data are based on the number of aircraft passing through an en route sector and require flight data to meet strict assumptions and conditions. While these methods are normally used in the actual operation of air traffic flow management departments in China, the results are not satisfactory due to the nonlinearity of traffic demand along en route sectors and the change in high-frequency noise. In order to refine aircraft control in airspace, it is necessary to predict traffic flow accurately. Thus, this paper proposes the quantile regression neural network and kernel density estimation method to obtain some quantiles of continuous traffic demand data in the future, which combines the strong nonlinear adaptive ability of neural networks with the ability of quantile regression to describe explanatory variables. By using these continuous conditional quantiles, we obtain the probability density function and probability density curve of the continuous traffic demand in the future using the kernel density estimation method. In this way, we can obtain not only a specific point prediction value and its change interval but also the probability of each value in the prediction change interval of traffic demand in the en route sector as well as a more accurate point prediction value for a specific day.

1. Introduction

With the sustained and rapid development of the civil aviation transport industry in China, air traffic flow management (ATFM) departments are facing a much more complex airspace structure, wider airline coverage, increasingly intensive airports, and more sophisticated flight activity management and control. Due to the impact of sudden and uncertain disturbance factors such as bad weather, military activities, major events, and facility failure, airspace in China has produced some world-class flight-busy areas such as the Yangtze River Delta and the Pearl River Delta in a short period. In order to alleviate congestion and balance the capacity and flow, it is necessary to accurately predict the air traffic flow in the en route sector. However, the ATFM departments in China usually use the deterministic prediction method in their actual operations, but as the result is not ideal, some scholars put forward predicting air traffic demand from the perspective of uncertainty.

The United States and Eurocontrol recognized the insufficiency of a deterministic air traffic demand prediction method as early as the beginning of this century and have introduced the probabilistic traffic demand prediction method to their Next-Generation Air Transport System (NextGen) program. Gradually, the uncertainty probabilistic air traffic demand prediction method has received extensive attention in recent years. This method involves different airspaces such as airports and the en route sector. Traditionally, the majority of research on the en route sector has focused on prediction methods showing how to obtain en route sector traffic demands based on historical data on the number of aircraft passing through these sectors. In 2019, Liu [1] proposed a modeling system to predict dynamic traffic demand uncertainty using RFID data and link traffic volume. In 2021, Xian [2] formulated and proposed a multivariate Poisson log-normal model with specific parameterization tailored to the traffic demand problem. In 2024, Khalesian et al. [3] developed an efficient method for traffic demand prediction to learn from an associated time series and produce reliable demand predictions. From this point of view, these methods mainly refer to the traffic flow prediction methods used in the field of ground transportation, including the autoregressive model, the differential autoregressive moving average model, the support vector machine model, Markov chains, gray prediction, the artificial neural network class method, fuzzy neural networks, the CARCH method, and so on. No matter which method is used above, most of the prediction results are numerical values which represent the traffic demand of sectors in a certain period in a deterministic way. Such prediction results can be used in practical operations for air traffic flow management; however, the prediction effect is not very satisfactory. The main reason for this problem is that the above methods require data to meet strict assumptions and conditions; that is, a large amount of aircraft operational data is required, and the data must be targeted to a specific set of aircraft, which is a large number of fixed points per aircraft in a fixed period of time per day. Regarding the operational data, any loose operational data of other aircraft not in the collection will be eliminated during the statistical analysis. But because of the nonlinearity and high frequency noise of the en route sector traffic demand changes, it is difficult to achieve a better prediction result.

Learning from the traffic flow combination prediction methods in the field of ground traffic, some scholars have begun to explore more complex models and methods to achieve a combination of two methods to predict changing traffic demands in en route sectors, for example, the nonparametric regression model [4,5], the support vector machine model [6,7], the neural network model [8,9], the Kalman filter model [10,11], and so on. These methods have achieved good prediction results, but as single-prediction methods, merely a point prediction value about the traffic demand is obtained without the probability density of the continuous traffic demand value at a certain period in time in the future, which makes it difficult to meet more uncertain airspace congestion control demands and for air traffic management personnel to make better traffic allocation decisions.

For a certain period of time in the future, the prediction of traffic demands in an en route sector may not be a simple deterministic value but an expected probability interval corresponding to a certain level. Interval prediction can convey more information than point prediction. In a given interval, a point prediction value can be obtained according to the midpoint value of the interval. Furthermore, a probability density function of future en route sector traffic demands can be obtained as a result of probability density prediction. Judging from the amount of information predicted, it is obvious that probability density prediction is the most detailed. If a continuous probability density function of future traffic demands can be obtained and the probability of the traffic demand distribution interval in en route sectors can be analyzed, air traffic management personnel can better understand the fluctuation range of future traffic demands in these en route sectors. And thus, they can better understand the uncertainties and risk factors that may exist in the future traffic demands of the en route sectors and make more reasonable decisions in time. However, the methods of probability density prediction rely on more theoretical derivation, involving more complex mathematical calculations. Therefore, there are few research studies on the prediction technology of en route sector traffic demand probability density [12].

The probability density of random variables can intuitively reflect the location, shape and other characteristics of their distribution. By observing the probability density curves of random variables, not only more accurate point predictions can be obtained but also more useful information about random variables can be obtained. It is well known that there are three methods to obtain the probability density function of a given random variable: the parameter estimation method, nonparametric estimation method and semi-parametric estimation method. The latter two methods assume that the sample data of a given random variable obey a known probability density function beforehand. Then, according to the assumed specific probability density function, the parameters in the probability density function are solved and correlation estimation is carried out. However, because of the difference between the actual probability density function and the assumed probability density function represented by the given sample size of a random variable, the probability density function obtained by parameter estimation and semi-parameter estimation is not so ideal. This provides a good background for the nonparametric estimation method, namely the kernel density estimation method. In 2009, Okabe et al. proposed a kernel density estimation method for estimating the density of points on a network and implemented the method in the GIS environment [13]. Subsequently, a large number of scholars have also conducted in-depth research on kernel density estimation methods. Among these scholars, what is more typical is that Chau et al. proposed a method combining the Stochastic Expectation–Maximization (SEM) algorithm and Sequential Monte Carlo (SMC) approaches for nonparametric estimation in state-space models in 2021 [14]. According to the probabilistic density estimation method, as long as the kernel function and optimal window width are appropriate, the real probabilistic density function of any random variable can be approximated without restriction. However, when large sample random variables appear, it will lead to a series of shortcomings such as long training time and complex operation. The common probability density estimation method is no longer easy to use; many scholars have put forward improvement measures, including Zwart’s efficient sample density estimation by combining gridding and an optimized kernel [15], Holmes’s fast nonparametric conditional density estimation [16], Rau’s accurate photometric redshift probability density estimation [17], Qifa Xu’s novel composite quantile regression neural network (CQRNN) model [18], Yaoyao He‘s hybrid wind power probability density prediction method based on quantile regression neural network and Epanechnikov kernel function using unbiased cross-validation [19], and so on. These methods can reduce disturbance or similar items in the observation samples when the kernel density estimation model is subjected to correlation processing, filtering, clustering or compressing subsets. Finally, the above methods can reduce the training time of probability density estimation and improve the training efficiency of the model.

In these previous studies, Koenker proposed quantile regression (QR) to solve the problems dealing with the asymmetric distribution of response variables or the scattered distribution of data using ordinary least square (OLS) [20]. Quantile regression (QR) can explain the effects of input variables on response variables at different quorum, and it can also obtain more information about the relationship between input variables and response variables. What is more, quantile regression can not only reveal the heterogeneous relationship between input variables and response variables at each quorum but also obtain the conditional quantile information of input variables at continuous quorum, which is convenient for scientific decision making. However, the QR proposed by Koenker is still based on linear regression. The result of this processing can describe and reflect the simple linear relationship between those variables, but it is difficult to deal with the complex nonlinear functions. Wibowo proposed a method that is a hybrid of KPCA, alpha-regression quantile and adaptive genetic algorithm (AGA). This method overcomes the limitation of Kernel principal component regression (KPCR) and achieves better prediction accuracy [21].

Building on the aforementioned research, most probabilistic forecasts of transportation demand assume that the timing errors of aircraft entering and exiting airspace or airports follow a normal distribution. Alternatively, historical statistical values of traffic flow are often defaulted to a certain normal distribution pattern from the perspective of uncertainty. However, an analysis of currently available actual operational data reveals that in many cases, this normal distribution either does not exist or is limited to only a few ideal operating days.

Therefore, to address potential shortcomings in traditional deterministic prediction methods, this paper proposes a probabilistic traffic demand prediction approach that integrates various methods from the perspective of demand uncertainty. The approach combines neural networks with the quantile regression method. By leveraging the extremely strong nonlinear adaptive ability of neural networks and the precision of quantile regression in describing explanatory variables, we can obtain 100 consecutive quantiles for the future. Subsequently, using these continuous conditional quantiles, the continuous probability density function and the probability density curve of a certain day in the future can be realized through the kernel density estimation method. This approach not only provides specific point prediction values and their change intervals but also calculates the probability of each value within the prediction change interval for en route sector traffic demand. It allows for a more accurate point prediction value for a given day, effectively correcting biases in the understanding of probabilistic traffic demand. Previous studies assumed a normal distribution pattern, but this approach extracts the stochastic characteristics of traffic demand predictions more accurately at a certain time scale. The structure of the paper is as follows:

In Section 2, we introduce the detailed process of quantile regression, laying the foundation for method improvement, data calculation, and verification in the following sections.

In Section 3, we provide a brief description of neural network theory. Subsequently, we combine the neural network and quantile regression models as the research method for this paper and use this approach to predict the probabilistic traffic demand in the aviation sector.

In Section 4, we introduce the method principle of kernel density estimation. Through the analysis of the expression of this method, we can obtain the probability density function and probability density graph of the future airway sector traffic demand.

Section 5 summarizes the content covered in Section 2, Section 3 and Section 4 and provides a detailed introduction to the application process of the three models mentioned above in the method presented in this paper. A clearer understanding of the method process will contribute to a better comprehension of the research ideas presented in Section 6.

In Section 6, we select air traffic flow data (based on empirical research in the central and southern regions of China) for a specific period. The neural network regression model is employed for prediction, and the quantiles of the predicted data are then substituted into the kernel density estimation model to obtain the probability density curve of traffic demand change. The validity of the method is verified through error analysis.

Section 7 discusses the value of the method proposed in this paper. The method is applied to the empirical study of air traffic flow in central and southern China, and the predicted value of the highest probability point, obtained through probability density integration, is compared with the prediction result of the traditional BP neural network.

2. Quantile Regression

Based on the conditional quantile regression of the explanatory variable X, quantile regression obtains a regression model for all quantiles. Therefore, the use of quantile regression can more accurately reflect the impact of input variables on the explanatory variable. Through the analysis of these regression models, we can obtain different results of input variables on the explanatory variables in various ranges, providing more information about the effects of input variables on the position, distribution and shape of the explanatory variables.

Quantile regression first defines the loss function; then, it estimates the optimal parameters to minimize the loss function and optimizes these parameters. The estimated parameters are constantly changing with different quantiles, allowing for a continuous reflection of the input variables on the response variables, providing a more comprehensive explanation and influence. The loss function is defined as:

$${\rho }_{\theta }\left(u\right)=u\left[\theta -I\left(u\right)\right]$$

(1)

In Formula (1), $I\left(u\right)=\left\{\begin{array}{cc}0,& u,0\\ 1,& u<0\end{array}\right.$ is an indicator function. The nature of this function is a piecewise linear convex function.

Quantile regression essentially assigns different weights to the absolute value of residuals through indicative functions, allowing for the acquisition of different parameter estimates by adjusting different quantiles. Through the analysis of these various parameter estimates, the influence of input variables on response variables under different quantiles can be obtained, providing more useful information for research. The quantile regression proposed by Koenker is still based on linear regression to analyze the relationship between input variables and response variables [20]. That is to say, if the response variable is Y, the input variables are a series of factors X₁, …, X_n; a linear quantile function model is established firstly. This linear quantile function model is as follows:

$$Q_Y(\theta \left|X\right.)={\beta }_0\left(\theta \right)+{\beta }_1\left(\theta \right)X_1+{\beta }_2\left(\theta \right)X_2+\cdots +{\beta }_n\left(\theta \right)X_n\equiv X^{\prime }\beta \left(\theta \right)$$

(2)

In Formula (2), X is a matrix consisting of explanatory variables; θ is quantile, and θ $\in$ (0,1); and $\beta \left(\theta \right)$ is a parameter matrix. Quantile regression uses a simplex method, interior point algorithm and other optimization methods to optimize Formula (3) and estimate the parameters.

$$\underset{\beta }{{\displaystyle \min }}{\displaystyle \sum _{i=1}^n{\rho }_{\theta }(Y_i-X_i^T\beta )}=\underset{\beta }{{\displaystyle \min }}{\displaystyle \sum _{i\in {\Phi }_1}\theta |Y_i-X_i^T\beta |}+\underset{\beta }{{\displaystyle \min }}{\displaystyle \sum _{i\in {\Phi }_2}(1-\theta )|Y_i-X_i^T\beta |}$$

(3)

In Formula (3), $X_i^{\mathrm{T}}$ is the transposition of the i-th component of the input variable X, Y_i is the i-th component of the response variable Y, ${\Phi }_1=\left\{i|i\in [1,n],Y_i-X_i^T\beta \ge 0\right\}$, ${\Phi }_2=\left\{i|i\in [1,n],Y_i-X_i^T\beta <0\right\}$, where n represents the number of components of the input variable X or the response variable Y.

There are several advantages of quantile regression. Firstly, it does not assume any distribution of the random perturbation residuals in the model, which significantly differs from least squares regression, providing the entire regression model with good robustness. Secondly, as it encompasses regression for all quantiles, quantile regression has the resilience to handle non-stationary data, such as abnormal or singular points, in the dataset, thereby better reflecting the explanatory variables. Finally, the parameters estimated by quantile regression exhibit asymptotic superiority under large sample theory.

3. Quantile Regression Neural Network

3.1. Neural Network Theory

An artificial neural network (ANN) is a complex network computing system composed of a large number of highly interrelated simple neurons. An artificial neural network is an active interdisciplinary subject. Its basic properties are highly nonlinear, self-learning, robustness, and generalization. At the same time, an artificial neural network also has the characteristics of computational uncertainty. The common forms of an artificial neural network are the RBF neural network, BP neural network, Hopfield neural network, wavelet neural network, and so on. The kernel function of the hidden layer of the neural network selected in this paper is the Sigmoid function. Using this function, highly complex data can be well fitted with nonlinear data, and a stable and better predictive ability of the nonlinear function can be established, providing a better way to improve the prediction accuracy of traffic demand [22]. The form of the hyperbolic tangent function is shown in Formula (4):

$$\tanh \left(x\right)=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$

(4)

In Formula (4), $\tanh \left(x\right)$ is the expected output value of the hidden layer of the neural network, and $x$ is the matrix composed of input variables.

Because the neural network is more suitable for stationary time series, it requires more data features when using this method to predict. However, predicting en route sector traffic demand in the air traffic network is closely related to air traffic flow management strategies. Traffic demand in the en route sector is usually considered a non-stationary time series, so it will produce a significant prediction error when only a neural network is used to predict the traffic demand in the en route sector. That is to say, the lower the stationarity of the data, the greater the prediction error that may be caused by using a neural network to predict.

3.2. Quantile Regression Based on Neural Network

A quantile regression neural network is employed to predict the quantile of traffic demand in the future en route sector. Subsequently, the Sigmoid function is utilized as the hidden layer function of the neural network, and the predicted quantile of traffic demand in the en route sector is used as the input variable for kernel density estimation to achieve the probability density prediction of traffic demand in the en route sector. The expression of the quantile regression model of the neural network is shown in Formula (5).

$$Q\left(\theta \left|X\right.\right)=f\left[x,u\left(\theta \right),v\left(\theta \right)\right]={\displaystyle \sum _{j=1}^J\left\{\frac{2v_j\left(\theta \right)}{1+e^{-2{\displaystyle \sum _{i=1}^n{u}_{ij}\left(\theta \right)X_i}}}-v_j\left(\theta \right)\right\}}$$

(5)

In Formula (5), $\theta$ is the quantile, and $u\left(\theta \right)={\left\{u_{ij}\left(\theta \right)\right\}}_{i=1, 2, \dots , n; j=1, 2\dots , J}$ is the weight matrix to be estimated between the input layer and the hidden layer; $v\left(\theta \right)={\left\{v_j\left(\theta \right)\right\}}_{j=1,2,\dots ,J}$ is the weight vector between the hidden layer and the output layer. In order to achieve the final parameter estimation of Formula (5), the objective function can be optimized using Formula (6):

$${\tilde{E}}_{\theta }=\frac{1}{N}{\displaystyle \sum _{i=1}^N{\rho }_{\theta }\left\{Y_i-f\left[x,u\left(\theta \right),v\left(\theta \right)\right]\right\}}$$

(6)

However, in order to keep the trained neural network from over-fitting, a penalty parameter term is added to the objective function, and a new objective function is obtained as shown in Formula (7).

$$E_{\theta }\left(u\left(\theta \right),v\left(\theta \right)\right)={\tilde{E}}_{\theta }+{\lambda }_1{\displaystyle \sum _{i,j}{u}_{ij}\left(\theta \right)}+{\lambda }_2{\displaystyle \sum _{i,j}{v}_j\left(\theta \right)}$$

(7)

In Formula (7), ${\lambda }_1$ and ${\lambda }_2$ are penalty parameters. By determining the optimal penalty parameters, the model can be effectively prevented from over-fitting empirical data, thereby reducing prediction errors and improving accuracy. Formula (7) can be optimized to obtain the optimal estimates of $\overline{u}\left(\theta \right)$ and $\overline{v}\left(\theta \right)$ for $u\left(\theta \right)$ and $v\left(\theta \right)$. Then, $\overline{u}\left(\theta \right)$ and $\overline{v}\left(\theta \right)$ are substituted into Formula (5) to obtain the conditional quantile estimation function of the response variable.

4. Kernel Density Estimation Probability Density Function

Kernel density estimation does not require any assumptions about the prior distribution of random variables but only necessitates determining the input variables, kernel functions, and the optimal window width [23]. By employing the kernel density estimation method, a continuous probability density curve can be obtained based on the predicted traffic demand for the future en route sector. The fundamental idea of kernel density estimation is to estimate a reasonable density function through this method. The kernel density estimator is shown in Formula (8).

$$\widehat{f}\left(x\right)=\frac{1}{nh}{\displaystyle \sum _{i=1}^{n}k\left(\frac{X_i-x_0}{h}\right)}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{k}_h\left(X_i-x_0\right)}$$

(8)

In Formula (8), $k\left(\right)$ is a kernel function, $k_h=k\left(x/h\right)/h$.

Compared with the density estimation method using the Gaussian kernel function, the Epanechnikov function also exhibits a bell-shaped profile, which significantly reduces the computational workload and enhances operational speed. The Epanechnikov kernel function is optimal in terms of mean square error with minimal efficiency loss. In this paper, the Epanechnikov kernel function is selected, and the optimal window width is determined through cross-validation during kernel density estimation. The form of the Epanechnikov kernel function is shown in Formula (9):

$$k\left(x\right)=\frac{3}{4}\left(1-x^2\right)I\left(\left|x\right|\le 1\right)$$

(9)

In Formula (9), $I\left(\right)$ is an indicative function. When the condition in brackets is true, the value of $I\left(\right)$ is 1, and when the condition is false, the value of $I\left(\right)$ is 0. Cross-validation function expression is shown in Formula (10).

$$C{V}_f\left(h\right)=\frac{1}{n{h}^2{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n\overline{k}\left(X_i-X_j\right)}}}-\frac{2}{n\left(n-1\right)}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j\ne i}^n{k}_h\left(X_i-X_j\right)}}$$

(10)

In Formula (10), $\overline{k}\left(v\right)={\displaystyle \int k\left(u\right)k\left(v-u\right)}du$ is a convolution kernel function derived from $k\left(\right)$. As long as the concrete form of $k\left(\right)$ is given, the concrete expression of $\overline{k}\left(v\right)$ can be obtained. In this way, the probability density function and the probability density curve of the future en route sector traffic demand can be realized by the above method.

5. Algorithm Process

Based on the analysis of each model above, this section will describe the method flow of this paper in detail, as shown in Figure 1.

The main steps are as follows:

Step 1: Because quantile regression has the advantage of providing a more detailed description of explanatory variables, this paper has chosen to utilize the quantile regression model for the variable analysis of existing data.

Step 2: The neural network possesses a robust nonlinear adaptive ability; therefore, combining quantile regression with the predicted results of the neural network model will enhance the accuracy of the predictions.

Step 3: By employing kernel density estimation to predict the quantiles obtained from the neural network quantile regression model, it is possible to derive the continuous probability density function and probability density curve for future traffic demand. This approach allows for obtaining not only the specific point predicted value and its variation interval but also the probability associated with each value within the change interval of the traffic demand prediction in the aviation sector. Consequently, a more accurate point prediction value for a given day can be obtained.

Step 4: The prediction results obtained through this method are compared with those obtained from the BP neural network model to validate its effectiveness.

6. Empirical Research

6.1. Data Sources and Sample Descriptions

This paper simulates and analyzes the historical operation data of AR05 in the central and southern regions of China, and the route distribution is shown in Figure 2. In Figure 2, the airspace enclosed by the white border is the en route sector. It can be seen that the density of the en routes (blue thick solid line) in the sector is higher; there are more intersections (green English name). The red font represents the name of the flight leg. And this sector is connected to four large busy airport terminals: namely, Guangzhou (ZGGG), Zhuhai (ZGUH), Hong Kong (VHH) and Macao (VMMC).

Based on the data of AR05 traffic flow from 1 May 2017 to 20 May 2017, we can find the daily periodic variation law of the traffic flow data, as shown in Figure 2. The nonparametric test of the data in Figure 3, using the run-length test method, indicates that the traffic demand sample used is a non-stationary time series [24].

Traffic flow prediction can be divided into long-term traffic flow prediction and short-term traffic flow prediction according to time span. Long-term traffic flow prediction is based on the hour, day, month or even year; however, short-term traffic flow prediction generally does not exceed 15 min, the short-term prediction is highly nonlinear and uncertain, and it also has strong correlation.

Using the traffic demand every 15 min on the sample day as the explained variable and the traffic flow during the same 15-min interval on the previous 20 days as the explanatory variable, we conducted a total of 1920 samples for rolling prediction. Firstly, according to the central limit theorem, it is assumed that the sample space obeys the normal distribution, and the Q-Q distribution of 1920 sets of sample data above is tested by the SPSS (27.0.1) data editor in Figure 4. In this figure, the black circles represent the scatter points plotted based on the quantiles of the sample data and the quantiles of the theoretical distribution, while the black line is the fitted line. It is found that the sample data do not conform to the normal distribution hypothesis. Therefore, the traffic demand is predicted based on the probability density distribution method. The 1920 sets of samples are fed into the quantile regression neural network model, where the model structure is established, and the neural network is trained to stabilize and meet the specified requirements. Subsequently, 100 consecutive conditional quantiles for each 15-min interval from 1 May 2017 to 20 May 2017 are obtained. Finally, these conditional quantiles are input into the kernel density estimation model to determine the probability density curve of traffic demand change in the en route sector every 15 min from 21 May 2017 to 30 May 2017.

6.2. Model Parameter Selection

The model constructed in this paper is based on the single hidden layer neural network quantile regression. The number of iterations is 5000, the input layer is 11, the hidden layer is 1, the output layer is 1, and the structure of the neural network is 11-1-1. At the same time, in order to prevent the quantile regression neural network into excessive fitting, the penalty parameters ${\lambda }_1$ and ${\lambda }_2$ are set to 0.1. In the model, the quantile points were selected from 0.0001 to 0.9999 with an interval of 0.01. A total of 100 quantile points were selected. This quantile regression neural network model parameters are determined. The kernel density estimation and the optimal window width are combined by the Epanechnikov kernel function and cross-validation; then, the final probability density curve and the point prediction value corresponding to the highest probability point are obtained.

6.3. Empirical Results and Analysis

According to the above contents, the samples obtained by the rolling method are brought into the quantile regression neural network, the neural network structure is trained, and 100 conditional quantiles of daily continuous traffic flow are brought into the kernel density estimation method; then, the complete probability density curve of the traffic demand of AR05 in the future daily 15 min is obtained. Because of the randomness of the neural network method, the traffic demand value in a 15-min interval of AR05 predicted by each time is different. Based on the data of 9:00–9:15 on 21 May 2017, nine probability density distributions of 100 predictions were selected randomly. The black line in Figure 5 represents the predicted values. As shown in Figure 5, it was found that the probability density distribution did not conform to a certain distribution law (such as normal distribution). This is because some traffic demand values correspond to a prediction probability of zero, resulting in discontinuities. This further confirms the correctness of the kernel density estimation method.

Figure 5 shows a situation that differs from the case where the probability density meets the normal distribution. In scenarios where the probability density of the traffic demand value in the en route sector lacks an obvious pattern, the true value (depicted by the red line at 28) does not appear near the highest point on the probability density plot. However, the predicted value (represented by the black line) corresponding to the maximum obtained after integrating the probability density shows little difference with the actual value. Meanwhile, a comparison is made with the results obtained by the traditional BP neural network prediction method and the actual value. The difference between the values obtained by the method in this paper and the true value is smaller, as shown in Figure 6.

In the left figure of Figure 6, the blue curve represents 100 predicted results using the BP neural network prediction method, and the red curve represents 100 predicted results using the probability density prediction method. The comparison between the two kinds of prediction results and the true value 28 (black dotted line) shows that the results of the BP neural network prediction method are more volatile, and the results of the probability density prediction method are more stable (always around the real value). If it is necessary to select the predicted value in the actual operation, the result obtained by the BP neural network method is likely to deviate too much from the true value, while the result obtained by the probability density prediction method is more accurate. Similarly, in the right figure of Figure 5, the blue shadow represents the prediction error corresponding to 100 predicted values obtained by the BP neural network prediction method, and the red shadow represents the prediction error corresponding to 100 predicted values obtained by the probability density prediction method. Comparing the two kinds of errors, it is found that the result error is very small through probability density prediction. On the contrary, the error deviation of the BP neural network prediction method is very large at the beginning; after 30 times of prediction, the result is more stable and close to the true value, but its accuracy still lags behind the probability density prediction. This shows that the BP neural network has a certain learning process, and this learning process will take a certain amount of time, which cannot fully meet the very strong practical timeliness requirements of ATC.

In order to further prove the accuracy of the proposed method, a more in-depth error analysis is made between the 100 times prediction result of the AR05 traffic demand value and the true value 28. The analysis is mainly carried out by means of root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE) [25], Hill inequality coefficient (TIC), deviation rate (BP) and variance rate (VP). The analysis results are shown in the left column diagram of each subgraph in Figure 6. From the figure, the following can be determined:

• The RMSE is 0.56, and the MAE is 0.45. That is, the error is only about 0.5. For actual operation, it can be regarded as no error.
• The MAPE is 1.59%, and it is generally believed that the prediction accuracy is higher if the MAPE is lower than 10%.
• The TIC is 0.01. When the TIC is between 0 and 1, the smaller the value, the less difference there is between the fitting value and the true value, and the higher the prediction accuracy. When the TIC equals 0, it signifies a 100% fit. Therefore, the TIC value obtained here is very small, indicating very high prediction accuracy.
• The BP is rounded to about 0, reflecting that there is almost no difference between the mean of predicted value and the mean of true value, that is, there is almost no systematic error.
• The VP is about 0.32, reflecting the small difference between the standard deviation of the predicted value and the standard deviation of the true value. Additionally, the fact that the true value always equals 28 does not fluctuate, which indicates that the predicted value fluctuates very little, which is also consistent with the information provided in Figure 7.

We also used the BP neural network method to predict the traffic demand value of AR05 from 9:00 to 9:15 on 21 May 2017 100 times. The root mean square error (RMSE), the mean absolute error (MAE), the mean absolute percentage error (MAPE), the Hill inequality coefficient (TIC), the deviation rate (BP) and the variance rate (VP) are shown in the middle column diagram of the subgraphs in Figure 7. Comparing the error parameters corresponding to the left column diagram obtained by the probability prediction method, it can be observed that the probabilistic prediction method exhibits better accuracy in prediction, system error, and prediction value fluctuation. The main reason for this difference is that the historical traffic flow data of en route sectors are not in a stationary sequence, making it less suitable for the BP neural network prediction method. Specifically, the VP obtained by the BP neural network is 19.80, which is larger, and the true value remains constant at 28 without fluctuation. This indicates that the predicted value of the BP neural network method fluctuates significantly. In Figure 7, we can see that the single prediction error of the probability prediction method is basically within 4%. Therefore, if this error value is extrapolated to the BP neural network method, more than 4% of the predicted values can be excluded, leaving 59 of the remaining effective predicted values, which is compared with the probability density prediction results as shown in Figure 8.

Analyzing these 59 prediction values, we obtain the right column diagram of Figure 6 (BP After). It is found that all the predictive error parameters are significantly improved, and compared with the predictive results obtained by the probabilistic prediction method, the following can be observed:

• The MAE is 0.30, which is less than the 0.45 value corresponding to the probabilistic prediction method, indicating that the overall prediction accuracy is better.
• The MAPE is 0.57%, which is lower than the 1.59% value corresponding to the probabilistic prediction method, which indicates that the prediction accuracy is higher.

Although some prediction error parameter values (after BP) are better than those of the probabilistic prediction methods, this improvement results from the elimination of data with large fluctuations. Therefore, if this method is chosen, determining how to eliminate invalid data without referencing real values or other prediction methods becomes a critical challenge.

The prediction for a 15 min period of one day will be changed into the consecutive days based on the data in 9:00–9:15 from 22 May 2017 to 30 May 2017; these two methods are used to predict 100 times per day. The prediction results are sorted from small to large and the corresponding data of the 50% quantile is selected as the final prediction data, as shown in Figure 9.

The error performance parameters of the above prediction results are compared on a daily basis, as shown in Figure 10. It can be seen that the performance of the probability density prediction method (red line) is significantly better than that of the BP neural network prediction method (blue line) over an extended period of time.

We will select another day from 22 May 2017 to 30 May 2017, such as May 23rd. The probability density prediction method and BP neural network prediction method are used to predict the traffic demand in the morning peak of 07:45:00–11:29:59 every 15 min. Finally, corresponding to these two methods, 15 probabilistic prediction results are obtained, and the probability density distribution is shown in Figure 11.

The 15 probabilistic prediction results are compared with the deterministic prediction results obtained by the BP neural network prediction method. As shown in Figure 12, it can be found that the accuracy of the probability prediction method under a continuous time condition is also better than that of the BP neural network prediction method.

We compare the 15 probabilistic predictions, 15 BP predictions and the sector’s airspace capacity (35 flights/15 min). As shown in Figure 13, according to the balance between the deterministic predictions and the sector’s airspace capacity, the interventional flow control measures should be taken during 09:15:00–09:29:59, 09:45:00–09:59:59, and 11:00:00–11:14:59. In fact, compared with more accurate probabilistic prediction results, air traffic flow management measurements should be appropriately intervened to avoid overload, because the flow demand is always at the limit of capacity during 08:45:00–09:29:59. By contrast, traffic demand is far less than sector’s airspace capacity during 11:00:00–11:14:59, so there is no need to intervene in air traffic flow management measurements.

From the further comparison in Figure 13, because the BP neural network prediction method is limited by the prediction scale, the accuracy of the prediction result is not as good as the probabilistic prediction method when the prediction scale is larger or numerous data samples. That is, the prediction results during the peak hours of flight operations are more accurate. The probabilistic prediction results accurately predict sector traffic overload during 09:00:00–09:14:59, which is conducive to the air traffic management department taking measures to ease congestion in advance. However, the probabilistic prediction results show that the traffic demand does not exceed the capacity, thus avoiding false alarms problem caused by BP neural network predictions method during 07:45:00–07:59:59 and 11:00:00–11:14:59.

7. Conclusions

The existing uncertain traffic demand prediction for the en route sector imposes high requirements on the accumulation and collation of historical data, including radar data, ADS-B data, etc. Specifically, historical data must be targeted at a specific set of aircraft, constituting a collection of aircraft within a fixed period of time every day, over a fixed en route point, with a substantial amount of operational data. However, given the current limitations of China’s air traffic flow management system and supported data management platform, obtaining historical data that fully meets the aforementioned conditions is currently unattainable. In light of the actual operation of China’s air traffic flow management system, we approach prediction from an alternative perspective, utilizing the available historical data of air traffic flow in the en route sector.

In this paper, we combine neural network and quantile regression methods to obtain quantiles of continuous traffic demand data in the future. This is achieved by leveraging the strong nonlinear adaptive ability of neural networks and the precise explanatory variable description provided by quantile regression. Subsequently, utilizing these continuous conditional quantiles, we derive the probability density function and probability density curve for future continuous traffic demand through kernel density estimation. This approach allows us not only to obtain specific point prediction values and their change intervals but also to determine the probability associated with each value within the prediction change interval for traffic demand in the en route sector. This results in obtaining more accurate point prediction values for the day. Based on empirical research in the central and southern regions of China, we compare the point prediction value with the highest probability, obtained by integrating the probability density, with the traditional BP neural network prediction result. The comparison results indicate that the probability prediction method proposed in this paper not only achieves more accurate results but also provides a continuous probability density curve for the future. This allows for a comprehensive reflection of the daily trend of traffic demand in the en route sector and its corresponding probability levels. If probabilistic prediction is applied to China’s air traffic flow management system, it can offer more useful information for air traffic managers to make informed decisions.

In this study, there are still some shortcomings that require further research and improvement. We used only 20 days of historical operational data as prediction samples may introduce sample specificity, so in subsequent studies, it would be beneficial to gather long-term historical data for prediction analysis. The probabilistic density prediction method employed is based on the combination of neural network quantile regression and kernel density estimation. In future research, other combinations of methods, such as error characteristic distribution, can be explored to further predict the uncertainty of traffic demand in the sector. It would be valuable to conduct comparative analyses of prediction results using various methods, expanding beyond the limitations of the BP neural network algorithm.