## 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, [PERSON] [1] proposed a modeling system to predict dynamic traffic demand uncertainty using RFID data and link traffic volume. In 2021, [PERSON] [2] formulated and proposed a multivariate Poisson log-normal model with specific parameterization tailored to the traffic demand problem. In 2024, [PERSON] 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.