## 1 Introduction
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, [PERSON] 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 [PERSON] 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.