AUTOREGRESSION MODELS OF SPACE OBJECTS MOVEMENT REPRESENTED BY TLE ELEMENTS

Annotation. The developed method, which is a modification of the previously developed methods for constructing autoregressive models, is used to simulate the motion of space objects in the time series of their TLE elements. The modeling system has been developed that includes: determining the optimal volume of training samples in modeling time series of TLE elements; determination of the autoregression order for each variable (TLE element); determination of the optimal structure and identification of the parameters of the autoregressive model for each variable; identification of patterns of evolution of the mean square error of autoregressive models in time based on the modeling of time series of TLE elements according to the principle of "moving interval".


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Introduction. The problem of improving the accuracy of forecasting the satellite position is relevant for the tasks of determining time of their existence, cataloging space debris, navigation, etc. To solve this problem, physical approaches are mainly used, but they require complete information about the space object at the beginning of the trajectory and environment calculations, as well as data on the maneuvers of the object under study [1]. In all cases, such data is not complete or is not updated regularly, and current observational capabilities are limited or costly.
The only open source of orbital data for solving such problem are the two-line elements (TLE), which regularly and promptly updated on the website of the American Space Monitoring System (SSS) [1]. 104 Space Perturbations -models for calculating the position and speed of the Earth's orbiting satellite, with an orbital period of more than 225 minutes) [2].
Several studies which represents different approaches to orbit propagation using TLE elements are investigated as well.
Various machine learning methods and hybrid methods are quite successfully used for solving problems of propagating and predicting the motion of space objects, among them neural network methods, Kalman filter and support vector method. In [3], a neural network was used to increase the accuracy of the analytical predictor. As a result, a combination of both methods reduces the error in calculating the position of a space object and improves the accuracy of predicting its motion. In [4], the main emphasis was placed on the Data Mining approach and the extraction of historical data on unknown disturbances using the extended Kalman filter. This approach improved the quality of filtration and prediction, but only for space objects in low Earth or- bits.
An approach based on distributed regression and transitional machine learning underlies the publication [5]. The result shows superiority over the extended Kalman filter. In addition, the method is able to evaluate significantly changing parameters of the orbit.
In [6]- [8], Keplerian orbital elements are used as initial estimates of TLE elements using differential corrections and nonlinear least squares methods. In [9], a Kalman filter was used to estimate TLE elements using GPS data.
The above methods look for a local minimum of the objective function, representing the sum of the squares of the position and velocity errors and depend on the initial estimate of the TLE elements. Methods such as genetic [10] and IWO (Invasive Weed Optimization) algorithms [11] do not require an initial assessment of TLE elements, but they are reportedly computationally expensive because they seek for a global optimum.
In the literature, there are two main approaches for evaluating TLE elements: machine-costing methods for searching for global optimum and methods for searching for local optimum, which depend on the initial estimate of TLE elements. The developed autoregressive models [18] offer an alternative, less machine-expensive method for improving the forecast, and The purpose of the work is to establish the possibility of using autoregressive models fofr orbit propagation of space objects represented by the time series of TLE elements (Two-Line Elements).

Major part
1 Modeling the motion of space objects represented by time series of TLE elements A distinctive feature of TLE-elements series is their time positioning not on a uniform temporal grid, but with irregular time intervals between observations, so-called "unequal observations". This feature was taken into account in the development of a method for estimating the parameters of autoregressive models with unequal-time observations for orbit propagation of large fragments of space debris [13]. The developed method, which is a modification of previously developed methods for constructing autoregressive models [14]- [16], is used to simulate the motion of space objects in the time series of their TLE elements.
The construction of statistical models was carried out on the basis of an iterative procedure for estimating the coefficients of beta-autoregressive models under conditions of unequally spaced observations [13]. The time series of TLE elements are represented by seven main and three additional variables (see Table 1). x -inclination had, as a rule, a constant value. Additional variables 8 x , 9 x were used to construct the figures, and the variable 10 x was used to calculate the degree of which the components of the coefficients are powered in the estimation procedure [13].
Trial calculations based on the iterative procedure [13] of the method of structural-parametric identification in the class of beta-autoregressive models found that in this problem the maximum number of previous values of output variables