The work summarizes the results obtained in the course of the implementation of Bachelor's final qualifying work and is devoted to the methods of simulating and applying the fractional Brownian motion in the problems of time series analysis. Software modules have been implemented to generate trajectories of fractal Brownian motion using the methods of stochastic representation, Cholesky decomposition and Davis-Hart. Algorithms vere compared in terms of their complexity and the quality of the resulting trajectories. The Hurst exponent was estimated by the Minkowski and R/S analysis methods. An approximation of time series by fractal Brownian motion using a power function is proposed and implemented for the subsequent application of a linear prediction algorithm based on the normal correlation theorem. It has been established that with the help of the presented approximation it is possible to achieve a satisfactory forecast of the exchange rate for several values ahead.
Облакова Т.В., Алексеев Д.С. Сравнительный анализ методов моделирования и прогнозирования временных рядов на основе теории фрактального броуновского движения. Математическое моделирование и численные методы, 2022, № 4, с. 48–62
The article presents a developed mathematical model of electric describing the coaxial electrogenerating elements (EGE) with isothermal cathode and a variety of ways for current collecting. To analyze their internal state and output parameters in the arc mode we used a two-parameter local linear current-voltage characteristic (CVC). It was shown that in the case of one-sided current collection maximum power of EGE and generated magnetic field asymptotically approach to their maximum values as the length of the electrodes goes into infinity. In the case of versatile current collection maximum values of these parameters can be achieved at the final length of the electrodes. In both methods of the current collection the acceptable value of EGE electrical power loss of 25% due to electrode non-equipotentionality was achieved at their universal critical length. The calculation of which is presented.
Loshkarev A., Oblakova T. Mathematical modeling of coaxial electrogenerating elements. Маthematical Modeling and Coтputational Methods, 2015, №1 (5), pp. 3-16
The study deals with the plane stationary problem of elasticity theory on the motion of a vertical concentrated load along the surface of an elastic half-space with a thin coating. The authors investigated modes in the surface layer at speeds close to the resonant speed of the surface wave. The research was done within the long-wave asymptotic model for the Rayleigh wave in the case of an elastic coated half-space. The modes are classified according to the ratio between the velocity of the load and the resonance speed and to the dispersion coefficient of linear coverage. The study discovers the modes having radiation from the source. The results obtained can be generalized to more complex physical properties of the coating material, including the effects of anisotropy, viscosity and prestraining.
Kaplunov J., Oblakova T., Prikazchikov D. Near-resonant modes of the moving load in the plane problem of elasticity theory for a half-space with a thin coating. Маthematical Modeling and Coтputational Methods, 2014, №1 (1), pp. 57-67
The work is devoted to methods for detecting long-term memory in financial time series. Using the method of analysis with the help of the original program code, a number of values of the real financial index S & P500 were studied, estimates of the Hurst index were obtained, and persistence was demonstrated. To solve the problem of predicting the future values of a series, the ARFIMA model is proposed, which is a generalization of the standard ARIMA model and involves the use of a fractional differentiation opera-tor. A two-stage algorithm for constructing a forecast for a series of logarithmic profits is presented and implemented. It is shown that the use of the ARFIMA model improves the quality of the forecast in comparison with ARIMA for all standard metrics.
Облакова Т.В., Касупович Э. Численное исследование персистентных временных рядов на основе модели ARFIMA. Математическое моделирование и численные методы, 2022, № 4, с. 114–125.