The forecast of the time series by approximating the fractal Brownian motion


  • V. V. Bondarenko


The problem of extrapolation (the forecast) of the observed time series, which are observed, is considered. The scheme for the solution of this problem is a two-stage operation (the definition of trend and functional transformation) over original series, which reduces it to a sequence, the parameters of which coincide with the parameters of Gaussian data. The completed actions allow to apply to the transformed time series linear forecasting procedure. The results of numerical experimentation, confirming the effectiveness of the proposed algorithm, which illustrates the quality of the functioning of proposed algorithms, in particular, computer simulations of random process -fractal Brownian motion, are presented. A fractal Brownian motion is considered in details, and considering four specific examples the satisfactory forecast is and its parameters are estimated.


SHiryayev А.N. Veroyatnost’. — M.: Nauka, 1989. — 640 s.

Bondarenko V.V. Аpproksimatsiya vremennogo ryada stepennoy funktsiyey fraktal’nogo brounovskogo dvizheniya // Problemy upravleniya i informatiki. — 2013. — № 3. — C. 113–116.

Bondarenko V.V. Iteratsionnyy algoritm otsenivaniya parametrov fraktal’nogo brounovskogo dvizheniya // Problemy upravleniya i informatiki. — № 4. — 2012. — C. 28–33.

Mandelbrot B.B., van Ness I.W. The Fractional Brownian motion, fractional noises and applications. — Philadelphia: SIAM Review, 1968. — 10, № 4. — 1325 р.

Coeurjolly J.-F. Simulation and identification of the fractional Brownian motion: A bibliographical and comparative study // Journal of statistical software. — 2000. — 5. — issue 7. — P. 1–52.

Федер Е. Фракталы. — М.: Мир, 1991. — 258 с.

Mandelbrot B.B , Hudson R.L. The Misbehavior of Market / A Fractal View of Risk, Ruin and Reward. — NY: Basic Books, 2006. — 368 p.

The National Geophysical Data Center. —

Sayt Natsional'noho banku Ukrayiny. —



Mathematical methods, models, problems and technologies for complex systems research