The Power Approximation of Time Series with Using Fractional Brownian Motion

We propose the approximating sequence and some of characteristics of this sequence to coincide with the increments of the fractional Brownian motion (fractional Browniannoise) for the observed time series. We study the Hurst parameter estimation algorithm and check the quality of the approximation. Citation: Bondarenko V (2017) The Power Approximation of Time Series with Using Fractional Brownian Motion. J Appl Computat Math 6: 353. doi: 10.4172/2168-9679.1000353


Introduction
We consider a mathematical model for the time series S 1 ..S n . The primary processing is smoothing, removing the trend -leads to improved time series x 1 ..x n -("initial"). Consider x 1 ,x 2 ..x n as the observed values of some quantity at some moments of time. Let us choose a random process X (t), where . This problem has a controversial solution, because different processes (with different distributions) may have the same trajectories. The subjective criteria for selection of X(t) is as follows.
1. Process should possess known characteristics, particularly the Gaussian process should be chosen.
2. X(t) should not be Markov, as the Markov communication does not provide an adequate description of real phenomena.
Fractional Brownian motion is defined as a Gaussian random process with characteristics [1]:
{y k } constitutes a Gaussian stationary sequence. Henceforth, the increments are going to be the subject of consideration. Consider the algorithm proposed for simultaneous estimation of parameters [2].
We check the method for simultaneous estimation of two unknown parameters fBm (H,σ) and propose a method for approximation of the time series by the power function from the increments of fractional Brownian motion.

Description of the Algorithm
Consider the absolute random moments of increments of fractional Brownian motion. 1 1 , Then calculate the mean ( ) The result was first proved in eqn. (6)

Theorem
With probability 1 In particular, when we obtain a consistent estimate for σ with the known value estimate for H: ε is the canonical Gaussian vector with the following characteristics: Now we prove consistency of this estimate. Let us introduce the notation: We use the formula for integration by parts for Gaussian measures that leads to the relation (by calculating the dispersion Equations (1) and (2) form a system, which is proposed to solve iteratively [3]. The essence of the algorithm is as follows: for an arbitrary value H∈(0;1) let us calculate the estimate 1n ( The values Ĥ of parameter H, which satisfies eqn. (2), is an estimate and 2ˆ2 1 n n σ σ σ + = We performed a numerical experiment, which implements the algorithm proposed.
We proved that the estimate of parameter σ

Approximation of Real Time Series
Let's S 1 ..S n be observed time series data of an arbitrary nature. Consider the procedure for its approximation of fractional Brownian motion. The first step of the algorithm -initial data processing is to bring them to a time series with zero trend X k =S k −M(k), M(k) is an approximation of the trend [4].
Let us select an adequate model of the corresponding random process for the actually observed time series: The criterion of the value of the Gaussian increments will be the "excess coefficient"   The general idea of approximation is in dimensional functional transformation g for each increment y k , g-is the increasing odd function.
Assume that a two-dimensional density distribution f k−j (x,y) of increments ) ; ( j k y y of converted time series satisfies the following conditions: Let's approximate f by the exponential function: To check the properties of increments Z k allow limit theorems for fractional Brownian motion proved in I. Nourdin.
-is positive here: In the studies mentioned above there have been proved the following limit relations      Let's consider the application of algorithm.
As a first example consider the data: the daily data of solar activity (366 data, ftp://ftp.ngdc.noaa.gov). We calculated the initial time series , a=0,085   The hypothesis T is accepted.
For the third example-the market rates of Bundesbank Germany. The hypothesis T is accepted.
In the fourth example, we have received the uncertain answer. We calculate the value of statistics