The Effectiveness Evaluation of Two Kinds of Fractal Sequences on Detrended Fluctuation Analysis
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Used of the fractional Brownian motion and fractional Gaussian noise sequence, the detrended fluctuation analysis (DFA) applied to estimate the Hurst exponent to verify the stability and dependability of the method by changing the data length and regression trend order. The result shows that the Hurst exponent estimate is stable and efficient with the length of data for fractional Brownian motion and fractional Gaussian noise sequence. The influence on the Hurst exponent is not obvious when the regression trend order was changed, and the estimate accuracy is improved with the increasing of Hurst exponent value.Keywords:
Hurst exponent
Fractional Brownian motion
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Gaussian Noise
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This article is devoted to the comparative analysis of four statistical methods: Variance-time, Rescaled Range (R/S), Periodogram and Detrended Fluctuation Analysis (DFA) to study the fractal time series that are simulated by applying an algorithm, based on the Fractional Gaussian Noise (FGN). The evaluation of the statistical methods studied was performed with respect to the precision parameter in determining the Hurst exponents of the simulated fractal time series, with DFA being determined as the most accurate method. The statistical methods studied are characterized by universal against the data examined. In the article, the DFA method is applied to the analysis of two independent types of real-time data representing continuous time series: in the field of cardiology in the study of heart intervals (RR time series) obtained from digital electrocardiographic (ECG) signals and in the field of the energy for the analysis of time series data for the hourly day-ahead spot prices of the Bulgarian electricity market for the period 2016-2019. The results of the application of fractal analysis in modern information technologies related to fractal theory presented in the article reveal great potential and new perspectives in modelling, processing and analysis of processes and signals in various scientific and technical fields.
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The multifractional model with random exponent (MPRE) is one of the most recent fractional models which extend the fractional Brownian motion (fBm). This paper is an empirical contribution to the justification of the MPRE. Working with several FX rates between 2006 and 2016, sampled every minute, we show the statistical significance of various fractional models applied to log-prices, from the fBm to the MPRE. We propose a method to extract realized Hurst exponents from log-prices. This provides us with a series of Hurst exponents on which we can estimate different models of dynamics. In the MPRE framework, the data justify using a fractional model for the dynamic of the Hurst exponent. We estimate and interpret the value of the key parameter of this model of nested fractality, which is the Hurst exponent of the Hurst exponents.
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Time series data of complex industrial system doesn't subject Gaussian distribution due to its sharp spikes and heavy-tailed characteristic. Finding the latent features and rules for these data is a meaningful topic. Here the industrial data analysis is discussed under the fractional thinking with an actual supermarket energy system as example. The cooling system in supermarket always exhibits many non-Gaussian behaviors which are hard to capture by traditional data analytic methods. This paper shows that the novel fractional-order perspective is suitable for the real industrial data. Hurst exponent and fractal theory are used to study the long-range dependency characteristic from the supermarket energy data. It is found that the α-stable distribution better match with the probability density of row data compared with the traditional Gaussian distribution (integer-order) firstly. Then rescaled range method (R/S) is adopted to get the Hurst exponent which can estimate the long-range dependency existing in these process variables. Furthermore, the fractal feature of time series is estimated by comparing the slope of different scaling function under different order according to multifractal detrended fluctuation analysis (MFDFA). The non-Gaussian statistical characteristics, Hurst exponents and fractal feature of supermarket data are derived by comparing the results under different parameters. The practical application results show that the fractional order thinking can deeply mine the latent information hidden in the process data and has a significant advantage in data analytics domain.
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Estimating of the Hurst exponent for experimental data plays a very important role in the research of processes which show properties of self-similarity. There are many methods for estimating the Hurst exponent using time series. The aim of this research is to carry out the comparative analysis of the statistical properties of the Hurst exponent estimators obtained by different methods using model stationary and nonstationary fractal time series. In this paper the most commonly used methods for estimating the Hurst exponents are examined. There are: / RS -analysis, variance-time analysis, detrended fluctuation analysis (DFA) and wavelet-based estimation. The fractal Brownian motion that is constructed using biorthogonal wavelets have been chosen as a model random process which exhibit fractal properties. In this paper, the results of a numerical experiment are represented where the fractal Brown motion was modelled for the specified values of the exponent H. The values of the Hurst exponent for the model realizations were varied within the whole interval of possible values 0 < H < 1.The lengths of the realizations were defined as 500, 1000, 2000 and 4000 values. For the nonstationary case model time series are presented by the sum of fractional noise and the trend component, which are a polynomial in varying degrees, irrational, transcendental and periodic functions. The estimates of H were calculated for each generated time series using the methods mentioned above. Samples of the exponent H estimates were obtained for each value of H and their statistical characteristics were researched. The results of the analysis have shown that the estimates of the Hurst exponent, which were obtained for the stationary realisations using the considered methods, are biased normal random variables. For each method the bias depends on the true value of the degrees self-similarity of the process and length of time series. Those estimates which are obtained by the DFA method and the wavelet transformation have the minimal bias. Standard deviations of the estimates depending on the estimation method and decrease, while the length of the series increases. Those estimates which are obtained by using the wavelet analysis have the minimal standard deviation. In the case of a nonstationary time series, represented by a trend and additive fractal noise, more accurate evaluation is obtained using the DFA method. This method allows estimating the Hurst exponent for experimental data with trend components of virtually any kind. The greatest difficulty in estimating, presents a series with a periodic trend component. It is desirable in addition to investigate the spectrum of the wavelet energy, which is demonstrated in the structure of the time series. In the presence of a slight trend, the wavelet-estimation is quite
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