Fractional Brownian Motions, Fractional Noises and Applications

Mandelbrot, Benoît B.; Ness, John W. Van

doi:10.1137/1010093

Cited by 6,631 publications

(4,235 citation statements)

References 10 publications

Supporting

Mentioning

4,135

Contrasting

Unclassified

Order By: Relevance

“…11, may be replaced by a noninteger, x, defined as true distance divided by the distance between centers of neighboring elements. The derivation is given by Bassingthwaighte and Beyer (5) following the presentation of the ideas by Mandelbrot and van Ness (19). We have not attempted in this study to make a direct comparison of the accuracy of the estimates of Ĥ from dispersion versus autocorrelation, but this needs doing.…”

Section: Discussionmentioning

confidence: 99%

Evaluation of the dispersional analysis method for fractal time series

1995

View full text Add to dashboard Cite

Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is <0.5 for rough anticorrelated series, >0.5 for positively correlated series, and =0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 − D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (>0.8) and low H (<0.5), and biases are greater for short time series than for long; and (iii) the addition of Gaussian noise (H = 0.5) degrades the signals: for those with negative correlation (H < 0.5) the degradation is great, the noise has only mild degrading effects on signals with H > 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H.Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.

show abstract

Section: Discussionmentioning

confidence: 99%

Evaluation of the dispersional analysis method for fractal time series

1995

View full text Add to dashboard Cite

show abstract

“…Fractional Gaussian noise has been proposed as a model (Mandelbrot and Ness, 1968) for long-range dependence in a variety of hydrological and geophysical time series. Nowadays, fGn is one of the most commonly used self-similar processes in network performance evaluation (Lopez-Ardao et al, 2000).…”

Section: Fluid Flow Analysis Of Pi α Controller Performancementioning

confidence: 99%

The use of a non-integer order PI controller with an active queue management mechanism

Domański

Domańska

Czachórski

et al. 2016

International Journal of Applied Mathematics and Computer Science

View full text Add to dashboard Cite

show abstract

“…Hurst exponent that represents the correlation properties of the time series based on its scaling behaviour is more useful for long-range correlated time series (Hurst, 1951). Hurst effect has been modelled by Mandelbrot (1968) as fractional Brownian motion (fBm) with associated properties as persistent, anti-persistent and random. The existence of persistence or long-range correlations in time series is helpful in making further projections as the measurements rely on the correlations with historical observations (Varotsos et al, 2005;Chelani, 2009).…”

Section: Introductionmentioning

confidence: 99%

Long-Range Correlations in Air Quality Time Series: Effect of Differencing and Shuffling

Chelani

2016

Aerosol Air Qual. Res.

View full text Add to dashboard Cite

Long-range correlations in the air quality index (AQI) are analysed using rescaled range analysis (R/S), detrended fluctuation analysis (DFA) and power spectral density analysis. Air quality index in five cities of India is considered for this purpose. Statistical transformations such as differencing and shuffling have been carried out to examine the effect of temporal correlations on long-range correlation property of the time series. All three methods indicated the presence of persistence in original AQI time series. After differencing, long-range correlation property is, however, observed to be distorted. R/S analysis did not show the similar results as DFA and power spectral density analysis. Shuffled time series is shown to possess persistence as in the original one by using R/S analysis, whereas other two methods showed random behaviour at most of the locations. This suggests that the persistence property is largely influenced by short-range correlations in the AQI time series. The incorporation of this information can enhance the performance of the models to forecast the air quality. The similarity in the results of DFA and power spectral density analysis suggests that both methods can be relied more than R/S analysis in studying the persistence property of the time series.

show abstract

Fractional Brownian Motions, Fractional Noises and Applications

Cited by 6,631 publications

References 10 publications

Evaluation of the dispersional analysis method for fractal time series

Evaluation of the dispersional analysis method for fractal time series

The use of a non-integer order PI controller with an active queue management mechanism

Long-Range Correlations in Air Quality Time Series: Effect of Differencing and Shuffling

Contact Info

Product

Resources

About