Physical Models for Long-Range Dependence and Self-Similarity

2017 ◽  
pp. 113-228
Author(s):  
Vladas Pipiras ◽  
Murad S. Taqqu
Keyword(s):  
Long Range ◽  
Physical Models ◽  
Long Range Dependence ◽  
Self Similarity

A study on the variations in long-range dependence of solar energetic particles during different solar cycles

2018 ◽  
Vol 13 (S340) ◽  
pp. 47-48
Author(s):  
V. Vipindas ◽  
Sumesh Gopinath ◽  
T. E. Girish
Keyword(s):  
Long Range ◽  
Long Memory ◽  
Hurst Exponent ◽  
High Energy ◽  
Energetic Particles ◽  
Solar Energetic Particles ◽  
Long Range Dependence ◽  
Solar Cycles ◽  
Self Similarity ◽  
High Energy Particles

AbstractSolar Energetic Particles (SEPs) are high-energy particles ejected by the Sun which consist of protons, electrons and heavy ions having energies in the range of a few tens of keVs to several GeVs. The statistical features of the solar energetic particles (SEPs) during different periods of solar cycles are highly variable. In the present study we try to quantify the long-range dependence (or long-memory) of the solar energetic particles during different periods of solar cycle (SC) 23 and 24. For stochastic processes, long-range dependence or self-similarity is usually quantified by the Hurst exponent. We compare the Hurst exponent of SEP proton fluxes having energies (>1MeV to >100 MeV) for different periods, which include both solar maximum and minimum years, in order to find whether SC-dependent self-similarity exist for SEP flux.

The nature of discrete second-order self-similarity

2003 ◽  
Vol 35 (02) ◽  
pp. 395-416 ◽  
Author(s):  
A. Gefferth ◽  
D. Veitch ◽  
I. Maricza ◽  
S. Molnár ◽  
I. Ruzsa
Keyword(s):  
Fixed Points ◽  
Long Range ◽  
Power Laws ◽  
Complete Classification ◽  
Second Order ◽  
General Definition ◽  
Long Range Dependence ◽  
Self Similarity ◽  
New Treatment ◽  
Definition Of

A new treatment of second-order self-similarity and asymptotic self-similarity for stationary discrete time series is given, based on the fixed points of a renormalisation operator with normalisation factors which are not assumed to be power laws. A complete classification of fixed points is provided, consisting of the fractional noise and one other class. A convenient variance time function approach to process characterisation is used to exhibit large explicit families of processes asymptotic to particular fixed points. A natural, general definition of discrete long-range dependence is provided and contrasted with common alternatives. The closely related discrete form of regular variation is defined, its main properties given, and its connection to discrete self-similarity explained. Folkloric results on long-range dependence are proved or disproved rigorously.

scholarly journals Fractional Laplace motion

2006 ◽  
Vol 38 (02) ◽  
pp. 451-464 ◽  
Author(s):  
T. J. Kozubowski ◽  
M. M. Meerschaert ◽  
K. Podgórski
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Hydraulic Conductivity ◽  
Fractional Brownian Motion ◽  
Long Range ◽  
Financial Time Series ◽  
Long Range Dependence ◽  
Self Similarity ◽  
One Dimensional ◽  
Financial Time

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

The nature of discrete second-order self-similarity

2003 ◽  
Vol 35 (2) ◽  
pp. 395-416 ◽  
Author(s):  
A. Gefferth ◽  
D. Veitch ◽  
I. Maricza ◽  
S. Molnár ◽  
I. Ruzsa
Keyword(s):  
Fixed Points ◽  
Long Range ◽  
Power Laws ◽  
Complete Classification ◽  
Second Order ◽  
General Definition ◽  
Long Range Dependence ◽  
Self Similarity ◽  
New Treatment ◽  
Definition Of

A new treatment of second-order self-similarity and asymptotic self-similarity for stationary discrete time series is given, based on the fixed points of a renormalisation operator with normalisation factors which are not assumed to be power laws. A complete classification of fixed points is provided, consisting of the fractional noise and one other class. A convenient variance time function approach to process characterisation is used to exhibit large explicit families of processes asymptotic to particular fixed points. A natural, general definition of discrete long-range dependence is provided and contrasted with common alternatives. The closely related discrete form of regular variation is defined, its main properties given, and its connection to discrete self-similarity explained. Folkloric results on long-range dependence are proved or disproved rigorously.

scholarly journals Fractional Laplace motion

2006 ◽  
Vol 38 (2) ◽  
pp. 451-464 ◽  
Author(s):  
T. J. Kozubowski ◽  
M. M. Meerschaert ◽  
K. Podgórski
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Hydraulic Conductivity ◽  
Fractional Brownian Motion ◽  
Long Range ◽  
Financial Time Series ◽  
Long Range Dependence ◽  
Self Similarity ◽  
One Dimensional ◽  
Financial Time

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

scholarly journals Application of Generalized Cauchy Process on Modeling the Long-Range Dependence and Self-Similarity of Sea Surface Chlorophyll Using 23 years of Remote Sensing Data

2021 ◽  
Vol 9 ◽  
Author(s):  
Junyu He
Keyword(s):  
Long Range ◽  
European Space Agency ◽  
Remote Sensing Data ◽  
Sea Surface ◽  
Long Range Dependence ◽  
Local Similarity ◽  
Self Similarity ◽  
Space Agency ◽  
Marine Environmental ◽  
Cauchy Model

Understanding the temporal characteristics of sea surface chlorophyll (SSC) is helpful for marine environmental management. This study chose 10 time series of remote daily sea surface chlorophyll products from the European Space Agency during the period from July 29, 1998 to December 31, 2020. A generalized Cauchy model was employed to capture the local and global behaviors of sea surface chlorophyll from a fractal perspective; the fractal dimension D measures the local similarity while the Hurst parameter H measures the global long-range dependence. The generalized Cauchy model was fitted to the empirical autocorrelation function values of each SSC series. The results showed that the sea surface chlorophyll was multi-fractal in both space and time with the D values ranging from 1.0000 to 1.7964 and H values ranging from 0.6757 to 0.8431. Specifically, regarding the local behavior, 9 of the 10 series had low D values (<1.5), representing weak self-similarity; on the other hand, regarding the global behavior, high H values represent strong long-range dependence that may be a general phenomenon of daily sea surface chlorophyll.

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