CHAOS, NOISE AND COMPLEX FRACTAL DIMENSIONS

Fractals ◽  
1993 ◽  
Vol 01 (01) ◽  
pp. 21-28 ◽  
Author(s):  
BRUCE J. WEST ◽  
X. FAN
Keyword(s):  
Time Series ◽  
Power Law ◽  
Colored Noise ◽  
Fractal Dimensions ◽  
Chaotic Time Series ◽  
Range Analysis ◽  
Rescaled Range ◽  
Rescaled Range Analysis ◽  
Power Law Index ◽  
Two Degree Of Freedom

Herein we propose a measure to distinguish a chaotic time series from one which is colored noise. We apply rescaled range analysis to the chaotic time series generated by a two-degree-of-freedom, nonintegrable Hamiltonian system, and find a modulated power-law for R/S. The combination of the power-law index and the period of modulation suggests a complex fractal dimension, which may be the signature of chaos. Some residual questions are collected.

SUNSPOTS DATA ANALYSIS USING TIME SERIES

Fractals ◽  
2008 ◽  
Vol 16 (03) ◽  
pp. 259-265 ◽  
Author(s):  
YUSUF H. SHAIKH ◽  
A. R. KHAN ◽  
M. I. IQBAL ◽  
S. H. BEHERE ◽  
S. P. BAGARE
Keyword(s):  
Time Series ◽  
Hurst Exponent ◽  
Sunspot Cycle ◽  
Fractal Dimensions ◽  
Data Sets ◽  
Range Analysis ◽  
Data Set ◽  
Rescaled Range ◽  
Rescaled Range Analysis ◽  
Sunspot Data

The record of the sunspot number visible on the sun is regularly collected over the centuries by various observatories for studying the different factors influencing the sunspot cycle and solar activity. Sunspots appear in cycles, and last several years. These cycles follow a certain pattern which is well known. We analyzed monthly and yearly averages of sunspot data observed from year 1818 to 2002 using rescaled range analysis. The Hurst exponent calculated for monthly data sets are 0.8899, 0.8800 and 0.8597 and for yearly data set is 0.7187. Fractal dimensions1 calculated are 1.1100, 1.1200, 1.1403 and 1.2813. From the study of Hurst exponent and fractal dimension, we conclude that time series of sunspots show persistent behavior. The fundamental tool of signal processing is the fast Fourier transform technique (FFT). The sunspot data is also analyzed using FFT. The power spectrum of monthly and yearly averages of sunspot shows distinct peaks at 11 years confirming the well known 11-year cycle. The monthly sunspot data is also analyzed using FFT to filter the noise in the data.

BOX DIMENSION OF COLORED NOISE AND DETERMINISTIC TIME SERIES IN HIGH DIMENSIONAL EUCLIDEAN SPACES

Fractals ◽  
1996 ◽  
Vol 04 (01) ◽  
pp. 91-95 ◽  
Author(s):  
WEN ZHANG ◽  
BRUCE J. WEST
Keyword(s):  
Time Series ◽  
Power Law ◽  
Colored Noise ◽  
Fractal Dimensions ◽  
Inverse Power ◽  
High Dimensional ◽  
Dimensional Euclidean Space ◽  
Box Dimension ◽  
Inverse Power Law ◽  
Lower Graph

We investigate the box dimension of a time series having an inverse power-law spectra in a high dimensional Euclidean space. The time series can be random (colored noise) or deterministic. Both isotropic and anisotropic cases are included in our investigation. We study both the graph dimension and trail dimension of the time series. We show that with the same inverse power-law spectra, the deterministic series has a lower graph dimension than that of the colored noise, though they both can have fractal dimensions. We also derive a sharp upper bound on the trial dimension of the time series.

scholarly journals The Nonlinear Time Sequence Analysis in the Alpine-Himalayan Earthquake Zone

2021 ◽  
Vol 299 ◽  
pp. 02001
Author(s):  
Jiemin Chen ◽  
Zelin Yan ◽  
Linfeng Xu ◽  
Zhixin Liu ◽  
Yan Liu ◽  
...  
Keyword(s):  
Time Series ◽  
Fractal Theory ◽  
Stable Region ◽  
Earthquake Activity ◽  
Range Analysis ◽  
Rescaled Range ◽  
Rescaled Range Analysis ◽  
Fractal Characteristics ◽  
Earthquake Zone ◽  
Himalayan Earthquake

The characteristics of the earthquake activity in the Eurasian earthquake zone, which is the second largest earthquake zone in the world, was investigated by researchers. The earthquake activity of the Eurasian earthquake zone was analysed in various disciplines, such as earth dynamics, rock mechanics, geology and tectonics. The emergence of fractal theory provided a new direction in exploring the characteristics of the earthquake activity in the Eurasian earthquake zone. This study processed the data on the earthquake activity in the Eurasian earthquake zone by self-similarity method and scaled invariant feature test and used the rescaled range analysis method to analyse the nonlinear time series fractal characteristics of the earthquake activity in the Eurasian earthquake zone. Results show that the time series of earthquake activity in the study area is not an independent Poisson process, which exhibits the characteristics of scale invariance and long-range correlation. Approximately 80% of the H values of the earthquake activity iteratively increase and decrease for moderate earthquakes, which is mainly concentrated during the increasing stage. The time difference of the H value between the two-neighbouring earthquake shows that the H value fluctuates in the active earthquake region and is stationary in the relatively stable region. Strong earthquakes will likely occur in the next few years because the H value fluctuates.

Detection of Dynamical Complexity Changes in Dst Time Series Using Entropy Concepts and Rescaled Range Analysis

2011 ◽  
pp. 211-220 ◽  
Author(s):  
Georgios Balasis ◽  
Ioannis A. Daglis ◽  
Anastasios Anastasiadis ◽  
Konstantinos Eftaxias
Keyword(s):  
Time Series ◽  
Range Analysis ◽  
Dynamical Complexity ◽  
Rescaled Range ◽  
Rescaled Range Analysis

scholarly journals Rescaled range analysis and conditional probability-based probe into the intrinsic pattern of rainfall over North Mountainous India

2021 ◽  
Author(s):  
Amith Sharma ◽  
Surajit Chattopadhyay
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Indian Summer Monsoon ◽  
Annual Rainfall ◽  
Conditional Probabilities ◽  
Autocorrelation Coefficient ◽  
Range Analysis ◽  
Rescaled Range ◽  
Rescaled Range Analysis ◽  
High Volatility

Abstract In work reported here, we have explored rainfall over North Mountainous India for pre-monsoon (MAM), Indian summer monsoon (JJAS), post-monsoon (OND) and Annual. The dependence of JJAS on MAM and OND on JJAS has been explored through conditional probabilities utilizing frequency distribution. An autocorrelation structure has shown that a low lag-1 autocorrelation coefficient characterizes all the time series. We have implemented rescaled range analysis. Through Hurst's exponent and fractal dimension, we have observed that the MAM time series of rainfall over North Mountainous India has a smooth trend and low volatility. We have further observed that for MAM and JJAS, we have , and D is closer to 1 than to 2. However, we have further observed that for OND and Annual rainfall over North Mountainous India and . Therefore, these two time series have been characterized by high volatility and randomness.

Evaluating rescaled range analysis for time series

1994 ◽  
Vol 22 (4) ◽  
pp. 432-444 ◽  
Author(s):  
James B. Bassingthwaighte ◽  
Gary M. Raymond
Keyword(s):  
Time Series ◽  
Range Analysis ◽  
Rescaled Range ◽  
Rescaled Range Analysis

CALCULATION OF THE POWER LAW INDEX FOR A TIME SERIES BY MEANS OF ITS FRACTAL DIMENSION

Fractals ◽  
1996 ◽  
Vol 04 (03) ◽  
pp. 265-271
Author(s):  
SHINICHI YASUE ◽  
KAZUOKI MUNAKATA ◽  
MASASHIGE KATO ◽  
SATORU MORI
Keyword(s):  
Time Series ◽  
Power Law ◽  
Present Method ◽  
Fractal Dimensions ◽  
Cosmic Ray ◽  
Statistical Fluctuations ◽  
Deep Underground ◽  
Integrated Time Series ◽  
Power Law Index ◽  
The Relationship

The relationship between the power-law index for a time series and the fractal dimensions for both the original and the integrated time series, is investigated by using a numerical experiment. This relationship is extended and applied to the cosmic ray time series recorded at the deep underground site at Matsushiro. It is shown that we can use the present method in order to obtain the stable and reliable value of the power-law index of a time series, even if the time series has relatively large statistical fluctuations.

ANALYSIS AND NUMERICAL COMPUTATION OF THE DIMENSION OF COLORED NOISE AND DETERMINISTIC TIME SERIES WITH POWER-LAW SPECTRA

Fractals ◽  
1994 ◽  
Vol 02 (01) ◽  
pp. 53-64
Author(s):  
WEN ZHANG ◽  
BRUCE J. WEST
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Numerical Computation ◽  
Trigonometric Series ◽  
Power Law ◽  
Colored Noise ◽  
Fractal Dimensions ◽  
Box Dimension ◽  
Inverse Power Law ◽  
High Degree

We investigate the box dimension of a graph of time series generated by trigonometric series with an inverse power-law spectrum. Such time series can either be random, in which case it is called colored noise, or deterministic in nature. We show analytically that both series can have fractal dimensions depending on the value of the exponent in the power-law. However, the fractal dimension of colored noise is 0.5 higher than that of the corresponding deterministic series. We comment on calculating fractal dimensions and present a reliable numerical algorithm which yields a high degree of consistency between experimental and analytical results.

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