Explore chapters and articles related to this topic
Correlation dimension as a measure of geophysical log chaos
Published in Vladimir Litvinenko, Topical Issues of Rational Use of Natural Resources 2019, 2019
Grassberger and Procaccia gave the algorithm of calculating correlation dimension based on a correlation sum (correlation integral) (Grassberger, Procaccia 1983). Mathematically it can be written: C(R)=limN→∞1N(N−1)∑i=1N∑j=1,j≠iNΘR−xi−xj
Autocorrelation Function, Mutual Information, and Correlation Dimension
Published in Nicholas Stergiou, Nonlinear Analysis for Human Movement Variability, 2018
A method to evaluate the number of degrees of freedom during posture is to determine the dimensionality of the center of pressure (COP) time series (Newell 1997). The correlation dimension is a measure of the dimensionality of a dynamical system. It measures how the data points in a time series from a dynamical system (e.g., COP time series from a swaying body during posture) are organized within a state space (Sprott and Rowlands 1995; Stergiou et al. 2004). The correlation dimension approximates the actual area that the dynamical system occupies in the state space. Small correlation dimension values can indicate a smaller number of the available degrees of freedom. In addition, small values (between 1.5 and 2.5) generally coincide with data that are deterministic in nature. Large correlation dimension values will characterize completely random data (between 4 and 6; Longstaff and Heath 1999; Sprott and Rowlands 1995; Stergiou et al. 2004). Cignetti et al. (2011) examined the difference in COP sitting data for infants at three stages of development (from the initial onset of sitting until completely independent sitting). Across the initial to the later developmental stages, the infants exhibited a reduction in correlation dimension in their COP data in the anterior/posterior direction, but not so in the medial/lateral direction. This indicated that the infants were increasing control of anterior/posterior sway by reducing the degrees of freedom involved.
Fractal Dimension of Biosignals
Published in Dinesh K. Kumar, Sridhar P. Arjunan, Behzad Aliahmad, Fractals, 2017
Dinesh K. Kumar, Sridhar P. Arjunan, Behzad Aliahmad
Correlation dimension is used as estimate to measure dimension of fractal objects [14]. It is computed using Grassberger-Procaccia algorithm [21] and based on the Theiler method [22]. This algorithm considers spatial correlation between pairs of points on a reconstructed attractor. Consider N number of points in a waveform denoted by x1 … … … …xN in some metric space with distances |xi − xj| between any pair of points. For any positive number r, the correlation sum C(r) is then defined as the fraction of pairs, whose distance is smaller than r, ()
Nonlinear Dynamic Analysis of the Transition from MILD Regime to Thermoacoustic Instability in a Reverse Flow Combustor
Published in Combustion Science and Technology, 2022
Atanu Dolai, Santanu Pramanik, Pabitra Badhuk, Ravikrishna RV
The correlation dimension indicates the topology of the attractor in the reconstructed phase space. For example, a correlation dimension of 1 signifies that the points in the phase space arrange themselves in one-dimensional structures such as lines or curves. The value of the correlation dimension depends on the embedding dimension of the system. This correlation dimension can also be used to differentiate between periodic, chaotic, and stochastic systems. In periodic and chaotic systems, as the embedding dimension is increased the correlation dimension saturates to a finite value. However, in stochastic systems the correlation dimension does not saturate, approaches infinity as the embedding dimension is increased. The correlation dimension is calculated using the Grassberger-Procassia (GP) algorithm (Grassberger and Procaccia 1983), (Grassberger and Procaccia 2004).
Chaotic characteristic analysis of short-term wind speed time series with different time scales
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2022
The correlation dimension is a non-linear measure of attractor correlation (Dlask and Kukal 2017). When appropriate delay time and embedding dimension m are selected to reconstruct the phase space of time series, the time series has obvious chaotic characteristics. The correlation dimension tends to be a finite fractal dimension with the increase of the embedding dimension. When the time series shows strong stochastic characteristics, the correlation dimension increases with the increase of embedding dimension, and will not reach saturation. Therefore, we can judge whether the short-term wind speed time series is chaotic or random according to whether the correlation dimension is saturated or not (Jiang and Kumar 2019). G-P method proposed by Grassberger and Procaccia can be introduced to calculate the correlation dimension (Zheng, Hu, and Zhang 2010).
Dynamical properties of partial-discharge patterns
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
Marek Lampart, Tomáš Vantuch, Ivan Zelinka, Stanislav Mišák
The correlation dimension is a measure of geometric structural complexity of strange attractors, by description of their static nature. The Grassberger–Procaccia algorithm was applied for calculating this measure [22], which can be described as follows. For time-series of N samples, it reconstructs the attractor dynamics by using delay coordinates to form multiple state space vectors, where and . is the reconstructed state space vector and m represents the embedding dimension with time lag .