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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
Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Grassberger and Procaccia (1983a,b) defined correlation dimension with respect to the correlation integral, or correlation sum in discrete form, which is () Cm(r)=1N(N−1)∑i,j=1;i≠j∞H(r−‖xi−xj‖);i≠j
Seizure Detection
Published in Andrea Varsavsky, Iven Mareels, Mark Cook, Epileptic Seizures and the EEG, 2016
Andrea Varsavsky, Iven Mareels, Mark Cook
The methods use to compute the correlation dimension of a signal can be found in Section 3.3.4. First a time-delay reconstruction as in Equation 3.35 is performed using τ = 30 samples (0.06 seconds at 512Hz sampling) and dimension n̂ = 4. The correlation sum is then computed (Equation 3.38) and the gradient in Equation 3.39 is used as an estimate of correlation dimension. ϵlower = 0.1 and ϵupper = 2 are used for computation, consistent across all data because it has been normalized to unit variance prior to analysis.
Recurrence determinism and Li–Yorke chaos for interval maps
Published in Dynamical Systems, 2019
Asymptotic recurrence m-determinism is (for typical x and ϵ) equal to the conditional probability, under the condition that the current state is a recurrence, that the following (m − 1) states are recurrences with the same time gap as the current one; here, by a recurrence, we mean that the distance of a state from some previous one is smaller than or equal to the precision ϵ. To be more precise, take an ergodic measure μ of the system (X, f). Then, by [11], for μ-a.e. x ∈ X and for all but countably many ϵ > 0, asymptotic correlation sum exists and is equal to the correlation integral of μ Thus, if Y, Z are independent X-valued random variables with distribution μ, then asymptotic correlation sum cm(x, ϵ) is (typically) equal to the probability that fi(Y) and fi(Z) are ϵ-close for every i ∈ [0, m). Consequently, asymptotic recurrence determinism is (typically) equal to the conditional probability that fi(Y) and fi(Z) are ϵ-close for every i ∈ [0, m), given that Y and Z are ϵ-close. For more details, see [8].