Elshorbagy et al. (2001) report the presence of chaos in the streamflow series in the Little River and Reed Creek based (only) on the low correlation dimensions of 1.19 and 1.07 obtained when the Grassberger-Procaccia correlation dimension algorithm (e.g. Grassberger & Procaccia, 1983) is employed. Whether or not a low correlation dimension is a proof of chaos is an important question (cf. Siva...

In this article, we study the scale-dependent dimensionality properties and overall structure of text data with a method that measures correlation dimension in different scales. As experimental results, we present the analysis of text data sets with the Reuters and Europarl corpora, which are also compared to artificially generated point sets. A comparison is also made with speech data. The res...

To process the vibration signal fault diagnosis automatically and effectively, a method to calculate the generalized fractal dimension (Dq) based on the traditional G-P correlation dimension algorithm is presented. First, the vibration signal, a kind of random fractal that meets the statistical self-similarity, is reconstructed according to the phase space reconstruction theory. Second, for var...

different discrete models of population dynamics of certain insects have been investigated under various feasible conditions within the framework of nonlinear dynamics. evolutionary phenomena are discussed through bifurcation analysis leading to chaos. some tools of nonlinear dynamics, such as lyapunov characteristic exponents (lce), lyapunov numbers, correlation dimension, etc. are calculated ...

Strong laws of large numbers are given for L-statistics (linear com­ binations of order statistics) and for U -statistics (averages of kernels of ran­ dom samples) for ergodic stationary processes, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisf...

We present a method to recover a fractal dimension of a multi-scale rough surface, the so-called correlation dimension, from the knowledge of the far-field scattered intensity. The results are validated by numerical experiments on Weierstrass-like surfaces.

For many chaotic systems, accurate calculation of the correlation dimension from measured data is difficult because of very slow convergence as the scale size is reduced. This problem is often caused by the highly nonuniform measure on the attractor. This paper proposes a method for collecting data at large scales and extrapolating to the limit of zero scale. The result is a vastly reduced requ...

where F (x) = t(F (x), x1, . . . , xd−1) for x = (x1, x2, . . . , xd). Assuming Y t ∈ Ω for closed Ω ⊆ R and also assuming the ergodicity of {Y t}, we formulate the system as (Ω,F , μ,F ), where F is the completion of the Borel σ-field with respect to μ, and μ is an invariant measure, i.e. μ(F−1A) = μ(A) for A ∈ F . See Carlsson (2002) for a description of the sufficient conditions for the exis...