The statistical properties of dimension calculations using small data sets

The statistical properties of estimates of pointwise dimension and their errors have been investigated for some maps and for some random variables. Substantial bias in the estimates is detected and modelled as a function of the sample size and the embedding dimension. The usual methods for calculating error bars are shown to underestimate the actual error bars by factors of ten and more. Procedures to improve the estimation of dimension in the cases studied are discussed, as are methods to improve the ability to distinguish noise from an attractor when using small data sets. Some idea of small is given when the attractors are similar to the ones studied.

[1]  P. Grassberger.  Finite sample corrections to entropy and dimension estimates , 1988 .

[2]  B. LeBaron,et al.  Nonlinear Dynamics and Stock Returns , 2021, Cycles and Chaos in Economic Equilibrium.

[3]  Hausdorff dimensions for sets with broken scaling symmetry , 1986 .

[4]  Leonard A. Smith,et al.  Lacunarity and intermittency in fluid turbulence , 1986 .

[5]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

[6]  Joachim Holzfuss,et al.  Approach to error-estimation in the application of dimension algorithms , 1986 .

[7]  R. L. Somorjai,et al.  Methods for Estimating the Intrinsic Dimsnionality of High-Dimensional Point Sets , 1986 .

[8]  C. L. Sayers.  WORK STOPPAGES: EXPLORING THE NONLINEAR DYNAMICS , 1988 .

[9]  Swinney,et al.  Strange attractors in weakly turbulent Couette-Taylor flow. , 1987, Physical review. A, General physics.

[10]  Peter Grassberger,et al.  Generalizations of the Hausdorff dimension of fractal measures , 1985 .

[11]  P. Atten,et al.  Determination of Attractor Dimension and Entropy for Various Flows: An Experimentalist’s Viewpoint , 1986 .

[12]  P. Grassberger,et al.  Characterization of experimental (noisy) strange attractors , 1984 .

[13]  H. G. E. Hentschel,et al.  The infinite number of generalized dimensions of fractals and strange attractors , 1983 .

[14]  A. Babloyantz,et al.  Evidence of Chaotic Dynamics of Brain Activity During the Sleep Cycle , 1985 .

[15]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[16]  G. T. Chapman,et al.  Characterizing Turbulent Channel Flow , 1986 .

[17]  J. Yorke,et al.  Strange attractors that are not chaotic , 1984 .

[18]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

[19]  A. Politi,et al.  Statistical description of chaotic attractors: The dimension function , 1985 .

[20]  J. Yorke,et al.  Dimension of chaotic attractors , 1982 .

[21]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[22]  Jensen,et al.  Erratum: Fractal measures and their singularities: The characterization of strange sets , 1986, Physical review. A, General physics.

[23]  P. Grassberger,et al.  Estimation of the Kolmogorov entropy from a chaotic signal , 1983 .

[24]  Z. Alexandrowicz,et al.  Fractal Dimension of Strange Attractors from Radius versus Size of Arbitrary Clusters , 1983 .

[25]  Shaun Lovejoy,et al.  Scale Invariance, Symmetries, Fractals, and Stochastic Simulations of Atmospheric Phenomena , 1986 .

[26]  A. Wolf,et al.  Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors , 1982 .

[27]  Biman Das,et al.  Calculating the dimension of attractors from small data sets , 1986 .

[28]  W. Brock,et al.  Is the business cycle characterized by deterministic chaos , 1988 .