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.