Correlation dimension and systematic geometric effects

Abstract

Correlation-dimension calculations have been widely undertaken in attempts to understand various physical and other natural phenomena. Some of these studies have perhaps been incautious in their claims. On the other hand, some authors have been sharply critical of such work. These criticisms hinge on doubts about data-set size. In particular, Smith [Phys. Lett. A 133, 283 (1988)] has suggested that ${42}^{\mathit{D}}$ points are a minimum number of data points, where D is the embedding dimension. He has suggested that, as a result, correlation-dimension calculations are limited to dimensions less than 5 or 6 even on supercomputers. By implication, many published results are placed into doubt, discouraging people from undertaking such calculations. We review the concept of critical embedding dimension and undertake a detailed analysis of data-size requirements, which culminates in tight error estimates for determination of correlation dimension. We conclude that, while data requirements are still substantial, they are not nearly so extreme as has been suggested, making the determination of correlation dimension from data sets feasible. Previously reported ``doubtful'' calculations of correlation dimension can be reviewed in the light of explicit error estimates.