Plateau onset for correlation dimension: When does it occur?

Chaotic experimental systems are often investigated using delay coordinates. Estimated values of the correlation dimension in delay coordinate space typically increase with the number of delays and eventually reach a plateau (on which the dimension estimate is relatively constant) whose value is commonly taken as an estimate of the correlation dimension ${\mathit{D}}_2$ of the underlying chaotic attractor. We report a rigorous result which implies that, for long enough data sets, the plateau begins when the number of delay coordinates first exceeds ${\mathit{D}}_2$. Numerical experiments are presented. We also discuss how lack of sufficient data can produce results that seem to be inconsistent with the theoretical prediction.