STATISTICAL ERROR IN A CHORD ESTIMATOR OF CORRELATION DIMENSION: THE “RULE OF FIVE”
James P. Theiler, Turab Lookman · International Journal of Bifurcation and Chaos · 1993
The statistical precision of a chord method for estimating dimension from a correlation integral is derived. The optimal chord length is determined, and a comparison is made with other estimators. The simple chord estimator is only 25% less precise than the optimal estimator which uses the full resolution and full range of the correlation integral. The analytic calculations are based on the hypothesis that all pairwise distances between the points in the embedding space are statistically independent. The adequacy of this approximation is assessed numerically, and a surprising result is observed in which dimension estimators can be anomalously precise for sets with reasonably uniform (nonfractal) distributions.