A fundamental bias in calculating dimensions from finite data sets

One bias inherent in calculating dimension for limited time-series data is investigated. The bias is derived from the fluctuation of the distribution of measures in the phase space and distorts the scaling with respect to eac h reference point to be concave up or down. These distortions are pronounce d for the experimental data whose number of points is not sufficient and wh ose scaling region is restricted to a relatively small interval. It is poss ible that the Grassberger-Procaccia algorithm and all its modified ones are affected by the bias. We evaluate the distortion quantitatively and show t he procedure required for the correction of the bias taking the case of an electroencephalogram (EEG). (C) 2000 Elsevier Science B.V. All rights reser ved.