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 each reference point to be concave up or down. These distortions are pronounced for the experimental data whose number of points is not sufficient and whose scaling region is restricted to a relatively small interval. It is possible that the Grassberger–Procaccia algorithm and all its modified ones are affected by the bias. We evaluate the distortion quantitatively and show the procedure required for the correction of the bias taking the case of an electroencephalogram (EEG).