Spurious dimension from correlation algorithms applied to limited time-series data

An algorithm devised for measuring the dimension of a strange attractor from a time series is applied both to autocorrelated Gaussian noise and to a dynamical system. It is analytically shown that a finite sequence of stochastic data—where by ‘‘finite’’ it is meant that N<2${\tau }^{m/2\mathrm{}}$, where N is the number of points in the sequence, τ is the autocorrelation time (in units of sampling period), and m is the embedding dimension—exhibits anomalous structure in its correlation integral. The anomaly is seen numerically in both stochastic and dynamical data. Unrecognized, it can lead to unnecessarily inaccurate and possibly spurious estimates of dimension. We propose a slight modification of the standard algorithm which eliminates this difficulty.