A posteriori tests to validate dimension estimates from time series

A posteriori tests are proposed to evaluate the degree of reliability of the estimate of dimension from a time series, using the method of correlation integrals. Although we consider in particular the correlation dimension (DI), our procedure can be applied to any generalized dimension as well as to the point-wise dimension. We propose the computation of two indices that can quantify to what degree the derivative of the log-log plot is constant with the correlation length and with the embedding dimension in the scaling region. An organized set of trials has been performed on time series from known model attractors, with different fractions of added measurement noise. On the basis of these trials, we found threshold values for the indices that discriminate between reliable and unreliable D-2 estimates. In the last part of the work we apply our procedure to real signals (electrocardiograms), finding good accordance between index values and the amount of noise in the time series. (C) 2000 Elsevier Science Ltd. All rights reserved.