Radius selection using kernel density estimation for the computation of nonlinear measures

When nonlinear measures are estimated from sampled temporal signals with finite-length, a radius parameter must be carefully selected to avoid a poor estimation. These measures are generally derived from the correlation integral, which quantifies the probability of finding neighbors, i.e., pair of points spaced by less than the radius parameter. While each nonlinear measure comes with several specific empirical rules to select a radius value, we provide a systematic selection method. We show that the optimal radius for nonlinear measures can be approximated by the optimal bandwidth of a Kernel Density Estimator (KDE) related to the correlation sum. The KDE framework provides non-parametric tools to approximate a density function from finite samples (e.g., histograms) and optimal methods to select a smoothing parameter, the bandwidth (e.g., bin width in histograms). We use results from KDE to derive a closed-form expression for the optimal radius. The latter is used to compute the correlation dimension and to construct recurrence plots yielding an estimate of Kolmogorov-Sinai entropy. We assess our method through numerical experiments on signals generated by nonlinear systems and experimental electroencephalographic time series.