Interval estimates of coupling delay using time series of oscillators

The need to estimate the delay times between elements of coupled oscillatory ensembles is frequently encountered in the analysis of systems of various natures, for which only separate records (time series) of the observed oscillations are available. The possibility of obtaining these estimates in the case of relatively short time series recorded in the presence of noise is considered. Using the well-known point estimator based on a model of the phase dynamics, an interval estimator is proposed, a correction that eliminates a bias of the point estimate is introduced, and a formula for the confidence interval is derived based on the maximum likelihood formalism. Possibilities of the proposed approach are illustrated by the results of numerical experiments in systems of coupled oscillators.