Quantitative causality analysis with coarsely sampled time series

The information flow-based quantitative causality analysis has been widely applied in different disciplines because of its origin from first principles, its concise form, and its computational efficiency. So far the algorithm for its estimation is based on differential dynamical systems, which, however, may make an issue for coarsely sampled time series. Here, we show that for linear systems, this is fine at least qualitatively; but for highly nonlinear systems, the bias increases significantly as the sampling frequency is reduced. This paper provides a partial solution to this problem, showing how causality analysis is assured faithful with coarsely sampled series when, of course, the statistics is sufficient. An explicit and concise formula has been obtained, with only sample covariances involved. It has been successfully applied to a system comprising of a pair of coupled R\"ossler oscillators. Particularly remarkable is the success when the two oscillators are nearly synchronized.