Using information to generate derivative coordinates from noisy time series

This paper describes an approach for recovering a signal, along with the derivatives of the signal, from a noisy time series. To mimic an experimental setting, noise was superimposed onto a deterministic time series. Data smoothing was then used to successfully recover the derivative coordinates; however, the appropriate level of data smoothing must be determined. To investigate the level of smoothing, an information theoretic is applied to show a loss of information occurs for increased levels of noise; conversely, we have shown data smoothing can recover information by removing noise. An approximate criterion is then developed to balance the notion of information recovery through data smoothing with the observation that nearly negligible information changes occur for a sufficiently smoothed time series.