ABSTRACT
Empirical time series in the life sciences are often nonstationary and have small signal-to-noise ratios, making it dif‡cult to accurately detect and characterize dynamical structure. The usual response to high noise is averaging, but time domain averaging is inappropriate, especially when the dynamics are nonlinear. We review alternative delay-space averaging methods based on the topology and short-term predictability of nonlinear dynamics and illustrate their application using the TISEAN software (Hegger, Kantz, & Schreiber, 1999). The methods were applied to a Lorenz series, which resembles the dynamics found by Kelly, Heathcote, Heath, and Longstaff (2001) in series of decision times. The Lorenz series was corrupted with up to 80% additive Gaussian noise, a lower signalto-noise ratio than has been used in any previous test of these methods,
CONTENTS
Nonlinear Dynamics ........................................................................................... 105 Methods for Dynamical Analysis ..................................................................... 106
Linear Dynamical Analysis ........................................................................... 109 Nonlinear Methods Based on Recurrence .................................................. 110 Nonlinear Methods Based on Sensitive Dependence ............................... 115 Surrogate Testing, Time Asymmetry, and Geometric Filtering ............... 116
Nonlinear Dynamical Analysis of Noisy Data................................................ 119 Testing for Stationarity .................................................................................. 119 Testing for Nonlinearity ................................................................................ 121 Noise Filtering ................................................................................................ 124
Discussion ............................................................................................................ 128 Acknowledgments .............................................................................................. 130 Notes ..................................................................................................................... 131 References ............................................................................................................. 131
but consistent with Kelly et al.’s data. Prediction methods performed the best for detecting nonstationarity and nonlinear dynamics, and optimal predictability provided an objective criterion for setting the parameters required by the analyses. Local linear ‡ltering methods preformed best for characterization, producing informative plots that revealed the nature of the underlying dynamics. These results suggest that a methodology based on delay-space averaging and prediction could be useful with noisy empirical data series.
