Fisher–Shannon Investigation of the Effect of Nonlinearity of Discrete Langevin Model on Behavior of Extremes in Generated Time Series

Abstract

Diverse forms of nonlinearity within stochastic equations give rise to varying dynamics in processes, which may influence the behavior of extreme values. This study focuses on two nonlinear models of the discrete Langevin equation: one with a fixed diffusion function (M1) and the other with a fixed marginal distribution (M2), both characterized by a nonlinearity parameter. Extremes are defined according to the run theory with thresholds based on percentiles. The behavior of inter-extreme times and run lengths is examined by employing Fisher’s Information Measure and the Shannon Entropy. Our findings reveal a clear relationship between the entropic and informational measures and the nonlinearity of model M1—these measures decrease as the nonlinearity parameter increases. Similar relationships are evident for the M2 model, albeit to a lesser extent, even though the background data’s marginal distribution remains unaffected by this parameter. As thresholds increase, both the values of Fisher’s Information Measure and the Shannon Entropy also increase.

Keywords

• nonlinear Langevin equation
• time series
• Fisher–Shannon plane
• run theory
• extremes