Varying-Coefficient Stochastic Differential Equations with Applications in Ecology

Abstract

Stochastic differential equations (SDEs) are popular tools to analyse time series data in many areas, such as mathematical finance, physics, and biology. They provide a mechanistic description of the phenomenon of interest, and their parameters often have a clear interpretation. These advantages come at the cost of requiring a relatively simple model specification. We propose a flexible model for SDEs with time-varying dynamics where the parameters of the process are nonparametric functions of covariates, similar to generalized additive models. Combining the SDE and nonparametric approaches allows for the SDE to capture more detailed, non-stationary, features of the data-generating process. We present a computationally efficient method of approximate inference, where the SDE parameters can vary according to fixed covariate effects, random effects, or basis-penalty smoothing splines. We demonstrate the versatility and utility of this approach with three applications in ecology, where there is often a modelling trade-off between interpretability and flexibility.Supplementary materials accompanying this paper appear online.

Spatio-temporal stochastic differential equations for crime incidence modeling

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Modeling noisy time-series data of crime with stochastic differential equations

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“…In recent years, there that been interest in modeling more complex SDE processes (e.g., Barndorff‐Nielsen & Shephard, 2001; Davis et al, 2020; Jasra et al, 2018; Maejima & Yamamoto, 2003; Mai, 2014; Roberts et al, 2004; Valdivieso et al, 2008; Voutilainen et al, 2021) or have a latent hierarchical structure that needs to be estimated (e.g., Delattre et al, 2012; Hughes et al, 2017; Maitra & Bhattacharya, 2018; Michelot et al, 2021; Tran et al, 2020). In many applications, processes are observed that evolve over space and time.…”