Statistical Methods for Stochastic Differential Equations

Kessler, Mathieu; Lindner, Alexander; Sørensen, Michael

doi:10.1201/b12126

Cited by 58 publications

(37 citation statements)

References 99 publications

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“…For the Langevin movement model based on a RSF, as defined in Section 2, the Euler approximation therefore provides explicit estimates and confidence intervals. Note that the Euler estimator is biased due to the approximation made in Equation (see Kessler et al, , Chapter 1). Therefore, both the estimate and the confidence interval must be interpreted with caution, as they depend on the quality of the Euler scheme.…”

Section: Inferencementioning

confidence: 99%

“…Moreover, let T Δ be the (2n) where E is a 2n-vector of independent N(0, γ 2 ) variables, and where ν = γ 2 β. The estimators for ν and γ 2 are derived from standard lin- Kessler et al, 2012, Chapter 1). Therefore, both the estimate and the confidence interval must be interpreted with caution, as they depend on the quality of the Euler scheme.…”

Section: Maximum Likelihood Estimationmentioning

confidence: 99%

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The Langevin diffusion as a continuous‐time model of animal movement and habitat selection

et al. 2019

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The utilization distribution of an animal describes the relative probability of space use. It is natural to think of it as the long‐term consequence of the animal's short‐term movement decisions: it is the accumulation of small displacements which, over time, gives rise to global patterns of space use. However, many estimation methods for the utilization distribution either assume the independence of observed locations and ignore the underlying movement (e.g. kernel density estimation), or are based on simple Brownian motion movement rules (e.g. Brownian bridges). We introduce a new continuous‐time model of animal movement, based on the Langevin diffusion. This stochastic process has an explicit stationary distribution, conceptually analogous to the idea of the utilization distribution, and thus provides an intuitive framework to integrate movement and space use. We model the stationary (utilization) distribution with a resource selection function to link the movement to spatial covariates, and allow inference about habitat preferences of animals. Standard approximation techniques can be used to derive the pseudo‐likelihood of the Langevin diffusion movement model, and to estimate habitat preference and movement parameters from tracking data. We investigate the performance of the method on simulated data, and discuss its sensitivity to the time scale of the sampling. We present an example of its application to tracking data of Steller sea lions Eumetopias jubatus. Due to its continuous‐time formulation, this method can be applied to irregular telemetry data. The movement model is specified using a habitat‐dependent utilization distribution, and it provides a rigorous framework to estimate long‐term habitat selection from correlated movement data. The Langevin movement model can be written as a linear model, which allows for very fast inference. Standard tools such as residuals can be used for model checking.

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Section: Inferencementioning

confidence: 99%

Section: Maximum Likelihood Estimationmentioning

confidence: 99%

The Langevin diffusion as a continuous‐time model of animal movement and habitat selection

et al. 2019

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“…Various statistical methods have been proposed to perform maximum likelihood estimation with unknown transition densities (see Iacus () and Kessler et al . ()). The most convenient procedures to estimate discretely observed diffusion processes rely on discretization schemes to approximate the SDE between two observations and use a surrogate of the likelihood function to compute an approximate maximum likelihood estimator of η .…”

Section: Introductionmentioning

confidence: 94%

“…More recently, Kessler (), Kessler et al . () and Uchida and Yoshida () introduced another Gaussian‐based approximation of the transition density between two consecutive observations by using higher order expansions of the conditional mean and variance of an observation given the observation at the previous time step. Another approach to approximate the likelihood of the observations based on Hermite polynomials expansion was introduced by Ait‐Sahalia () and extended in several directions recently; see Li () and all the references therein.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Differential Equation Based on a Multimodal Potential to Model Movement Data in Ecology

Gloaguen

Étienne

Corff

2017

Journal of the Royal Statistical Society Series C: Applied Statistics

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Abstract. This paper proposes a new model for individuals movement in ecology. The movement process is defined as a solution to a stochastic differential equation whose drift is the gradient of a multimodal potential surface. This offers a new flexible approach among the popular potential based movement models in ecology. To perform parameter inference, the widely used Euler method is compared with two other pseudo-likelihood procedures and with a Monte Carlo Expectation Maximization approach based on exact simulation of diffusions. Performances of all methods are assessed with simulated data and with a data set of fishing vessels trajectories. We show that the usual Euler method performs worse than the other procedures for all sampling schemes.

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“…Secondly, they account for model uncertainty or environmental fluctuations by their inherent stochasticity. Lastly, they remedy the otherwise omnipresent issue of the inconsistent drift estimator (Kessler et al ., ) in plain SDEs (only fixed effects), when the observation time horizon is finite, because the mixed effects approach facilitates pooling of data across subjects, which leads to unbiasedness of the drift estimator as the number of subjects approaches ∞.…”

Section: Introductionmentioning

confidence: 99%

Inference for Biomedical Data by Using Diffusion Models with Covariates and Mixed Effects

Ruse

Samson

Ditlevsen

2019

Journal of the Royal Statistical Society Series C: Applied Statistics

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Summary Neurobiological data such as electroencephalography measurements pose a statistical challenge due to low spatial resolution and poor signal‐to‐noise ratio, as well as large variability from subject to subject. We propose a new modelling framework for this type of data based on stochastic processes. Stochastic differential equations with mixed effects are a popular framework for modelling biomedical data, e.g. in pharmacological studies. Whereas the inherent stochasticity of diffusion models accounts for prevalent model uncertainty or misspecification, random‐effects model intersubject variability. The two‐layer stochasticity, however, renders parameter inference challenging. Estimates are based on the discretized continuous time likelihood and we investigate finite sample and discretization bias. In applications, the comparison of, for example, treatment effects is often of interest. We discuss hypothesis testing and evaluate by simulations. Finally, we apply the framework to a statistical investigation of electroencephalography recordings from epileptic patients. We close the paper by examining asymptotics (the number of subjects going to ∞) of maximum likelihood estimators in multi‐dimensional, non‐linear and non‐homogeneous stochastic differential equations with random effects and included covariates.

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Statistical Methods for Stochastic Differential Equations

Cited by 58 publications

References 99 publications

The Langevin diffusion as a continuous‐time model of animal movement and habitat selection

The Langevin diffusion as a continuous‐time model of animal movement and habitat selection

Stochastic Differential Equation Based on a Multimodal Potential to Model Movement Data in Ecology

Inference for Biomedical Data by Using Diffusion Models with Covariates and Mixed Effects

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