Hypoelliptic diffusions: filtering and inference from complete and partial observations

TY - JOUR

T1 - Hypoelliptic diffusions

T2 - filtering and inference from complete and partial observations

AU - Ditlevsen, Susanne

AU - Samson, Adeline

PY - 2019/4

Y1 - 2019/4

N2 - The statistical problem of parameter estimation in partially observed hypoelliptic diffusion processes is naturally occurring in many applications. However, because of the noise structure, where the noise components of the different co-ordinates of the multi-dimensional process operate on different timescales, standard inference tools are ill conditioned. We propose to use a higher order scheme to approximate the likelihood, such that the different timescales are appropriately accounted for. We show consistency and asymptotic normality with non-typical convergence rates. When only partial observations are available, we embed the approximation in a filtering algorithm for the unobserved co-ordinates and use this as a building block in a stochastic approximation expectation–maximization algorithm. We illustrate on simulated data from three models: the harmonic oscillator, the FitzHugh–Nagumo model used to model membrane potential evolution in neuroscience and the synaptic inhibition and excitation model used for determination of neuronal synaptic input.

AB - The statistical problem of parameter estimation in partially observed hypoelliptic diffusion processes is naturally occurring in many applications. However, because of the noise structure, where the noise components of the different co-ordinates of the multi-dimensional process operate on different timescales, standard inference tools are ill conditioned. We propose to use a higher order scheme to approximate the likelihood, such that the different timescales are appropriately accounted for. We show consistency and asymptotic normality with non-typical convergence rates. When only partial observations are available, we embed the approximation in a filtering algorithm for the unobserved co-ordinates and use this as a building block in a stochastic approximation expectation–maximization algorithm. We illustrate on simulated data from three models: the harmonic oscillator, the FitzHugh–Nagumo model used to model membrane potential evolution in neuroscience and the synaptic inhibition and excitation model used for determination of neuronal synaptic input.

KW - 1.5 strong order discretization scheme

KW - Approximate maximum likelihood

KW - Hypoelliptic diffusion

KW - Parameter estimation

KW - Particle filter

KW - Stochastic approximation expectation–maximization algorithm

UR - http://www.scopus.com/inward/record.url?scp=85058976830&partnerID=8YFLogxK

U2 - 10.1111/rssb.12307

DO - 10.1111/rssb.12307

M3 - Journal article

AN - SCOPUS:85058976830

SN - 1369-7412

VL - 81

SP - 361

EP - 384

JO - Journal of the Royal Statistical Society, Series B (Statistical Methodology)

JF - Journal of the Royal Statistical Society, Series B (Statistical Methodology)

IS - 2

ER -