880 research outputs found
Estimation in the partially observed stochastic Morris-Lecar neuronal model with particle filter and stochastic approximation methods
Parameter estimation in multidimensional diffusion models with only one
coordinate observed is highly relevant in many biological applications, but a
statistically difficult problem. In neuroscience, the membrane potential
evolution in single neurons can be measured at high frequency, but biophysical
realistic models have to include the unobserved dynamics of ion channels. One
such model is the stochastic Morris-Lecar model, defined by a nonlinear
two-dimensional stochastic differential equation. The coordinates are coupled,
that is, the unobserved coordinate is nonautonomous, the model exhibits
oscillations to mimic the spiking behavior, which means it is not of
gradient-type, and the measurement noise from intracellular recordings is
typically negligible. Therefore, the hidden Markov model framework is
degenerate, and available methods break down. The main contributions of this
paper are an approach to estimate in this ill-posed situation and nonasymptotic
convergence results for the method. Specifically, we propose a sequential Monte
Carlo particle filter algorithm to impute the unobserved coordinate, and then
estimate parameters maximizing a pseudo-likelihood through a stochastic version
of the Expectation-Maximization algorithm. It turns out that even the rate
scaling parameter governing the opening and closing of ion channels of the
unobserved coordinate can be reasonably estimated. An experimental data set of
intracellular recordings of the membrane potential of a spinal motoneuron of a
red-eared turtle is analyzed, and the performance is further evaluated in a
simulation study.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS729 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
AReS and MaRS - Adversarial and MMD-Minimizing Regression for SDEs
Stochastic differential equations are an important modeling class in many
disciplines. Consequently, there exist many methods relying on various
discretization and numerical integration schemes. In this paper, we propose a
novel, probabilistic model for estimating the drift and diffusion given noisy
observations of the underlying stochastic system. Using state-of-the-art
adversarial and moment matching inference techniques, we avoid the
discretization schemes of classical approaches. This leads to significant
improvements in parameter accuracy and robustness given random initial guesses.
On four established benchmark systems, we compare the performance of our
algorithms to state-of-the-art solutions based on extended Kalman filtering and
Gaussian processes.Comment: Published at the Thirty-sixth International Conference on Machine
Learning (ICML 2019
Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals
Estimation of parameters of a diffusion based on discrete time observations
poses a difficult problem due to the lack of a closed form expression for the
likelihood. From a Bayesian computational perspective it can be casted as a
missing data problem where the diffusion bridges in between discrete-time
observations are missing. The computational problem can then be dealt with
using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown
parameters appear in the diffusion coefficient, direct implementation of
data-augmentation results in a Markov chain that is reducible. Furthermore,
data-augmentation requires efficient sampling of diffusion bridges, which can
be difficult, especially in the multidimensional case.
We present a general framework to deal with with these problems that does not
rely on discretisation. The construction generalises previous approaches and
sheds light on the assumptions necessary to make these approaches work. We
define a random-walk type Metropolis-Hastings sampler for updating diffusion
bridges. Our methods are illustrated using guided proposals for sampling
diffusion bridges. These are Markov processes obtained by adding a guiding term
to the drift of the diffusion. We give general guidelines on the construction
of these proposals and introduce a time change and scaling of the guided
proposal that reduces discretisation error. Numerical examples demonstrate the
performance of our methods
Markov chain Monte Carlo for exact inference for diffusions
We develop exact Markov chain Monte Carlo methods for discretely-sampled,
directly and indirectly observed diffusions. The qualification "exact" refers
to the fact that the invariant and limiting distribution of the Markov chains
is the posterior distribution of the parameters free of any discretisation
error. The class of processes to which our methods directly apply are those
which can be simulated using the most general to date exact simulation
algorithm. The article introduces various methods to boost the performance of
the basic scheme, including reparametrisations and auxiliary Poisson sampling.
We contrast both theoretically and empirically how this new approach compares
to irreducible high frequency imputation, which is the state-of-the-art
alternative for the class of processes we consider, and we uncover intriguing
connections. All methods discussed in the article are tested on typical
examples.Comment: 23 pages, 6 Figures, 3 Table
Langevin and Hamiltonian based Sequential MCMC for Efficient Bayesian Filtering in High-dimensional Spaces
Nonlinear non-Gaussian state-space models arise in numerous applications in
statistics and signal processing. In this context, one of the most successful
and popular approximation techniques is the Sequential Monte Carlo (SMC)
algorithm, also known as particle filtering. Nevertheless, this method tends to
be inefficient when applied to high dimensional problems. In this paper, we
focus on another class of sequential inference methods, namely the Sequential
Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising
alternative to SMC methods. After providing a unifying framework for the class
of SMCMC approaches, we propose novel efficient strategies based on the
principle of Langevin diffusion and Hamiltonian dynamics in order to cope with
the increasing number of high-dimensional applications. Simulation results show
that the proposed algorithms achieve significantly better performance compared
to existing algorithms
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
- …