134 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
Extension of the SAEM algorithm for nonlinear mixed models with two levels of random effects
This article focuses on parameter estimation of multi-levels nonlinear mixed
effects models (MNLMEMs). These models are used to analyze data presenting
multiple hierarchical levels of grouping (cluster data, clinical trials with
several observation periods,...). The variability of the individual parameters
of the regression function is thus decomposed as a between-sub ject variability
and higher levels of variability (for example within-sub ject variability). We
propose maximum likelihood estimates of parameters of those MNLMEMs with two
levels of random effects, using an extension of the SAEM-MCMC algorithm. The
extended SAEM algorithm is split into an explicit direct EM algorithm and a
stochastic EM part. Compared to the original algorithm, additional sufficient
statistics have to be approximated by relying on the conditional distribution
of the second level of random effects. This estimation method is evaluated on
pharmacokinetic cross-over simulated trials, mimicking theophyllin
concentration data. Results obtained on those datasets with either the SAEM
algorithm or the FOCE algorithm (implemented in the nlme function of R
software) are compared: biases and RMSEs of almost all the SAEM estimates are
smaller than the FOCE ones. Finally, we apply the extended SAEM algorithm to
analyze the pharmacokinetic interaction of tenofovir on atazanavir, a novel
protease inhibitor, from the ANRS 107-Puzzle 2 study. A significant decrease of
the area under the curve of atazanavir is found in patients receiving both
treatments
Parametric estimation of complex mixed models based on meta-model approach
Complex biological processes are usually experimented along time among a
collection of individuals. Longitudinal data are then available and the
statistical challenge is to better understand the underlying biological
mechanisms. The standard statistical approach is mixed-effects model, with
regression functions that are now highly-developed to describe precisely the
biological processes (solutions of multi-dimensional ordinary differential
equations or of partial differential equation). When there is no analytical
solution, a classical estimation approach relies on the coupling of a
stochastic version of the EM algorithm (SAEM) with a MCMC algorithm. This
procedure needs many evaluations of the regression function which is clearly
prohibitive when a time-consuming solver is used for computing it. In this work
a meta-model relying on a Gaussian process emulator is proposed to replace this
regression function. The new source of uncertainty due to this approximation
can be incorporated in the model which leads to what is called a mixed
meta-model. A control on the distance between the maximum likelihood estimates
in this mixed meta-model and the maximum likelihood estimates obtained with the
exact mixed model is guaranteed. Eventually, numerical simulations are
performed to illustrate the efficiency of this approach
Parameter estimation and treatment optimization in a stochastic model for immunotherapy of cancer
Adoptive Cell Transfer therapy of cancer is currently in full development and
mathematical modeling is playing a critical role in this area. We study a
stochastic model developed by Baar et al. in 2015 for modeling immunotherapy
against melanoma skin cancer. First, we estimate the parameters of the
deterministic limit of the model based on biological data of tumor growth in
mice. A Nonlinear Mixed Effects Model is estimated by the Stochastic
Approximation Expectation Maximization algorithm. With the estimated
parameters, we head back to the stochastic model and calculate the probability
that the T cells all get exhausted during the treatment. We show that for some
relevant parameter values, an early relapse is due to stochastic fluctuations
(complete T cells exhaustion) with a non negligible probability. Then, focusing
on the relapse related to the T cell exhaustion, we propose to optimize the
treatment plan (treatment doses and restimulation times) by minimizing the T
cell exhaustion probability in the parameter estimation ranges.Comment: major reorganisation of the paper and the reformulation of many
substantial part
Parametric inference for mixed models defined by stochastic differential equations
International audienceNon-linear mixed models defined by stochastic differential equations (SDEs) are consid- ered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measure- ment instants. A tuned hybrid Gibbs algorithm based on conditional Brownian bridges simulations of the unobserved process paths is included in this algorithm. The convergence is proved and the error induced on the likelihood by the Euler-Maruyama approximation is bounded as a function of the step size of the approximation. Results of a pharmacokinetic simulation study illustrate the accuracy of this estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach
Estimation of parameters in incomplete data models defined by dynamical systems.
International audienceParametric incomplete data models defined by ordinary differential equa- tions (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Ac- curate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochas- tic approximation EM algorithm (SAEM) in the maximum likelihood es- timation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings-Metropolis (H-M) algorithms. A modified H-M algorithm, including an original Local Linearization scheme to solve the ODEs, is pro- posed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacoki- netic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates
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