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Inference for Convolutionally Observed Diffusion Processes.

ABSTRACT: We propose a new statistical observation scheme of diffusion processes named convolutional observation, where it is possible to deal with smoother observation than ordinary diffusion processes by considering convolution of diffusion processes and some kernel functions with respect to time parameter. We discuss the estimation and test theories for the parameter determining the smoothness of the observation, as well as the least-square-type estimation for the parameters in the diffusion coefficient and the drift one of the latent diffusion process. In addition to the theoretical discussion, we also examine the performance of the estimation and the test with computational simulation, and show an example of real data analysis for one EEG data whose observation can be regarded as smoother one than ordinary diffusion processes with statistical significance.

SUBMITTER: Nakakita SH

PROVIDER: S-EPMC7597089 | biostudies-literature | 2020 Sep

REPOSITORIES: biostudies-literature

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Inference for Convolutionally Observed Diffusion Processes.

Nakakita Shogo H SH Uchida Masayuki M

Entropy (Basel, Switzerland) 20200915 9

We propose a new statistical observation scheme of diffusion processes named convolutional observation, where it is possible to deal with smoother observation than ordinary diffusion processes by considering convolution of diffusion processes and some kernel functions with respect to time parameter. We discuss the estimation and test theories for the parameter determining the smoothness of the observation, as well as the least-square-type estimation for the parameters in the diffusion coefficien ...[more]

PMID: 33286801

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Project description:Approximate Bayesian computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time-dependent, real-world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise: First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. To obtain summaries that are less sensitive to the intrinsic stochasticity of the model, we propose to build up the statistical method (e.g. the choice of the summary statistics) on the underlying structural properties of the model. Here, we focus on the existence of an invariant measure and we map the data to their estimated invariant density and invariant spectral density. Then, to ensure that these model properties are kept in the synthetic data generation, we adopt measure-preserving numerical splitting schemes. The derived property-based and measure-preserving ABC method is illustrated on the broad class of partially observed Hamiltonian type SDEs, both with simulated data and with real electroencephalography data. The derived summaries are particularly robust to the model simulation, and this fact, combined with the proposed reliable numerical scheme, yields accurate ABC inference. In contrast, the inference returned using standard numerical methods (Euler-Maruyama discretisation) fails. The proposed ingredients can be incorporated into any type of ABC algorithm and directly applied to all SDEs that are characterised by an invariant distribution and for which a measure-preserving numerical method can be derived.

Project description:BackgroundStochastic effects can be important for the behavior of processes involving small population numbers, so the study of stochastic models has become an important topic in the burgeoning field of computational systems biology. However analysis techniques for stochastic models have tended to lag behind their deterministic cousins due to the heavier computational demands of the statistical approaches for fitting the models to experimental data. There is a continuing need for more effective and efficient algorithms. In this article we focus on the parameter inference problem for stochastic kinetic models of biochemical reactions given discrete time-course observations of either some or all of the molecular species.ResultsWe propose an algorithm for inference of kinetic rate parameters based upon maximum likelihood using stochastic gradient descent (SGD). We derive a general formula for the gradient of the likelihood function given discrete time-course observations. The formula applies to any explicit functional form of the kinetic rate laws such as mass-action, Michaelis-Menten, etc. Our algorithm estimates the gradient of the likelihood function by reversible jump Markov chain Monte Carlo sampling (RJMCMC), and then gradient descent method is employed to obtain the maximum likelihood estimation of parameter values. Furthermore, we utilize flux balance analysis and show how to automatically construct reversible jump samplers for arbitrary biochemical reaction models. We provide RJMCMC sampling algorithms for both fully observed and partially observed time-course observation data. Our methods are illustrated with two examples: a birth-death model and an auto-regulatory gene network. We find good agreement of the inferred parameters with the actual parameters in both models.ConclusionsThe SGD method proposed in the paper presents a general framework of inferring parameters for stochastic kinetic models. The method is computationally efficient and is effective for both partially and fully observed systems. Automatic construction of reversible jump samplers and general formulation of the likelihood gradient function makes our method applicable to a wide range of stochastic models. Furthermore our derivations can be useful for other purposes such as using the gradient information for parametric sensitivity analysis or using the reversible jump samplers for full Bayesian inference. The software implementing the algorithms is publicly available at http://cbcl.ics.uci.edu/sgd.

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Inference for Convolutionally Observed Diffusion Processes.

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Inference for Convolutionally Observed Diffusion Processes.

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