Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media

Abstract

We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded. To explore whether these data can be used to identify the hopping rates in each layer, we compute various profile likelihoods using two methods: first, an exact likelihood is evaluated using a relatively expensive Markov chain approach; and, second, we form an approximate likelihood by assuming the distribution of exit times is given by a Gamma distribution whose first two moments match the moments from the continuum limit description of the stochastic model. Using the exact and approximate likelihoods, we construct various profile likelihoods for a range of problems. In cases where parameter values are not identifiable, we make progress by re-interpreting those data with a reduced model with a smaller number of layers.

Profile-Wise Analysis: A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models

“…In the case of a simulation-based stochastic model, we assume an auxiliary likelihood function [14], also known as a synthetic or surrogate likelihood [18,73], is available. For example, our recent work [59] considered a random walk simulation-based model of diffusive heat transfer across layered heterogeneous materials, and we generated data in terms of breakthrough-type time series to mimic experiments involving heat transfer across layered skin. For identifiability analysis and inference, we made use of an approximate likelihood function based on moment matching, written in terms of a Gamma distribution, and showed that this led to accurate inference at a fraction of the computational cost of working directly with the stochastic random walk simulations.…”

“…We describe our workflow for dynamic, differential equation-based models for simplicity. However, the same ideas carry over to other models, e.g., stochastic differential equations [4], spatiotemporal models such as partial differential equations (PDEs) [58] or simulation-based stochastic models [59].…”

Particle-Environment Interactions In Arbitrary Dimensions: A Unifying Analytic Framework To Model Diffusion With Inert Spatial Heterogeneities

“…Directions for future applications span across spatial and temporal scales: the role of a building geometry or floor plan on infection dynamics in hospital wards and supermarkets [86][87][88]; the prediction of search pattern behaviour of animals in different types of vegetation cover [89,90]; the heat transfer through layers of skin with differing thermal properties [91]; and the influence of topological defects on the diffusive properties in crystals [92,93] and territorial systems [94][95][96].…”

Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics

“…Another feature of our modelling approach is that we focus on deterministic ODE models with a separate noise model. While this is an important class of widely-used models, other approaches such as working with stochastic process models are also of high interest for different applications [40] and an interesting extension of the current work would be to extend the current framework so that we can deal with stochastic process models. An intermediate approach would be to use a deterministic ODE model but correct the noise model to account for missing stochastic process model effects [41].…”

“…Note that if we consider ψ to be a scalar interest parameter then ( ψ, ω * ( ψ )) defines a univariate function that we can visualise as a curve. In contrast, if ψ is a pair of interest parameters then ( ψ, ω * ( ψ )) is a function of two variables that is often called a bivariate profile that we can visualise as a heat map or contour plot [40].…”