Abstract
The transition density of a stochastic, logistic population growth model with multiplicative intrinsic noise is analytically intractable. Inferring model parameter values by fitting such stochastic differential equation (SDE) models to data therefore requires relatively slow numerical simulation. Where such simulation is prohibitively slow, an alternative is to use model approximations which do have an analytically tractable transition density, enabling fast inference. We introduce two such approximations, with either multiplicative or additive intrinsic noise, each derived from the linear noise approximation (LNA) of a logistic growth SDE. After Bayesian inference we find that our fast LNA models, using Kalman filter recursion for computation of marginal likelihoods, give similar posterior distributions to slow, arbitrarily exact models. We also demonstrate that simulations from our LNA models better describe the characteristics of the stochastic logistic growth models than a related approach. Finally, we demonstrate that our LNA model with additive intrinsic noise and measurement error best describes an example set of longitudinal observations of microbial population size taken from a typical, genome-wide screening experiment.
Highlights
Stochastic models simultaneously describe dynamics and noise or heterogeneity in real systems (Chen et al, 2010)
We have presented two new diffusion processes for modelling logistic growth data where fast inference is required: the linear noise approximation (LNA) of the stochastic logistic growth model (SLGM) with multiplicative noise and the LNA of the SLGM with additive intrinsic noise
Both the LNAM and LNAA are derived from the linear noise approximation of the stochastic logistic growth model (SLGM)
Summary
Stochastic models simultaneously describe dynamics and noise or heterogeneity in real systems (Chen et al, 2010). For SDE models where no explicit expression for the transition density is available, it is possible to infer parameter values by simulating a latent process using a data augmentation approach (Golightly and Wilkinson, 2005). This method is computationally intensive and not practical for all applications. We derive transition densities for the two approximate models and construct a Kalman filter by choosing measurement noise to be either multiplicative or additive to retain linear Gaussian structure. Román-Román and Torres-Ruiz (2012) present a logistic growth diffusion process (RRTR) which has a transition density that can be written explicitly, allowing inference of model parameter values from discrete sampling trajectories. We chose a lognormal (multiplicative) measurement error model in order to construct a linear Gaussian structure, enabling fast inference through the use of a Kalman filter for marginal likelihood computation
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