Statistical inference for noisy nonlinear ecological dynamic systems

Abstract

Chaotic ecological dynamic systems defy conventional statistical analysis. Systems with near-chaotic dynamics are little better. Such systems are almost invariably driven by endogenous dynamic processes plus demographic and environmental process noise, and are only observable with error. Their sensitivity to history means that minute changes in the driving noise realization, or the system parameters, will cause drastic changes in the system trajectory. This sensitivity is inherited and amplified by the joint probability density of the observable data and the process noise, rendering it useless as the basis for obtaining measures of statistical fit. Because the joint density is the basis for the fit measures used by all conventional statistical methods, this is a major theoretical shortcoming. The inability to make well-founded statistical inferences about biological dynamic models in the chaotic and near-chaotic regimes, other than on an ad hoc basis, leaves dynamic theory without the methods of quantitative validation that are essential tools in the rest of biological science. Here I show that this impasse can be resolved in a simple and general manner, using a method that requires only the ability to simulate the observed data on a system from the dynamic model about which inferences are required. The raw data series are reduced to phase-insensitive summary statistics, quantifying local dynamic structure and the distribution of observations. Simulation is used to obtain the mean and the covariance matrix of the statistics, given model parameters, allowing the construction of a 'synthetic likelihood' that assesses model fit. This likelihood can be explored using a straightforward Markov chain Monte Carlo sampler, but one further post-processing step returns pure likelihood-based inference. I apply the method to establish the dynamic nature of the fluctuations in Nicholson's classic blowfly experiments.

Reviews and syntheses: parameter identification in marine planktonic ecosystem modelling

“…The optimization of parameter values only ensures that simulation results remain dynamically and ecologically consistent, which is comparable with those DA approaches in physical oceanography that produce dynamically and kinematically consistent solutions of ocean circulation (e.g. Wunsch and Heimbach, 2007;Wunsch et al, 2009). Thorough reviews of common DA methods applied in marine biogeochemical modelling are given by Robinson and Lermusiaux (2002) and by Matear and Jones (2011).…”

“…A major challenge in calibrating biogeochemical models on global scale is that the simulations require many millennia until tracer distributions are in equilibrium with the given circulation field and the biogeochemical processes (Wunsch and Heimbach, 2008). Equilibrium solutions are usually achieved by integrating tracer fields for several thousand years in a so-called model spin-up, based on some seasonally cycling climatological circulation fields.…”

“…Equilibrium solutions are usually achieved by integrating tracer fields for several thousand years in a so-called model spin-up, based on some seasonally cycling climatological circulation fields. Convergence to steady state conditions depends on the region, tracer type, and form of boundary condition (Wunsch and Heimbach, 2008;Primeau and Deleersnijder, 2009;Siberlin and Wunsch, 2011). It also depends on the values assigned to the parameters of the biogeochemical model, and it is not necessarily a monotonic function of time, but can exhibit inflection points that reflect the interaction of diverse processes happening on different timescales (Kriest and Oschlies, 2015).…”

“…A better approach here might be to use a "synthetic likelihood" (Wood, 2010), whereby the raw data and model output are replaced with a carefully chosen, informative set of phase-insensitive summary statistics (e.g. means, standard deviations, and lag correlations; cf.…”

Bayesian hierarchical statistical SIRS models

“…More recent Bayesian hierarchical models, which are widely used for mapping non-infectious diseases, aim to capture the true spatial process (e.g., Besag et al, 1991;Carlin and Banerjee, 2002), but their process models and parameter models are not appropriate for epidemics. Those that do have a dynamical spatial statistical component have not generally been parameterized in terms of the interpretable components of the epidemic (e.g., Mugglin et al, 2002;Wood, 2010).…”