Source-linked AI summary
Time series analysis via mechanistic models
Carles Bretó, Daihai He, Edward L. Ionides, Aaron A. King
TL;DR
The paper addresses how to perform formal inference for nonlinear mechanistic time-series models, including implicitly specified stochastic systems. It develops plug-and-play models and iterated-filtering inference, then applies them to measles and competing cholera strains, finding evidence relevant to environmental stochasticity and model improvement.
Problem
Formal inference for scientifically motivated nonlinear, partially observed dynamical systems remains difficult, especially when models are specified through simulation rather than closed-form transitions.
Method
The paper develops implicitly specified Markov chains with stochastic transition rates and applies plug-and-play iterated filtering for likelihood-based inference.
Results
The framework is demonstrated on measles transmission and cholera incidence involving two competing pathogen strains, with the measles analysis favoring environmental stochasticity over its restricted alternative.
Takeaways & Limitations
Simulation-based mechanistic inference can analyze biological time series while accommodating nonlinear dynamics, partial observation, and stochastic transition rates.
Takeaways & Limitations
The measles analysis cannot distinguish environmental stochasticity from model misspecification, while the cholera models omit epidemiological features such as waning immunity.
Abstract
from arXiv · showhide
The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae.
1. Introduction.
The paper develops mechanistic time-series analysis for drawing inference about dynamical systems from observations, while preserving scientifically motivated model structure. It targets nonlinear, partially observed stochastic systems through simulation-based, plug-and-play inference.
- Mechanistic models describe how a system evolves and relate its latent state to available observations over time.
- Key questions include whether data are consistent with a model, which parameter values are plausible, and whether one mechanistic model fits better than another.
- The framework addresses nonlinear systems for which formal inference is difficult and qualitative simulation comparisons are otherwise common.
- Implicit models require only a procedure that generates sample paths, allowing plug-and-play inference without closed-form transition probabilities.
- The paper introduces implicitly specified stochastic dynamic models and demonstrates them through measles and cholera analyses.
2. Compartment models with stochastic rates.
The paper constructs implicitly specified stochastic compartment models as continuous-time Markov chains with random transition rates. The framework uses flow-based population representations and Euler simulation, while extending Poisson systems to accommodate overdispersion.
- A compartment model represents counts across categories, with flows between compartments satisfying a conservation-of-mass identity.
- Each flow has a state- and time-dependent rate, interpreted as the rate at which individuals move between compartments.
- The model introduces integrated noise processes so transition rates acquire multiplicative white-noise variation.
- The continuous-time model is constructed as the limit of coupled discrete-time multinomial processes with random rates and can be simulated using an Euler scheme.
- With zero noise, the model reduces to the Poisson system; nonzero rate noise permits infinitesimal variance to exceed the mean and supports overdispersed dynamics.
- Adding white noise to rates yields a novel Markov-process construction motivated by the need for flexible models compatible with plug-and-play inference.
3. Plug-and-play inference methodology.
The paper applies iterated filtering to likelihood-based inference for partially observed nonlinear Markov models using only simulation and measurement procedures. The method introduces parameter perturbations during filtering, combines local estimates, and supports likelihood-based uncertainty and hypothesis testing.
- Inference uses parameterized transition and measurement rules with observations collected at discrete times from a partially observed stochastic process.
- Iterated filtering maximizes likelihood through sequential Monte Carlo while requiring only the ability to simulate sample paths.
- The methodology provides maximum-likelihood estimates, profile-likelihood, bootstrap or Fisher-information confidence intervals, and likelihood-ratio tests.
- Parameter perturbations create time-varying estimates, whose combination through the updating rule is the procedure's main innovative component.
- The implementation includes a sequential Monte Carlo filter and requires choices of algorithmic parameters such as J, L, M, a, and b.
- Parameter estimation remains insufficiently automated because selecting algorithmic parameters requires trial and error.
4. Time series analysis for biological systems.
The paper applies stochastic mechanistic modeling and plug-and-play inference to biological time series, using measles and competing-strain cholera as examples. The analyses show how environmental stochasticity and alternative epidemiological assumptions affect statistical conclusions.
- 4. Time series analysis for biological systems.: Biological population models must capture both qualitative dynamics and the quantitative statistical behavior of available observations.Environmental stochasticity is important for representing unpredictability relevant to forecasting, intervention effects, and unobserved system components.
- 4. Time series analysis for biological systems.: The framework is demonstrated through measles transmission and a more complex model of competing cholera strains.The cholera analysis concerns interactions between Inaba and Ogawa serotypes using 30 years of biweekly incidence records.
- 4.1. Environmental stochasticity in measles epidemics.: Environmental stochasticity in the fitted measles model reproduces irregular epidemic cycles that demographic stochasticity alone could not explain.The historical deviations included one-, two-, or three-year cycles depending on population size and birth rate.
- 4.2. A mechanistic model for competing strains of cholera.: For cholera, regime B was better supported than regime A, with p < 10−6, while cross-immunity was lower in regime A and poorly identified in regime B.Regime B estimated complete cross-immunity, γ = 1, but its standard error was large because the higher reporting rate implied fewer individuals exposed to both serotypes.
- 4.2. A mechanistic model for competing strains of cholera.: The cholera regimes expose a limitation of basic compartment models: continuous disease severity and infectiousness are represented as discrete or binary categories.The authors suggest that modeling differing severity levels could combine regime B’s data matching with regime A’s scientific interpretation, while noting omitted epidemiological factors may affect conclusions.
5. Discussion.
The discussion situates stochastic-rate compartment models within a broader mechanistic framework, emphasizing overdispersion, discrete population dynamics, and plug-and-play likelihood inference. It also identifies data availability as a boundary on model complexity and parameter estimation.
- Discussion: Compartment models extend mechanistic time-series analysis beyond ODEs and SDEs to discrete stochastic population processes.Their relevance also extends to chemodynamic systems involving molecular counts and chemical transformations.
- Discussion: Discrete Markov-chain models can be preferable to SDEs for populations because exact simulation, fade-outs, and nonnegativity are more naturally handled.These advantages matter especially when populations become small or disease transmission temporarily disappears.
- Discussion: Overdispersed Markov models allow simultaneous transitions of multiple individuals, representing events such as contaminated meals or shared exposures.The Poisson system permits only single-individual transitions and is therefore equidispersed.
- Discussion: White noise in transition rates offers parsimonious parameterization, preserves the Markov property, and can represent high-frequency infection-rate variability.For the measles and cholera examples, the discussion reports evidence supporting environmental stochasticity or high-frequency rate variability.
- Discussion: The paper uses likelihood-based, non-Bayesian inference and presents iterated filtering as routine for some situations challenging for Bayesian methodology.Bayesian and non-Bayesian analyses are described as complementary approaches.
- Discussion: Model extensions should be guided by available data, which limit estimable parameters and the questions that observations can reasonably answer.The discussion highlights assessing whether additional components, such as age structure, improve the statistical description.
APPENDIX A: THEOREMS CONCERNING COMPARTMENT MODELS WITH STOCHASTIC RATES
The appendix constructs compartment-model Markov chains with stochastic transition rates and states theorems characterizing their transition probabilities. Individual transition clocks are coupled through the evolving population state and shared rate noise.
- Theorems concerning compartment models with stochastic rates: Theorem A.1 constructs the process by labeling individuals and generating exponential transition clocks for possible compartment changes.The construction recursively defines event times, compartments, and clock variables from the initial population state.
- Theorems concerning compartment models with stochastic rates: At each individual’s earliest scheduled event, the construction updates its compartment and regenerates independent exponential clocks for the remaining destinations.The event time is the minimum of destination-specific event times.
- Theorems concerning compartment models with stochastic rates: The resulting state process is a Markov chain whose infinitesimal transition probabilities satisfy the model specification in equation (2).The state counts are updated through incoming and outgoing transition processes.
- Theorems concerning compartment models with stochastic rates: Transition clocks are equivalent to restarting clocks after every transition because exponential random variables are memoryless.The appendix calls these variables transition clocks and explains their interpretation through integrated transition rates.
- Theorems concerning compartment models with stochastic rates: Individual trajectories are coupled because transition rates depend on the shared state and noise processes can be shared across individuals or compartments.Evaluating the integrated rates therefore requires tracking all individuals simultaneously.
- Theorems concerning compartment models with stochastic rates: Theorem A.2 gives limiting transition probabilities under its stated assumptions, while broader dependence among rate processes can remove the simple simultaneous-transition structure.The full-independence assumption permits multiple individuals to move between one compartment pair but excludes simultaneous transitions between different pairs.
- Theorems concerning compartment models with stochastic rates: The construction and subsequent analysis use assumptions that simplify theorems, while the model formulation is intended to support processes beyond currently available mathematical analysis.One such simplification concerns resolving ties between simultaneous event times.
APPENDIX B: EQUIDISPERSION OF POISSON SYSTEMS
The appendix analyzes infinitesimal count increments in the Poisson system and extends the calculation to stochastic transition rates under a uniform rate bound. It also identifies that bound as necessary for consistency with the stated results.
- Equidispersion of Poisson systems: For a finite-state Poisson system, a uniform bound on µij(t,x)xi ensures uniform control of the small-time increment probabilities.The bound yields a tail-probability comparison involving a Poisson distribution.
- Equidispersion of Poisson systems: The conditional variance of an increment satisfies Var(∆Nij|X(t)=x) = µijxiδ + o(δ).The same infinitesimal rate determines the leading-order mean and variance.
- Equidispersion of Poisson systems: The variance calculation extends to stochastic rate functions when X(t) is conditionally Markov given the rates and the same uniform bound ν exists.The appendix states that a similar calculation applies under this condition.
- Equidispersion of Poisson systems: The necessity of ν follows from an inconsistency between the appendix equations and Theorem A.2 when white noise is added to the rates.The limitation concerns the validity of the stated infinitesimal results without the uniform bound.
Theorems concerning compartment models with stochastic rates
The supplementary material contains proofs of Theorems A.1 and A.2, which were stated in Appendix A.
- Theorems concerning compartment models with stochastic rates: The supplementary PDF provides proofs for Theorems A.1 and A.2 stated in Appendix A.The passage identifies the supplement by DOI 10.1214/08-AOAS201SUPP.