Inference for nonlinear dynamical systems - PubMed (original) (raw)

Inference for nonlinear dynamical systems

E L Ionides et al. Proc Natl Acad Sci U S A. 2006.

Abstract

Nonlinear stochastic dynamical systems are widely used to model systems across the sciences and engineering. Such models are natural to formulate and can be analyzed mathematically and numerically. However, difficulties associated with inference from time-series data about unknown parameters in these models have been a constraint on their application. We present a new method that makes maximum likelihood estimation feasible for partially-observed nonlinear stochastic dynamical systems (also known as state-space models) where this was not previously the case. The method is based on a sequence of filtering operations which are shown to converge to a maximum likelihood parameter estimate. We make use of recent advances in nonlinear filtering in the implementation of the algorithm. We apply the method to the study of cholera in Bangladesh. We construct confidence intervals, perform residual analysis, and apply other diagnostics. Our analysis, based upon a model capturing the intrinsic nonlinear dynamics of the system, reveals some effects overlooked by previous studies.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.

Fig. 1.

Diagrammatic representation of a model for cholera population dynamics. Each individual is in S (susceptible), I (infected), or one of the classes Rj (recovered). Compartments B, C, and D allow for birth, cholera mortality, and death from other causes, respectively. The arrows show rates, interpreted as described in the text.

Fig. 2.

Fig. 2.

Data and simulation. (A) One realization of the model using the parameter values in Table 1. (B) Historic monthly cholera mortality data for Dhaka, Bangladesh. (C) Southern oscillation index (SOI), smoothed with local quadratic regression (33) using a bandwidth parameter (span) of 0.12.

Fig. 3.

Fig. 3.

Diagnostic plots. (A–C) Convergence plots for four MIFs, shown for three parameters. The dotted line shows θ*. The parabolic lines give the sliced likelihood through θ̂, with the axis scale at the top right. (D–F) Corresponding close-ups of the sliced likelihood. The dashed vertical line is at θ̂.

Fig. 4.

Fig. 4.

Profile likelihood for the environmental reservoir parameter. The larger of two MIF replications was plotted at each value of ω (circles), maximizing over the other parameters. Local quadratic regression (33, 35) with a bandwidth parameter (span) of 0.5 was used to estimate the profile likelihood (solid line). The dotted lines construct an approximate 99% confidence interval (ref. and Supporting Text) of (75 × 10−6, 210 × 10−6).

Fig. 5.

Fig. 5.

Superimposed annual cycles of cholera mortality in Dhaka, 1891–1940.

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