# Inference for nonlinear dynamical systems

@article{Ionides2006InferenceFN, title={Inference for nonlinear dynamical systems}, author={Edward L. Ionides and Carles Bret{\'o} and Aaron A. King}, journal={Proceedings of the National Academy of Sciences}, year={2006}, volume={103}, pages={18438 - 18443} }

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…

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## References

SHOWING 1-10 OF 63 REFERENCES

### Theory and Methods

- Engineering
- 1998

Abstract A self-organizing filter and smoother for the general nonlinear non-Gaussian state-space model is proposed. An expanded state-space model is defined by augmenting the state vector with the…

### Fitting population dynamic models to time-series data by gradient matching

- Mathematics
- 2002

We describe and test a method for fitting noisy differential equation models to a time series of population counts, motivated by stage-structured models of insect and zooplankton populations. We…

### A self-organizing state-space model

- Engineering
- 1998

A self-organizing filter and smoother for the general nonlinear non-Gaussian state-space model is proposed. An expanded state-space model is defined by augmenting the state vector with the unknown…

### Inference in hidden Markov models

- Computer Science, MathematicsSpringer series in statistics
- 2005

This book is a comprehensive treatment of inference for hidden Markov models, including both algorithms and statistical theory, and builds on recent developments to present a self-contained view.

### POPULATION TIME SERIES: PROCESS VARIABILITY, OBSERVATION ERRORS, MISSING VALUES, LAGS, AND HIDDEN STATES

- Environmental Science
- 2004

Population sample data are complex; inference and prediction require proper accommodation of not only the nonlinear interactions that determine the expected future abundance, but also the…

### On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm

- Mathematics, Computer Science
- 2001

A new Markov chain Monte Carlo approach to Bayesian analysis of discretely observed diffusion processes and shows that, because of full dependence between the missing paths and the volatility of the diffusion, the rate of convergence of basic algorithms can be arbitrarily slow if the amount of the augmentation is large.

### Disentangling Extrinsic from Intrinsic Factors in Disease Dynamics: A Nonlinear Time Series Approach with an Application to Cholera

- Environmental ScienceThe American Naturalist
- 2004

A nonlinear time series model with two related objectives: the reconstruction of immunity patterns from data on cases and population sizes and the identification of the respective roles of extrinsic and intrinsic factors in the dynamics.

### Stochastic models for cell motion and taxis

- BiologyJournal of mathematical biology
- 2004

An Ornstein- Uhlenbeck model for cell velocity is found to compare favorably with a nonlinear diffusion model, and this situation, involving the effect of an electric field on cell behavior, is considered in detail.

### Asymptotic normality of the maximum likelihood estimator in state space models

- Mathematics, Computer Science
- 1999

This paper generalizes the results of Bickel, Ritov and Ryden to state space models, where the latent process is a continuous state Markov chain satisfying regularity conditions, which are fulfilled if the latentprocess takes values in a compact space.