Statistical inference for noisy nonlinear ecological dynamic systems

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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 trajectory1. 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 methods2, 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 experiments3, 4, 5.

At a glance


  1. Measuring fit of the Ricker model.
    Figure 1: Measuring fit of the Ricker model.

    a, Population data simulated from the Ricker model in the text, observed under Poisson sampling (log(r) = 3.8, σ = 0.3, ϕ = 10). b, The log joint probability density, log(fθ(y, e)), of data, y, and random process noise terms, e, plotted against the value of the first process noise deviate, e1, with the rest of e and y held fixed. c, Log(fθ(y, e)) plotted against model parameter r, again with e and y held fixed. d, The log synthetic likelihood, ls, plotted against log(r) for the Ricker model and the data given in a (Nr = 500).

  2. Synthetic likelihood evaluation.
    Figure 2: Synthetic likelihood evaluation.

    Starting at the top, we wish to evaluate the fit of the model with parameter vector θ to the raw data vector y. Replicate data vectors are simulated from the model, given the value of θ. Each replicate, and the raw data, is converted into a vector of statistics, or s, in the same way. The are used to estimate the mean vector, , and covariance matrix, , of s, according to the model with parameters θ. We use , and s respectively as the mean vector, the covariance matrix and the argument of the log multivariate normal (MVN) probability density function, to evaluate the log synthetic likelihood, ls.

  3. Blowfly data and model runs.
    Figure 3: Blowfly data and model runs.

    a, b, Two laboratory adult populations of sheep blowfly maintained under adult food limitation4, 5. c, d, As in a and b but maintained under moderate and more severe juvenile food limitation4. eh, Two replicates (one solid, one dashed) from the full model (equation (4)) fitted separately to the data shown in each of panels ad, immediately above. il, As in eh for the model with demographic stochasticity only. The observations are made every second day. The simulation phase is arbitrary. Notice the qualitatively good match of the dynamics (eh) of the full model (equation (4)) to the data, relative to the insufficiently variable dynamics of the model with demographic stochasticity only (il).

  4. Blowfly model stability diagram.
    Figure 4: Blowfly model stability diagram3, 14.

    The coloured points are samples from the stability-controlling parameter combinations δτ and , plotted (with matching colour coding) for each experimental run shown in Fig. 3. The open and filled circles show stability properties for alternative chain starting conditions: they give indistinguishable results, although the conditions marked by the filled circle lie in the plausible range for external noise-driven dynamics14. The dynamics comprise limit cycles perturbed by noise but not driven by noise. The fluctuations are driven by the intrinsic population-dynamic processes, not by random variation exciting a resonance in otherwise stable dynamics.


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  1. Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

    • Simon N. Wood

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Supplementary information

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  1. Supplementary Information (2.7M)

    This file contains Supplementary Information comprising: 1 Method Implementation and MCMC output; 2 Further examples; 3 Software: the sl package for R and References.

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  1. Supplementary Data 1 (42K)

    This file is an R source package (suitable for use with R on unix like operating systems), implementing the examples in the paper and supplementary material, as well as the providing some routines for rapid computation of summary statistics, and robust evaluation. R is a free statistical language and environment available from

  2. Supplementary Data 2 (105K)

    This file contains the same R package as in the Supplementary Data 1 file, but for the Windows version of R.

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