Adaptive Mixture Modelling Metropolis Methods for Bayesian Analysis of Non-linear State-Space Models.

We describe a strategy for Markov chain Monte Carlo analysis of non-linear, non-Gaussian state-space models involving batch analysis for inference on dynamic, latent state variables and fixed model parameters. The key innovation is a Metropolis-Hastings method for the time series of state variables based on sequential approximation of filtering and smoothing densities using normal mixtures. These mixtures are propagated through the non-linearities using an accurate, local mixture approximation method, and we use a regenerating procedure to deal with potential degeneracy of mixture components. This provides accurate, direct approximations to sequential filtering and retrospective smoothing distributions, and hence a useful construction of global Metropolis proposal distributions for simulation of posteriors for the set of states. This analysis is embedded within a Gibbs sampler to include uncertain fixed parameters. We give an example motivated by an application in systems biology. Supplemental materials provide an example based on a stochastic volatility model as well as MATLAB code.

Discharge competence and pattern formation in peatlands: a meta-ecosystem model of the Everglades ridge-slough landscape.

Regular landscape patterning arises from spatially-dependent feedbacks, and can undergo catastrophic loss in response to changing landscape drivers. The central Everglades (Florida, USA) historically exhibited regular, linear, flow-parallel orientation of high-elevation sawgrass ridges and low-elevation sloughs that has degraded due to hydrologic modification. In this study, we use a meta-ecosystem approach to model a mechanism for the establishment, persistence, and loss of this landscape. The discharge competence (or self-organizing canal) hypothesis assumes non-linear relationships between peat accretion and water depth, and describes flow-dependent feedbacks of microtopography on water depth. Closed-form model solutions demonstrate that 1) this mechanism can produce spontaneous divergence of local elevation; 2) divergent and homogenous states can exhibit global bi-stability; and 3) feedbacks that produce divergence act anisotropically. Thus, discharge competence and non-linear peat accretion dynamics may explain the establishment, persistence, and loss of landscape pattern, even in the absence of other spatial feedbacks. Our model provides specific, testable predictions that may allow discrimination between the self-organizing canal hypotheses and competing explanations. The potential for global bi-stability suggested by our model suggests that hydrologic restoration may not re-initiate spontaneous pattern establishment, particularly where distinct soil elevation modes have been lost. As a result, we recommend that management efforts should prioritize maintenance of historic hydroperiods in areas of conserved pattern over restoration of hydrologic regimes in degraded regions. This study illustrates the value of simple meta-ecosystem models for investigation of spatial processes.

Nonholonomically Constrained Dynamics and Optimization of Rolling Isolation Systems

Rolling Isolation Systems provide a simple and effective means for protecting components from horizontal floor vibrations. In these systems a platform rolls on four steel balls which, in turn, rest within shallow bowls. The trajectories of the balls is uniquely determined by the horizontal and rotational velocity components of the rolling platform, and thus provides nonholonomic constraints. In general, the bowls are not parabolic, so the potential energy function of this system is not quadratic. This thesis presents the application of Gauss's Principle of Least Constraint to the modeling of rolling isolation platforms. The equations of motion are described in terms of a redundant set of constrained coordinates. Coordinate accelerations are uniquely determined at any point in time via Gauss's Principle by solving a linearly constrained quadratic minimization. In the absence of any modeled damping, the equations of motion conserve energy. This mathematical model is then used to find the bowl profile that minimizes response acceleration subject to displacement constraint.