Ecological non-linear state space model selection via adaptive particle Markov chain Monte Carlo (AdPMCMC)

We develop a novel advanced Particle Markov chain Monte Carlo algorithm that is capable of sampling from the posterior distribution of non-linear state space models for both the unobserved latent states and the unknown model parameters. We apply this novel methodology to five population growth models, including models with strong and weak Allee effects, and test if it can efficiently sample from the complex likelihood surface that is often associated with these models. Utilising real and also synthetically generated data sets we examine the extent to which observation noise and process error may frustrate efforts to choose between these models. Our novel algorithm involves an Adaptive Metropolis proposal combined with an SIR Particle MCMC algorithm (AdPMCMC). We show that the AdPMCMC algorithm samples complex, high-dimensional spaces efficiently, and is therefore superior to standard Gibbs or Metropolis Hastings algorithms that are known to converge very slowly when applied to the non-linear state space ecological models considered in this paper. Additionally, we show how the AdPMCMC algorithm can be used to recursively estimate the Bayesian Cram\'er-Rao Lower Bound of Tichavsk\'y (1998). We derive expressions for these Cram\'er-Rao Bounds and estimate them for the models considered. Our results demonstrate a number of important features of common population growth models, most notably their multi-modal posterior surfaces and dependence between the static and dynamic parameters. We conclude by sampling from the posterior distribution of each of the models, and use Bayes factors to highlight how observation noise significantly diminishes our ability to select among some of the models, particularly those that are designed to reproduce an Allee effect

Metropolized Randomized Maximum Likelihood for sampling from multimodal distributions

This article describes a method for using optimization to derive efficient independent transition functions for Markov chain Monte Carlo simulations. Our interest is in sampling from a posterior density $\pi(x)$ for problems in which the dimension of the model space is large, $\pi(x)$ is multimodal with regions of low probability separating the modes, and evaluation of the likelihood is expensive. We restrict our attention to the special case for which the target density is the product of a multivariate Gaussian prior and a likelihood function for which the errors in observations are additive and Gaussian

Stochastic Weighted Graphs: Flexible Model Specification and Simulation

In most domains of network analysis researchers consider networks that arise in nature with weighted edges. Such networks are routinely dichotomized in the interest of using available methods for statistical inference with networks. The generalized exponential random graph model (GERGM) is a recently proposed method used to simulate and model the edges of a weighted graph. The GERGM specifies a joint distribution for an exponential family of graphs with continuous-valued edge weights. However, current estimation algorithms for the GERGM only allow inference on a restricted family of model specifications. To address this issue, we develop a Metropolis--Hastings method that can be used to estimate any GERGM specification, thereby significantly extending the family of weighted graphs that can be modeled with the GERGM. We show that new flexible model specifications are capable of avoiding likelihood degeneracy and efficiently capturing network structure in applications where such models were not previously available. We demonstrate the utility of this new class of GERGMs through application to two real network data sets, and we further assess the effectiveness of our proposed methodology by simulating non-degenerate model specifications from the well-studied two-stars model. A working R version of the GERGM code is available in the supplement and will be incorporated in the gergm CRAN package.Comment: 33 pages, 6 figures. To appear in Social Network

Monte Carlo techniques for real-time quantum dynamics

The stochastic-gauge representation is a method of mapping the equation of motion for the quantum mechanical density operator onto a set of equivalent stochastic differential equations. One of the stochastic variables is termed the "weight", and its magnitude is related to the importance of the stochastic trajectory. We investigate the use of Monte Carlo algorithms to improve the sampling of the weighted trajectories and thus reduce sampling error in a simulation of quantum dynamics. The method can be applied to calculations in real time, as well as imaginary time for which Monte Carlo algorithms are more-commonly used. The method is applicable when the weight is guaranteed to be real, and we demonstrate how to ensure this is the case. Examples are given for the anharmonic oscillator, where large improvements over stochastic sampling are observed.Comment: 28 pages, submitted to J. Comp. Phy

Phase transition between synchronous and asynchronous updating algorithms

We update a one-dimensional chain of Ising spins of length $L$ with algorithms which are parameterized by the probability $p$ for a certain site to get updated in one time step. The result of the update event itself is determined by the energy change due to the local change in the configuration. In this way we interpolate between the Metropolis algorithm at zero temperature for $p$ of the order of 1/L and for large $L$, and a synchronous deterministic updating procedure for $p=1$. As function of $p$ we observe a phase transition between the stationary states to which the algorithm drives the system. These are non-absorbing stationary states with antiferromagnetic domains for $p>p_c$, and absorbing states with ferromagnetic domains for $p\leq p_c$. This means that above this transition the stationary states have lost any remnants to the ferromagnetic Ising interaction. A measurement of the critical exponents shows that this transition belongs to the universality class of parity conservation.Comment: 5 pages, 3 figure

Wang-Landau sampling in three-dimensional polymers

Monte Carlo simulations using Wang-Landau sampling are performed to study three-dimensional chains of homopolymers on a lattice. We confirm the accuracy of the method by calculating the thermodynamic properties of this system. Our results are in good agreement with those obtained using Metropolis importance sampling. This algorithm enables one to accurately simulate the usually hardly accessible low-temperature regions since it determines the density of states in a single simulation.Comment: 5 pages, 9 figures arch-ive/Brazilian Journal of Physic