SMCTC : sequential Monte Carlo in C++

Sequential Monte Carlo methods are a very general class of Monte Carlo methods for sampling from sequences of distributions. Simple examples of these algorithms are used very widely in the tracking and signal processing literature. Recent developments illustrate that these techniques have much more general applicability, and can be applied very effectively to statistical inference problems. Unfortunately, these methods are often perceived as being computationally expensive and difficult to implement. This article seeks to address both of these problems. A C++ template class library for the efficient and convenient implementation of very general Sequential Monte Carlo algorithms is presented. Two example applications are provided: a simple particle filter for illustrative purposes and a state-of-the-art algorithm for rare event estimation

SMCTC: Sequential Monte Carlo in C++

Sequential Monte Carlo methods are a very general class of Monte Carlo methods for sampling from sequences of distributions. Simple examples of these algorithms are used very widely in the tracking and signal processing literature. Recent developments illustrate that these techniques have much more general applicability, and can be applied very effectively to statistical inference problems. Unfortunately, these methods are often perceived as being computationally expensive and difficult to implement. This article seeks to address both of these problems. A C++ template class library for the efficient and convenient implementation of very general Sequential Monte Carlo algorithms is presented. Two example applications are provided: a simple particle filter for illustrative purposes and a state-of-the-art algorithm for rare event estimation.

Pointwise Convergence in Probability of General Smoothing Splines

Establishing the convergence of splines can be cast as a variational problem which is amenable to a $\Gamma$-convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, $n$, as $\lambda_n=n^{-p}$. Using standard theorems from the $\Gamma$-convergence literature, we prove that the general spline model is consistent in that estimators converge in a sense slightly weaker than weak convergence in probability for $p\leq \frac{1}{2}$. Without further assumptions we show this rate is sharp. This differs from rates for strong convergence using Hilbert scales where one can often choose $p>\frac{1}{2}$

Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using kk-Means Type Optimization

We address the task of estimating multiple trajectories from unlabeled data. This problem arises in many settings, one could think of the construction of maps of transport networks from passive observation of travellers, or the reconstruction of the behaviour of uncooperative vehicles from external observations, for example. There are two coupled problems. The first is a data association problem: how to map data points onto individual trajectories. The second is, given a solution to the data association problem, to estimate those trajectories. We construct estimators as a solution to a regularized variational problem (to which approximate solutions can be obtained via the simple, efficient and widespread $k$-means method) and show that, as the number of data points, $n$, increases, these estimators exhibit stable behaviour. More precisely, we show that they converge in an appropriate Sobolev space in probability and with rate $n^{-1/2}$

A Simple Approach to Maximum Intractable Likelihood Estimation

Approximate Bayesian Computation (ABC) can be viewed as an analytic approximation of an intractable likelihood coupled with an elementary simulation step. Such a view, combined with a suitable instrumental prior distribution permits maximum-likelihood (or maximum-a-posteriori) inference to be conducted, approximately, using essentially the same techniques. An elementary approach to this problem which simply obtains a nonparametric approximation of the likelihood surface which is then used as a smooth proxy for the likelihood in a subsequent maximisation step is developed here and the convergence of this class of algorithms is characterised theoretically. The use of non-sufficient summary statistics in this context is considered. Applying the proposed method to four problems demonstrates good performance. The proposed approach provides an alternative for approximating the maximum likelihood estimator (MLE) in complex scenarios

The Cuntz splice does not preserve ∗*-isomorphism of Leavitt path algebras over Z\mathbb{Z}

We show that the Leavitt path algebras $L_{2,\mathbb{Z}}$ and $L_{2-,\mathbb{Z}}$ are not isomorphic as $*$-algebras. There are two key ingredients in the proof. One is a partial algebraic translation of Matsumoto and Matui's result on diagonal preserving isomorphisms of Cuntz--Krieger algebras. The other is a complete description of the projections in $L_{\mathbb{Z}}(E)$ for $E$ a finite graph. This description is based on a generalization, due to Chris Smith, of the description of the unitaries in $L_{2,\mathbb{Z}}$ given by Brownlowe and the second named author. The techniques generalize to a slightly larger class of rings than just $\mathbb{Z}$.Comment: 17 pages. Since version 2 we extended the arguments from Z to more general ring

The iterated auxiliary particle filter

We present an offline, iterated particle filter to facilitate statistical inference in general state space hidden Markov models. Given a model and a sequence of observations, the associated marginal likelihood L is central to likelihood-based inference for unknown statistical parameters. We define a class of "twisted" models: each member is specified by a sequence of positive functions psi and has an associated psi-auxiliary particle filter that provides unbiased estimates of L. We identify a sequence psi* that is optimal in the sense that the psi*-auxiliary particle filter's estimate of L has zero variance. In practical applications, psi* is unknown so the psi*-auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate psi*, and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the bootstrap particle filter in challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm

Maximum likelihood parameter estimation for latent variable models using sequential Monte Carlo

We present a sequential Monte Carlo (SMC) method for maximum likelihood (ML) parameter estimation in latent variable models. Standard methods rely on gradient algorithms such as the Expectation- Maximization (EM) algorithm and its Monte Carlo variants. Our approach is different and motivated by similar considerations to simulated annealing (SA); that is we propose to sample from a sequence of artificial distributions whose support concentrates itself on the set of ML estimates. To achieve this we use SMC methods. We conclude by presenting simulation results on a toy problem and a nonlinear non-Gaussian time series model

On blocks, tempering and particle MCMC for systems identification

The widespread use of particle methods for addressing the filtering and smoothing problems in state-space models has, in recent years, been complemented by the development of particle Markov Chain Monte Carlo (PMCMC) methods. PMCMC uses particle filters within offline systems-identification settings. We develop a modified particle filter, based around block sampling and tempering, intended to improve their exploration of the state space and the associated estimation of the marginal likelihood. The aim is to develop particle methods with improved robustness properties, particularly for parameter values which are not able to explain observed data well, for use within PMCMC algorithms. The proposed strategies do not require a substantial analytic understanding of the model structure, unlike most techniques for improving particle-filter performance