Mixture Kalman filters

In treating dynamic systems, sequential Monte Carlo methods use discrete samples to represent a complicated probability distribution and use rejection sampling, importance sampling and weighted resampling to complete the on-line 'filtering' task. We propose a special sequential Monte Carlo method, the mixture Kalman filter, which uses a random mixture of the Gaussian distributions to approximate a target distribution. It is designed for on-line estimation and prediction of conditional and partial conditional dynamic linear models, which are themselves a class of widely used non-linear systems and also serve to approximate many others. Compared with a few available filtering methods including Monte Carlo methods, the gain in efficiency that is provided by the mixture Kalman filter can be very substantial. Another contribution of the paper is the formulation of many non-linear systems into conditional or partial conditional linear form, to which the mixture Kalman filter can be applied. Examples in target tracking and digital communications are given to demonstrate the procedures proposed.     

New sequential Monte Carlo methods for nonlinear dynamic systems

In this paper we present several new sequential Monte Carlo (SMC) algorithms for online estimation (filtering) of nonlinear dynamic systems. SMC has been shown to be a powerful tool for dealing with complex dynamic systems. It sequentially generates Monte Carlo samples from a proposal distribution, adjusted by a set of importance weight with respect to a target distribution, to facilitate statistical inferences on the characteristic (state) of the system. The key to a successful implementation of SMC in complex problems is the design of an efficient proposal distribution from which the Monte Carlo samples are generated. We propose several such proposal distributions that are efficient yet easy to generate samples from. They are efficient because they tend to utilize both the information in the state process and the observations. They are all Gaussian distributions hence are easy to sample from. The central ideas of the conventional nonlinear filters, such as extended Kalman filter, unscented Kalman filter and the Gaussian quadrature filter, are used to construct these proposal distributions. The effectiveness of the proposed algorithms are demonstrated through two applications—real time target tracking and the multiuser parameter tracking in CDMA communication systems.This work was supported in part by the U.S. National Science Foundation (NSF) under grants CCR-9875314, CCR-9980599, DMS-9982846, DMS-0073651 and DMS-0073601.     

Density-Tempered Marginalized Sequential Monte Carlo Samplers

We propose a density-tempered marginalized sequential Monte Carlo (SMC) sampler, a new class of samplers for full Bayesian inference of general state-space models. The dynamic states are approximately marginalized out using a particle filter, and the parameters are sampled via a sequential Monte Carlo sampler over a density-tempered bridge between the prior and the posterior. Our approach delivers exact draws from the joint posterior of the parameters and the latent states for any given number of state particles and is thus easily parallelizable in implementation. We also build into the proposed method a device that can automatically select a suitable number of state particles. Since the method incorporates sample information in a smooth fashion, it delivers good performance in the presence of outliers. We check the performance of the density-tempered SMC algorithm using simulated data based on a linear Gaussian state-space model with and without misspecification. We also apply it on real stock prices using a GARCH-type model with microstructure noise.     

Fitting finite mixture models using iterative Monte Carlo classification

Parameters of a finite mixture model are often estimated by the expectation–maximization (EM) algorithm where the observed data log-likelihood function is maximized. This paper proposes an alternative approach for fitting finite mixture models. Our method, called the iterative Monte Carlo classification (IMCC), is also an iterative fitting procedure. Within each iteration, it first estimates the membership probabilities for each data point, namely the conditional probability of a data point belonging to a particular mixing component given that the data point value is obtained, it then classifies each data point into a component distribution using the estimated conditional probabilities and the Monte Carlo method. It finally updates the parameters of each component distribution based on the classified data. Simulation studies were conducted to compare IMCC with some other algorithms for fitting mixture normal, and mixture t, densities.     

The Monte Carlo EM method for estimating multivariate tobit latent variable models

We propose a multivariate tobit (MT) latent variable model that is defined by a confirmatory factor analysis with covariates for analysing the mixed type data, which is inherently non-negative and sometimes has a large proportion of zeros. Some useful MT models are special cases of our proposed model. To obtain maximum likelihood estimates, we use the expectation maximum algorithm with its E-step via the Gibbs sampler made feasible by Monte Carlo simulation and its M-step greatly simplified by a sequence of conditional maximization. Standard errors are evaluated by inverting a Monte Carlo approximation of the information matrix using Louis's method. The methodology is illustrated with a simulation study and a real example.     

Some contributions to sequential Monte Carlo methods for option pricing

Pricing options is an important problem in financial engineering. In many scenarios of practical interest, financial option prices associated with an underlying asset reduces to computing an expectation w.r.t. a diffusion process. In general, these expectations cannot be calculated analytically, and one way to approximate these quantities is via the Monte Carlo (MC) method; MC methods have been used to price options since at least the 1970s. It has been seen in Del Moral P, Shevchenko PV. [Valuation of barrier options using sequential Monte Carlo. 2014. arXiv preprint] and Jasra A, Del Moral P. [Sequential Monte Carlo methods for option pricing. Stoch Anal Appl. 2011;29:292–316] that Sequential Monte Carlo (SMC) methods are a natural tool to apply in this context and can vastly improve over standard MC. In this article, in a similar spirit to Del Moral and Shevchenko (2014) and Jasra and Del Moral (2011), we show that one can achieve significant gains by using SMC methods by constructing a sequence of artificial target densities over time. In particular, we approximate the optimal importance sampling distribution in the SMC algorithm by using a sequence of weighting functions. This is demonstrated on two examples, barrier options and target accrual redemption notes (TARNs). We also provide a proof of unbiasedness of our SMC estimate.     

An efficient Monte Carlo EM algorithm for Bayesian lasso

The lasso is a popular technique of simultaneous estimation and variable selection in many research areas. The marginal posterior mode of the regression coefficients is equivalent to estimates given by the non-Bayesian lasso when the regression coefficients have independent Laplace priors. Because of its flexibility of statistical inferences, the Bayesian approach is attracting a growing body of research in recent years. Current approaches are primarily to either do a fully Bayesian analysis using Markov chain Monte Carlo (MCMC) algorithm or use Monte Carlo expectation maximization (MCEM) methods with an MCMC algorithm in each E-step. However, MCMC-based Bayesian method has much computational burden and slow convergence. Tan et al. [An efficient MCEM algorithm for fitting generalized linear mixed models for correlated binary data. J Stat Comput Simul. 2007;77:929–943] proposed a non-iterative sampling approach, the inverse Bayes formula (IBF) sampler, for computing posteriors of a hierarchical model in the structure of MCEM. Motivated by their paper, we develop this IBF sampler in the structure of MCEM to give the marginal posterior mode of the regression coefficients for the Bayesian lasso, by adjusting the weights of importance sampling, when the full conditional distribution is not explicit. Simulation experiments show that the computational time is much reduced with our method based on the expectation maximization algorithm and our algorithms and our methods behave comparably with other Bayesian lasso methods not only in prediction accuracy but also in variable selection accuracy and even better especially when the sample size is relatively large.     

Optimal SIR algorithm vs. fully adapted auxiliary particle filter: a non asymptotic analysis

Particle filters (PF) and auxiliary particle filters (APF) are widely used sequential Monte Carlo (SMC) techniques. In this paper we comparatively analyse, from a non asymptotic point of view, the Sampling Importance Resampling (SIR) PF with optimal conditional importance distribution (CID) and the fully adapted APF (FA). We compute the (finite samples) conditional second order moments of Monte Carlo (MC) estimators of a moment of interest of the filtering pdf, and analyse under which circumstances the FA-based estimator outperforms (or not) the optimal Sequential Importance Sampling (SIS)-based one. Our analysis is local, in the sense that we compare the estimators produced by one time step of the different SMC algorithms, starting from a common set of weighted points. This analysis enables us to propose a hybrid SIS/FA algorithm which automatically switches at each time step from one loop to the other. We finally validate our results via computer simulations.     

Computational Aspects Related to Martingale Estimating Functions for a Discretely Observed Diffusion

Martingale estimating functions for a discretely observed diffusion have turned out to provide estimators with nice asymptotic properties. However, their expression usually involves some conditional expectation that has to be evaluated through Monte Carlo simulations giving rise to an approximated estimator. In this work we study, for ergodic models, the asymptotic properties of the approximated estimator and describe the influence of the number of independent simulated trajectories involved in the Monte Carlo method as well as of the approximation scheme used. Our results are of practical relevance to assess the implementation of martingale estimating functions for discretely observed diffusions.     

On Mixture Double Autoregressive Time Series Models

This article proposes a mixture double autoregressive model by introducing the flexibility of mixture models to the double autoregressive model, a novel conditional heteroscedastic model recently proposed in the literature. To make it more flexible, the mixing proportions are further assumed to be time varying, and probabilistic properties including strict stationarity and higher order moments are derived. Inference tools including the maximum likelihood estimation, an expectation–maximization (EM) algorithm for searching the estimator and an information criterion for model selection are carefully studied for the logistic mixture double autoregressive model, which has two components and is encountered more frequently in practice. Monte Carlo experiments give further support to the new models, and the analysis of an empirical example is also reported.     

A pseudo-marginal sequential Monte Carlo algorithm for random effects models in Bayesian sequential design

Motivated by the need to sequentially design experiments for the collection of data in batches or blocks, a new pseudo-marginal sequential Monte Carlo algorithm is proposed for random effects models where the likelihood is not analytic, and has to be approximated. This new algorithm is an extension of the idealised sequential Monte Carlo algorithm where we propose to unbiasedly approximate the likelihood to yield an efficient exact-approximate algorithm to perform inference and make decisions within Bayesian sequential design. We propose four approaches to unbiasedly approximate the likelihood: standard Monte Carlo integration; randomised quasi-Monte Carlo integration, Laplace importance sampling and a combination of Laplace importance sampling and randomised quasi-Monte Carlo. These four methods are compared in terms of the estimates of likelihood weights and in the selection of the optimal sequential designs in an important pharmacological study related to the treatment of critically ill patients. As the approaches considered to approximate the likelihood can be computationally expensive, we exploit parallel computational architectures to ensure designs are derived in a timely manner.     

Robust Inference in Conditionally Linear Nonlinear Regression Models

Abstract.  We consider robust methods of likelihood and frequentist inference for the nonlinear parameter, say α , in conditionally linear nonlinear regression models. We derive closed-form expressions for robust conditional, marginal, profile and modified profile likelihood functions for α under elliptically contoured data distributions. Next, we develop robust exact-F confidence intervals for α and consider robust Fieller intervals for ratios of regression parameters in linear models. Several well-known examples are considered and Monte Carlo simulation results are presented.     

Filtering via approximate Bayesian computation

Approximate Bayesian computation (ABC) has become a popular technique to facilitate Bayesian inference from complex models. In this article we present an ABC approximation designed to perform biased filtering for a Hidden Markov Model when the likelihood function is intractable. We use a sequential Monte Carlo (SMC) algorithm to both fit and sample from our ABC approximation of the target probability density. This approach is shown to, empirically, be more accurate w.r.t.?the original filter than competing methods. The theoretical bias of our method is investigated; it is shown that the bias goes to zero at the expense of increased computational effort. Our approach is illustrated on a constrained sequential lasso for portfolio allocation to 15 constituents of the FTSE 100 share index.     

Nonlinear and non-gaussian state estimation: A quasi-optimal estimator

The rejection sampling filter and smoother, proposed by Tanizaki (1996, 1999), Tanizaki and Mariano (1998) and Hiirzeler and Kiinsch (1998), take a lot of time computationally. The Markov chain Monte Carlo smoother, developed by Carlin, Poison and StofFer (1992), Carter and Kohn (1994, 1996) and Geweke and Tanizaki (1999a, 1999b), does not show a good performance depending on noniinearity and nonnormality of the system in the sense of the root mean square error criterion, which reason comes from slow convergence of the Gibbs sampler. Taking into account these problems, we propose the nonlinear and non-Gaussian filter and smoother which have much less computational burden and give us relatively better state estimates, although the proposed estimator does not yield the optimal state estimates in the sense of the minimum mean square error. The proposed filter and smoother are called the quasi-optimal filter and quasi-optimal smoother in this paper. Finally, through some Monte Carlo studies, the quasi-optimal filter and smoother are compared with the rejection sampling procedure and the Markov chain Monte Carlo procedure.     

Value at risk estimation under stochastic volatility models using adaptive PMCMC methods

In this paper, we propose a value-at-risk (VaR) estimation technique based on a new stochastic volatility model with leverage effect, nonconstant conditional mean and jump. In order to estimate the model parameters and latent state variables, we integrate the particle filter and adaptive Markov Chain Monte Carlo (MCMC) algorithms to develop a novel adaptive particle MCMC (A-PMCMC) algorithm. Comprehensive simulation experiments based on three stock indices and two foreign exchange time series show effectiveness of the proposed A-PMCMC algorithm and the VaR estimation technique.     

Monte Carlo Kalman filter and smoothing for multivariate discrete state space models

The author studies state space models for multivariate binomial time series, focussing on the development of the Kalman filter and smoothing for state variables. He proposes a Monte Carlo approach employing the latent variable representation which transplants the classical Kalman filter and smoothing developed for Gaussian state space models to discrete models and leads to a conceptually simple and computationally convenient approach. The method is illustrated through simulations and concrete examples.     

Sequential Bayesian inference in hidden Markov stochastic kinetic models with application to detection and response to seasonal epidemics

We study sequential Bayesian inference in stochastic kinetic models with latent factors. Assuming continuous observation of all the reactions, our focus is on joint inference of the unknown reaction rates and the dynamic latent states, modeled as a hidden Markov factor. Using insights from nonlinear filtering of continuous-time jump Markov processes we develop a novel sequential Monte Carlo algorithm for this purpose. Our approach applies the ideas of particle learning to minimize particle degeneracy and exploit the analytical jump Markov structure. A motivating application of our methods is modeling of seasonal infectious disease outbreaks represented through a compartmental epidemic model. We demonstrate inference in such models with several numerical illustrations and also discuss predictive analysis of epidemic countermeasures using sequential Bayes estimates.     

Inequality Constrained State-Space Models

ABSTRACT

The standard Kalman filter cannot handle inequality constraints imposed on the state variables, as state truncation induces a nonlinear and non-Gaussian model. We propose a Rao-Blackwellized particle filter with the optimal importance function for forward filtering and the likelihood function evaluation. The particle filter effectively enforces the state constraints when the Kalman filter violates them. Monte Carlo experiments demonstrate excellent performance of the proposed particle filter with Rao-Blackwellization, in which the Gaussian linear sub-structure is exploited at both the cross-sectional and temporal levels.     

Nonlinear filters based on taylor series expansions ∗

The nonlinear filters based on Taylor series approximation are broadly used for computational simplicity, even though their filtering estimates are clearly biased. In this paper, first, we analyze what is approximated when we apply the expanded nonlinear functions to the standard linear recursive Kalman filter algorithm. Next, since the state variable α_t and α_t-t are approximated as a conditional normal distribution given information up to time t - 1 (i.e., I_t-1) in approximation of the Taylor series expansion, it might be appropriate to evaluate each expectation by generating normal random numbers of α_t and α_t-1 given I_t-1 and those of the error terms θ and η_t. Thus, we propose the Monte-Carlo simulation filter using normal random draws. Finally we perform two Monte-Carlo experiments, where we obtain the result that the Monte-Carlo simulation filter has a superior performance over the nonlinear filters such as the extended Kalman filter and the second-order nonlinear filter.     

Testing for Granger-causality in quantiles

This paper proposes a consistent parametric test of Granger-causality in quantiles. Although the concept of Granger-causality is defined in terms of the conditional distribution, most articles have tested Granger-causality using conditional mean regression models in which the causal relations are linear. Rather than focusing on a single part of the conditional distribution, we develop a test that evaluates nonlinear causalities and possible causal relations in all conditional quantiles, which provides a sufficient condition for Granger-causality when all quantiles are considered. The proposed test statistic has correct asymptotic size, is consistent against fixed alternatives, and has power against Pitman deviations from the null hypothesis. As the proposed test statistic is asymptotically nonpivotal, we tabulate critical values via a subsampling approach. We present Monte Carlo evidence and an application considering the causal relation between the gold price, the USD/GBP exchange rate, and the oil price.