Coupling stochastic EM and approximate Bayesian computation for parameter inference in state-space models

We study the class of state-space models and perform maximum likelihood estimation for the model parameters. We consider a stochastic approximation expectation–maximization (SAEM) algorithm to maximize the likelihood function with the novelty of using approximate Bayesian computation (ABC) within SAEM. The task is to provide each iteration of SAEM with a filtered state of the system, and this is achieved using an ABC sampler for the hidden state, based on sequential Monte Carlo methodology. It is shown that the resulting SAEM-ABC algorithm can be calibrated to return accurate inference, and in some situations it can outperform a version of SAEM incorporating the bootstrap filter. Two simulation studies are presented, first a nonlinear Gaussian state-space model then a state-space model having dynamics expressed by a stochastic differential equation. Comparisons with iterated filtering for maximum likelihood inference, and Gibbs sampling and particle marginal methods for Bayesian inference are presented.     

Local non-stationarity test in mean for Markov switching GARCH models: an approximate Bayesian approach

In order to exploit mean-reverting behavior among the price differential between two markets, one can use unit root tests to determine which pairs of assets appear to exhibit mean-reverting behavior. Since nonlinear mean reversion shares the same meaning as local stationarity, this paper proposes a Bayesian hypothesis testing to detect the presence of a local unit root in the mean equation using Markov switching GARCH models. This model incorporates a fat-tailed error distribution to analyze asymmetric effects on both the conditional mean and conditional volatility of financial time series. To implement the test, we propose a numerical approximation of the marginal likelihoods to posterior odds by using an adaptive Markov Chain Monte Carlo scheme. Our simulation study demonstrates that the approximate Bayesian test performs properly. The proposed method utilizes the daily basis between the FTSE 100 Index and Index Futures as an illustration.     

Efficient computation of sparse approximate inverses

We investigate different methods for computing a sparse approximate inverse M for a given sparse matrix A by minimizing ∥AM − E∥ in the Frobenius norm. Such methods are very useful for deriving preconditioners in iterative solvers, especially in a parallel environment. We compare different strategies for choosing the sparsity structure of M and different ways for solving the small least squares problem that are related to the computation of each column of M. Especially we show how we can take full advantage of the sparsity of A. © 1998 John Wiley & Sons, Ltd.     

Bayesian policy support for adaptive strategies using computer models for complex physical systems

In this paper, we discuss combining expert knowledge and computer simulators in order to provide decision support for policy makers managing complex physical systems. We allow future states of the complex system to be viewed after initial policy is made, and for those states to influence revision of policy. The potential for future observations and intervention impacts heavily on optimal policy for today and this is handled within our approach. We show how deriving policy dependent system uncertainty using computer models leads to an intractable backwards induction problem for the resulting decision tree. We introduce an algorithm for emulating an upper bound on our expected loss surface for all possible policies and discuss how this might be used in policy support. To illustrate our methodology, we look at choosing an optimal CO₂ abatement strategy, combining an intermediate complexity climate model and an economic utility model with climate data.     

Adaptive computation for convection dominated diffusion problems

We derive sharp L~∞(L~1) a posteriori error estimate for the convection dominated diffusion equations of the formThe derived estimate is insensitive to the diffusion parameter ε→0. The problem is discretized implicitly in time via the method of characteristics and in space via continuous     

Adaptive computation for convection dominated diffusion problems

We derive sharp L∞(L ¹ ) a posteriori error estimate for the convection dominated diffusion equations of the form

$$\frac{{\partial u}}{{\partial t}} + div(vu) - \varepsilon \Delta u = g.$$

The derived estimate is insensitive to the diffusion parameter ε → 0. The problem is discretized implicitly in time via the method of characteristics and in space via continuous piecewise linear finite elements. Numerical experiments are reported to show the competitive behavior of the proposed adaptive method.

     

On some approximate methods for nonlinear models

We discuss some recent inadequate applications of the homotopy perturbation method, the Adomian decomposition method and the variational iteration method to nonlinear problems.     

An efficient approximate solution for stochastic Lanchester models

Combat modeling is one of the essential topics for military decision making. The Lanchester equation is a classic method for modeling warfare, and many variations have extended its limitations and relaxed its assumptions. As a model becomes more complex, solving it analytically becomes intractable or computationally expensive. Hence, we propose two approximation methods: moment-matching scheme and a supporting method called battle-end approximation. These methods give an approximate solution in a short amount of time, while maintaining a high level of accuracy in simulation results in terms of hypothesis testing and numerical verification. They can be applied to computationally intensive problems, such as optimal resource allocation and analysis with asymmetric power like snipers or stealth aircrafts.     

Bayesian multiperiod forecasts for ARX models

Bayestian muliperiod forecasts for AR models with random independent exogenous variables under normal-gamma and normal-inverted Wishart prior assumptions are investigated. By suitably arranging the integration order of the model's parameters, at-density mixture approximation is analytically derived to provide an estimator of the posterior predictive density for any future observation. In particular, a suitablet-density is proposed by a convenient closed form. The precision of the discussed methods is examined by using some simulated data and one set of real data up to lead-six-ahead forecasts. It is found that the numerical results of the discussed methods are rather close. In particular, when sample sizes are sufficiently large, it is encouraging to apply a convenientt-density in practical usage. In fact, thist-density estimator asymptotically converges to the true density.This research was supported by the National Science Council, Republic of China under contract #NSC82-0208-M-008-086.     

Objective Bayesian analysis for CAR models

Objective priors, especially reference priors, have been studied extensively for spatial data in the last decade. In this paper, we study objective priors for a CAR model. In particular, the properties of the reference prior and the corresponding posterior are studied. Furthermore, we show that the frequentist coverage probabilities of posterior credible intervals depend only on the spatial dependence parameter $\rho $ , and not on the regression coefficient or the error variance. Based on the simulation study for comparing the reference and Jeffreys priors, the performance of two reference priors is similar and better than the Jeffreys priors. One spatial dataset is used for illustration.     

Adaptive greedy techniques for approximate solution of large RBF systems

For the solution of large sparse linear systems arising from interpolation problems using compactly supported radial basis functions, a class of efficient numerical algorithms is presented. They iteratively select small subsets of the interpolation points and refine the current approximative solution there. Convergence turns out to be linear, and the technique can be generalized to positive definite linear systems in general. A major feature is that the approximations tend to have only a small number of nonzero coefficients, and in this sense the technique is related to greedy algorithms and best n-term approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Decision diagrams for the approximate analysis of Markov models

Decision diagrams of various types can be used to encode the exact state space and transition rate matrix of large Markov models. However, the exact solution of such models still requires to store at least one real vector with one entry per reachable state, a formidable limitation to the practical use of these encodings. Thus, we discuss automatic techniques for the approximate computation of performance measures when the Markov model can be compactly encoded but not exactly solved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive eigenvalue computation: complexity estimates

This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on ℓ ₂, the space of square summable sequences, the problem becomes sufficiently well conditioned so that a gradient type iteration can be shown to reduce the error by some fixed factor per step. It then remains to realize these (ideal) iterations within suitable dynamically updated error tolerances. It is shown under which circumstances the adaptive scheme exhibits in some sense asymptotically optimal complexity. This research was supported in part by the Leibniz Programme of the DFG, by the SFB 401 funded by DFG, the DFG Priority Program SPP1145 and by the EU NEST project BigDFT.     

A hierarchy of approximate models for the Maxwell equations

Summary. In this paper we perform an asymptotic study of the Maxwell equations with respect to the small parameter where is the characteristic velocity associated with the physical problem and is the speed of light. This enables us to derive the quasistatic and Darwin models as respectively first and second order approximations of the Maxwell equations. Moreover, an interpretation of the obtained variational formulations gives us the appropriate boundary conditions for these models. Received May 18, 1995     

Bayesian multivariate Poisson models for insurance ratemaking

When actuaries face the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or a homeowner’s insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic: several studies have shown that there is a positive correlation between types of claim. Here we introduce different multivariate Poisson regression models in order to relax the independence assumption, including zero-inflated models to account for excess of zeros and overdispersion. These models have been largely ignored to date, mainly because of their computational difficulties. Bayesian inference based on MCMC helps to resolve this problem (and also allows us to derive, for several quantities of interest, posterior summaries to account for uncertainty). Finally, these models are applied to an automobile insurance claims database with three different types of claim. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models together with their zero-inflated versions.     

Bayesian estimation and inference for log-ACD models

This paper adapts Bayesian Markov chain Monte Carlo methods for application to some auto-regressive conditional duration models. Subsequently, the properties of these estimators are examined and assessed across a range of possible conditional error distributions and dynamic specifications, including under error mis-specification. A novel model error distribution, employing a truncated skewed Student-t distribution is proposed and the Bayesian estimator assessed for it. The results of an extensive simulation study reveal that favourable estimation properties are achieved under a range of possible error distributions, but that the generalised gamma distribution assumption is most robust and best preserves these properties, including when it is incorrectly specified. The results indicate that the powerful numerical methods underlying the Bayesian estimator allow more efficiency than the (quasi-) maximum likelihood estimator for the cases considered.     

Kalman filtering on approximate state-space models

Several state-space models for estimating a second-order stochastic process are proposed in this paper on the basis of the approximate Karhunen-Loève expansion. Properties of these models are studied and then the Kalman filtering method is applied. The accuracy of the models on the basis of two different situations, deterministic or random inputs, is studied by means of a simulation of a Brownian motion.This work was supported in part by DGICYT, Project No. PS93-0201.