Uniform stability of a particle approximation of the optimal filter derivative

Sequential Monte Carlo methods, also known as particle methods, are a widely used set of computational tools for inference in non-linear non-Gaussian state-space models. In many applications it may be necessary to compute the sensitivity, or derivative, of the optimal filter with respect to the static parameters of the state-space model; for instance, in order to obtain maximum likelihood model parameters of interest, or to compute the optimal controller in an optimal control problem. In Poyiadjis et al. [2011] an original particle algorithm to compute the filter derivative was proposed and it was shown using numerical examples that the particle estimate was numerically stable in the sense that it did not deteriorate over time. In this paper we substantiate this claim with a detailed theoretical study. Lp bounds and a central limit theorem for this particle approximation of the filter derivative are presented. It is further shown that under mixing conditions these Lp bounds and the asymptotic variance characterized by the central limit theorem are uniformly bounded with respect to the time index. We demon- strate the performance predicted by theory with several numerical examples. We also use the particle approximation of the filter derivative to perform online maximum likelihood parameter estimation for a stochastic volatility model.

FIRST PASSAGE APPROXIMATION FOR NORMAL STATIONARY RANDOM PROCESSES.

A new approximate solution for the first passage probability of a stationary Gaussian random process is presented which is based on the estimation of the mean clump size. A simple expression for the mean clump size is derived in terms of the cumulative normal distribution function, which avoids the lengthy numerical integrations which are required by similar existing techniques. The method is applied to a linear oscillator and an ideal bandpass process and good agreement with published results is obtained. By making a slight modification to an existing analysis it is shown that a widely used empirical result for the asymptotic form of the first passage probability can be deduced theoretically.

Vortex breaking and cutting in type II superconductors.

In technological superconductors, the Lorentz force on the flux vortices is opposed by inhomogeneous pinning and so the critical current may be controlled by a combination of vortex entanglement, cutting, and cross-joining. To understand the roles of these processes we report measurements of structures in which a weak pinning layer is sandwiched between two strongly pinning leads. Quantitative modeling of the results demonstrates that in such systems the critical current is limited by the deformation of individual vortices and not by subsequent cross-joining processes.

Insulin analog preparations and their use in children and adolescents with type 1 diabetes mellitus.

Standard or 'traditional' human insulin preparations such as regular soluble insulin and neutral protamine Hagedorn (NPH) insulin have shortcomings in terms of their pharmacokinetic and pharmacodynamic properties that limit their clinical efficacy. Structurally modified insulin molecules or insulin 'analogs' have been developed with the aim of delivering insulin replacement therapy in a more physiological manner. In the last 10 years, five insulin analog preparations have become commercially available for clinical use in patients with type 1 diabetes mellitus: three 'rapid' or fast-acting analogs (insulin lispro, aspart, and glulisine) and two long-acting analogs (insulin glargine and detemir). This review highlights the specific pharmacokinetic properties of these new insulin analog preparations and focuses on their potential clinical advantages and disadvantages when used in children and adolescents with type 1 diabetes mellitus. The fast-acting analogs specifically facilitate more flexible insulin injection timing with regard to meals and activities, whereas the long-acting analogs have a more predictable profile of action and lack a peak effect. To date, clinical trials in children and adolescents have been few in number, but the evidence available from these and from other studies carried out in adults with type 1 diabetes suggest that they offer significant benefits in terms of reduced frequency of nocturnal hypoglycemia, better postprandial blood glucose control, and improved quality of life when compared with traditional insulins. In addition, insulin detemir therapy is unique in that patients may benefit from reduced risk of excessive weight, particularly during adolescence. Evidence for sustained long-term improvements in glycosylated hemoglobin, on the other hand, is modest. Furthermore, alterations to insulin/insulin-like growth factor I receptor binding characteristics have also raised theoretical concerns that insulin analogs may have an increased mitogenic potential and risk of tumor development, although evidence from both in vitro and in vivo animal studies do not support this assertion. Long-term surveillance has been recommended and further carefully designed prospective studies are needed to evaluate the overall benefits and clinical efficacy of insulin analog therapy in children and adolescents with type 1 diabetes.

Asymptotic behaviour of approximate Bayesian estimators

Although approximate Bayesian computation (ABC) has become a popular technique for performing parameter estimation when the likelihood functions are analytically intractable there has not as yet been a complete investigation of the theoretical properties of the resulting estimators. In this paper we give a theoretical analysis of the asymptotic properties of ABC based parameter estimators for hidden Markov models and show that ABC based estimators satisfy asymptotically biased versions of the standard results in the statistical literature.