Uncertainties for recursive estimators in nonlinear state-space models, with applications to epidemiology

Consider a nonlinear dynamic system where one wishes to estimate a state vector using noisy measurements. Many algorithms have been proposed to address this problem, among them the extended Kalman filter (and its variants) and constant-gain stochastic approximation. To quantify the efficacy of these algorithms, it is necessary to describe the distribution of the state estimation error. Typically, performance has been measured by the estimation error covariance alone, which does not provide enough information to probabilistically quantify the estimation accuracy. By casting the estimation error in an autoregressive-type form, this paper addresses the broader question of the distribution of the error for a general class of recursive algorithms. We illustrate the distributional results in an epidemiological problem of disease monitoring.     

Use of the Kalman filter for inference in state-space models with unknown noise distributions

The Kalman filter is frequently used for state estimation in state-space models when the standard Gaussian noise assumption does not apply. A problem arises, however, in that inference based on the incorrect Gaussian assumption can lead to misleading or erroneous conclusions about the relationship of the Kalman filter estimate to the true (unknown) state. This note shows how inequalities from probability theory associated with the probabilities of convex sets have potential for characterizing the estimation error of a Kalman filter in such a non-Gaussian (distribution-free) setting.     

Short communication: Screening stable flies and house flies as potential vectors of digital dermatitis in dairy cattle

Digital dermatitis is a bacterial infection that causes lesions above the heel bulbs on cattle hooves, and several bacterial species from the genus Treponema are suspected to be causative agents of this polymicrobial condition. Transmission of the bacteria to healthy cows is understudied, particularly with regard to potential insect vectors. Therefore, the objective of this research was to determine if flies captured from a dairy farm known to have digital dermatitis are contaminated with Treponema bacteria. The DNA-based assays did not detect any Treponema phagedenis from stable flies and house flies collected at a dairy experiencing an outbreak of digital dermatitis. Other potential means of bacterial transmission are discussed.     

Global Random Optimization by Simultaneous Perturbation Stochastic Approximation

We examine the theoretical and numerical global convergence properties of a certain ldquogradient freerdquo stochastic approximation algorithm called the ldquosimultaneous perturbation stochastic approximation (SPSA)rdquo that has performed well in complex optimization problems. We establish two theorems on the global convergence of SPSA, the first involving the well-known method of injected noise. The second theorem establishes conditions under which ldquobasicrdquo SPSA without injected noise can achieve convergence in probability to a global optimum, a result with important practical benefits.