The Tobit Kalman filter: an estimator for censored data

Abstract

The Kalman Filter has become ubiquitous in tracking and estimation. Many estimation applications, especially those using low cost commercial of-the-shelf sensors (COTS), are subject to a special type of measurement nonlinearity called censoring. Censoring frequently takes the form of sensor saturation, occlusion regions, and limit-of-detection. These forms of censoring are known as Tobit model Type 1 censoring. Introduction of censored measurements into the Kalman filter results in biased estimates of the underlying states. In this dissertation, we present the first formulation of the Kalman filter capable of estimating state variables from censored data without bias. We refer to this formulation as the Tobit Kalman filter. Previous work on Kalman filtering with measurement nonlinearities or sensor faults includes a Kalman filter for intermittent measurements, the particle filter, the unscented Kalman filter (UKF) and the extended Kalman filter (EKF). Intermittent measurement nonlinearity is similar to the censored measurement model; with the exception that censored data measurements are correlated with the state values. Previous work for intermittent measurements in estimation reduces the Kalman filter to a linear predictor when the measurement is missing. Use of either this formulation or a standard Kalman filter as an estimator in a censored data example will result in a biased estimate of the state. The particle filter is able to estimate the state values when the measurements are subject to censoring under certain cases, but comes with a substantial computational burden. The UKF is a less computationally expensive approach that proves to be non-robust when the measurements are near a censoring region. The EKF suffers from an undefined Jacobian at the threshold itself, and the Jacobian is zero in the censored region. On the other hand, the Tobit Kalman filter provides unbiased recursive estimates of latent state variables in or near saturated regions. This results by properly accounting for the statistics in the Tobit model, and using them to adapt the Kalman filter error terms and state measurement updates. The Tobit Kalman filter is completely recursive and computationally inexpensive while previous attempts at Tobit based state estimators where not recursive and are not computationally feasible. The many applications to the Tobit Kalman filter include MEMS sensor based tracking with saturation, visual tracking with camera frame censoring and biological measurements with limit of detection saturation's. In addition to the theoretical work for the Tobit Kalman filter, this dissertation discusses the applications of the Tobit Kalman filter by presenting simulations.