Nonlinear Estimators for Censored Data: A Comparison of the EKF, the UKF and the Tobit Kalman Filter

Measurement censoring, or Tobit model censoring, is common in many engineering applications. It arises from limits in sensor dynamic range, and may be exacerbated by poor calibration of sensors. Censoring is often referred to as a clipped measurement or limit-of-detection discontinuity, and is represented as a piecewise-linear transform of the output variable. The slope of the piecewise-linear transform is zero in the censored region. This form of nonlinearity presents significant challenges when a nonlinear approximation to the Kalman filter is to be used as an estimator. The Tobit Kalman filter is a new method that is a computationally efficient, unbiased estimator for linear dynamical systems with censored output. In this paper, we use Monte Carlo methods to compare the performance of the Tobit Kalman Filter to the performance of the Extended Kalman Filter and the Unscented Kalman Filter. We show that the Tobit Kalman Filter reliably provides accurate estimates of the state and state error covariance with censored measurement data, while both the EKF and the UKF provide unreliable estimates in censored data conditions.