Outliers Rejection in Kalman Filtering—Some New Observations

A standard outlier-rejection scheme applied in Kalman filtering, based on the acceptance/rejection gate for measurement innovation, is discussed in this paper. The main idea behind this approach is based on assumptions that measurements can be "normal", as described in the measurement model and "abnormal" outliers that are generated by a totally different model. The goal of the acceptance/rejection gate is to accept normal measurements and reject abnormal ones. A concrete and simple case of range estimation in the presence of multipath outliers is thoroughly analyzed. The results are both nontrivial (even surprising) and important for designers of such rejection schemes who may use them as guidance for efficient design. The first observation is that the outlier-rejection scheme may provide worse results than the scheme with no rejection at all. This is because there is a positive, albeit relatively low probability that the system will enter and remain in a mode in which outliers are accepted and normal measurements are rejected. In this case, the estimation errors become very big and have a significant influence on the total standard deviations (even if their probability of occurrence is low). The main and very important conclusion is that outlier-rejection schemes cannot be applied without a proper recovery scheme that prevents the system from remaining "stuck" in normal-measurement rejection mode. In this paper, three different recovery schemes are proposed:  a one-sided rejection scheme (only applicable to multipath-type outliers)  a Kalman-filter reset scheme  a set of parallel Kalman filters, where the output is provided by the filter with minimal innovation size. The design and performance analysis of each recovery scheme are described. The conclusion is that the performance of the recovery schemes is very close to the case without any outliers at all, up to very high (0.45) multipath-occurrence probability. Keywords—Kalman filter, outlier, measurement rejection I. THE OUTLIER REJECTION SCHEME—AN OVERVIEW In today's advanced navigation systems, proper sensor integration seems to be the key to success. Traditionally, Kalman filters developed for linear, stochastic systems were used for sensor integration. In practice, however, sensors can produce unexpected anomalies in their measurements, for example, in the GNSS (due to sporadic interference or significant multipath) magnetometers affected by environmental magnetic variance, false fixes from featurematching systems and many others. The common engineering procedure used to overcome such phenomena is outlier rejection (also called innovation filtering, spike filtering, measurement gating, reasonability testing or prefiltering; for overview and details see Chapter 17 in [1]). The main idea for outlier rejection is to define an acceptance/rejection gate for every measurement. The goal is to reject measurements created by outliers and accept normal measurements (created by the assumed nominal measurement model). In this section, the outlier-rejection scheme is presented in the context of Kalman-filter implementation. Every Kalman filter is based on two equations: propagation (1) and measurement (2): ( 1) ( ) ( ) ( ) X k A k X k w k    (1) ( ) ( ) ( ) ( ) z k H k X k v k   (2) where ( ) X k is the state vector, ( ) A k is the transition matrix, ( ) w k is the process noise, ( ) z k is the measurement, ( ) H k is the measurement matrix, and ( ) v k is the measurement noise. In 1960, Rudolf Kalman proposed a recursive; optimal estimation algorithm for this problem (with several additional assumptions) called the Kalman filter [2]. This algorithm has been well documented (see, for example, [3],[4],[5]). Aside from the model described in (1) and (2), the Kalman filter requires three additional statistical quantities: 0 P – the covariance of the state vector (0) X , ( ) Q k – the covariance of the process noise ( ) w k and ( ) R k – the covariance of the measurement noise ( ) v k . In addition, zero mean, independence and Gaussian distribution are assumed for all random variables. The equations of the Kalman filter are described below (assuming that the propagation rate and measurement rate are not necessarily equal). Initialization