A Normal Variance Mixture Model for Robust Kalman Filtering

The Kalman filter is ubiquitous for state space models because of its desirable statistical properties, ease of implementation, and generally good performance. However, it can perform poorly in the presence of outliers, or measurements with noise variances much greater than those assumed by the filter. An algorithm that is similar to the Kalman filter but robust to outliers is derived in this report. This algorithm -- called the normal variance mixture filter (NVMF) -- replaces the Gaussian distribution for the noise in the Kalman filter measurement model with a normal variance mixture distribution that admits heavier tails. Choice of the mixing density determines the complexity and performance of the NVMF. When the mixing density is the Dirac delta function, the NVMF is equivalent to the Kalman filter. Choice of an inverse gamma mixing density leads to closed-form recursions for the state estimate and its error covariance matrix that are robust to outliers. The NVMF is compared to the benchmark probabilistic data association filter (PDAF), as well as two other robust filters from the recent literature, for a simulated example. While all four robust filters outperform the Kalman filter when outliers are present, the NVMF provides the most consistent performance across all simulations.