Robust Filtering

Abstract

Filtering methods are powerful tools to estimate the hidden state of a state-space model from observations available in real time. However, they are known to be highly sensitive to the presence of small misspecifications of the underlying model and to outliers in the observation process. In this article, we show that the methodology of robust statistics can be adapted to sequential filtering. We define a filter as being robust if the relative error in the state distribution caused by misspecifications is uniformly bounded by a linear function of the perturbation size. Since standard filters are nonrobust even in the simplest cases, we propose robustified filters which provide accurate state inference in the presence of model misspecifications. The robust particle filter naturally mitigates the degeneracy problems that plague the bootstrap particle filler (Gordon, Salmond, and Smith) and its many extensions. We illustrate the good properties of robust filters in linear and nonlinear state-space examples. Supplementary materials for this article are available online

Fused Graphical Lasso for Brain Networks with Symmetries

“…Specifically, we consider the first-order dynamic conditional score (DCS) model for the location discussed by Harvey and Luati (2014), where each x (t) is assumed to be conditionally distributed as a Student-t random variable with ν degrees of freedom, x (t) | ℱ t−1 ∼ t ( (t) , 2 ), with the filtration ℱ s representing the information set up to time s. The signal (t) is estimated based on an autoregressive mechanism, where û (t) is a realization of a martingale difference sequence, that is, E(u (t) | ℱ t−1 ) = 0, proportional to the score of the conditional likelihood of the time-varying location, that is, (0) is set equal to a fixed value. In this framework, the dynamic BOLD signal is updated by a filter that is robust with respect to extreme values (Calvet et al, 2015). The robustness comes from the properties of the martingale difference sequence u (t) : if the data arise from a heavy tail distribution, then the score û (t) is less sensitive to extreme values than the score of a Gaussian distribution or than the inno-…”

Particle Filters and Data Assimilation

“…Robust Filters. Calvet et al (2015) propose a robust filter to address the issue of observation outliers. They modify the likelihood g(yt | xt) in order to reduce the impact of an observation yt that comes from a distribution other than the nominal g(yt | xt)dν(yt).…”

Robust statistics: a selective overview and new directions

“…Challenging problems remain for nonstationary series, including unit roots and cointegration (see for a start, Ref ). Moreover, although robust versions of the Kalman filter are available (see Refs ), the investigation of the robustness properties of the entire filtering distribution has been tackled only recently (see Ref 81 and 82).…”