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.

• Kalman filter
• Particle filter
• Robust statistics
• State-space model
• Weight degeneracy

Additional information

Notes on contributors

Laurent E. Calvet

Laurent E. Calvet is Professor, Department of Finance, HEC Paris, 1 rue de la Libération, 78351 Jouy-en-Josas, France (E-mail: [email protected]). Veronika Czellar is Associate Professor, Department of Finance, Economics and Control, EMLYON Business School, 23 avenue Guy de Collongue, 69134 Ecully, France (E-mail: [email protected]). Elvezio Ronchetti is Professor, Research Center for Statistics and Geneva School of Economics and Management, University of Geneva, Blv. Pont d’Arve 40 CH-1211 Geneva, Switzerland (E-mail: [email protected]). The authors thank the Editor, the Associate Editor, and the Referees for useful comments which improved the original version of the article. The first two authors gratefully acknowledge financial support from the HEC Foundation and the Europlace Institute of Finance. The research of the third author was partially supported by a Swiss National Science Foundation grant 100018–140295.

Veronika Czellar

Laurent E. Calvet is Professor, Department of Finance, HEC Paris, 1 rue de la Libération, 78351 Jouy-en-Josas, France (E-mail: [email protected]). Veronika Czellar is Associate Professor, Department of Finance, Economics and Control, EMLYON Business School, 23 avenue Guy de Collongue, 69134 Ecully, France (E-mail: [email protected]). Elvezio Ronchetti is Professor, Research Center for Statistics and Geneva School of Economics and Management, University of Geneva, Blv. Pont d’Arve 40 CH-1211 Geneva, Switzerland (E-mail: [email protected]). The authors thank the Editor, the Associate Editor, and the Referees for useful comments which improved the original version of the article. The first two authors gratefully acknowledge financial support from the HEC Foundation and the Europlace Institute of Finance. The research of the third author was partially supported by a Swiss National Science Foundation grant 100018–140295.

Elvezio Ronchetti

Laurent E. Calvet is Professor, Department of Finance, HEC Paris, 1 rue de la Libération, 78351 Jouy-en-Josas, France (E-mail: [email protected]). Veronika Czellar is Associate Professor, Department of Finance, Economics and Control, EMLYON Business School, 23 avenue Guy de Collongue, 69134 Ecully, France (E-mail: [email protected]). Elvezio Ronchetti is Professor, Research Center for Statistics and Geneva School of Economics and Management, University of Geneva, Blv. Pont d’Arve 40 CH-1211 Geneva, Switzerland (E-mail: [email protected]). The authors thank the Editor, the Associate Editor, and the Referees for useful comments which improved the original version of the article. The first two authors gratefully acknowledge financial support from the HEC Foundation and the Europlace Institute of Finance. The research of the third author was partially supported by a Swiss National Science Foundation grant 100018–140295.