Robust Parameter Estimation of Non-Linear State Space Models Using a Divergence-Based Estimator

In this paper, we consider parameter estimation of Non-linear State Space Models (NSSMs) in the presence of outliers. The parameters of the NSSMs are typically estimated using maximum likelihood estimation with Gaussian error assumption, which is vulnerable to outliers. To mitigate impact of outliers, existing methods adopt a Student $ t $ distribution or a combination of an outlier-robust loss function with a Gaussian approximation. However, robustness of both approaches is insufficient. In addition, the $ t $ distribution approach is less applicable compared to the Gaussian distribution due to lack of parameter identifiability. Furthermore, the Gaussian approximation in the second approach typically discards higher-order moments of posteriors of unobserved states, leading to an inaccurate inference. To address these issues, we propose a novel robust estimator for the NSSMs based on the $\gamma$-divergence (Fujisawa and S. Eguchi, 2018) and an iterative algorithm to obtain the proposed estimate. This estimator achieves stronger robustness than the existing approaches, while maintaining the Gaussian assumption. We show asymptotic normality of the proposed estimator. The iterative algorithm alternates between posterior inference of the unobserved states and parameter updates. For the posterior inference, we use a robust Monte-Carlo approximation, which retains all-orders of moments unlike the Gaussian approximation. For the parameter updates, we derive an efficient algorithm that provides closed-form update formulae for specific models. Simulation and real-data experiments demonstrate that the proposed method estimated the parameters of the NSSMs more accurately than the existing methods in the presence of outliers.