Kalman Filter, Sensor Fusion, and Constrained Regression: Equivalences and Insights
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The Kalman filter (KF) is one of the most widely used tools for data assimilation and sequential estimation. In this paper, we show that the state estimates from the KF in a standard linear dynamical system setting are exactly equivalent to those given by the KF in a transformed system, with infinite process noise (a "flat prior") and an augmented measurement space. This reformulation--which we refer to as augmented measurement sensor fusion (SF)--is conceptually interesting, because the transformed system here is seemingly static (as there is effectively no process model), but we can still capture the state dynamics inherent to the KF by folding the process model into the measurement space. Apart from being interesting, this reformulation of the KF turns out to be useful in problem settings in which past states are eventually observed (at some lag). In such problems, when we use the empirical covariance to estimate the measurement noise covariance, we show that the state predictions from augmented measurement SF are exactly equivalent to those from a regression of past states on past measurements, subject to particular linear constraints (reflecting the relationships encoded in the measurement map). This allows us to port standard ideas (say, regularization methods) in regression over to dynamical systems. For example, we can posit multiple candidate process models, fold all of them into the measurement model, transform to the regression perspective, and apply ℓ_1 penalization to perform process model selection. We give various empirical demonstrations, and focus on an application to nowcasting the weekly incidence of influenza in the US.
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