A New Approach to Linear Filtering and Prediction Problems
📜 Abstract
The classical filtering and prediction problem is re-examined using the Bode-Shannon representation of random processes and the “state transition” method of analysis of dynamic systems. New results are: (1) The formulation and methods of solution of the problem apply without modification to stationary and nonstationary statistics and to growing-memory and infinite-memory filters. (2) A nonlinear difference (or differential) equation is derived for the covariance matrix of the optimal estimation error. From the solution of this equation the coefficients of the difference (or differential) equation of the optimal linear filter are obtained without further calculations. (3) The filtering problem is shown to be the dual of the noise-free regulator problem. The new method developed here is applied to two well-known problems, confirming and extending earlier results. The discussion is largely self-contained and proceeds from first principles; basic concepts of the theory of random processes are reviewed in the Appendix.
✨ Summary
- The paper reformulates Wiener filtering and prediction in a state-transition framework for linear dynamical systems driven by independent Gaussian disturbances.
- It identifies the optimal estimate with an orthogonal projection under Gaussian assumptions or linear least-squares loss, and represents the estimator as a recursive feedback system driven by the innovation—the portion of the latest observation not predictable from earlier observations.
- Its central computational result is a nonlinear recursion for the estimation-error covariance matrix, now recognized as the discrete Riccati recursion. Once the covariance is computed, the filter gain and estimator transition matrix follow directly.
- The formulation covers stationary and nonstationary models, finite, growing, and infinite memory, and filtering, prediction, and smoothing problems. It also establishes a precise duality between optimal filtering and the noise-free linear-quadratic regulator problem.
- Subsequent research developed the Kalman and Kalman–Bucy filtering theory, studied the associated Riccati equations and their stability, and extended the framework to smoothing, nonlinear estimation, and other variants. Later literature explicitly treats the paper as foundational for Kalman filtering and Riccati-based filtering/control theory. (epubs.siam.org)
- The approach also had direct engineering influence. NASA reports that Kalman-filter methods were applied in the early 1960s to Apollo spacecraft navigation and later adapted into extended and square-root forms for practical aerospace computation. NASA subsequently documented applications to spacecraft navigation and air-traffic-management systems. (ntrs.nasa.gov)