A modified displacment rank and some applications

Recursive algorithms for linear least squares estimation problems have been based mainly on state-space models. Recently, some new recursive solutions were obtained for processes classified in terms of their "index of nonstationarity" or equivalently-- the displacement rank of their covariance functions. While this definition provides a natural explanation of the properties of constant-coefficient state-space models, it is not satisfactory for time-variant models. However a modified definition of the displacement rank makes it possible to imbed time-varying state-space models in the more general input-output framework. In so doing, we are able to show the mutual relationships of the Kalman filter Riccati equation, the time-varying Chandrasekhar equations and the Krein-Levinson equations.