RLS Framework with Segmentation of the Forgetting Profile and Low Rank Updates

This report describes a new regularization approach based on segmentation of the forgetting profile in sliding window least squares estimation. Each segment is designed to enforce specific desirable properties of the estimator such as rapidity, desired condition number of the information matrix, accuracy, numerical stability, etc. The forgetting profile is divided in three segments, where the speed of estimation is ensured by the first segment, which employs rapid exponential forgetting of recent data.The second segment features a decline in the profile and marks the transition to the third segment, characterized by slow exponential forgetting to reduce the condition number of the information matrix using more distant data. Condition number reduction mitigates error propagation, thereby enhancing accuracy and stability. This approach facilitates the incorporation of a priori information regarding signal characteristics (i.e., the expected behavior of the signal) into the estimator. Recursive and computationally efficient algorithm with low rank updates based on new matrix inversion lemma for moving window associated with this regularization approach is developed. New algorithms significantly improve the approximation accuracy of low resolution daily temperature measurements obtained at the Stockholm Old Astronomical Observatory, thereby enhancing the reliability of temperature predictions.