On exact and optimal recovering of missing values for sequences

TL;DR

This paper investigates conditions under which missing values in sequences can be exactly and optimally recovered without relying on probabilistic models, introducing new criteria based on Z-transform properties.

Contribution

It provides new sufficient conditions for recoverability of sequence missing values using Z-transform degeneracy and proposes an optimal recovery method with robustness analysis.

Findings

Sequences are recoverable if their Z-transforms have certain degeneracy.

Optimal recovery can be achieved via projection methods.

Robustness to noise and truncation is demonstrated.

Abstract

The paper studies recoverability of missing values for sequences in a pathwise setting without probabilistic assumptions. This setting is oriented on a situation where the underlying sequence is considered as a sole sequence rather than a member of an ensemble with known statistical properties. Sufficient conditions of recoverability are obtained; it is shown that sequences are recoverable if there is a certain degree of degeneracy of the Z-transforms. We found that, in some cases, this degree can be measured as the number of the derivatives of Z-transform vanishing at a point. For processes with non-degenerate Z-transform, an optimal recovering based on the projection on a set of recoverable sequences is suggested. Some robustness of the solution with respect to noise contamination and truncation is established.