In this paper we offer a review and bibliography of work on Hankel low-rank approximation and completion, with particular emphasis on how this methodology can be used for time series analysis and forecasting. We begin by describing possible formulations of the problem and offer commentary on related topics and challenges in obtaining globally optimal solutions. Key theorems are provided, and the paper closes with some expository examples.

This work uses a data-driven approach to analyse how the resource requirements of patients with chronic obstructive pulmonary disease (COPD) may change, quantifying how those changes impact the hospital system with which the patients interact. This approach is composed of a novel combination of often distinct modes of analysis: segmentation, operational queuing theory, and the recovery of parameters from incomplete data. By combining these methods as presented here, this work demonstrates that potential limitations around the availability of fine-grained data can be overcome. Thus, finding useful operational results despite using only administrative data. The paper begins by finding a useful clustering of the population from this granular data that feeds into a multi-class M/M/c model, whose parameters are recovered from the data via parameterisation and the Wasserstein distance. This model is then used to conduct an informative analysis of the underlying queuing system and the needs of the population under study through several what-if scenarios. The analyses used to form and study this model consider, in effect, all types of patient arrivals and how those types impact the system. With that, this study finds that there are no quick solutions to reduce the impact of COPD patients on the system, including adding capacity to the system. In this analysis, the only effective intervention to reduce the strain caused by those presenting with COPD is to enact external policies which directly improve the overall health of the COPD population before they arrive at the hospital.

In this paper we consider the low-rank matrix completion problem with specific application to forecasting in time series analysis. Briefly, the low-rank matrix completion problem is the problem of imputing missing values of a matrix under a rank constraint. We consider a matrix completion problem for Hankel matrices and a convex relaxation based on the nuclear norm. Based on new theoretical results and a number of numerical and real examples, we investigate the cases when the proposed approach can work. Our results highlight the importance of choosing a proper weighting scheme for the known observations.