The paper contains several theoretical results related to the weighted nonlinear least-squares problem for low-rank signal estimation, which can be considered as a Hankel structured low-rank approximation problem. A parameterization of the subspace of low-rank time series connected with generalized linear recurrence relations (GLRRs) is described and its features are investigated. It is shown how the obtained results help to describe the tangent plane, prove optimization problem features and construct stable algorithms for solving low-rank approximation problems. For the latter, a stable algorithm for constructing the projection onto a subspace of time series that satisfy a given GLRR is proposed and justified. This algorithm is used for a new implementation of the known Gauss-Newton method using the variable projection approach. The comparison by stability and computational cost is performed theoretically and with the help of an example.

Local linear kernel fitting is a popular nonparametric technique for modelling nonlinear time series data. Investigations into it, although extensively made for continuousvalued case, are still rare for the time series that are discrete-valued. In this paper, we propose and develop the uniform consistency of local linear maximum likelihood (LLML) fitting for time series regression allowing response to be discrete-valued under β-mixing dependence condition. Specifically, the uniform consistency of LLML estimators is established under time series conditional exponential family distributions with aid of a beta-mixing empirical process through local estimating equations. The rate of convergence is also provided under mild conditions. Performances of the proposed method are demonstrated by a Monte-Carlo simulation study and an application to COVID-19 data. There is a huge potential for the developed theory contributing to further development of discrete-valued response semiparametric time series models.

International Press of Boston - publishers of scholarly mathematical and scientific journals and books

This article presents a novel approach to clustering and feature selection for categorical time series via interpretable frequency-domain features. A distance measure is introduced based on the spectral envelope and optimal scalings, which parsimoniously characterize prominent cyclical patterns in categorical time series. Using this distance, partitional clustering algorithms are introduced for accurately clustering categorical time series. These adaptive procedures offer simultaneous feature selection for identifying important features that distinguish clusters and fuzzy membership when time series exhibit similarities to multiple clusters. Clustering consistency of the proposed methods is investigated, and simulation studies are used to demonstrate clustering accuracy with various underlying group structures. The proposed methods are used to cluster sleep stage time series for sleep disorder patients in order to identify particular oscillatory patterns associated with sleep disruption.