Graduation Term

Summer 2026

Degree Name

Master of Science (MS)

Department

Department of Mathematics

Committee Chair

Sunil Chebolu

Committee Member

Xiaotian Dai

Abstract

Polynomial-preserving linear filters are fundamental tools for smoothing and trend extraction in noisy time series, particularly when the objective is to reduce short-run variation without distorting underlying low-degree structure. This thesis develops a general framework for such filters based on local polynomial approximation and least-squares estimation. In particular, we establish that the resulting estimators can be represented as symmetric linear filters whose weights satisfy moment conditions that ensure exact polynomial reproduction up to a specified degree. Building on this framework, we derive explicit coefficient vectors for several classical smoothing filters, including the Savitzky-Golay filters SG(5,2) and SG(7,3), as well as the Spencer 15-point moving average. A central contribution of this work is a detailed, self-contained derivation of the Spencer 15-point filter, whose construction is often presented only in an abbreviated form in the literature. For each filter, polynomial reproduction properties are verified analytically through the associated moment conditions. The thesis further demonstrates that, despite their distinct constructions, Savitzky-Golay filters arising from local least squares and Spencer filters arising from successive averaging and correction both satisfy the same polynomial reproduction constraints. This provides a unified interpretation of these methods within a common mathematical framework. The theoretical results are complemented by numerical experiments using both synthetic and real-world time series data. These experiments illustrate the tradeoff between smoothness and responsiveness across filters. Overall, the work clarifies how polynomial reproduction, symmetry, and window length jointly determine filter performance and provides a coherent pathway from general existence results to explicit, implementable smoothing rules used in practice.

Access Type

Thesis-Open Access

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