Spline and Wavelet Smoothing Techniques for Functional Data
Mohammad Mahdi Eslami · 2024
Advancements in statistical technologies have led to the collection of increasingly sophisticated data, necessitating new tools and techniques for extracting information from underlying data patterns. Functional Data Analysis (FDA) addresses this challenge by examining dataset variability when observations are curves, enabling analysis of derivative information. Central to the FDA framework is the process of curve fitting, where mathematical functions or curves are crafted to approximate the observed data. This paper focuses on two key smoothing techniques in FDA: Splines and Wavelets. Splines utilize knots and piecewise polynomial functions to approximate curves, while Wavelets decompose data into frequency sub-bands, filtering out rapid changes for smoother estimation. The paper delves into fundamental spline concepts, including spline tuning parameters and knot placement strategies, and explores wavelet bases. Wavelets are ideal for capturing dynamic frequency behaviors and transient features in signals, while splines excel in providing smooth and continuous representations of underlying patterns in functional data, offering local control over the fitted curve. Furthermore, we review the practical application of these smoothing methods in diverse datasets, including Dublin bike usage data, electrocardiogram (ECG) signals, and magnetic resonance imaging (MRI) datasets. This paper contributes to a deeper understanding of advanced smoothing techniques, emphasizing their practical relevance in extracting insights from complex datasets across diverse domains.