Optimally weighted L(2) distance for functional data.

Area of Science:

• Statistics
• Functional Data Analysis
• Machine Learning

Background:

• Functional data analysis (FDA) commonly uses L2 distance for statistical methods.
• Choosing an appropriate distance measure is crucial for method performance in FDA.

Purpose of the Study:

• To investigate the impact of weighted L2 distance on FDA methods.
• To introduce and evaluate novel weight functions for improved performance.

Main Methods:

• Implemented weighted L2 distance with spline-based functional data representations.
• Considered design density, inverse-variance, and a novel coefficient-of-variation-minimizing weight function.
• Applied methods to k-medoids clustering, nonparametric classification, and permutation testing.

Main Results:

• Weighted L2 distance significantly enhances the performance of various FDA techniques.
• The proposed novel weight function demonstrates superior results in simulations and real-world data.
• Improvements were observed in clustering, classification, and permutation testing.

Conclusions:

• Weighted L2 distance is a valuable enhancement for functional data analysis.
• The novel weight function offers a robust and effective approach for improving statistical method performance in FDA.