Some Paradoxical Perspectives on Approaches and Structures in Functional Data Analysis

Abstract

Developments in functional data analysis have attracted considerable attention in recent literature. This study aims to provide general information about functional data analysis and demonstrate how it can be enriched with auxiliary tools. When the article is considered as a whole, the results predominantly pertain to functional linear models. The first section discusses the estimation of regression model parameters using the Least Squares method. It explains how the Least Squares method is applied within a functional framework and incorporates auxiliary calculations as part of the modeling process. At this stage, the Error Sum of Squares, which forms the basis of the Least Squares method, is represented as a vector field. The second section addresses the interim estimation problem. In this part, the Bernstein polynomial is combined with the wavelet transform to address the interim estimation challenge. The final section introduces various types of functional data analysis. Specifically, the Bernstein polynomial is used in estimating a functional linear model with functional coefficients. Employing the Bernstein polynomial as a model component in the linear model offers a simpler and more innovative approach compared to traditional functional linear model structures. The methods proposed in this study are generally practical and compatible with the classical framework of functional data analysis.