Introdução à análise de dados funcionais com ênfase em splines e b-splines

Abstract

The development of non parametric curve estimation, which from the second half of the 20th century onward has been closely linked to the advances in computational science, has opened many paths to fields previously deemed ‘high specialty’, given their proximity to functional analysis and usually computationally costly estimation techniques. In these areas, the elements of interest cease to be limited to finite quantities of values, ‘discrete’ so to speak, and begin to encompass objects such as curves and surfaces, all of which assume values in functional spaces of infinite dimension. Through this lens, Functional Data Analysis (FDA) can be considered the progression of Multivariate Datas Analysis (MDA), whence MDA originally encompassed samples composed of vectors and other finite dimensional objects. It is therefore necessary to comprehend which developments in non parametric curve estimation are applied within this new field of functional data analysis, and how: that is the purpose of this work. Initially, the topic of functional data analysis is introduced, covering its summary statistics, as well as the covariance function; thereafter, there is a brief introduction to data interpolation and smoothing, concluding in the presentation of splines and B-splines as useful functions for this purpose, as well as roughness penalties. Functional clustering and functional regression models are thereafter introduced, to serve as foundation to the concluding applications. The fda package, as developed by Ramsay et al (2009, (4)), is utilized throughout.