pablomm / tfg

Bachelor's thesis: Functional data analysis: interpolation, registration, and nearest neighbors in scikit-fda

Functional data analysis: interpolation, registration, and nearest neighbors in scikit-fda

Bachelor's thesis to obtain a double degree in Computer Science and Mathematics at the Autonomous University of Madrid. This repository contains the files used to generate the undergraduate thesis document. The contributions made in this work can be found in the scikit-fda project repository.

Abstract

Functional Data Analysis (FDA) is a branch of Statistics devoted to the study of random quantities that depend on a continuous parameter, such as time series or curves in space. In FDA the data instances can be viewed as random functions sampled from an underlying stochastic process.

In this work we consider three different tasks in FDA: the use of interpolation techniques to estimate the values of the functions at unobserved points, the registration of these type of data, and the solution of classification and regression problems in which the instances are characterized by functional attributes. In particular, in this project the scikit-fda package for FDA in Python has been extended with functionality in these areas.

Generally, the data instances considered in FDA consist of a collection of observations at a discrete values of the parameter on which they depend (e.g. time or space). For some applications it is convenient, and in some cases necessary, to estimate the value of these functions at unobserved points. This can be achieved through the use of interpolation from the available measurements.

In some applications, the functions observed have similar shapes, but exhibit variability whose origin can be traced to distortions in the scale of the continuous parameter on which the data depend. Registration consists in characterizing this variability and eliminating it from the sample considered.

In this work we also address classification and regression problems with data that are characterized by functions. Specifically, we design nearest neighbors estimators based on the notion of closeness among samples.

Specifically, in this work the scikit-fda package has been extended to include interpolation methods based on splines. The package has also been endowed with tools for data registration using either shifts, landmark alignment, or elastic registration, which makes use of the Fisher-Rao metric to align the functions in a sample. In addition, models based on nearest neighbors have been included to carry out regression, with both scalar and functional response, and classification.