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Roughness regularization for functional data analysis with free knots spline estimation

Anna De Magistris, Valentina De Simone, Elvira Romano, Gerardo Toraldo

TL;DR

This work proposes a free knots spline estimation method for functional data with two penalty terms and demonstrates its performance by comparing the results of several clustering methods on simulated and real data.

Abstract

In the era of big data, an ever-growing volume of information is recorded, either continuously over time or sporadically, at distinct time intervals. Functional Data Analysis (FDA) stands at the cutting edge of this data revolution, offering a powerful framework for handling and extracting meaningful insights from such complex datasets. The currently proposed FDA me\-thods can often encounter challenges, especially when dealing with curves of varying shapes. This can largely be attributed to the method's strong dependence on data approximation as a key aspect of the analysis process. In this work, we propose a free knots spline estimation method for functional data with two penalty terms and demonstrate its performance by comparing the results of several clustering methods on simulated and real data.

Roughness regularization for functional data analysis with free knots spline estimation

TL;DR

This work proposes a free knots spline estimation method for functional data with two penalty terms and demonstrates its performance by comparing the results of several clustering methods on simulated and real data.

Abstract

In the era of big data, an ever-growing volume of information is recorded, either continuously over time or sporadically, at distinct time intervals. Functional Data Analysis (FDA) stands at the cutting edge of this data revolution, offering a powerful framework for handling and extracting meaningful insights from such complex datasets. The currently proposed FDA me\-thods can often encounter challenges, especially when dealing with curves of varying shapes. This can largely be attributed to the method's strong dependence on data approximation as a key aspect of the analysis process. In this work, we propose a free knots spline estimation method for functional data with two penalty terms and demonstrate its performance by comparing the results of several clustering methods on simulated and real data.
Paper Structure (8 sections, 1 theorem, 25 equations, 8 figures, 7 tables)

This paper contains 8 sections, 1 theorem, 25 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Smoothing Functional Data
  3. Smoothing functional data with a roughness penalty
  4. Free knots spline
  5. Computational experiments
  6. Synthetic datasets
  7. Application: Real Datasets
  8. Conclusions

Key Result

Proposition 1

If the $\boldsymbol{\Phi}(\textbf{k})$ is full rank, the matrix $\textbf{H(k)}$ is SPD and let $\sigma(\textbf{H})$ be the set of eigenvalues of matrix $\textbf{H(k)}$ we have $\sigma(\textbf{H})\subseteq [\sigma^-, \sigma^+]$, with .

Figures (8)

  • Figure 1: Simulated data.
  • Figure 2: Comparison of FS0,FS1 and FS2 for data approximation in four case.
  • Figure 3: Evaluation of the number of clusters in the simulation dataset. The optimal number is indicated from the dotted line.
  • Figure 4: Clustering structure: Cluster 1 in red, Cluster 2 in blue, Cluster 3 in green, and Cluster 4 in orange. First column represents synthetic datasets,the second column represents the data approximated using FS0 segmented into clusters, the third column shows the data approximated using FS1 segmented into clusters, the last column illustrates the data approximated using FS2 segmented into clusters.
  • Figure 5: New Cases of Covid19 in different Country from 2020 to 2021.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof