# Functional Data Analysis for Imaging Mean Function Estimation: Computing Times and Parameter Selection

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## Abstract

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## 1. Introduction

#### 1.1. Functional Data Analysis

#### 1.2. Applicability of FDA to Imaging Data

#### 1.3. Objectives

## 2. Materials and Methods

#### 2.1. Imaging Data

#### 2.2. Delaunay Triangulations

#### 2.3. Mean Function and SCC for One-Group Setup

#### 2.4. Mean Function and SCC for Two-Group Setup

## 3. Results

#### 3.1. Delaunay Triangulations

#### 3.2. One-Group Mean Function and SCC Estimation

#### 3.3. Two-Group Mean Function and SCC Estimation

## 4. Discussion

## 5. Computer Specifications

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FDA | Functional Data Analysis |

SCC | Simultaneous Confidence Corridor |

PET | Positron Emissionn Tomography |

18F-FDG | 18-Fluorodeoxyglucose |

AD | Alzheimer’s Disease |

SPM | Statistical Parametric Mapping |

## References

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**Figure 1.**Delaunay triangulations produced for this practical case with real brain imaging data. Increasing triangulation’s degree of fineness is measured by parameter N. (

**a**) N = 10. (

**b**) N = 25. (

**c**) N = 50.

**Figure 2.**(

**a**) Scale; (

**b**) Lower SCC; (

**c**) Mean Function; and (

**d**) Upper SCC for brain imaging data. SCCs calculated for $\alpha =0.05$ using Delaunay triangulations (fineness degree $N=10$).

**Figure 3.**Example of results for a two-sample approach comparing two sets of images: one conformed by control patients and another by pathological (AD) patients. Blue indicates detected hypo-activity while orange indicates hyper-activity. Delaunay triangulations’ fineness degree $N=10$. (

**a**) $\alpha =0.1$. (

**b**) $\alpha =0.05$. (

**c**) $\alpha =0.01$.

**Figure 4.**Computing times for Delaunay triangulations for complex neuroimaging data structures with growing fineness degree values. Curve fitted with local (LOESS) regression.

**Figure 5.**Computing times for one-group mean function and SCC estimation for neuroimaging data with growing value of triangulation fineness degree. Curve fitted using local (LOESS) regression.

**Figure 6.**Computing times for two-group mean function and SCC estimation for the differences between groups with growing value of triangulation fineness degree. Curve fitted using local (LOESS) regression.

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**MDPI and ACS Style**

Arias-López, J.A.; Cadarso-Suárez, C.; Aguiar-Fernández, P.
Functional Data Analysis for Imaging Mean Function Estimation: Computing Times and Parameter Selection. *Computers* **2022**, *11*, 91.
https://doi.org/10.3390/computers11060091

**AMA Style**

Arias-López JA, Cadarso-Suárez C, Aguiar-Fernández P.
Functional Data Analysis for Imaging Mean Function Estimation: Computing Times and Parameter Selection. *Computers*. 2022; 11(6):91.
https://doi.org/10.3390/computers11060091

**Chicago/Turabian Style**

Arias-López, Juan A., Carmen Cadarso-Suárez, and Pablo Aguiar-Fernández.
2022. "Functional Data Analysis for Imaging Mean Function Estimation: Computing Times and Parameter Selection" *Computers* 11, no. 6: 91.
https://doi.org/10.3390/computers11060091