# Diurnal Physical Activity Patterns across Ages in a Large UK Based Cohort: The UK Biobank Study

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data and Preprocessing

#### UK Biobank: Accelerometry Sub-Study

#### 2.2. Methodology and Analysis

#### Physical Activity Acceleration Profiles

#### Curve Registration

#### Functional Regression

#### Computation, Software, and Reproducibility

`mgcv::bam()`function in the

`mgcv`[34] package, and registration is performed using the

`registr`package [35].

`R`package

`rnhanesdata`[7,37]. Our analysis is reproduced using the NHANES data via a supplementary markdown file to be uploaded to Github upon publication.

#### 2.3. Populations of Comparison

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BLSA | Baltimore Longitudinal Study of Aging |

ENMO | Euclidean Norm Minus One |

FDA | Functional Data Analysis |

FoSR | Function on Scalar Regression |

FPCA | Functional Principal Component Analysis |

NHANES | National Health and Nutrition Examination Survey |

PA | Physical Activity |

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**Figure 1.**Toy Registration Example. Black and gray colors denote two separate subjects, and panels represent (

**A**) binary activity profiles, (

**B**) unregistered probability curves, (

**C**) warping functions, and (

**D**) binary activity profiles and probability curves that have been aligned using the warping functions in panel (

**C**).

**Figure 2.**Estimated population average acceleration (ENMO) by age and time of day for Males (

**panel A**), Females (

**panel B**), and the estimated difference between Males and Females (

**panel C**). Color intensity in panels A/B indicates lower (purple) versus higher (green) levels of activity. Color intensity in panel C indicates Males are less (purple), the same (white), or more (green) active as compared to Females, with a solid black line demarcating the transitions between regions where males are more or less active than females (i.e., the difference is 0). Estimates of these surfaces in panels A/B are obtained by fitting a function-on-scalar regression (FoSR) model separately by sex where the population average ENMO is allowed to vary smoothly in both time of day and age. The FoSR model is fit by modelling the average ENMO as a tensor product smooth of marginal spline bases, with cyclic cubic regression splines used in the time domain to respect the cyclic nature of time. Panel C is obtained by taking the difference of panels A and B.

**Figure 3.**Estimated population cumulative average acceleration (ENMO) by time of day for Males (

**panel A**), Females (

**panel B**), and the estimated difference between Males and Females (

**panel C**). separately for ages 50 (black), 60 (red), and 70 (blue) years old. Solid lines denote point estimates and dashed lines represent point-wise confidence intervals. Estimates of the curves presented in panels A/B are obtained by fitting a function-on-scalar regression (FoSR) model separately by sex where the population average activity count is allowed to vary smoothly in both time of day and age, then numerically integrating the estimated population average activity count over the 12 a.m.–12 a.m. period. The FoSR model is fit by modelling the average ENMO as a tensor product smooth of marginal spline bases, with cyclic cubic regression splines used in the time domain to respect the cyclic nature of time. Panel C is obtained by taking the difference of panels A and B.

**Figure 4.**Estimated population average probability of being active (or, more specifically, generating ENMO greater than or equal to 30 milli-gravity units) by age and time of day for Males (

**panel A**), Females (

**panel B**), and the estimated difference between Males and Females (

**panel C**). Color intensity in panels A/B indicates lower (purple) versus higher (green) probability of being active. Color intensity in panel C indicates Males are less (purple), the same (white), or more (green) likely to be active as compared to Females, with a solid black line demarcating the transitions between regions where males are more or less likely to be active than females (i.e., the difference is 0). Estimates of these surfaces in panels A/B are obtained by fitting a generalized function-on-scalar regression (FoSR) model separately by sex where the log odds of being active is allowed to vary smoothly in both time of day and age. The FoSR model is fit by modelling the log odds of being active as a tensor product smooth of marginal spline bases, with cyclic cubic regression splines used in the time domain to respect the cyclic nature of time using binary active/inactive profiles. Panel C is obtained by taking the difference of panels A and B.

**Figure 5.**Estimated population average warping functions for Males (

**panel A**), Females (

**panel B**), and the estimated difference between Males and Females (

**panel C**). separately for ages 50 (black), 60 (red), and 70 (blue) years old. Estimates of the lines presented in panels A/B are obtained by fitting a function-on-scalar regression (FoSR) model separately by sex where the population average warping function is allowed to vary smoothly in both time of day and age. The FoSR model is fit by modelling the average warping function as a tensor product smooth of marginal spline bases with cubic regression splines used in both the time and age domains. Panel C is obtained by taking the difference of panels A and B.

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**MDPI and ACS Style**

Wrobel, J.; Muschelli, J.; Leroux, A.
Diurnal Physical Activity Patterns across Ages in a Large UK Based Cohort: The UK Biobank Study. *Sensors* **2021**, *21*, 1545.
https://doi.org/10.3390/s21041545

**AMA Style**

Wrobel J, Muschelli J, Leroux A.
Diurnal Physical Activity Patterns across Ages in a Large UK Based Cohort: The UK Biobank Study. *Sensors*. 2021; 21(4):1545.
https://doi.org/10.3390/s21041545

**Chicago/Turabian Style**

Wrobel, Julia, John Muschelli, and Andrew Leroux.
2021. "Diurnal Physical Activity Patterns across Ages in a Large UK Based Cohort: The UK Biobank Study" *Sensors* 21, no. 4: 1545.
https://doi.org/10.3390/s21041545