Presentation of the project

Cycling

Power Data

Mean Maximal Power Curves

• Consider some generic exercise which last $T$ seconds.

• Let $Z=\{z_t,t=1,2,\dots ,T\}$ a sequence of observation of the power output.

• Assume that, given $t_1,t_2$ two time stamps such that $t_2>t_1$, $t_2-t_1$ is constant.

• An MMP curve is defined as $$X\left(t\right)=\underset{t_2-t_1=t}{max}\frac{z_{t_1}+\cdots +z_{t_2}}{t_2-t_1}.$$

Mean Maximal Power Curves

Mean per phase

Standard deviation per phase

Tests for the difference of the means

Model

• MMP curves consist of random realizations from a stochastic process $X=\left\{X\right(t):t\in [1,7200\left]\right\}$ with continuous trajectories.

• We consider the following model $$X_{ijl}\left(t\right)={\mu }_j\left(t\right)+B_{jl}\left(t\right)+E_{ijl}\left(t\right)$$ where

  • $X_{ijl}\left(t\right)$: MMP output for a particular observation.
  • ${\mu }_j\left(t\right)$: fixed effect for the phase of the menstrual cycle.
  • $B_{jl}\left(t\right)$: phase-specific functional random intercept for training.
  • $E_{ijl}\left(t\right)$: smooth error term accounting for observation-specific variability.

Eigenfunctions

Takeaway ideas

• Power output data exhibits the variable nature of performance in women's professional cycling.

• May help the athletes to have a better understanding of their performance regarding their menstrual cycle.

• Hard to give a definitive conclusion of the influence of menstrual cycle on the performance here!

• Add more menstrual cycles, determine a better classification of the trainings and include a random effect for athletes.

References

Cederbaum, J. (2017). “Functional linear mixed models for complex correlation structures and general sampling grids”. Text.PhDThesis.

Liebl, D. and M. Reimherr (2022). Fast and Fair Simultaneous Confidence Bands for Functional Parameters. DOI: 10.48550/arXiv.1910.00131. arXiv: 1910.00131 [math, stat].

McNulty, K. L., K. J. Elliott-Sale, E. Dolan, et al. (2020). “The Effects of Menstrual Cycle Phase on Exercise Performance in Eumenorrheic Women: A Systematic Review and Meta-Analysis”. In: Sports Medicine 50.10, pp. 1813–1827. ISSN: 1179-2035. DOI: 10.1007/s40279-020-01319-3.