Functional Data Analysis for Predicting Pediatric Failure to Complete Ten Brief Exercise Bouts

Abstract

Physiological response to physical exercise through analysis of cardiopulmonary measurements has been shown to be predictive of a variety of diseases. Nonetheless, the clinical use of exercise testing remains limited because interpretation of test results requires experience and specialized training. Additionally, until this work no methods have identified which dynamic gas exchange or heart rate responses influence an individual's decision to start or stop physical activity. This research examines the use of advanced machine learning methods to predict completion of a test consisting of multiple exercise bouts by a group of healthy children and adolescents. All participants could complete the ten bouts at low or moderate-intensity work rates, however, when the bout work rates were high-intensity, 50% refused to begin the subsequent exercise bout before all ten bouts had been completed (task failure). We explored machine learning strategies to model the relationship between the physiological time series, the participant's anthropometric variables, and the binary outcome variable indicating whether the participant completed the test. The best performing model, a generalized spectral additive model with functional and scalar covariates, achieved 93.6% classification accuracy and an F1 score of 93.5%. Additionally, functional analysis of variance testing showed that participants in the 'failed' and 'success' groups have significantly different functional means in three signals: heart rate, oxygen uptake rate, and carbon dioxide uptake rate. Overall, these results show the capability of functional data analysis with generalized spectral additive models to identify key differences in the exercise-induced responses of participants in multiple bout exercise testing.

Fig. 7.

Ten bouts of Heart Rate data, originally observed breath-by-breath and time interpolated to a second-by-second representation. Each participant’s observations are shown with a unique color.

Fig. 8.

Ten bouts of Respiratory Rate data, originally observed breath-by-breath and time interpolated to a second-by-second representation.

Fig. 9.

Ten bouts of O₂ Uptake Rate data, originally observed breath-by-breath and time interpolated to a second-by-second representation.

Fig. 10.

Ten bouts of CO₂ Uptake Rate data, originally observed breath-by-breath and time interpolated to a second-by-second representation.

Fig. 11.

Four bouts of Heart Rate after converting the discrete time series to 78 smoothed and continuously registered functional data objects.

Fig. 12.

Four bouts of Respiratory Rate after converting the discrete time series to 78 smoothed and continuously registered functional data objects.

Fig. 13.

Four bouts of O₂ uptake rate after converting the discrete time series to 78 smoothed and continuously registered functional data objects.

Fig. 14.

Four bouts of CO₂ uptake rate after converting the discrete time series to 78 smoothed and continuously registered functional data objects.

Fig. 15.

Comparison of functional means for the RR signal during the first four exercise bouts. p-value = 0.186. Participants who failed to complete ten bouts during MBEB are labelled as ‘1’ and colored green. The red line depicts the functional mean for ‘task-completers.’ The black line indicates the mean trajectory for all 78 participants. The plot on the right shows Heart Rate curves for MBEB ‘task-failures’ (green) and ‘task-completers’ (red), bootstrapped 500 times. The black line indicates the mean trajectory for all 78 participants. Notice that there is substantial overlap between the two groups’ signals; quitters and non-quitters have virtually indistinguishable respiratory rates.

Fig. 16.

Comparison of functional means for the $\dot{V}{\text{O}}_2$ signal during the first four exercise bouts. p-value = 0. Participants who failed to complete ten bouts during MBEB are labelled as ‘1’ and colored green. The red line depicts the functional mean for ‘task-completers.’ The black line indicates the mean trajectory for all 78 participants. The plot on the right shows Heart Rate curves for MBEB ‘task-failures’ (green) and ‘task-completers’ (red), bootstrapped 500 times. The black line represents the bootstrapped mean function for 78 participants.

Fig. 17.

Comparison of functional means for the $\dot{V}{\text{CO}}_2$ signal during the first four exercise bouts. p-value = 0. Participants who failed to complete ten bouts during MBEB are labelled as ‘1’ and colored green. The red line depicts the functional mean for ‘task-completers.’ The black line indicates the mean trajectory for all 78 participants. The plot on the right shows Heart Rate curves for MBEB ‘task-failures’ (green) and ‘task-completers’ (red), bootstrapped 500 times. The black line represents the bootstrapped mean function for 78 participants.

Fig. 1.

One participant’s second-by-second heart rate for the full MBEB session. In general, HR was the signal that contained the least noise in our data set; individual exercise bouts are very easily discerned.

Fig. 2.

One participant’s second-by-second respiratory rate for the full MBEB session. In general, RR was the signal that contained the most noise in our data set; individual exercise bouts are difficult to discern.

Fig. 3.

Example estimation of the smoothing parameter λ. An appropriate level of smoothing was determined by visual inspection of the relationship between GCV and DoF in the smoothed model. This procedure is explained in depth in [21]. This figure shows a minimal GCV when the model contains 350 DoF, which corresponds to a λ near 200. Thus, 200 was chosen as the smoothing penalty for the set of HR curves, and the fit was validated after visual inspection of the smoothness (see Fig. 4). This process was repeated for all variables.

Fig. 4.

Heart Rate data after converting the discrete time series to 78 smoothed and registered curves. Each participant’s time series is represented as an individually colored function.

Fig. 5.

Visual output of the functional permutation t-test between Early- and Late-puberty males. The blue curve shows the t-statistic for the observed values. The green curve represents the 95% quantiles, and the dashed red line is the 95% quantile of the maximum of null distribution t-statistics. The t-test confirms that the derivatives are indeed different except in the regions of overlap (the first few moments of exercise). This could signify a fundamental difference in the physiology between puberty groups when holding gender status constant.

Fig. 6.

Comparison of functional means for the Heart Rate signal [X(t)] during the first four exercise bouts. Task-failures are labelled as ‘1’ with a solid green mean function. ‘Task-completers’ are labelled ‘0’ with a solid red mean function. The black line indicates the mean trajectory for all participants.