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Summary Object-oriented data analysis is a fascinating and evolving field in modern statistical science, with the potential to make significant contributions to biomedical applications. This statistical framework facilitates the development of new methods to analyze complex data objects that capture more information than traditional clinical biomarkers. This paper applies the object-oriented framework to analyze physical activity levels, measured by accelerometers, as response objects in a regression model. Unlike traditional summary metrics, we utilize a recently proposed representation of physical activity data as a distributional object, providing a more nuanced and complete profile of individual energy expenditure across all ranges of monitoring intensity. A novel hybrid Fréchet regression model is proposed and applied to US population accelerometer data from National Health and Nutrition Examination Survey (NHANES) 2011 to 2014. The semi-parametric nature of the model allows for the inclusion of nonlinear effects for critical variables, such as age, which are biologically known to have subtle impacts on physical activity. Simultaneously, the inclusion of linear effects preserves interpretability for other variables, particularly categorical covariates such as ethnicity and sex. The results obtained are valuable from a public health perspective and could lead to new strategies for optimizing physical activity interventions in specific American subpopulations. 

A novel non-parametric estimator of the correlation between grouped measurements of a quantity is proposed in the presence of noise. The main motivation is functional brain network construction from fMRI data, where brain regions correspond to groups of spatial units, and correlation between region pairs defines the network. The challenge resides in the fact that both noise and intra-regional correlation lead to inconsistent inter-regional correlation estimation using classical approaches. While some existing methods handle either one of these issues, no nonparametric approaches tackle both simultaneously. To address this problem, a trade-off between two procedures is proposed: correlating regional averages, which is not robust to intra-regional correlation; and averaging pairwise inter-regional correlations, which is not robust to noise. To that end, the data is projected onto a space where Euclidean distance is used as a proxy for sample correlation. Hierarchical clustering is then leveraged to gather together highly correlated variables within each region prior to inter-regional correlation estimation. The convergence of the proposed estimator is analyzed, and the proposed approach is empirically shown to surpass several other popular methods in terms of quality. Illustrations on real-world datasets that further demonstrate its effectiveness are provided.