Modeling Physical Activity Profiles in COPD Patients: A Basis Expansion Approach to Variable Domain Functional Regression Models

Physical activity plays an important role in the well-being of patients with Chronic Obstructive Pulmonary Disease (COPD). The telEPOC program, developed at the Galdakao-Usansolo University Hospital (Vizcaya, Spain), provided a telemonitoring data set resulting in a variable domain functional data set. In this sense, the aim is to study the impact of physical activity (measured as daily steps collected in a time interval delimited by the patient's participation in the experiment) on the rate of hospitalizations. To address this challenge, a novel basis expansion approach is proposed to fit a variable-domain functional regression model, which assumes the functional nature of the covariate (daily steps) and incorporates the sample data into the model without alignment. This will enrich the interpretability of the model by incorporating the information provided by the domain through a two-dimensional regression coefficient. In addition, a penalized estimation of the model is proposed in terms of a 2D P-spline penalty, which allows considering different degree of smoothness for each dimension of the functional regression coefficient (anisotropic penalty). The performance of our methodology is also tested and compared with state-of-the-art methods by means of a simulation study. Software for our method can be found in the VDPO R package available in CRAN.