Modeling spatially dependent functional data via regression with differential regularization

Abstract

We propose a method for modeling spatially dependent functional data, based on regression with differential regularization. The regularizing term enables to include problem-specific information about the spatio-temporal variation of the phenomenon under study, formalized in terms of a time-dependent partial differential equation. The method is implemented using a discretization based on finite elements in space and finite differences in time. This non-tensor product basis allows to handle efficiently data distributed over complex domains and where the shape of the domain influences the phenomenon's behavior. Moreover, the method can comply with specific conditions at the boundary of the domain of interest. Simulation studies compare the proposed model to available techniques for spatio-temporal data. The method is also illustrated via an application to the study of blood-flow velocity field in a carotid artery affected by atherosclerosis, starting from echo-color doppler and magnetic resonance imaging data.

A novel approach to the analysis of spatial and functional data over complex domains

“…Considering the example of buoy data, we can for instance describe the Gulf stream by a diffusion-transport differential equation, and use this PDE in the estimation functional (2): the resulting estimator will hence appropriately account for the fact than sea temperatures at two nearby buoys, lying in the direction of the current, are more strongly associated that sea temperature at two buoys, that have the same reciprocal distance, but lie transversely with respect to the current. Another example is offered by Azzimonti et al [2015] and Arnone et al [2019], and concerns the study of blood flow velocity within arteries, starting from ecocolor doppler acquisitions. In this application the PDE is based upon extensive problem-specific knowledge about fluid-dynamics, and specifically about heamodynamics, and formalizes the main features of the complex physics of the phenomenon under study.…”

“…These conditions may concern the value of f and/or the value of the normal derivative of f at the boundary of the domain. This permits a very flexible modeling of the behavior of the field at the boundaries of the domain, and is crucial in many applications to obtain meaningful estimates; see, e.g., Azzimonti et al [2015], Arnone et al [2019]. ement analysis or isogeometric analysis can be used to obtain an approximate solution.…”

“…In engineering problems and in many applications in the physical sciences and biosciences, the source of this complexity is the complex physics of the phenomenon under study. One example is offered by Azzimonti et al [2015] and Arnone et al [2019], that study blood flow velocity in human arteries, starting from eco-color doppler data.…”

“…Moreover, it is often the case that the phenomenon under study is characterized by some specific conditions at the boundaries of the domain of interest. For instance, in the study of blood-flow velocity, detailed by Azzimonti et al [2015] and Arnone et al [2019], the blood-flow must be zero at the arterial walls, that constitutes the boundary of the domain, due to fric- Figure 2: Left: profile of SOAR shuttle, described by Non-Uniform Rational B-Splines [Courtesy of Swiss Space Systems Holding SA]; the winglet is highlighted in yellow. Center: measurements of pressure coefficient obtained through pressure probes on the shuttle winglet.…”

“…In our experience, one key to face the challenges posed by the analysis of data characterized by complex spatial dependencies consists in developing methods that merge ideas and approaches from dif-ferent scientific disciplines, with an intense interplay of statistics, applied mathematics and engineering. This work in particular offers an expository overview of an innovative class of models, named Spatial Regression with Partial Differential Equation regularization, SR-PDE [Sangalli et al, 2013, Azzimonti et al, 2014, 2015, Ettinger et al, 2016, Dassi et al, 2015, Wilhelm et al, 2016, Lila et al, 2016a, Wilhelm and Sangalli, 2016, Bernardi et al, 2017, Arnone et al, 2019. These are regression methods with regularization terms that involves a Partial Differential Equation (PDE).…”