Optimally weighted L2 distances for spatially dependent functional data

In recent years, in many application fields, extracting information from data in the form of functions is of most interest rather than investigating traditional multivariate vectors. Often these functions have complex spatial dependences that need to be accountied for using appropriate statistical analysis. Spatial Functional Statistics presents a fruitful analytics framework for this analysis. The definition of a distance measure between spatially dependent functional data is critical for many functional data analysis tasks such as clustering and classification. For this reason, and based on the specific characteristics of functional data, several distance measures have been proposed in the last few years. In this work we develop a weighted distance for spatially dependent functional data, with an optimized weight function. Assuming a penalized basis representation for the functional data, we consider weight functions depending also on the spatial location in two different situations: a classical georeferenced spatial structure and a connected network one. The performance of the proposed distances are compared using standard metrics applied to both real and simulated data analysis.