The presence of curves that deviate markedly from the core of a set of curves can greatly affect inference and forecasting in a functional regression model. Thus their detection is key to increase the accuracy of the required estimates. This work introduces the concepts of high leverage in general functional regression models with independent and spatially correlated errors. The projection matrix, also known as Hat matrix, plays a crucial role in classical model diagnosis, since it provides a measure of leverage. We propose a generalisation of the projection matrix in both the functional and the spatial functional frameworks under two settings, when the response variable is a scalar, and when it is a function itself, the so-called total model. Commonly used influence measures are also proposed as functions of the generalised functional leverages and residuals. An application of the proposed procedures for investigating the effect of outliers on the relationship between transformation of the banking industry and the size of cooperative banks in Italy over a period of 14 years is presented.

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 L 2 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.