Abstract
This paper gives an overview presentation of spatial regression with differential regularization, a novel class of models for the accurate estimation of surfaces and spatial fields, that merges advanced statistical methodology with numerical analysis techniques. Thanks to the combination of potentialities from these two scientific areas, the proposed class of models has important advantages with respect to the classical statistical techniques used to analyze spatially distributed data. The models here described are able to efficiently deal with data distributed over irregularly shaped domains, including non-planar domains, only few methods existing in literature for this type of data structures. Moreover, they can incorporate problem-specific prior information about the spatial structure of the phenomenon under study, with a very flexible modeling of space variation, allowing naturally for anisotropy and non-stationarity. The models have a generalized additive framework with a regularizing term involving a differential quantity of the spatial field. The estimators have good inferential properties; moreover, thanks to the use of numerical analysis techniques, they are computationally highly efficient. The method is illustrated in various applied contexts, including demographic data and medical imaging data.
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Acknowledgements
This paper reviews joint works with Laura Azzimonti, Bree Ettinger, Fabio Nobile, Simona Perotto, Jim Ramsay and Piercesare Secchi. This research has been funded by the research program Dote Ricercatore Politecnico di Milano—Regione Lombardia, project “Functional data analysis for life sciences”, and by the starting grant FIRB Futuro in Ricerca, MIUR Ministero dell’Istruzione dell’Università e della Ricerca, research project “Advanced statistical and numerical methods for the analysis of high dimensional functional data in life sciences and engineering” (http://mox.polimi.it/users/sangalli/firbSNAPLE.html).
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Sangalli, L.M. (2015). Estimating Surfaces and Spatial Fields via Regression Models with Differential Regularization. In: Paganoni, A., Secchi, P. (eds) Advances in Complex Data Modeling and Computational Methods in Statistics. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-11149-0_13
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