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Space time data over complex domains
Spatial regression with differential regularization is a novel class of models that merges advanced statistical methodology with scientific computing techniques. Thanks to the combination of potentialities from these two scientific areas, these methods have important advantages with respect to classical techniques used in spatial data analysis. Spatial regression with differential regularization is able to efficiently handle data distributed over irregularly shaped domains and can comply with specific conditions at the domain boundaries. Moreover, these models have the capacity to incorporate problem-specific a priori information about the phenomenon under study, formalized in terms of partial differential equations. The use of numerical analysis techniques, and specifically of Finite Elements, makes the models computationally very efficient.
For reference on the methods, see
- Sangalli, L.M., Ramsay, J.O., Ramsay, T.O. (2013), Spatial spline regression models, Journal of the Royal Statistical Society Ser. B, Statistical Methodology, 75, 4, 681–703 PDF
- Azzimonti, L., Sangalli, L.M., Secchi, P., Domanin, M., Nobile, F. (2015), Blood flow velocity field estimation via spatial regression with PDE penalization, Journal of the American Statistical Association, Theory and Methods, 110 (511), 1057-1071 PDF
Spatial regression with differential regularization is currently implemented in the R package fdaPDE. The package cannot currently handle space-time data. This project aims to extend the current functions to the analysis of data with a spatio-temporal dependence. This will lead to the new version of the existing R package, that will be submitted to CRAN. For reference on the extension to spatio-temporal data, see:
- Bernardi, M.S., Sangalli, L.M., Mazza, G., Ramsay, J.O. (2017), A penalized regression model for spatial functional data with application to the analysis of the production of waste in Venice province, Stochastic Environmental Research and Risk Assessment, 31, 1, 23-38 PDF
The project aims to extend the current fdaPDE functions to the analysis of data distributed over spatio-temporal domains. The core of the fdaPDE package is implemented in C++ and interfaces with R. Therefore, the project requires programming in both languages. Besides the extension of the main algorithms in C++, the new functionalities will require:
- proper documentation to be included in fdaPDE manual and in R vignettes,
- extension of the R interface to handle the case of space-time data,
- development of appropriated test cases.
The new and enhanced fdaPDE package, that will include the functions for space-time data, will enable the analysis of spatio-temporal data over complex domains. This will offer a new and important alternative with respect to classical approaches to spatio-temporal data analysis, that mostly work on regular domains, such as rectangular and tensorized domains.
The mentors for this project will be Laura M. Sangalli (laura.sangalli@polimi.it, http://www1.mate.polimi.it/~sangalli), Luca Formaggia (luca.formaggia@polimi.it, http://www1.mate.polimi.it/~forma/), MOX - Laboratory for Modeling and Scientific Computing, Department of Mathematics, Politecnico di Milano, and Eardi Lila (e.lila@maths.cam.ac.uk), Cambridge Centre for Analysis, University of Cambridge.