Spatial regression with differential regularizations is a novel class of models for the analysis of spatially (and spacetime) distributed data,
that merges advanced statistical methodology and 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 classical spatial data analysis techniques. Spatial Regression
with differential regularizations is able to efficiently deal with data distributed
over irregularly shaped domains, with complex boundaries, strong concavities
and interior holes [1]. Moreover, it can comply with specific conditions at
the boundaries of the problem domain [1, 2, 3], which is fundamental in many applications
to obtain meaningful estimates. The proposed models have the capacity to incorporate problemspecific prior
information about the spatial structure of the phenomenon under study, formalized in terms of a governing partial differential equation; this very flexible modeling of spacevariation allows to naturally account for anisotropy and nonstationarity [2, 3, 10, 11]. Spacevarying covariate information is also included in the models via a semiparametric
framework. A generalized linear setting allows for response variables having any distribution within the exponential family [8], including also Poisson count data. Density estimation from pointpattern data is described in [18, 19]. Spatial regression with differential regularizations can also deal with data scattered over nonplanar domains, specifically over Riemannian manifold domains, including surface domains with nontrivial geometries [4, 5, 6, 7, 13]. This has fascinating applications in the earthsciences, in the lifesciences [4, 5, 6, 13] and in engineering [7]. The models can also be extended to spacetime data and spatially dependent functional data [9, 11, 12, 13]. Moreover, building on these regression models, it is possible to define a principal component analysis technique for signals spatially distributed over complex domains [6], thus enabling population studies. The use of advanced numerical analysis techniques, and in particular of the finite element method [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 15, 18, 19, 20] and of isogeometric analysis [7], makes the models computationally very efficient. Some first results on the consistency of the estimators are given in [17] and on inferential methods for the linear part of the models in [16]. Introductions to this line of reserach are given in [14, 15]. R/C++ code implementing these models is released via The Comprehensive R Archive Network [20].
CODE: fdaPDE package
References
*  indicates Corresponding Author
#  indicates a supervised Master student, PhD student or Postdoctoral collaborator
 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, 681703. PDF Postprint Code
 Azzimonti#, L., Nobile, F., Sangalli*, L.M., Secchi, P. (2014),
Mixed Finite Elements for spatial regression with PDE penalization,
SIAM/ASA Journal on Uncertainty Quantification, 2 (1), 305335. PDF Postprint Code
 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), 10571071. PDF Postprint Supplementary material
Code
 Dassi#, F., Ettinger#, B., Perotto, S., Sangalli, L.M. (2015),
A mesh simplification strategy for a spatial regression analysis over the cortical surface of the brain,
Applied Numerical Mathematics, Vol. 90, pp. 111131. PDF Postprint Code
 Ettinger#, B., Perotto, S., Sangalli*, L.M. (2016),
Spatial regression models over twodimensional manifolds,
Biometrika, 103 (1), 7188. PDF Postprint Supplementary material Code
 Lila#, E., Aston, J.A.D., Sangalli, L.M. (2016),
Smooth Principal Component Analysis over twodimensional manifolds with an application to Neuroimaging, Annals of Applied Statistics, 10 (4), 18541879. PDF Postprint Code
 Wilhelm#, M., Dede', L., Sangalli, L.M., Wilhelm, P. (2016),
IGS: an IsoGeometric approach for Smoothing on surfaces,
Computer Methods in Applied Mechanics and Engineering, 302, 7089. PDF Postprint
 Wilhelm#, M., Sangalli*, L.M. (2016),
Generalized Spatial Regression with Differential Regularization,
Journal of Statistical Computation and Simulation, 86 (13), 24972518. PDF Postprint Code
 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), 2338. PDF Postprint Code
 Bernardi#, M.S., Carey, M., Ramsay, J.O., and Sangalli*, L.M. (2018),
Modeling spatial anisotropy via regression with partial differential regularization,
Journal of Multivariate Analysis, 167, 1530.
PDF Postprint Code
 Arnone#, E., Azzimonti#, A., Nobile, F., and Sangalli*, L.M. (2019),
Modelling spatially dependent functional data via regression with differential regularization,
Journal of Multivariate Analysis, 170, 275295.
PDF Postprint Code
 Arnone#, E., Sangalli*, L.M. and Vicini, A. (2021),
Smoothing spatiotemporal data with complex missing data patterns,
Statistical Modelling, DOI: 10.1177/1471082X211057959.
PDF Postprint Code
 Ponti#, L., Perotto, S. and Sangalli, L.M. (2022),
A PDEregularized smoothing method for spacetime data over manifolds with application to medical data,
International Journal for Numerical Methods in Biomedical Engineering, to appear.
Code
 Sangalli*, L.M. (2020),
A novel approach to the analysis of spatial and functional data over complex domains,
Quality Engineering, 32, 2, 181190, followed by discussions and a rejoinder by the author.
PDF Postprint Code
Sangalli*, L.M. (2020), Rejoinder,
Quality Engineering, 32, 2, 197198.
PDF
 Sangalli*, L.M. (2021),
Spatial regression with partial differential equation regularization,
International Statistical Review, 89, 3, 505531.
PDF
Code
 Arnone#, E., Kneip, A., Nobile, F. and Sangalli*, L.M. (2022),
Some first results on the consistency of spatial regression with partial differential equation regularization, Statistica Sinica, 32, 209238.
PDF Postprint
 Ferraccioli#, F., Sangalli*, L.M., Finos, L. (2022),
Some first inferential tools for spatial regression with differential regularization,
Journal of Multivariate Analysis, DOI: 10.1016/j.jmva.2021.104866.
PDF
Postprint
 Ferraccioli#, F., Arnone#, E., Finos, L., Ramsay, J.O., Sangalli*, L.M. (2021),
Nonparametric density estimation over
complicated domains,
Journal of the Royal Statistical Society Ser. B, Statistical Methodology, 83, 346368.
PDF Code
 Arnone#, E., Ferraccioli#, F., Pigolotti#, C., Sangalli*, L.M. (2022),
A roughness penalty approach to estimate densities over twodimensional manifolds,
Computational Statistics and Data Analysis, 174, 107527.
PDF
Code
 Arnone#, E., Sangalli, L. M., Lila#, E., Ramsay, J., and Formaggia, L. (2022),
fdaPDE: StatisticalAnalysis of Functional and Spatial Data, Based on Regression with PDE Regularization. R package version 1.18.
Other publications by L.M. Sangalli
