Spatial regression with differential regularizations


Spatial regression with differential regularizations is a novel class of models for the analysis of spatially (and space-time) 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 problem-specific 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 space-variation allows to naturally account for anisotropy and non-stationarity [2, 3]. Space-varying 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 [10], including also Poisson count data. Spatial regression with differential regularizations can also deal with data scattered over non-planar domains, specifically over Riemannian manifold domains, including surface domains with non-trivial geometries [4, 6, 7, 9]. This has fascinating applications in the earth-sciences, in the life-sciences [4, 6, 7] and in engineering [9]. The model can also be extended to space-time data [5]. Moreover, building on these regression models, it is possible to define a principal component analysis technique for signals spatially distributed over complex domains [7], 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, 7, 8, 10] and of isogeometric analysis [9], makes the models computationally very efficient. R/C++ code implementing these models is released via The Comprehensive R Archive Network [8].

CODE: fdaPDE package

References

  1. 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.
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  2. 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), 305-335.
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  3. 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.
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  4. 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. 111-131.
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  5. 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.
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  6. Ettinger, B., Perotto, S., Sangalli, L.M. (2016),
    Spatial regression models over two-dimensional manifolds,
    Biometrika, 103 (1), 71-88.
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  7. Lila, E., Aston, J.A.D., Sangalli, L.M. (2016),
    Smooth Principal Component Analysis over two-dimensional manifolds with an application to Neuroimaging,
    Annals of Applied Statistics, 10 (4), 1854-1879.
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  8. Lila, E., Sangalli, L.M., Ramsay, J.O., Formaggia, L. (2016),
    fdaPDE: functional data analysis and Partial Differential Equations; statistical analysis of functional and spatial data, based on regression with partial differential regularizations,
    R package version 0.1-4.

  9. 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, 70-89.
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  10. Wilhelm, M., Sangalli, L.M. (2016),
    Generalized Spatial Regression with Differential Regularization,
    Journal of Statistical Computation and Simulation, 86 (13), 2497-2518.
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Other publications by L.M. Sangalli



brain meshbrain meshbrain mesh
   
Study of population density over the Island of Montreal.
brain mesh   neuroimaging signal
Study of hemodynamical signals associated to neural activity over the cerebral cortex, the highly convoluted thin sheet of neural tissue that constitutes the outermost part of the brain.
artery mesh   hemodynamic signal
Study of hemodynamical forces exerted by blood flow over the wall of a carotid artery affected by an aneurysm.

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spatial, regression, spatial regression, spatial regression with differential regularization, differential regularization, PDE, partial differential equation, Laura Sangalli