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Advances in Complex Data Modeling and Computational Methods in Statistics pp 191–209Cite as

Estimating Surfaces and Spatial Fields via Regression Models with Differential Regularization

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Part of the book series: Contributions to Statistics ((CONTRIB.STAT.))

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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References

  1. Augustin, N.H., Trenkel, V.M., Wood, S.N., Lorance, P.: Space-time modelling of blue ling for fisheries stock management issue. Environmetrics 24(Part 2), 109–119 (2013)

    Google Scholar 

  2. Azzimonti, L.: Blood flow velocity field estimation via spatial regression with PDE penalization. Ph.D. Thesis, Politecnico di Milano (2013)

    Google Scholar 

  3. Azzimonti, L., Nobile, F., Sangalli, L.M., Secchi, P.: Mixed finite elements for spatial regression with PDE penalization. SIAM/ASA J. Uncertain. Quantif. 2(Part 1), 305–335 (2013)

    Google Scholar 

  4. Azzimonti, L., Sangalli, L.M., Secchi, P., Domanin, M., Nobile, F.: Blood flow velocity field estimation via spatial regression with PDE penalization. J. Am. Stat. Assoc. Theory Methods. (2014) doi:10.1080/01621459.2014.946036

    Google Scholar 

  5. Baramidze, V., Lai, M.-J., Shum, C.K.: Spherical splines for data interpolation and fitting. SIAM J. Sci. Comput. 28, 241–259 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Buja, A., Hastie, T., Tibshirani, R.: Linear smoothers and additive models. Ann. Stat. 17, 453–555 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chung, M.K., Robbins, S.M., Dalton, K.M., Davidson, R.J., Alexander, A.L., Evans, A.C.: Cortical thickness analysis in autism with heat kernel smoothing. NeuroImage 25, 1256–1265 (2005)

    Article  Google Scholar 

  8. Dassi, F., Ettinger, B., Perotto, S., Sangalli, L.M.: A mesh simplification strategy for a spatial regression analysis over the cortical surface of the brain. Technical Report MOX 31/2013, Dipartimento di Matematica, Politecnico di Milano (2013)

    Google Scholar 

  9. D’Elia, M., Perego, M., Veneziani, A.: A variational data assimilation procedure for the incompressible Navier-Stokes equations in hemodynamics. SIAM J. Sci. Comput. 52(Part 4), 340–359 (2012)

    Google Scholar 

  10. Dierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal Surfaces, vol. 1, 2nd edn. Springer, Heidelberg (2010)

    Google Scholar 

  11. Ettinger, B., Guillas, S., Lai, M.-J.: Bivariate splines for ozone concentration forecasting. Environmetrics 23(Part 4), 317–328 (2012)

    Google Scholar 

  12. Ettinger, B., Perotto, S., Sangalli, L.M.: Spatial regression models over two-dimensional manifolds. Technical Report MOX 54/2012, Dipartimento di Matematica, Politecnico di Milano (2012)

    Google Scholar 

  13. Formaggia, L., Quarteroni, A., Veneziani, A.: Cardiovascular mathematics: modeling and simulation of the circulatory system. Springer, Berlin (2009)

    Book  Google Scholar 

  14. Gneiting, T.: Strictly and non-strictly positive definite functions on spheres. Bernoulli 19, 1087–1500 (2013)

    Article  MathSciNet  Google Scholar 

  15. Guillas, S., Lai, M.: Bivariate splines for spatial functional regression models. J. Nonparametr. Stat. 22(Part 4), 477–497 (2010)

    Google Scholar 

  16. Hagler, D.J., Saygin, A.P., Sereno, M.I.: Smoothing and cluster thresholding for cortical surface-based group analysis of fMRI data. NeuroImage 33, 1093–1103 (2006)

    Article  Google Scholar 

  17. Jun, M.: Non-stationary cross-covariance models for multivariate processes on a globe. Scand. J. Stat. 38, 726–747 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lindgren, F., Rue, H., Lindström, J.: An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J. R. Stat. Soc. Ser. B Stat. Methodol. 73, 423–498, with discussions and a reply by the authors (2011)

    Google Scholar 

  19. Marra, G., Miller, D., Zanin, L.: Modelling the spatiotemporal distribution of the incidence of resident foreign population. Stat. Neerl. 66, 133–160 (2012)

    Article  MathSciNet  Google Scholar 

  20. Quarteroni, A.: Numerical Models for Differential Problems. Springer, Milan (2013)

    Google Scholar 

  21. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2011)

    Google Scholar 

  22. Ramsay, T.: Spline smoothing over difficult regions. J. R. Stat. Soc. Ser. B Stat. Methodol. 64, 307–319 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ramsay, J.O., Silverman, B.W.: Functional Data Analysis, 2nd edn. Springer, Berlin (2005)

    Google Scholar 

  24. Sangalli, L.M., Ramsay, J.O., Ramsay, T.O.: Spatial spline regression models. J. R. Stat. Soc. Ser. B Stat. Methodol. 75(Part 4), 681–703 (2013)

    Google Scholar 

  25. Stuart, A.: Inverse problems: a Bayesian perspective. Acta Numer. 19, 451–559 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wahba, G.: Spline interpolation and smoothing on the sphere. SIAM J. Sci. Stat. Comput. 2, 5–16 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  27. Wood, S.N., Bravington, M.V., Hedley, S.L.: Soap film smoothing. J. R. Stat. Soc. Ser. B Stat. Methodol. 70, 931–955 (2008)

    Article  MathSciNet  Google Scholar 

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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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Authors and Affiliations

  1. MOX - Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Laura M. Sangalli

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  1. Laura M. Sangalli
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Correspondence to Laura M. Sangalli .

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  1. Dipartimento di Matematica, Politecnico di Milano, Milano, Milano, Italy

    Anna Maria Paganoni

  2. Dipartimento di Matematica, Politecnico di Milano, Milano, Milano, Italy

    Piercesare Secchi

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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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