A penalized regression model for spatial functional data with application to the analysis of the production of waste in Venice province


We propose a method for the analysis of functional data with complex dependencies, such as spatially dependent curves or time dependent surfaces, over highly textured domains. The models are based on the idea of regression with partial differential regularizations. In particular, we consider here two roughness penalties that account separately for the regularity of the field in space and in time. Among the various modelling features, the proposed method is able to deal with spatial domains featuring peninsulas, islands and other complex geometries. Space-time varying covariate information is included in the model via a semi-parametric framework. The proposed method is compared via simulation studies to other spatio-temporal techniques and it is applied to the analysis of the annual production of waste in the towns of Venice province.

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We thank Alessandra Menafoglio for comments on this work. We are also grateful to the Associate Editor and three anonymous referees, whose suggestions greatly improved the presentation of this work. L.M. Sangalli acknowledges funding by MIUR Ministero dell’Istruzione dell’Università e della Ricerca, FIRB Futuro in Ricerca Starting Grant 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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Correspondence to Laura M. Sangalli.

Spatio-temporal test function

Spatio-temporal test function

The spatio-temporal test function f(xyt), defined over the C-shaped domain, used in the simulation studies, is constructed as:

$$\begin{aligned} \begin{array}{lr} \cos (t)(q+x)+(y-r)^2 &{}\text {if}\ \ \ x\ge 0 \ \ \& \ \ y>0 \\ \cos (2t)(-q-x)+(-y-r)^2 &{}\text {if}\ \ \ x\ge 0 \ \ \& \ \ y\le 0 \\ \cos (t)(-\arctan \left(\frac{y}{x}\right)r)+(\sqrt{x^2+y^2}-r)^2 K(x,y) &{}\text {if}\ \ \ x< 0 \ \ \& \ \ y>0 \\ \cos (2t)(-\arctan \left(\frac{y}{x}\right)r)+(\sqrt{x^2+y^2}-r)^2 K(x,y) &{}\text {if}\ \ \ x< 0 \ \ \& \ \ y\le 0 \end{array}, \end{aligned}$$

where \(K(x,y)=\left(\frac{y}{r_0}{\mathbbm{1}}_{|y|\le r_0 \& x>-r}+{\mathbbm{1}}_{|y|> r_0 || x\le -r}\right)^2\)\({\mathbbm{1}}_A\) denotes the indicator function of the subset A, \(r_0 = 0.1\), \(r = 0.5\) and \(q=\frac{\pi r}{2}\).

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Bernardi, M.S., Sangalli, L.M., Mazza, G. et al. A penalized regression model for spatial functional data with application to the analysis of the production of waste in Venice province. Stoch Environ Res Risk Assess 31, 23–38 (2017). https://doi.org/10.1007/s00477-016-1237-3

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  • Space-time model
  • Differential regularization
  • Finite elements