Kriging with external drift for functional data for air quality monitoring
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Functional data featured by a spatial dependence structure occur in many environmental sciences when curves are observed, for example, along time or along depth. Recently, some methods allowing for the prediction of a curve at an unmonitored site have been developed. However, the existing methods do not allow to include in a model exogenous variables that, for example, bring meteorology information in modeling air pollutant concentrations. In order to introduce exogenous variables, potentially observed as curves as well, we propose to extend the so-called kriging with external drift—or regression kriging—to the case of functional data by means of a three-step procedure involving functional modeling for the trend and spatial interpolation of functional residuals. A cross-validation analysis allows to choose smoothing parameters and a preferable kriging predictor for the functional residuals. Our case study considers daily PM10 concentrations measured from October 2005 to March 2006 by the monitoring network of Piemonte region (Italy), with the trend defined by meteorological time-varying covariates and orographical constant-in-time variables. The performance of the proposed methodology is evaluated by predicting PM10 concentration curves on 10 validation sites, even with simulated realistic datasets on a larger number of spatial sites. In this application the proposed methodology represents an alternative to spatio-temporal modeling but it can be applied more generally to spatially dependent functional data whose domain is not a time interval.
KeywordsFunctional data modeling Functional linear model Residual kriging Particulate matter Spatial dependence
The authors are grateful to the Associate Editor and two anonymous referees whose comments and suggestions improved the reading and quality of the manuscript. Rosaria Ignaccolo’s work was partially supported by Regione Piemonte and FIRB 2012 Grant (project no. RBFR12URQJ) provided by the Italian Ministry of Education, Universities and Research. This research was also supported in part by the Spanish Ministry of Education and Science and Bancaja through Grants MTM2010-14961 and P1-1B2012-52.
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