Visual tools to identify influential observations in spatial data

Abstract

We adapted the hair-plot, proposed by Genton and Ruiz-Gazen (2010), to identify and vi- sualize influential observations in spatial data. Three graphic tools were created: the bihair-plot, the principal components hair-plot and functional hair-plot. The first tool depict trajectories of the values of a spatial semivariance estimator when adding a perturbation to each observation of a vector of spatial data observed considering two lags. The second describes trajectories of the principal components of a spatial semivariance estimator values for all lags when each observation of data is perturbed, making it possible to identify influential observations in spa- tial data containing as much information as possible from the data set. The third is obtained from the values of the trace-semivariogram estimator when the data receive a disturbance. The estimators considered in the study were the sample semivariogram for univariate case, sample cross-semivariogram for bivariate case and sample trace-semivariogram for functional data. Another method used to obtain the cross-semivariogram was Minimum Volume Ellipsoid, which is more sensitive to outliers. Based on this, we observed that it is not possible to detect influential observations. We defined the quadratic form of the estimators and the influence function, in order to understand their behavior and properties. Finally, we make an application with these tools in the pollution data for the univariate case, complementing the results shown in Genton and Ruiz-Gazen (2010), the meuse data from the sp package for the bivariate case and average temperatures from the geofd package for the functional case.