Visual Tools to Identify Influential Observations in Spatial Data

Cross-semivariogram, influential spatial data, principal components.

We adapted the hair-plot, proposed by Genton e Ruiz-Gazen (2010), to identify and visualize 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 is 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 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 spatial 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. Through the influence analysis of this estimator, 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 e Ruiz-Gazen (2010), the meuse data from the sp package for the bivariate case, and the maritimes data from package geofd in R.