Kriging and Semivariogram Deconvolution in the Presence of Irregular Geographical Units

Abstract

This paper presents a methodology to conduct geostatistical variography and interpolation on areal data measured over geographical units (or blocks) with different sizes and shapes, while accounting for heterogeneous weight or kernel functions within those units. The deconvolution method is iterative and seeks the pointsupport model that minimizes the difference between the theoretically regularized semivariogram model and the model fitted to areal data. This model is then used in areato-point (ATP) kriging to map the spatial distribution of the attribute of interest within each geographical unit. The coherence constraint ensures that the weighted average of kriged estimates equals the areal datum. This approach is illustrated using health data (cancer rates aggregated at the county level) and population density surface as a kernel function. Simulations are conducted over two regions with contrasting county geographies: the state of Indiana and four states in the Western United States. In both regions, the deconvolution approach yields a point support semivariogram model that is reasonably close to the semivariogram of simulated point values. The use of this model in ATP kriging yields a more accurate prediction than a naïve point kriging of areal data that simply collapses each county into its geographic centroid. ATP kriging reduces the smoothing effect and is robust with respect to small differences in the point support semivariogram model. Important features of the point-support semivariogram, such as the nugget effect, can never be fully validated from areal data. The user may want to narrow down the set of solutions based on his knowledge of the phenomenon (e.g., set the nugget effect to zero). The approach presented avoids the visual bias associated with the interpretation of choropleth maps and should facilitate the analysis of relationships between variables measured over different spatial supports.

Spatial scale and small area population statistics for England and Wales

“…age ranges) and number of variables are limited since the analysis is intended to be illustrative. Table 2 (1998), Goovaerts (2008) and Nagle et al (2011). However, population-weighted centroids of OAs, LSOAs and MSOAs were accepted in this analysis since, for these zones, random reassignment of centroids within zones would make minimal difference to the results.…”

“…The point support variogram can then be regularised to determine how the variogram changes as the spatial resolution decreases (cells become larger) and an optimal cell size could be identified in this way. Variogram deconvolution is possible where the data are for irregularly shaped zones rather than cells (see Goovaerts 2008 and also Lloyd 2014, Zhang et al 2014. However, in the present study, the aim is not to identify an optimal size (and shape) of zones but to assess how much information is contained within each set of zones assessed (the index of dissimilarity and geographical variances are computed to determine how much variation is associated with each scale, as represented by the nested zonal systems).…”

Application of Multivariate Geostatistics in Environmental Epidemiology: Case Study from Houston, Texas

“…These are the ratios of the cross versus the direct variograms which can capture spatial association that is not possible to calculate following conventional methods. Therefore, we can assume the presence of intrinsic correlation whereby the model of intrinsic correlation is entirely specified by its spatial structure, and by the variance-covariance matrix [3] [20] [21]. Tests of normality lead to rejection of the null hypothesis of normality for CVD and COPD ER visits.…”

Proposal of a Geostatistical Procedure for Transportation Planning Field

“…After choosing an initial point of a support model 0 (ℎ), the deconvolution method is iterative and seeks to calculate the support model that minimizes the difference between the theoretical model of a regularized semivariogram and the point-support semivariogram model to be estimated (Goovaerts, 2008). The optimization criterion is the relative difference ( ) between both semivariogram models.…”