3 - Two techniques for the assessment of the positional uncertainty derived from the polygon-to-point collapse operation
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One of the most resorted techniques for cartographic generalization is that of collapse, the operation by which a geometric feature is transformed into another lower-dimensional geometry. Issues in this respect are:Solving the location of the lower-dimensional geometry for each feature.Approximating the set of alternative spatial configurations that may be defined by the derived patterns as a consequence of the the spatial extent of the higher-dimensional geometries, since alternative locations for the lower-dimensional features will hence result in different spatial patterns.From the analytical viewpoint, the effect of this variability on the signal of a measure that uses the spatial structure of the lower-dimensional primitive depends on the size of the window of analysis and on the size and shape of the higher-dimensional geometries, as the interrelation between both conditions makes the extent of the positional uncertainty be resized with respect to the extent of the window of analysis.Such a cartographic problem is typical in landscape studies involving area features. In our study case disconnected scatters of lithic artefacts recorded on the terrain surface are modelled as polygon objects and these in turn collapsed into point objects located at the mean centres of the polygons. To assess the degree of positional uncertainty of the data set caused by the collapse operation, under the spatial domain of the specified window of analysis W, two exploratory techniques are applied:The first uses the length of the links that connect all possible pairs of points in a point realization. For each alternative point configuration of collapsed polygons a_i, a_j,... , the list of interpoint distances t_xy is measured for every pair of points {x,y}. Then for each pair {a_i,a_j} the absolute difference between each t_xy(a_i) in a_i and its equivalent t_xy(a_j) in a_j is computed. The resultant differences are next standardized by dividing them by the square root of W (in order that the standardization result from measures with the same dimensions). The frequency distribution of the ratio signals for each {a_i,a_j} can finally be summarized. In this respect the median and extreme quantiles provide robust measures of the interpoint distance discrepancy signals.The second involves the transformation of the point pattern into a scalar model and the use of measures of association. Now, for each a_i, a_j,… , the density surface of each a, say alpha, is estimated and structured in a grid dataset. Next, for each pair {alpha_i,alpha_j} the bivariate association of the scalar values at colocalized grid cells is estimated. In this regard the following techniques have been applied:Bivariate plotting of colocalized cells. Individual plots can be accumulated in order to jointly explore the dispersion of the density patterns in the sample of alternative point realizations.The Spearman's rank correlation coefficient, as its improves robustness against bias generated by non-Gaussian density distributions. Evaluation of significance should take account of autocorrelation processes and be associated to specific support sizes (due to the sample size variability introduced by the modifiable blocking of the grid).