Regionalized Variables take on values according to spatial location. Given: Where: A “structural” coarse scale forcing or trend A random” Local spatial dependency error variance (considered normally distributed) Usually removed by detrending What we are interested in Then: Sometime useful Notation:, “if this is true then”, “the distance between point a and b”, “for all, by definition”, “Z at this location is a member of the set of locations call set D”

Measurements of location (center of distribution –mean (m µ x ) –median –mode Measurements of spread (variability) –variance –standard deviation –interquartile range Measurements of shape (symmetry & length –coefficient of skewness –coefficient of variation Summary of a histogram

In the same way that we take traditional descriptive statistics to characterize the frequency distribution of the observations measurement variable, we can also make use of the frequency distribution of the x-coordinate and y-coordinate of the location observations to characterize the spatial pattern of observation. These statistics may also be weighted relative to the measurement variable.

Just as the traditional standard deviation describes the variation of a distributions; the standard distance characterizes the variation of distances from the mean distance in a set of spatial observations. When the variation of all observations from the mean location in the “x” dimension is calculated separately from the variation in the “y” dimension the two terms for the standard deviation ellipse is found. The angle of rotaion for the major and minor axis of the ellipse is determined for the “y” dimensional axis such that the distance for all observations in the “y” dimension to the “y” axis is minimized. This angle of rotation is then applied to the “x” axis.

When considering line features as the spatial unit of analysis, the line’s location for distance calculation is represented by the center point between the starting and ending nodes. The “Orientation” or “Direction” of the line is determined for each line in it’s individual “graph space” (origins at 0,0 coordinates for each line feature). As unit vectors these representations may be summarized and the variance in distribution of their orientation (or directions) angles from the mean of the distribution of angles is characterized as the circular variance which ranges from “0” when all lines are pointing the same way, and “1” when lines point in opposite directions.