Clarifying the standard deviational ellipse

Abstract

For a set of geographical units in the Cartesian coordinate system, the locus of the standard deviation of the × coordinates of the set forms a closed curve as the system is rotated about the origin. This curve, often referred to as "standard deviational ellipse" (SDE), is not in fact an ellipse. The actual shape of the curve has remained unclear since the issue was mentioned initially by Lefever in 1926. In the present paper this closed curve, referred to as "standard deviation curve" (SDC), is clarified mathematically, and some of its applications in spatial analysis are discussed. The shape of SDC changes from a single circle to double circles when the distribution of the set of geographical units changes from an even condition to a straight line. The shape of SDC is determined explicitly by the ratio of its minor axis to its major axis. This ratio, therefore, is a useful index to show to what extent the distribution of a set of geographical units is circular, or linear. In addition, the size and radius of SDC can be used to indicate the distribution density of geographical units. The major axis of SDC, whose angle is determined explicitly for the first time, indicates the major orientation of geographical units. A program has been developed to apply SDC to spatial analysis (mean center, major orientation, distribution density, circular condition, etc.). The program is available from jxοngηotmail.com. It is written in the MapBasic language, and runs under MapInfo.