Mathematical analysis of synthetic measures based on radar charts

Abstract

This work contains a description of a technique for constructing two synthetic indicators (measures) using a graphical presentation in the form of radar maps. The paper presents the structure and properties of indicators and their formal notation specially created for this purpose using the analogon of a scalar product of vectors. In particular, it proves the theorem on polygon fields, induced by radar maps, prepared for structural vectors, which allows to build concentration indicators. In order to demonstrate the usefulness of tools constructed by such means, the example shows how significant structural changes can be imperceptible when utilizing only the GINI concentration indicator’s value, but are noticeable when using the concentration indicator developed by the authors. In addition, it illustrates the change in the value of concentration indicators (GINI and the indicator developed by the authors) on two families of Lorenz curves, together with changes in concentration. The practical application of this technique for constructing indicators that create rankings is presented on empirical data on the level of material deprivation in the countries that joined the EU in 2004 and 2007. These data have also been annotated (for comparison purposes) using the so-called overrepresentation maps (Grade Correspondence Analysis method).

Keyword : radar chart, concentration meter, S-shift operator