The correlation between two vectors is the cosine of the angle between the centered data. While the cosine is a measure of association, the literature has spent little attention to the use of the sine as a measure of distance. A key application of the sine is a new "sine-diagonal inequality / disproportionality" (SDID) measure for votes and their assigned seats for parties for Parliament. This application has nonnegative data and uses regression through the origin (RTO) with non-centered data. Textbooks are advised to discuss this case because the geometry will improve the understanding of both regression and the distinction between descriptive statistics and statistical decision theory. Regression may better be introduced and explained by looking at the angles relevant for a vector and its estimate rather than looking at the Euclidean distance and the sum of squared errors. The paper provides an overview of the issues involved. A new relation between the sine and the Euclidean distance is derived. Equal or Proportional Representation (EPR) scales down from electorate to Parliament while District Representation (DR) confuses elections with contests (USA, UK).
The notebook with its included packages is available at:
https://www.wolframcloud.com/objects/thomas-cool/Voting/2018-02-20-Overview-r-squared.nb
NB. This relates to http://community.wolfram.com/groups/-/m/t/1267872 All this forms part of a larger framework given by my other paper: "One woman, one vote. Though not in the USA, UK and France". My diagnosis is that "political science on electoral systems" is still in the Humanities and pre-science, notably by relying more upon common language instead of sharp definitions that are relevant for empirics. On the other hand, there are also mathematicians who deal with their definitions abstractly, without a proper grounding in empirical research. My invitation to empirical researchers is to help make a difference, notably in re-engineering the theory on electoral systems.
PM. Updated April 16 2018
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