Geografic-Information-System/lib/arraystats/discont.README.txt

Discont
Jean-Pierre Grimmeau - Université Libre de Bruxelles <grimmeau@ulb.ac.be>

The discont algorithm systematically searches discontinuities in the slope
of the cumulated frequencies curve, by approximating this curve through
straight line segments whose vertices define the class breaks. This
algorithm is inspired by techniques of automatic line generalization used
in cartography [1]. The first approximation is a straight line which links
the two end nodes of the curve. This line is then replaced by a
two-segmented polyline whose central node is the point on the curve which
is farthest from the preceding straight line. The point on the curve
furthest from this new polyline is then chosen as a new node to create
break up one of the two preceding segments, and so forth. The problem of
the difference in terms of units between the two axes is solved by
rescaling both amplitudes to an interval between 0 and 1. In the original
algorithm, the process is stopped when the difference between the slopes
of the two new segments is no longer significant. As the slope is the
ratio between the frequency and the amplitude of the corresponding
interval, i.e. its density, this effectively tests whether the frequencies
of the two newly proposed classes are different from those obtained by
simply distributing the sum of their frequencies amongst them in
proportion to the class amplitudes. 

The algorithm described above creates class breaks which each are
identical to a specific observation. It is thus necessary to decide to
which class these observations should be attributed. It seems logical to
prefer the densest, i.e. the one with the strongest slope. The
automatisation of this method allows to distinguish classes with high
frequencies from those with low frequencies, but also to introduce
subtleties and to delimit transition classes.

This method, inspired by Jenks' algorithm [2], provides a good analysis of
the distribution, but not necessarily cartographically satisfying class
breaks. It is thus up to the cartographer to judge whether all the
identified breaks are cartographically useful (or whether some should be
combined) and whether any of the class amplitudes is too large. In the
latter case, the class should be subdivided into equal intervals
(arithmetic progression) as by definition, the classes resulting from the
discont algorithm have a homogeneous interior distribution. If the general
distribution of the data is close to the normal distribution, it is also
possible to combine equiprobable class breaks [3] , with their advantage
of regularity, with discont class breaks for the extremes which often have
large amplitudes when using equiprobable class breaks.

[1] Douglas, D.H. & Peucker, T.K. (1973) Algorithms for the reduction of the number of points required to represent a digitized line or its caricature, The Canadian Cartographer, 10, pp. 112-122.

[2] Jenks, G.F. (1963) Generalisation in statistical mapping, Annals of the Association of American Geographers, 53, pp.15-26.

[3] Grimmeau, J.P. (1977) Cartographie par plages et discontinuités spatiales, Paris, Espace géographique, VI, pp.49-58.