# Compositions and Their Sample Space: From Proportions to Cartesian Coordinates

4 March 2016
12:00 pm
Compositions and Their Sample Space: From Proportions to Cartesian Coordinates
Professor Juan José Egozcue
Dept. of Applied Mathematics III, Universidad Politécnica de Cataluña
Professor Vera Pawlowsky-Glahn
Dept. of Computer Science and Applied Mathematics, University of Girona

Compositions are equivalence classes of proportional vectors with
positive components, commonly represented as vectors of constant
sum. Typical examples are sets of proportions, percentages,
concentrations or shares. Ratios between parts convey all the
information. Some basic principles configure the characteristics of
an appropriate sample space, the simplex, and its structure,
including operations and metrics. This leads to a Euclidean
structure of the simplex, with linear relations, bases,
coordinates, distances and projections, known as the Aitchison
geometry. Most standard statistical methods can be safely applied
to compositional Cartesian coordinates.

The design of a breakwater under ocean wave action requires to
model its vulnerability. Vulnerability can be viewed as the
conditional probabilities of responses of the breakwater, given a
design and a wave action. These probabilities are considered as a
composition and its estimation provides an example of regression of
a composition on real explanatory variables using simulated
responses.