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.