Compositional data

In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, percentages, and ppm can all be thought of as compositional data. In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, percentages, and ppm can all be thought of as compositional data. In three variables, compositional data in three variables can be plotted via ternary plots. The use of a barycentric plot on three variables graphically depicts the ratios of the three variables as positions in an equilateral triangle. In general, John Aitchison defined compositional data to be proportions of some whole in 1982. In particular, a compositional data point (or composition for short) can be represented by a positive real vector. The sample space of compositional data is a simplex: The only information is given by the ratios between components, so the information of a composition is preserved under multiplication by any positive constant. Therefore the sample space of compositional data can always be assumed to be a standard simplex, i.e. κ = 1 {displaystyle kappa =1} . In this context, normalization to the standard simplex is called closure and is denoted by C [ ⋅ ] {displaystyle scriptstyle {mathcal {C}}} : where D is the number of parts (components) and [ ⋅ ] {displaystyle } denotes a row vector. The simplex can be given the structure of a real vector space in several different ways. The following vector space structure is called Aitchison geometry or the Aitchison simplex and has the following operations: