Hyperelliptic curve

In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying exclusively relative information.

This definition, given by John Aitchison (1986) has several consequences:

• A compositional data point, or composition for short, can be represented by a positive real vector with as many parts as considered. Sometimes, if the total amount is fixed and known, one component of the vector can be omitted.

• As compositions only carry relative information, the only information is given by the ratios between components. Consequently, a composition multiplied by any positive constant contains the same information as the former. Therefore, proportional positive vectors are equivalent when considered as compositions.

• As usual in mathematics, equivalent classes are represented by some element of the class, called a representative. Thus, equivalent compositions can be represented by positive vectors whose components add to a given constant $\scriptstyle \kappa$. The vector operation assigning the constant sum representative is called closure and is denoted by $\scriptstyle {\mathcal {C}}[\cdot ]$:

${\mathcal {C}}[x_{1},x_{2},\dots ,x_{D}]=\left[{\frac {x_{1}}{\sum _{i=1}^{D}x_{i}}},{\frac {x_{2}}{\sum _{i=1}^{D}x_{i}}},\dots ,{\frac {x_{D}}{\sum _{i=1}^{D}x_{i}}}\right],\$

where D is the number of parts (components) and $[\cdot ]$ denotes a row vector.

• Compositional data can be represented by constant sum real vectors with positive components, and this vectors span a simplex, defined as

${\mathcal {S}}^{D}=\left\{\mathbf {x} =[x_{1},x_{2},\dots ,x_{D}]\in \mathbb {R} ^{D}\left|x_{i}>0,i=1,2,\dots ,D;\sum _{i=1}^{D}x_{i}=\kappa \right.\right\}.\$

This is the reason why $\scriptstyle {\mathcal {S}}^{D}$ is considered to be the sample space of compositional data. The positive constant $\scriptstyle \kappa$ is arbitrary. Frequent values for $\scriptstyle \kappa$ are 1 (per unit), 100 (percent, %), 1000, 10⁶ (ppm), 10⁹ (ppb), ...

• In statistics, compositional data is frequently considered to be data in which each data point is an D-tuple of nonnegative numbers whose sum is 1. Typically each of the D components x_i of each data point [x₁, ..., x_D] says what proportion (or "percentage") of a statistical unit falls into the ith category in a list of D categories. Very often ternary plots are used in analysis of compositional data to represent a three part composition.

• An alternative nomenclatures for compositional analysis is simplicial analysis, motivated by the concept of simplicial sets.

Remarks on the definition of the simplex:

• In mathematical frameworks, the superscript of $\scriptstyle {\mathcal {S}}^{D}$, accounting for the number of parts, is often changed to D − 1, describing the dimension.

• The components of the vector are assumed to be positive. However, in some definitions of the simplex, non-negative components are admitted. Here null components are avoided, because ratios between components of which some are zero are meaningless.

Examples

• Each data point may correspond to a rock composed of three different minerals; a rock of which 10% is the first mineral, 30% is the second, and the remaining 60% is the third would correspond to the triple [0.1, 0.3, 0.6]; a data set would contain one such triple for each rock in a sample of rocks.

• Each data point may correspond to a town; a town in which 35% of the people are Christians, 55% are Muslims, 6% are Jews, and the remaining 4% are others would correspond to the quadruple [0.35, 0.55, 0.06, 0.04]; a data set would correspond to a list of towns.

• In chemistry, compositions can be expressed as molar concentrations of each component. As the sum of all concentrations is not determined, the whole composition of D parts is needed and thus expressed as a vector of D molar concentrations. These compositions can be translated into weight per cent multiplying each component by the appropriated constant.

• In a survey, the proportions of people positively answering some different items can be expressed as percentages. As the total amount is identified as 100, the compositional vector of D components can be defined using only D − 1 components, assuming that the remaining component is the percentage needed for the whole vector to add to 100.

• In probability and statistics, a partition of the sampling space into disjoint events is described by the probabilities assigned to such events. The vector of D probabilities can be considered as a composition of D parts. As they add to one, one probability can be suppressed and the composition is completely determined.

External links

• CoDaWeb - Compositional Data Website
• Pawlowsky-Glahn, V., Egozcue, J.J., Tolosana-Delgado, R. (2007), Lecture Notes on Compositional Data Analysis.
• Why, and How, Should Geologists Use Compositional Data Analysis (wikibook)

References

• J. Aitchison, 1986: The Statistical Analysis of Compositional Data, Chapman & Hall, reprinted in 2003 with additional material by The Blackburn Press