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 kappa. The vector operation assigning the constant sum representative is called closure, where D is the number of parts (components) and denotes a row vector.
• Compositional data can be represented by constant sum real vectors with positive components, and this vectors span a simplex. …
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