18 JOHN AITCHISON follows: (12.1) pr(t ~ llx) ~ I - pr(t ~ Olx) ~ F ( ao + t, a logx ) , whe.e t, a ~ 0. Hypotheses that the categorization depends only on a subcomposition, for exam- ple on the subcomposition formed from parts 1, ... , C is then simply specified by ac+I = · · · = an = 0, and so the whole lattice of subcompositional hypotheses can be readily and systematically investigated. A striking example of the use of this technique is to be found in discriminating between two types of limestone. Thomas and Aitchison [T-A] show that of the 17- part (major-oxide, trace element) composition a simple major-oxide subcomposition (CaO, Fe20a, MgO) provides excellent discrimination, equal to that of the full composition. 13. Convex linear mixing problems By a convex linear mixing problem we mean one in which some target D- composition y is visualized as arising from some convex linear combination 1r = {1r11 ••• , 7rc) of C source or end member compositions x 1 , ••• , Xc, as (13.1) One such problem is where information is available only on the target in the form of a compositional data set and the objective is to attempt to find fixed end member source compositions from which the target compositions could have arisen through varying mixtures of these source compositions. For successful approaches to, and algorithms for, this difficult problem, see [R] and [We]. A completely different problem arises if we have some information about the sources. If there are C sources with precise compositions x1 , ... , xc the problem commonly posed is to what extent a target composition y can be expressed as a convex linear combination {13.1) of the source compositions. The problem is often made more complicated by the following features which increase substantially its statistical nature. (a) The source compositions are not precise but are defined only as an ob- served cluster of compositions. (b) The target composition is not precise but is defined only as a cluster of compositions. If we denote by y and x 1 , ••• , xc generic target and source compositions, then the question of interest is the extent to which the data may support a relationship of the form (13.1). A reasonable technique for resolving this problem [A-B] is to use the ith source samples to assess in terms of .CP(~i• 'li) the compositional variability of the ith source. Although the distributional problem of determining the consequent variability of y in (13.1) cannot be resolved exactly, there are excellent approximations in terms of logistic normal and skew logistic normal distributions which allow the evaluation of the likelihood function for 7r and hence the testing of various hypotheses concerning 1r. Aitchison and Bacon-Shone [A-B] also consider other models of convex linear mixing, for example where 7r may be variable and where there may extra variability characterized by a perturbation of (13.1). Such models are of interest in investigating the sources of pollution see [A-B] for one such application.
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