Compound compositional data processes

Abstract

Compositional data is non-negative data subject to the unit sum constraint. The logistic normal distribution provides a framework for compositional data when it satisfies sub-compositional coherence in that the inference from a sub- composition should be the same based on the full composition or the sub-composition alone. However, in many cases sub-compositions are not coherent because of additional structure on the compositions, which can be modelled as process(es) inducing change. Sometimes data are collected with a model already well validated and hence with the focus on estimation of the model parameters. Alternatively, sometimes the appropriate model is unknown in advance and it is necessary to use the data to identify a suitable model. In both cases, a hierarchy of possible structure(s) is very helpful. This is evident in the evaluation of, for example, geochemical and household expenditure data. In the case of geochemical data, the structural process might be the stoichiometric constraints induced by the crystal lattice sites, which ensures that amalgamations of some elements are constant in molar terms. The choice of units (weight percent oxide or moles) has an impact on how the data can be modelled and interpreted. For simple igneous systems (e.g. Hawaiian basalt) mineral modes can be calculated from which a valid geochemical interpretation can be obtained. For household expenditure data, the structural process might be how teetotal households have distinct spending patterns on discretionary items from non-teetotal households. Measurement error is an example of another underlying process that reflects how an underlying discrete distribution (e.g. for the number of molecules in a sample) is converted using a linear calibration into a non-negative measurement, where measurements below the stated detection limit are reported as zero. Compositional perturbation involves additive errors on the log-ratio space and is the process that does show sub-compositional coherence. The mixing process involves the combination of compositions into a new composition, such as minerals combining to form a rock, where there may be considerable knowledge about the set of possible mixing processes. Finally, recording error may affect the composition, such as recording the components to a specified number of decimal digits, implying interval censoring, which implies error is close to uniform on the simplex.