Time series analysis of compositional data

Abstract

In recent years various methods have been developed for modelling multivariate (or vector) time series. However if each vector consists of proportions so that elements must sum to unity these methods break down. Data with this sum-constraint are termed compositional data. It is the aim of this thesis to propose a possible approach to such data. The method applied is to find a function that will map the sum-constrained data onto an unconstrained space, that is to map the spherical simplex onto the real plane. Two specific mappings are investigated. These turn out to be multivariate generalizations of the well known logistic transformation. However, both of these functions are asymmetrical. For the first this asymmetry is induced by the choice of one of the variables in the vector series, as a reference variable. It is shown that the model is invariant under this choice. For the second, a specific order to the variables must be imposed. However, this is seen to be useful in examining a type of compositional independence known as neutrality. Methods for using the resulting two models for forecasting are discussed. There are two main problems that occur. The first is that the moments of the underlying distributions corresponding to these models cannot be evaluated algebraically. This means that the minimum mean square error forecast cannot be evaluated. The second is that these distributions are not necessarily uni-modal, which may make the use of the minimum mean square error forecast nonsensical. Various solutions are suggested, and these are compared in a short numerical study. The final part of the thesis examines the relationships between the components of the proportions. This utilizes time series methods for examining Wiener-Granger causality, and combines them with various concepts of compositional independence. These latter concepts include neutrality as mentioned above, and have been developed to deal with the sum-constraint. (D80179)

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