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Compositional data, which is data consisting of fractions or probabilities,is common in many fields including ecology, economics, physical science andpolitical science. If these data would otherwise be normally distributed, theirspread can be conveniently represented by a multivariate normal distributiontruncated to the non-negative space under a unit simplex. Here thisdistribution is called the simplex-truncated multivariate normal distribution.For calculations on truncated distributions, it is often useful to obtain rapidestimates of their integral, mean and covariance; these quantitiescharacterising the truncated distribution will generally possess differentvalues to the corresponding non-truncated distribution. In this paper, threedifferent approaches that can estimate the integral, mean and covariance of anysimplex-truncated multivariate normal distribution are described and compared.These three approaches are (1) naive rejection sampling, (2) a method describedby Gessner et al. that unifies subset simulation and the Holmes-Diaconis-Rossalgorithm with an analytical version of elliptical slice sampling, and (3) asemi-analytical method that expresses the integral, mean and covariance interms of integrals of hyperrectangularly-truncated multivariate normaldistributions, the latter of which are readily computed in modern mathematicaland statistical packages. Strong agreement is demonstrated between all threeapproaches, but the most computationally efficient approach depends stronglyboth on implementation details and the dimension of the simplex-truncatedmultivariate normal distribution. For computations in low-dimensionaldistributions, the semi-analytical method is fast and thus should beconsidered. As the dimension increases, the Gessner et al. method becomes theonly practically efficient approach of the methods tested here.

Integral, mean and covariance of the simplex-truncated multivariate normal distribution