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Mixed-membership unsupervised clustering is widely used to extract informative patterns from data in many application areas. For a shared dataset, the stochasticity and unsupervised nature of clustering algorithms can cause difficulties in comparing clustering results produced by different algorithms, or even multiple runs of the same algorithm, as outcomes can differ owing to permutation of the cluster labels or genuine differences in clustering results. Here, with a focus on inference of individual genetic ancestry in population-genetic studies, we study the cost of misalignment of mixed-membership unsupervised clustering replicates under a theoretical model of cluster memberships. Using Dirichlet distributions to model membership coefficient vectors, we provide theoretical results quantifying the alignment cost as a function of the Dirichlet parameters and the Hamming permutation difference between replicates. For fixed Dirichlet parameters, the alignment cost is seen to increase with the Hamming distance between permutations. Datasets with low variance across individuals of membership coefficients for specific clusters generally produce high misalignment costs—so that a single optimal permutation has far lower cost than suboptimal permutations. Higher variability in data, as represented by greater variance of membership coefficients, generally results in alignment costs that are similar between the optimal permutation and suboptimal permutations. We demonstrate the application of the theoretical results to data simulated under the Dirichlet model, as well as to membership estimates from inference of human-genetic ancestry. The results can contribute to improving cluster alignment algorithms that seek to find optimal permutations of replicates. Supplementary materials for this article are available online. 

Abstract Allele-sharing statistics for a genetic locus measure the dissimilarity between two populations as a mean of the dissimilarity between random pairs of individuals, one from each population. Owing to within-population variation in genotype, allele-sharing dissimilarities can have the property that they have a nonzero value when computed between a population and itself. We consider the mathematical properties of allele-sharing dissimilarities in a pair of populations, treating the allele frequencies in the two populations parametrically. Examining two formulations of allele-sharing dissimilarity, we obtain the distributions of within-population and between-population dissimilarities for pairs of individuals. We then mathematically explore the scenarios in which, for certain allele-frequency distributions, the within-population dissimilarity – the mean dissimilarity between randomly chosen members of a population – can exceed the dissimilarity between two populations. Such scenarios assist in explaining observations in population-genetic data that members of a population can be empirically more genetically dissimilar from each other on average than they are from members of another population. For a population pair, however, the mathematical analysis finds that at least one of the two populations always possesses smaller within-population dissimilarity than the value of the between-population dissimilarity. We illustrate the mathematical results with an application to human population-genetic data. 

Abstract High-dimensional datasets on cultural characters contribute to uncovering insights about factors that influence cultural evolution. Because cultural variation in part reflects descent processes with a hierarchical structure – including the descent of populations and vertical transmission of cultural traits – methods designed for hierarchically structured data have potential to find applications in the analysis of cultural variation. We adapt a network-based hierarchical clustering method for use in analysing cultural variation. Given a set of entities, the method constructs a similarity network, hierarchically depicting community structure among them. We illustrate the approach using four datasets: pronunciation variation in the US mid-Atlantic region, folklore variation in worldwide cultures, phonemic variation across worldwide languages and temporal variation in first names in the US. In these examples, the method provides insights into processes that affect cultural variation, uncovering geographic and other influences on observed patterns and cultural characters that make important contributions to them.