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Population structure is a confounding factor in genome-wide association studies, increasing the rate of false positive associations. To correct for it, several model-based algorithms such as ADMIXTURE and STRUCTURE have been proposed. These tend to suffer from the fact that they have a considerable computational burden, limiting their applicability when used with large datasets, such as those produced by next generation sequencing techniques. To address this, nonmodel based approaches such as sparse nonnegative matrix factorization (sNMF) and EIGENSTRAT have been proposed, which scale better with larger data. Here we present a novel nonmodel-based approach, population structure inference using kernel-PCA and optimization (PSIKO), which is based on a unique combination of linear kernel-PCA and least-squares optimization and allows for the inference of admixture coefficients, principal components, and number of founder populations of a dataset. PSIKO has been compared against existing leading methods on a variety of simulation scenarios, as well as on real biological data. We found that in addition to producing results of the same quality as other tested methods, PSIKO scales extremely well with dataset size, being considerably (up to 30 times) faster for longer sequences than even state-of-the-art methods such as sNMF. PSIKO and accompanying manual are freely available at https://www.uea.ac.uk/computing/psiko.

Introgression is an evolutionary process which provides an important source of innovation for evolution. Although various methods have been used to detect introgression, very few methods are currently available for constructing evolutionary histories involving introgression. In this article, we propose a new method for constructing such evolutionary histories whose starting point is a species forest (consisting of a collection of lineage trees, usually arising as a collection of clades or monophyletic groups in a species tree), and a gene tree for a specific allele of interest, or allele tree for short. Our method is based on representing introgression in terms of a certain “overlay” of the allele tree over the lineage trees, called an overlaid species forest (OSF). OSFs are similar to phylogenetic networks although a key difference is that they typically have multiple roots because each monophyletic group in the species tree has a different point of origin. Employing a new model for introgression, we derive an efficient algorithm for building OSFs called OSF-Builder that is guaranteed to return an optimal OSF in the sense that the number of potential introgression events is minimized. As well as using simulations to assess the performance of OSF-Builder, we illustrate its use on a butterfly data set in which introgression has been previously inferred. The OSF-Builder software is available for download from https://www.uea.ac.uk/computing/software/OSF-Builder.