A Genealogical Interpretation of Principal Components Analysis

Principal components analysis, PCA, is a statistical method commonly used in population genetics to identify structure in the distribution of genetic variation across geographical location and ethnic background. However, while the method is often used to inform about historical demographic processes, little is known about the relationship between fundamental demographic parameters and the projection of samples onto the primary axes. Here I show that for SNP data the projection of samples onto the principal components can be obtained directly from considering the average coalescent times between pairs of haploid genomes. The result provides a framework for interpreting PCA projections in terms of underlying processes, including migration, geographical isolation, and admixture. I also demonstrate a link between PCA and Wright's fst and show that SNP ascertainment has a largely simple and predictable effect on the projection of samples. Using examples from human genetics, I discuss the application of these results to empirical data and the implications for inference.Author Summary: Genetic variation in natural populations typically demonstrates structure arising from diverse processes including geographical isolation, founder events, migration, and admixture. One technique commonly used to uncover such structure is principal components analysis, which identifies the primary axes of variation in data and projects the samples onto these axes in a graphically appealing and intuitive manner. However, as the method is non-parametric, it can be hard to relate PCA to underlying process. Here, I show that the underlying genealogical history of the samples can be related directly to the PC projection. The result is useful because it is straightforward to predict the effects of different demographic processes on the sample genealogy. However, the result also reveals the limitations of PCA, in that multiple processes can give the same projections, it is strongly influenced by uneven sampling, and it discards important information in the spatial structure of genetic variation along chromosomes.