The site frequency spectrum for general coalescents

Jeffrey P. Spence, John A. Kamm, Yun S. Song

We present an efficient method for computing the expected site frequency spectrum (SFS) for general Λ- and Ξ-coalescents. For time-homogeneous coalescents, the runtime of our algorithm is O(n^2), where n is the sample size. This is a factor of n^2 faster than the state-of-the-art method. Furthermore, in contrast to existing methods, our method generalizes to time-inhomogeneous Λ- and Ξ-coalescents with measures that factorize as Λ(dx)/ζ(t) and Ξ(dx)/ζ(t), respectively, where ζ denotes a strictly positive function of time. The runtime of our algorithm in this setting is O(n^3). We also obtain general theoretical results for the identifiability of the Λ measure when ζ is a constant function, as well as for the identifiability of the function ζ under a fixed Ξ measure.