Admixture Inference under HI, GA-I, and CGF-I Models

GA and CGF models assume that the admixture is strictly continuous from the beginning of admixture to present. This assumption seems too strong to be valid in empirical studies. Here, we extended GA model and CGF model to GA-I model and CGF-I model respectively, by considering continuous admixture followed by isolation. In this case, the admixture event lasts from G_start generations ago to G_end generations ago. Similar to the previous case, the coefficients of polynomial functions can be represented as a vector of length G_start for each model, whose first G_end − 1 entries are filled with zeros. Suppose the admixture lasted for n = G_start − G_end + 1 generations, then

In this case, we can also try to find the parameters that minimize the objective function (eq. 2) under new models. By examining all possible pairs of (G_end, G_start), it is possible to determine the global minimum of the objective function, but this might not be computationally efficient. Here, we used a faster algorithm (Algorithm 1) to determine the starting and ending time points of admixture.

Let E and S be the ending and starting time points (in generations, prior to the present) of the admixture, which we wanted to search for to minimize the objective function. The search starts from (E⁰, S⁰) = (1, J), where J is the upper bound for the beginning of the admixture event, which can be set to be a large integer to seek for a relatively ancient admixture event. In our analysis of recent admixed populations, we set J = 500. For k = 1, 2, …, (E^k, S^k) is updated from (E^k−1, S^k−1) by two alternative proposals denoted by and . For convenience, we defined

where θ₀, θ₁ can be determined by ordinary least squares.

We chose the proposal that resulted in a smaller value for f. The search stopped when the value of f with (E^k−1, S^k−1) was no larger than that of either proposal or E^k = S^k. In this way, we could readily estimate the time interval of the admixture event (G_end, G_start) quickly.