bnpsd (version 1.1.1)

admix_prop_1d_circular: Construct admixture proportion matrix for circular 1D geography

Description

Assumes \(k\) intermediate subpopulations placed along a circumference (the \([0, 2\pi]\) line that wraps around) with even spacing spread by random walks (see details below), then \(n\) individuals sampled equally spaced in \([a,b]\) (default \([0, 2\pi]\) with a small gap so first and last individual do not overlap) draw their admixture proportions relative to the Von Mises density that models the random walks of each of these intermediate subpopulations. The spread of the random walks (the \(\sigma=1/\sqrt{\kappa}\) of the Von Mises densities) is set to sigma if not missing, otherwise \(\sigma\) is found numerically to give the desired bias coefficient bias_coeff, the coancestry matrix of the intermediate subpopulations coanc_subpops (up to a scalar factor), and the final \(F_{ST}\) of the admixed individuals (see details below).

Usage

admix_prop_1d_circular(n_ind, k_subpops, sigma = NA,
  coord_ind_first = 2 * pi/(2 * n_ind), coord_ind_last = 2 * pi * (1 -
  1/(2 * n_ind)), bias_coeff, coanc_subpops, fst)

Arguments

n_ind

Number of individuals

k_subpops

Number of intermediate subpopulations

sigma

Spread of intermediate subpopulations (approximate standard deviation of Von Mises densities, see above) The edge cases sigma = 0 and sigma = Inf are handled appropriately!

coord_ind_first

Location of first individual

coord_ind_last

Location of last individual

OPTIONS FOR BIAS COEFFICIENT VERSION

bias_coeff

The desired bias coefficient, which specifies \(\sigma\) indirectly. Required if sigma is missing.

coanc_subpops

The length-\(k\) vector of inbreeding coefficients (or \(F_{ST}\)'s) of the intermediate subpopulations, up to a scaling factor (which cancels out in calculations). Required if sigma is missing.

fst

The desired final \(F_{ST}\) of the admixed individuals. Required if sigma is missing.

Value

If sigma was provided, the \(n \times k\) admixture proportion matrix. If sigma is missing, a named list is returned containing admix_proportions, the rescaled coanc_subpops, and the sigma that together give the desired \(bias_coeff\) and final \(F_{ST}\) of the admixed individuals.

Details

Assuming the full range of \([0, 2\pi]\) is considered, and the first and last individuals do not overlap, the gap between individuals is \(\Delta = 2 \pi / n\). To not have any individuals on the edge, we place the first individual at \(\Delta / 2\) and the last at \(2 \pi - \Delta / 2\). The location of subpopulation \(j\) is $$\Delta / 2 + (j-1/2)/k (2 \pi - \Delta),$$ chosen to agree with the default correspondence between individuals and subpopulations of the linear 1D geography admixture scenario (admix_prop_1d_linear).

When sigma is missing, the function determines its value using the desired bias_coeff, coanc_subpops up to a scalar factor, and fst. Uniform weights for the final generalized \(F_{ST}\) are assumed. The scaling factor of the input coanc_subpops is irrelevant because it cancels out in bias_coeff; after sigma is found, coanc_subpops is rescaled to give the desired final \(F_{ST}\). However, the function stops with a fatal error if the rescaled coanc_subpops takes on any values greater than 1, which are not allowed since coanc_subpops are IBD probabilities.

Examples

# NOT RUN {
# admixture matrix for 1000 individuals drawing alleles from 10 subpops
# and a spread of about 2 standard deviations along the circular 1D geography
admix_proportions <- admix_prop_1d_circular(n_ind = 1000, k_subpops = 10, sigma = 2)

# a similar model but with a bias coefficient of exactly 1/2
k_subpops <- 10
# FST vector for intermediate independent subpops, up to a factor (will be rescaled below)
coanc_subpops <- 1 : k_subpops
obj <- admix_prop_1d_circular(
    n_ind = 1000,
    k_subpops = k_subpops,
    bias_coeff = 0.5,
    coanc_subpops = coanc_subpops,
    fst = 0.1 # desired final FST of admixed individuals
)

# in this case return value is a named list with three items:
admix_proportions <- obj$admix_proportions

# rescaled coancestry data (matrix or vector) for intermediate subpops
coanc_subpops <- obj$coanc_subpops

# and the sigma that gives the desired bias_coeff and final FST
sigma <- obj$sigma

# }