Simulate Admixed Populations with bnpsd
Author: Alejandro Ochoa and John D. Storey
Date: 2021-08-23

# Simulate Admixed Populations with `bnpsd`

#### Alejandro Ochoa and John D. Storey

#### 2021-08-23

\[ \newcommand{\Fst}{F_{\text{ST}}} \newcommand{\ft}[1][j]{f_{#1}^T} \]

# Introduction

The `bnpsd` package simulates the genotypes of an admixed population. In the Pritchard-Stephens-Donnelly (PSD) model (Pritchard, Stephens, and Donnelly 2000), admixed individuals draw their alleles with individual-specific probabilities (admixture proportions) from \(K\) intermediate subpopulations. We impose the Balding-Nichols (BN) model (Balding and Nichols 1995) to the intermediate subpopulation allele frequency, which thus evolve independently with subpopulation-specific inbreeding coefficients (\(\Fst\) values) from a common ancestral population \(T\). The kinship coefficients and generalized \(\Fst\) of the admixed individuals were derived in recent work (Ochoa and Storey 2016). A simulated admixed population generated using this package was used to benchmark kinship and \(\Fst\) estimators in the accompanying paper (Ochoa and Storey 2021). Here we briefly summarize the notation and intuition behind the key parameters (see (Ochoa and Storey 2016) for precise definitions).

There are conceptually two parts of the BN-PSD model, one that defines relatedness between individuals (under the admixture framework), while the other simulates allele frequencies and genotypes conditional on the specified parameters of relatedness/admixture.

## The BN-PSD population structure

The population structure determines how individuals are related to each other. The key parameters are the coancestry coefficients of the intermediate subpopulations and the admixture proportions of each individual for each subpopulation (\(q_{ju}\)), which are treated as fixed variables.

The `bnpsd` functions that model population structure (not drawing allele frequencies or genotypes) admit arbitrary coancestry matrices for the intermediate subpopulations. However, the code that draws allele frequencies and genotypes requires these subpopulations to be independent, which means that the coancestry matrix must have zero values off-diagonal. In this setting—the only case we’ll considered in this vignette—the diagonal values of the coancestry matrix of the intermediate subpopulations are inbreeding coefficients (\(\ft[S_u]\) below), which correspond to subpopulation-specific \(\Fst\) values.

Each intermediate subpopulation \(S_u\) (\(u \in \{1, ..., K\}\)) evolved independently from a shared ancestral population \(T\) with an inbreeding coefficient denoted by \(\ft[S_u]\). Note that \(T\) is a superscript, and does not stand for exponentiation or matrix transposition. Each admixed individual \(j \in \{1, ..., n\}\) draws each allele from \(S_u\) with probability given by the admixture proportion \(q_{ju}\) (\(\sum_{u=1}^K q_{ju} = 1 \forall j\)). In this case the coancestry coefficients \(\theta^T_{jk}\) between individuals \(j,k\) (including \(j=k\) case) and the \(\Fst\) of the admixed individuals are given by: \[ \theta^T_{jk} = \sum_{u=1}^K q_{ju} q_{ku} \ft[S_u], \quad \quad \Fst = \sum_{j=1}^n \sum_{u=1}^K w_j q_{ju}^2 \ft[S_u], \] where \(0 < w_j < 1, \sum_{j=1}^n w_j = 1\) are user-defined weights for individuals (default \(w_j=\frac{1}{n} \forall j\)). Note \(\theta^T_{jk}\) equals the kinship coefficient for \(j \ne k\) and the inbreeding coefficient for \(j=k\).

The bias coefficient \(s\) is defined by \[ s = \frac{\bar{\theta}^T}{\Fst} \] where \(\bar{\theta}^T = \sum_{j=1}^n \sum_{k=1}^n w_j w_k \theta_{jk}^T\). This \(0 < s \le 1\) approximates the proportional bias of \(\Fst\) estimators that assume independent subpopulations, and one `bnpsd` function below fits its parameters to yield a desired \(s\).

## Random allele frequencies and genotypes

This section details the distributions of the allele frequencies and genotypes of the various populations or individuals of the BN-PSD model.

Every biallelic locus \(i\) in the ancestral population \(T\) has an ancestral reference allele frequency denoted by \(p_i^T\). By default the `bnpsd` code draws \[ p_i^T \sim \text{Uniform}(a, b) \] with \(a=0.01, b=0.5\), but the code accepts \(p_i^T\) from arbitrary distributions (see below).

The distribution of the allele frequency at locus \(i\) in subpopulation \(S_u\), denoted by \(p_i^{S_u}\), is the BN distribution: \[ p_i^{S_u} | T \sim \text{Beta} \left( \nu_s p_i^T, \nu_s \left( 1-p_i^T \right) \right), \] where \(\nu_s = \frac{1}{\ft[S_u]}-1\). Allele frequencies for different loci and different subpopulations (\(S_u,S_v, u \ne v\)) are drawn independently.

Each admixed individual \(j\) at each locus \(i\) draws alleles from a mixture of Bernoulli distributions from each intermediate subpopulation, which is mathematically equivalent to assigning what we call individual-specific allele frequencies \(\pi_{ij}\) constructed as: \[ \pi_{ij} = \sum_{u=1}^K p_i^{S_u} q_{ju}. \] The unphased genotype \(x_{ij}\) (encoded to count the number of reference alleles) is drawn as: \[ x_{ij}|\pi_{ij} \sim \text{Binomial}(2, \pi_{ij}). \]

# Simulation examples

## Population structure: 1D geography

Let’s generate the same population structure used in the simulation of (Ochoa and Storey 2021).

    # this package
    library(bnpsd)
    # for nice colors
    library(RColorBrewer)
    # for visualizing coancestry matrix with plot_popkin, and other handy functions
    library(popkin)

    # dimensions of data/model
    # number of individuals (NOTE this is 10x less than in publication!)
    n_ind <- 100
    # number of intermediate subpops
    k_subpops <- 10

    # define population structure
    # subpopulation FST vector, up to a scalar
    inbr_subpops <- 1 : k_subpops
    # desired bias coefficient
    bias_coeff <- 0.5
    # desired FST for the admixed individuals
    Fst <- 0.1
    # admixture proportions from 1D geography
    obj <- admix_prop_1d_linear(
        n_ind = n_ind,
        k_subpops = k_subpops,
        bias_coeff = bias_coeff,
        coanc_subpops = inbr_subpops,
        fst = Fst
    )
    admix_proportions <- obj$admix_proportions
    # rescaled inbreeding vector for intermediate subpopulations
    inbr_subpops <- obj$coanc_subpops

    # get pop structure parameters of the admixed individuals
    coancestry <- coanc_admix(admix_proportions, inbr_subpops)

    # verify that we got the desired FST!
    fst_admix(admix_proportions, inbr_subpops)
    #> [1] 0.1

    # the mean of the diagonal of the coancestry matrix also equals FST
    inbr <- diag(coancestry)
    fst(inbr) # `fst` is a function in the popkin package
    #> [1] 0.1

    # verify that we got the desired `bias_coeff` too!
    mean(coancestry) / Fst
    #> [1] 0.5

    # visualize the per-subpopulation inbreeding coefficients (FSTs)
    # shrink default margins
    par(mar = c(4, 4, 0, 0) + 0.2)
    # colors for independent subpopulations
    col_subpops <- brewer.pal(k_subpops, "Paired")

    barplot(inbr_subpops, col = col_subpops, names.arg = 1 : k_subpops, ylim = c(0, 1),
        xlab = 'Subpopulation', ylab = 'Inbreeding coeff.')

    # visualize the admixture proportions
    # shrink default margins
    par(mar = c(1, 4, 0, 0) + 0.2)
    barplot(
        t(admix_proportions),
        col = col_subpops,
        border = NA,
        space = 0,
        ylab = 'Admixture prop.'
    )
    mtext('Individuals', 1)

    # Visualize the coancestry matrix using "popkin"!
    plot_popkin(
        coancestry,
        # zero inner margin (plus padding) because we have no labels
        mar = 0,
        leg_title = 'Coancestry'
    )

## Draw random allele frequencies and genotypes

Now let’s draw all the random allele frequencies and genotypes from the population structure. The easiest way is to use `draw_all_admix`:

    # number of loci in simulation (NOTE this is 30x less than in publication!)
    m_loci <- 10000
    # draw all random Allele Freqs (AFs) and genotypes
    # reuse the previous inbr_subpops, admix_proportions
    out <- draw_all_admix(
        admix_proportions = admix_proportions,
        inbr_subpops = inbr_subpops,
        m_loci = m_loci,
        # NOTE by default p_subpops and p_ind are not returned, but here we will ask for them
        want_p_subpops = TRUE,
        # NOTE: asking for `p_ind` increases memory usage substantially,
        # so don't ask for it unless you're sure you want it!
        want_p_ind = TRUE
    )
    # genotypes
    X <- out$X
    # ancestral AFs
    p_anc <- out$p_anc
    # intermediate independent subpopulation AFs
    p_subpops <- out$p_subpops
    # individual-specific AFs
    p_ind <- out$p_ind

    # inspect distribution of ancestral AFs (~ Uniform(0.01, 0.5))
    # shrink default margins for these figures
    par(mar = c(4, 4, 0, 0) + 0.2)
    hist(p_anc, xlab = 'Ancestral AF', main = '', xlim = c(0, 1))

    # distribution of intermediate population AFs
    # (all subpopulations combined)
    # (will be more dispersed toward 0 and 1 than ancestral AFs)
    hist(p_subpops, xlab = 'Intermediate Subpopulation AF', main = '', xlim = c(0, 1))

    # distribution of individual-specific AFs (admixed individuals)
    # (admixture reduces differentiation, so these resemble ancestral AFs a bit more)
    hist(p_ind, xlab = 'Individual-specific AF', main = '', xlim = c(0, 1))

    # genotype distribution of admixed individuals
    barplot(table(X), xlab = 'Genotypes', ylab = 'Frequency', col = 'white')

Let’s verify that the correlation structure of the genotypes matches the theoretical coancestry matrix we constructed earlier. For this we use the `popkin` function of the package with the same name.

    # for best estimates, group individuals into subpopulations using the geography
    # this averages more individuals in estimating the minimum kinship
    subpops <- ceiling( ( 1 : n_ind ) / n_ind * k_subpops )
    table(subpops) # got k_subpops = 10 with 100 individuals each
    #> subpops
    #>  1  2  3  4  5  6  7  8  9 10
    #> 10 10 10 10 10 10 10 10 10 10
    # now estimate kinship using popkin
    kinship_estimate <- popkin(X, subpops)
    # replace diagonal with inbreeding coeffs. to match coancestry matrix
    coancestry_estimate <- inbr_diag(kinship_estimate)

    # Visualize the coancestry matrix using "popkin"!
    plot_popkin(
        list( coancestry, coancestry_estimate ),
        titles = c('Truth', 'Estimate'),
        # second value is top margin, for panel titles
        mar = c(0, 2.5),
        leg_title = 'Coancestry'
    )

We can also look at the structure of the two allele frequency matrices, `p_subpops` and `p_ind`, using `popkin_af`, which estimates coancestry from allele frequency matrices (in contrast, `popkin` estimates kinship from genotype matrices, and is preferred when genotypes are available).

    coancestry_subpops_estimate <- popkin_af( p_subpops )
    coancestry_ind_estimate <- popkin_af( p_ind )

The first coancestry matrix is small, and matches the expectation of independent subpopulations (zero values off-diagonal) with the subpopulation inbreeding coefficients as the diagonal coancestry values:

    plot_popkin(
        list( diag( inbr_subpops ), coancestry_subpops_estimate ),
        titles = c('Truth', 'Estimate'),
        mar = c(0, 2.5),
        leg_title = 'Coancestry'
    )

The second coancestry matrix, estimated from individual-specific allele frequencies, is the same in expectation to the true coancestry matrix of the model:

    plot_popkin(
        list( coancestry, coancestry_ind_estimate ),
        titles = c('Truth', 'Estimate'),
        mar = c(0, 2.5),
        leg_title = 'Coancestry'
    )

The last coancestry matrix also equals in expectation the transformed kinship matrix estimated earlier, but this estimate from allele frequencies is less noisy. However, in real data such allele frequencies are not observed, and their estimates can vary in quality and result in biases, so the above is more a validation of the simulation than a practical estimation approach.

## Customizing the allele frequency and genotype pipeline

The random variables generated by `draw_all_admix` above can also be generated separately using the following functions (where \(p\) is the usual variable symbol for allele frequencies):

- `draw_p_anc` (drawn ancestral allele frequencies)
- `draw_p_subpops` (draw independent subpopulation allele frequencies)
- `make_p_ind_admix` (construct individual-specific allele frequencies under the admixture model)
- `draw_genotypes_admix` (draw genotypes under the admixture model)

These functions are provided for greater flexibility (examples follow further below). However, the joint function `draw_all_admix` has the advantage of removing fixed loci (loci that were randomly set to 0 or 2 for all individuals, despite non-zero allele frequencies), which are re-drawn from scratch (starting from the ancestral allele frequencies).

Here is the step-by-step procedure for drawing AFs and genotypes in the default `draw_all_admix` (except for the re-drawing of fixed loci):

    # reuse the previous m_loci, inbr_subpops, admix_proportions
    # draw ancestral AFs
    p_anc <- draw_p_anc(m_loci)
    # draw intermediate independent subpopulations AFs
    p_subpops <- draw_p_subpops(p_anc, inbr_subpops)
    # draw individual-specific AFs
    p_ind <- make_p_ind_admix(p_subpops, admix_proportions)
    # draw genotypes
    X <- draw_genotypes_admix(p_ind)

We provide functions for these separate steps to allow for more flexibility. It is worth noting that `draw_p_anc` can be used to draw from a Symmetric Beta distribution if the `beta` parameter is set. In the example below, a small shape value of `beta` \(< 1\) is used to draw a distribution with more rare minor variants.

    # this increases the proportion of rare alleles
    p_anc_beta <- draw_p_anc(m_loci, beta = 0.1)
    # shrink default margins for these figures
    par(mar = c(4, 4, 0, 0) + 0.2)
    hist(p_anc_beta, xlab = 'Ancestral AF', main = '', xlim = c(0, 1))

However, if you want any other distribution, that can be implemented at this step. For a ridiculous example, the ancestral allele frequencies could be drawn from a (non-symmetric) Beta instead of using `draw_p_anc`:

    # this increases the proportion of rare alleles
    p_anc_beta <- rbeta(m_loci, shape1 = 0.1, shape2 = 1)
    # shrink default margins for these figures
    par(mar = c(4, 4, 0, 0) + 0.2)
    hist(p_anc_beta, xlab = 'Ancestral AF', main = '', xlim = c(0, 1))

You could also draw genotypes from the ancestral population or the intermediate populations:

    # draw genotypes for one individual from the ancestral population
    # use "cbind" to turn the vector p_anc into a column matrix
    # ("draw_genotypes_admix" expects a matrix)
    X_anc <- draw_genotypes_admix( cbind( p_anc ) )
    # returns a column matrix:
    dim(X_anc)
    #> [1] 10000     1

    # draw genotypes from intermediate populations
    # draws one individual per intermediate population
    X_subpops <- draw_genotypes_admix(p_subpops)

# Additional population structures

Here we show examples for functions that create admixture matrices for various simple population structures. The admixture scenarios implemented in `bnpsd` are generated by these functions:

- `admix_prop_1d_linear`: Linear 1D geography
- `admix_prop_1d_circular`: Circular 1D geography
- `admix_prop_indep_subpops`: Independent Subpopulations

The main example (above) was for `admix_prop_1d_linear`, the rest follow.

## Circular 1D geography

This is a twist on the earlier 1D geography where subpopulations and individuals are placed on a circumference, so random walks wrap around and the appropriate density is the Von Misses distribution.

Let’s generate an analogous population structure to the original “linear” example.

    # data dimensions
    # number of individuals
    n_ind <- 100
    # number of intermediate subpops
    k_subpops <- 10

    # define population structure
    # subpopulation FST vector, up to a scalar
    inbr_subpops <- 1 : k_subpops
    # desired bias coefficient
    bias_coeff <- 0.5
    # desired FST for the admixed individuals
    Fst <- 0.1

    # admixture proportions from *circular* 1D geography
    obj <- admix_prop_1d_circular(
        n_ind = n_ind,
        k_subpops = k_subpops,
        bias_coeff = bias_coeff,
        coanc_subpops = inbr_subpops,
        fst = Fst
    )
    admix_proportions <- obj$admix_proportions
    inbr_subpops <- obj$coanc_subpops

    # get pop structure parameters of the admixed individuals
    coancestry <- coanc_admix(admix_proportions, inbr_subpops)

    # verify that we got the desired FST!
    fst_admix(admix_proportions, inbr_subpops)
    #> [1] 0.1

    # verify that we got the desired bias_coeff too!
    mean(coancestry) / Fst
    #> [1] 0.5

    # visualize the per-subpopulation inbreeding coefficients (FSTs)
    # tweak margins/etc
    par(mar = c(2.5, 2.5, 0.3, 0) + 0.2, lab = c(2, 1, 7), mgp = c(1.5, 0.5, 0))
    # colors for independent subpopulations
    col_subpops <- brewer.pal(k_subpops, "Paired")
    barplot(inbr_subpops, col = col_subpops, names.arg = colnames(admix_proportions), xlab = 'Subpopulation', ylab = 'Inbr')

    # visualize the admixture proportions
    # tweak margins/etc
    par(mar = c(1, 4, 0.4, 0) + 0.2, lab = c(2, 2, 7))
    barplot(
        t(admix_proportions),
        col = col_subpops,
        border = NA,
        space = 0,
        ylab = 'Admix prop'
    )
    mtext('Individuals', 1)

    # Visualize the coancestry matrix using "popkin"!
    plot_popkin(
        coancestry,
        leg_n = 3,
        mar = c(0, 0.4),
        leg_title = 'Coancestry'
    )

## Independent subpopulations

The independent subpopulations model, where individuals are actually unadmixed, is the most trivial form of the BN-PSD admixture model.

    # define population structure
    # we'll have k_subpops = 3, each subpopulation with these sizes:
    n1 <- 100
    n2 <- 50
    n3 <- 20

    # here's the labels (for simplicity, list all individuals of S1 first, then S2, then S3)
    labs <- c(
        rep.int('S1', n1),
        rep.int('S2', n2),
        rep.int('S3', n3)
    )
    # data dimensions infered from labs:

    # number of individuals "n_ind"
    length(labs)
    #> [1] 170

    # number of subpopulations "k_subpops"
    k_subpops <- length(unique(labs))
    k_subpops
    #> [1] 3

    # desired admixture matrix
    admix_proportions <- admix_prop_indep_subpops(labs)

    # got a numeric matrix with a single 1 value per row
    # (denoting the sole subpopulation from which each individual draws its ancestry)
    head(admix_proportions, 2)
    #>      S1 S2 S3
    #> [1,]  1  0  0
    #> [2,]  1  0  0

    # construct the intermediate subpopulation FST vector
    # the desired final FST
    Fst <- 0.2
    # subpopulation FST vector, unnormalized so far
    inbr_subpops <- 1 : k_subpops
    # normalized to have the desired FST
    # NOTE fst is a function in the `popkin` package
    inbr_subpops <- inbr_subpops / fst(inbr_subpops) * Fst
    # verify FST for the intermediate subpopulations
    fst(inbr_subpops)
    #> [1] 0.2

    # get coancestry of the admixed individuals
    coancestry <- coanc_admix(admix_proportions, inbr_subpops)
    # before getting FST for individuals, weigh then inversely proportional to subpop sizes
    weights <- weights_subpops(labs) # function from `popkin` package
    # verify FST for individuals (same as for intermediate subpops for this pop structure)
    fst_admix(admix_proportions, inbr_subpops, weights)
    #> [1] 0.2

    # visualize the per-subpopulation inbreeding coefficients (FSTs)
    # tweak margins/etc
    par(mar = c(2.5, 2.5, 0, 0) + 0.2, lab = c(2, 1, 7), mgp = c(1.5, 0.5, 0))
    # colors for independent subpopulations
    col_subpops <- brewer.pal(k_subpops, "Paired")
    barplot(inbr_subpops, col = col_subpops, names.arg = colnames(admix_proportions), xlab = 'Subpopulation', ylab = 'Inbr')

    # visualize the admixture proportions
    # tweak margins/etc
    par(mar = c(1, 4, 0.4, 0) + 0.2, lab = c(2, 2, 7))
    barplot(
        t(admix_proportions),
        col = col_subpops,
        border = NA,
        space = 0,
        ylab = 'Admix prop'
    )
    mtext('Individuals', 1)

    # Visualize the coancestry matrix using "popkin"!
    plot_popkin(
        coancestry,
        leg_n = 3,
        mar = c(0, 0.4),
        leg_title = 'Coancestry'
    )

## Intermediate subpopulations related by a tree

In all of the previous examples we simulated from independent subpopulations first, then admixed individuals in various ways (1D linear and 1D circular scenarios, as well as “independent subpopulations” which meant there was no admixture). Here we use an extension of the approach to simulate the intermediate subpopulations related by a tree, which allows for correlated subpopulations.

First let’s construct the tree of interest. An easy-to-understand example is a serial founder effect, where there are consecutive splits on one branch, which somewhat resembles the human tree. For simplicity we model only 5 “tip” subpopulations and shall assume that there is equal inbreeding on every branch, equal to 0.1. This tree, in the standard (Newick) character format, is:

    tree_str <- '(S1:0.1,(S2:0.1,(S3:0.1,(S4:0.1,S5:0.1)N3:0.1)N2:0.1)N1:0.1)T;'

The tips (leaf nodes) are labeled as S1 to S5, the root node is labeled as T, and the (internal) nodes are N1 to N3. Before proceeding, we parse and process this tree using the `read.tree` function from the package `ape`, which creates the desired tree object of class `phylo` that the our functions require.

    # base package for phylogenetics
    library(ape)
    # parses tree, creates object of class "phylo"
    tree_subpops <- read.tree( text = tree_str )
    # plot tree
    par( mar = c(3,0,0,0) + 0.2 )
    plot( tree_subpops, show.node.label = TRUE )
    # add axis and label
    axisPhylo( backward = FALSE )
    mtext( 'Coancestry', side = 1, line = 2 )

The values of the edges in this tree are taken to be inbreeding values, and we draw allele frequencies for each subpopulation sequentially from its parent population using the Balding-Nichols distribution, which is achieved with the function `draw_p_subpops_tree`:

    p_subpops_tree <- draw_p_subpops_tree(
        p_anc = p_anc,
        tree_subpops = tree_subpops
    )

Let’s validate the correlation structure of these allele frequencies:

    # estimate coancestry
    coancestry_subpops_tree_estimate <- popkin_af( p_subpops_tree )
    # expected coancestry according to model
    coancestry_subpops_tree <- coanc_tree( tree_subpops )
    # plot
    plot_popkin(
        list( coancestry_subpops_tree, coancestry_subpops_tree_estimate ),
        titles = c('Truth', 'Estimate'),
        mar = c(2, 2.5),
        leg_title = 'Coancestry',
        names = TRUE
    )

Note that the coancestry values in the tree are not additive in the final coancestry matrix. While the tree edges go up to 0.4 (0.1 added 4 times), the maximum coancestry (equal to the total self-coancestry for node `S5` relative to `T`) is actually equal to:

    1 - (1 - 0.1)^4
    #> [1] 0.3439

The discrepancy is that the coancestry values in the matrix are all relative to the root node `T`, while the edges in the plot of the tree are each relative to their parent node (the edge `N3`-`S5` is relative to `N3`). The coancestry between a pair of different subpopulations is given by the total coancestry of the most recent common ancestor node of the two subpopulations (so the coancestry of `S4` and `S5` is the self-coancestry of `N3`).

### Admixture with intermediate tree structure

Now we repeat our previous simulation of the admixture of individuals, but this time replacing the independent subpopulation structure with our previous tree structure for the intermediate subpopulations. In our previous code, it suffices to replace the previous `inbr_subpops` with `coancestry_subpops_tree` which we estimated from `tree_subpops` in the last section.

    # number of admixed individuals
    n_ind <- 100
    # number of subpopulations, extracted from true coancestry
    k_subpops <- nrow( coancestry_subpops_tree )

    # define population structure
    # for simplicity here we set spread parameter `sigma` directly
    # admixture proportions from 1D geography
    admix_proportions <- admix_prop_1d_linear(
        n_ind = n_ind,
        k_subpops = k_subpops,
        sigma = 0.5
    )

    # get pop structure parameters of the admixed individuals
    coancestry <- coanc_admix(admix_proportions, coancestry_subpops_tree)

    # calculate true FST of simulation
    fst_admix(admix_proportions, coancestry_subpops_tree)
    #> [1] 0.2078772

These admixture proportions resemble our previous simulation, except we reduced the number of intermediate subpopulations and the spread parameter `sigma` was picked manually just above:

    # visualize the admixture proportions
    # colors for these independent subpopulations
    col_subpops <- brewer.pal(k_subpops, "Paired")
    # shrink default margins
    par(mar = c(1, 4, 0, 0) + 0.2)
    barplot(
        t(admix_proportions),
        col = col_subpops,
        border = NA,
        space = 0,
        ylab = 'Admixture prop.'
    )
    mtext('Individuals', 1)

The coancestry matrix looks a bit more blocky because there are fewer intermediate subpopulations and there is less spread, and also resembles the coancestry of the tree subpopulations we visualized earlier:

    # Visualize the coancestry matrix using "popkin"!
    plot_popkin(
        coancestry,
        # zero inner margin (plus padding) because we have no labels
        mar = 0,
        leg_title = 'Coancestry'
    )

### Draw random allele frequencies and genotypes for admixture with intermediate tree structure

The function `draw_all_admix` accepts `tree_subpops` in place of `inbr_subpops` to perform the entire simulation of genotypes and intermediate quantities for intermediate subpopulations related by a tree. However, unlike functions for the true structure (like `coanc_admix` and `fst_admix` earlier) which accept arbitrary intermediate coancestry matrices, the functions for drawing intermediate subpopulation allele frequencies requires trees or vectors (independent subpopulations) and do not support arbitrary coancestry matrices.

For simplicity we ask for genotypes only and skip analyzing the other allele frequency matrices (`p_subpops` here is simulated from `draw_p_subpops_tree`, which we analyzed earlier).

    # number of loci in simulation (NOTE this is 30x less than in publication!)
    m_loci <- 10000
    # draw all random Allele Freqs (AFs) and genotypes
    # reuse the previous inbr_subpops, admix_proportions
    out <- draw_all_admix(
        admix_proportions = admix_proportions,
        tree_subpops = tree_subpops,
        m_loci = m_loci
    )
    # genotypes
    X <- out$X

Let’s verify that the correlation structure of the genotypes matches the theoretical coancestry matrix we constructed earlier.

    # estimate kinship using popkin
    # here we didn't use subpopulation labels for simplicity
    kinship_estimate <- popkin(X)
    # replace diagonal with inbreeding coeffs. to match coancestry matrix
    coancestry_estimate <- inbr_diag(kinship_estimate)

    # Visualize the coancestry matrix using "popkin"!
    plot_popkin(
        list( coancestry, coancestry_estimate ),
        titles = c('Truth', 'Estimate'),
        # second value is top margin, for panel titles
        mar = c(0, 2.5),
        leg_title = 'Coancestry'
    )

# References

Balding, D. J., and R. A. Nichols. 1995. “A Method for Quantifying Differentiation Between Populations at Multi-Allelic Loci and Its Implications for Investigating Identity and Paternity.” Genetica 96 (1-2): 3–12.

Ochoa, Alejandro, and John D. Storey. 2016. “FST and Kinship for Arbitrary Population Structures I: Generalized Definitions.” bioRxiv doi:10.1101/083915. https://doi.org/10.1101/083915.

———. 2021. “Estimating FST and Kinship for Arbitrary Population Structures.” PLoS Genet 17 (1): e1009241. https://doi.org/10.1371/journal.pgen.1009241.

Pritchard, J. K., M. Stephens, and P. Donnelly. 2000. “Inference of Population Structure Using Multilocus Genotype Data.” Genetics 155 (2): 945–59.