validation.py

#!/usr/bin/env python3

import os
import sys
import pathlib
import argparse
import random
import functools
import itertools
import concurrent.futures
import time

import numpy as np
import tskit
import msprime
# Work around an issue on systems with large numbers of cores.
# https://github.com/cggh/scikit-allel/issues/285
os.environ["NUMEXPR_MAX_THREADS"] = f"{os.cpu_count()}"  # NOQA
import allel

import matplotlib
matplotlib.use('Agg')  # NOQA # don't try to use $DISPLAY
import matplotlib.pyplot as plt
from matplotlib.backends.backend_pdf import PdfPages

import stdpopsim
import stdpopsim.cli


def warning(msg):
    """
    Print a warning, with output less ugly than that of warnings.warn().
    """
    print(f"WARNING: {msg}", file=sys.stderr)


def irradiate(contig, x=20):
    """
    Increase mutation rate by a factor of `x`.
    """
    return stdpopsim.Contig(
            recombination_map=contig.recombination_map,
            mutation_rate=x*contig.mutation_rate,
            genetic_map=contig.genetic_map)

#
# Simulation functions.
#


def _onepop_PC(engine_id, out_dir, seed, N0=1000, *size_changes, **sim_kwargs):
    species = stdpopsim.get_species("CanFam")
    contig = species.get_contig("chr35", length_multiplier=0.01)  # ~265 kb
    contig = irradiate(contig)
    model = stdpopsim.PiecewiseConstantSize(N0, *size_changes)
    model.generation_time = species.generation_time
    samples = model.get_samples(100)
    engine = stdpopsim.get_engine(engine_id)
    t0 = time.perf_counter()
    ts = engine.simulate(model, contig, samples, seed=seed, **sim_kwargs)
    t1 = time.perf_counter()
    out_file = out_dir / f"{seed}.trees"
    ts.dump(out_file)
    return out_file, t1 - t0


def onepop_constantN_msprime1(out_dir, seed):
    """
    Single population with constant population size.
    """
    return _onepop_PC("msprime", out_dir, seed)


def onepop_constantN_slim1(out_dir, seed):
    """
    Single population with constant population size.
    """
    return _onepop_PC("slim", out_dir, seed)


def onepop_constantN_slim2(out_dir, seed):
    """
    Single population with constant population size.
    Burn-in is disabled and since there are no demographic_events, SLiM exits
    immediately. Tree sequences are constructed via recapitation.
    """
    return _onepop_PC("slim", out_dir, seed, slim_burn_in=0)


def onepop_constantN_slim3(out_dir, seed):
    """
    Single population with constant population size.
    Time and Ne are rescaled by a factor of 10.
    """
    return _onepop_PC("slim", out_dir, seed, slim_scaling_factor=10)


def onepop_bottleneck_msprime1(out_dir, seed):
    """
    Single population with bottleneck and recovery.
    """
    return _onepop_PC("msprime", out_dir, seed, 5000, (800, 100), (1000, 1000))


def onepop_bottleneck_slim1(out_dir, seed):
    """
    Single population with bottleneck and recovery.
    """
    return _onepop_PC("slim", out_dir, seed, 5000, (800, 100), (1000, 1000))


def onepop_bottleneck_slim2(out_dir, seed):
    """
    Single population with bottleneck and recovery.
    Burn-in is disabled.
    """
    return _onepop_PC(
            "slim", out_dir, seed, 5000, (800, 100), (1000, 1000),
            slim_burn_in=0)


def onepop_bottleneck_slim3(out_dir, seed):
    """
    Single population with bottleneck and recovery.
    Time and Ne are rescaled by a factor of 10.
    """
    return _onepop_PC(
            "slim", out_dir, seed, 5000, (800, 100), (1000, 1000),
            slim_scaling_factor=10)


class _PiecewiseSize(stdpopsim.DemographicModel):
    """
    A copy of stdpopsim.PiecewiseConstantSize that permits growth rates.
    """
    id = "Piecewise"
    description = "Piecewise size population model over multiple epochs."
    citations = []
    populations = [stdpopsim.Population(id="pop0", description="Population 0")]
    author = None
    year = None
    doi = None

    def __init__(self, N0, growth_rate, *args):
        self.population_configurations = [
            msprime.PopulationConfiguration(
                initial_size=N0, growth_rate=growth_rate,
                metadata=self.populations[0].asdict())
        ]
        self.migration_matrix = [[0]]
        self.demographic_events = []
        for t, initial_size, growth_rate in args:
            self.demographic_events.append(msprime.PopulationParametersChange(
                time=t, initial_size=initial_size, growth_rate=growth_rate,
                population_id=0))


def _onepop_expgrowth(
        engine_id, out_dir, seed, N0=5000, N1=500, T=1000, **sim_kwargs):
    growth_rate = - np.log(N1 / N0) / T
    species = stdpopsim.get_species("DroMel")
    contig = species.get_contig("chr2R", length_multiplier=0.01)  # ~250 kb
    contig = irradiate(contig)
    model = _PiecewiseSize(N0, growth_rate, (T, N1, 0))
    model.generation_time = species.generation_time
    samples = model.get_samples(100)
    engine = stdpopsim.get_engine(engine_id)
    t0 = time.perf_counter()
    ts = engine.simulate(model, contig, samples, seed=seed, **sim_kwargs)
    t1 = time.perf_counter()
    out_file = out_dir / f"{seed}.trees"
    ts.dump(out_file)
    return out_file, t1 - t0


def onepop_expgrowth_msprime1(out_dir, seed):
    """
    Single population with exponential population size growth.
    """
    return _onepop_expgrowth("msprime", out_dir, seed)


def onepop_expgrowth_slim1(out_dir, seed):
    """
    Single population with exponential population size growth.
    """
    return _onepop_expgrowth("slim", out_dir, seed)


def onepop_expgrowth_slim2(out_dir, seed):
    """
    Single population with exponential population size growth.
    Burn-in is disabled.
    """
    return _onepop_expgrowth("slim", out_dir, seed, slim_burn_in=0)


def onepop_expgrowth_slim3(out_dir, seed):
    """
    Single population with exponential population size growth.
    Time and Ne are rescaled by a factor of 10.
    """
    return _onepop_expgrowth("slim", out_dir, seed, slim_scaling_factor=10)


def _twopop_IM(
        engine_id, out_dir, seed,
        NA=1000, N1=500, N2=5000, T=1000, M12=0, M21=0, pulse=None, samples=None,
        **sim_kwargs):
    species = stdpopsim.get_species("AraTha")
    contig = species.get_contig("chr5", length_multiplier=0.01)  # ~270 kb
    contig = irradiate(contig)
    model = stdpopsim.IsolationWithMigration(
            NA=NA, N1=N1, N2=N2, T=T, M12=M12, M21=M21)
    if pulse is not None:
        model.demographic_events.append(pulse)
        model.demographic_events.sort(key=lambda x: x.time)
    # XXX: AraTha has species.generation_time == 1, but there is the potential
    # for this to mask bugs related to generation_time scaling, so we use 3 here.
    model.generation_time = 3
    if samples is None:
        samples = model.get_samples(50, 50, 0)
    engine = stdpopsim.get_engine(engine_id)
    t0 = time.perf_counter()
    ts = engine.simulate(model, contig, samples, seed=seed, **sim_kwargs)
    t1 = time.perf_counter()
    out_file = out_dir / f"{seed}.trees"
    ts.dump(out_file)
    return out_file, t1 - t0


def twopop_no_migration_msprime1(out_dir, seed):
    """
    Two populations with different sizes and no migrations.
    """
    return _twopop_IM("msprime", out_dir, seed)


def twopop_no_migration_slim1(out_dir, seed):
    """
    Two populations with different sizes and no migrations.
    """
    return _twopop_IM("slim", out_dir, seed)


def twopop_no_migration_slim2(out_dir, seed):
    """
    Two populations with different sizes and no migrations.
    Burn-in is disabled. Time and Ne are rescaled by a factor of 10.
    """
    return _twopop_IM(
            "slim", out_dir, seed, slim_burn_in=0, slim_scaling_factor=10)


def twopop_asymmetric_migration_msprime1(out_dir, seed):
    """
    Two populations with different sizes and migrations from pop2 to pop1.
    """
    return _twopop_IM("msprime", out_dir, seed, M12=0, M21=0.001)


def twopop_asymmetric_migration_slim1(out_dir, seed):
    """
    Two populations with different sizes and migrations from pop2 to pop1.
    """
    return _twopop_IM("slim", out_dir, seed, M12=0, M21=0.001)


def twopop_asymmetric_migration_slim2(out_dir, seed):
    """
    Two populations with different sizes and migrations from pop2 to pop1.
    Burn-in is disabled. Time and Ne are rescaled by a factor of 10.
    """
    return _twopop_IM(
            "slim", out_dir, seed, M12=0, M21=0.001,
            slim_burn_in=0, slim_scaling_factor=10)


_pulse_m21 = msprime.MassMigration(
        time=20, proportion=0.1, source=1, destination=0)


def twopop_pulse_migration_msprime1(out_dir, seed):
    """
    Two populations with different sizes and introgression from pop2 to pop1.
    """
    return _twopop_IM("msprime", out_dir, seed, pulse=_pulse_m21)


def twopop_pulse_migration_slim1(out_dir, seed):
    """
    Two populations with different sizes and introgression from pop2 to pop1.
    """
    return _twopop_IM("slim", out_dir, seed, pulse=_pulse_m21)


def twopop_pulse_migration_slim2(out_dir, seed):
    """
    Two populations with different sizes and introgression from pop2 to pop1.
    Burn-in is disabled. Time and Ne are rescaled by a factor of 10.
    """
    return _twopop_IM(
            "slim", out_dir, seed, pulse=_pulse_m21,
            slim_burn_in=0, slim_scaling_factor=10)


_ancient_samples = 50 * [msprime.Sample(0, time=0), msprime.Sample(1, time=500)]


def twopop_ancient_samples_msprime1(out_dir, seed):
    """
    Two populations, with ancient sampling of the second population.
    """
    return _twopop_IM("msprime", out_dir, seed, samples=_ancient_samples)


def twopop_ancient_samples_slim1(out_dir, seed):
    """
    Two populations, with ancient sampling of the second population.
    """
    return _twopop_IM("slim", out_dir, seed, samples=_ancient_samples)


def twopop_ancient_samples_slim2(out_dir, seed):
    """
    Two populations, with ancient sampling of the second population.
    Burn-in is disabled. Time and Ne are rescaled by a factor of 10.
    """
    return _twopop_IM(
            "slim", out_dir, seed, samples=_ancient_samples,
            slim_burn_in=0, slim_scaling_factor=10)


def do_cmd(cmd, out_dir, seed):
    cmd = cmd.split()
    assert "-o" not in cmd and "--output" not in cmd
    assert "-s" not in cmd and "--seed" not in cmd
    out_file = out_dir / f"{seed}.trees"
    full_cmd = cmd + f" -q -o {out_file} -s {seed}".split()
    t0 = time.perf_counter()
    stdpopsim.cli.stdpopsim_main(full_cmd)
    t1 = time.perf_counter()
    assert os.path.exists(out_file)
    return out_file, t1 - t0


_homsap_250k = " HomSap -c chr1 -l 0.001 "


def Africa_1T12_msprime1(out_dir, seed):
    cmd = "-e msprime" + _homsap_250k + "-d Africa_1T12 100"
    return do_cmd(cmd, out_dir, seed)


def Africa_1T12_slim1(out_dir, seed):
    cmd = "-e slim" + _homsap_250k + "-d Africa_1T12 100"
    return do_cmd(cmd, out_dir, seed)


def OutOfAfrica_3G09_msprime1(out_dir, seed):
    samples = 3 * " 33"
    cmd = "-e msprime" + _homsap_250k + "-d OutOfAfrica_3G09" + samples
    return do_cmd(cmd, out_dir, seed)


def OutOfAfrica_3G09_slim1(out_dir, seed):
    samples = 3 * " 33"
    cmd = "-e slim" + _homsap_250k + "-d OutOfAfrica_3G09" + samples
    return do_cmd(cmd, out_dir, seed)


def AmericanAdmixture_4B11_msprime1(out_dir, seed):
    samples = 4 * " 25"
    cmd = "-e msprime" + _homsap_250k + "-d AmericanAdmixture_4B11" + samples
    return do_cmd(cmd, out_dir, seed)


def AmericanAdmixture_4B11_slim1(out_dir, seed):
    samples = 4 * " 25"
    cmd = "-e slim" + _homsap_250k + "-d AmericanAdmixture_4B11" + samples
    return do_cmd(cmd, out_dir, seed)


def AncientEurasia_9K19_msprime1(out_dir, seed):
    samples = 8 * " 12"
    cmd = "-e msprime" + _homsap_250k + "-d AncientEurasia_9K19" + samples
    return do_cmd(cmd, out_dir, seed)


def AncientEurasia_9K19_slim1(out_dir, seed):
    samples = 8 * " 12"
    cmd = "-e slim" + _homsap_250k + "-d AncientEurasia_9K19" + samples
    return do_cmd(cmd, out_dir, seed)


#
# Stats functions.
#


def tmrca(ts):
    """
    Time to most recent common ancestor of sample, aka tree height.
    """
    tmrcas = [tree.time(tree.root) for tree in ts.trees()]
    min_, median, max_ = np.quantile(tmrcas, (0, 0.5, 1))
    return {"min(tmrca)": min_,
            "median(tmrca)": median,
            "max(tmrca)": max_,
            }


def ts_properties(ts):
    """
    TreeSequence properties.
    """
    return {"num_trees": ts.num_trees,
            "num_edges": ts.num_edges,
            "num_nodes": ts.num_nodes,
            "num_sites": ts.num_sites,
            }


def pooled_pop_stats(ts):
    """
    Population statistics, with samples pooled from all populations.
    """
    n = ts.num_samples // 2
    samples = list(itertools.chain(*(ts.samples(i) for i in range(ts.num_populations))))
    sample_sets = [samples[:n], samples[n:]]
    return {"diversity": ts.diversity(),
            "Tajimas_D": ts.Tajimas_D(),
            "$f_2$": ts.f2(sample_sets),
            "$Y_2$": ts.Y2(sample_sets),
            "segregating_sites": ts.segregating_sites(),
            }


def pairwise_pop_stats(ts):
    """
    Pairwise population statistics, calculated for all pairs of populations.
    """
    pops = [i for i in range(ts.num_populations) if len(ts.samples(i)) > 0]
    if len(pops) < 2:
        return None
    sample_sets = [ts.samples(i) for i in pops]
    indexes = list(itertools.combinations(range(len(pops)), 2))
    f2 = ts.f2(sample_sets, indexes)
    Y2 = ts.Y2(sample_sets, indexes)
    stats = dict()
    for i, (j, k) in enumerate(indexes):
        stats[f"$f_2$[{pops[j]},{pops[k]}]"] = f2[i]
        stats[f"$Y_2$[{pops[j]},{pops[k]}]"] = Y2[i]
    return stats


def linkage_disequilibrium(
        ts, span=40000, bins=20, min_obs_per_bin=8, max_sequence_length=1e6):
    """
    R^2 as a function of site-separation distance, for `bins` bins up to a
    site-separation distance of `span` bp.
    """
    if ts.sequence_length > max_sequence_length:
        ts = ts.keep_intervals([(0, max_sequence_length)], record_provenance=False)
    position = [site.position for site in ts.sites()]
    num_sites = len(position)
    assert num_sites == int(ts.num_sites)

    nans = np.full(bins, np.nan)
    if num_sites >= min_obs_per_bin:
        gts = np.expand_dims(ts.genotype_matrix(), axis=-1)
        gn = allel.GenotypeArray(gts, dtype='i1').to_n_alt()
        ld = allel.rogers_huff_r(gn)**2
        assert len(ld) == num_sites * (num_sites - 1) // 2

        # Bin the pairwise site R^2 in `ld` by site separation distance.
        r2 = np.zeros(bins)
        n = np.zeros(bins)
        i = 0
        for j in range(num_sites):
            for k in range(j+1, num_sites):
                distance = position[k] - position[j]
                if distance >= span:
                    break
                index = int(distance * bins / span)
                if not np.isnan(ld[i]):
                    r2[index] += ld[i]
                    n[index] += 1
                i += 1
        # Divide `r2` by `n`, but return NaN where n has insufficient observations.
        r2 = np.divide(r2, n, out=nans, where=n >= min_obs_per_bin)
    else:
        # Too few segregating sites to do anything meaningful.
        # LD plots may be blank.
        r2 = nans

    return {f"$\Delta$bp$\in[{span*k/bins/1000:.0f}\,$k$,"  # NOQA
            f"{span*(k+1)/bins/1000:.0f}\,$k$)$": r2[k]     # NOQA
            for k in range(bins)}


def allele_frequency_spectrum(ts, bins=20):
    """
    Allele frequency spectrum for `bins` allele frequency bins.
    Values are log(1+counts) for each bin.
    """
    full_afs = ts.allele_frequency_spectrum(span_normalise=False, polarised=True)
    afs = np.zeros(bins)
    for j in range(1, len(full_afs)):
        index = int((j - 1) * bins / (len(full_afs) - 1))
        afs[index] += full_afs[j]
    afs = np.log(1+afs)
    return {f"AF$\in$[{k/bins:.2f},{(k+1)/bins:.2f})": afs[k]  # NOQA
            for k in range(bins)}


def node_arity(ts):
    """
    The number of children for internal nodes of each marginal tree.
    In msprime with the hudson simulation model, this is always 2.
    In SLiM, this might be more than 2, particularly with small population sizes.
    """
    max_arity = 0
    non_binary = 0
    for tree in ts.trees():
        for node in tree.nodes():
            if tree.is_internal(node):
                num_children = len(tree.children(node))
                if num_children > max_arity:
                    max_arity = num_children
                if num_children > 2:
                    non_binary += 1

    return {"max(node_arity)": max_arity,
            "count(node_arity>2)": non_binary,
            }


_simulation_functions = [
    onepop_constantN_msprime1,
    onepop_constantN_slim1,
    onepop_constantN_slim2,
    onepop_constantN_slim3,
    onepop_bottleneck_msprime1,
    onepop_bottleneck_slim1,
    onepop_bottleneck_slim2,
    onepop_bottleneck_slim3,
    onepop_expgrowth_msprime1,
    onepop_expgrowth_slim1,
    onepop_expgrowth_slim2,
    onepop_expgrowth_slim3,

    twopop_no_migration_msprime1,
    twopop_no_migration_slim1,
    twopop_no_migration_slim2,
    twopop_asymmetric_migration_msprime1,
    twopop_asymmetric_migration_slim1,
    twopop_asymmetric_migration_slim2,
    twopop_pulse_migration_msprime1,
    twopop_pulse_migration_slim1,
    twopop_pulse_migration_slim2,
    twopop_ancient_samples_msprime1,
    twopop_ancient_samples_slim1,
    twopop_ancient_samples_slim2,

    Africa_1T12_msprime1,
    Africa_1T12_slim1,
    OutOfAfrica_3G09_msprime1,
    OutOfAfrica_3G09_slim1,
    AmericanAdmixture_4B11_msprime1,
    AmericanAdmixture_4B11_slim1,
    AncientEurasia_9K19_msprime1,
    AncientEurasia_9K19_slim1,
]

_stats_functions = [
    ts_properties,
    tmrca,
    pooled_pop_stats,
    pairwise_pop_stats,
    linkage_disequilibrium,
    allele_frequency_spectrum,

    # Node arity stats are disabled as they're only relevant in special cases.
    #node_arity,
]

_default_comparisons = [
    (onepop_constantN_msprime1, onepop_constantN_slim1),
    (onepop_constantN_msprime1, onepop_constantN_slim2),
    (onepop_constantN_msprime1, onepop_constantN_slim3),
    (onepop_bottleneck_msprime1, onepop_bottleneck_slim1),
    (onepop_bottleneck_msprime1, onepop_bottleneck_slim2),
    (onepop_bottleneck_msprime1, onepop_bottleneck_slim3),
    (onepop_expgrowth_msprime1, onepop_expgrowth_slim1),
    (onepop_expgrowth_msprime1, onepop_expgrowth_slim2),
    (onepop_expgrowth_msprime1, onepop_expgrowth_slim3),

    (twopop_no_migration_msprime1, twopop_no_migration_slim1),
    (twopop_no_migration_msprime1, twopop_no_migration_slim2),
    (twopop_asymmetric_migration_msprime1, twopop_asymmetric_migration_slim1),
    (twopop_asymmetric_migration_msprime1, twopop_asymmetric_migration_slim2),
    (twopop_pulse_migration_msprime1, twopop_pulse_migration_slim1),
    (twopop_pulse_migration_msprime1, twopop_pulse_migration_slim2),
    (twopop_ancient_samples_msprime1, twopop_ancient_samples_slim1),
    (twopop_ancient_samples_msprime1, twopop_ancient_samples_slim2),

    (Africa_1T12_msprime1, Africa_1T12_slim1),
    (OutOfAfrica_3G09_msprime1, OutOfAfrica_3G09_slim1),
    (AmericanAdmixture_4B11_msprime1, AmericanAdmixture_4B11_slim1),
    (AncientEurasia_9K19_msprime1, AncientEurasia_9K19_slim1),
]

stats_functions = {f.__name__: f for f in _stats_functions}
simulation_functions = {f.__name__: f for f in _simulation_functions}
default_comparisons = [(t[0].__name__, t[1].__name__)
                       for t in _default_comparisons]


def do_simulations(rng, path, num_replicates, executor, key):
    out_dir = path / "trees" / key
    out_dir.mkdir(parents=True, exist_ok=True)
    func = functools.partial(simulation_functions[key], out_dir)
    seeds = (rng.randrange(1, 2**32) for _ in range(num_replicates))
    res = list(executor.map(func, seeds))
    files, times = zip(*res)
    # dump timing info to a file
    np.savetxt(out_dir / "times.txt", times)
    return files, times


def find_simulations(path, key):
    out_dir = path / "trees" / key
    files = list(out_dir.glob("*.trees"))
    if len(files) == 0:
        raise RuntimeError(f"{out_dir}: no *.trees found.")

    times_file = out_dir / "times.txt"
    if times_file.exists():
        times = np.loadtxt(times_file)
    else:
        warning(f"No times.txt found for {key}")
        times = []

    return files, times


def compute_stats(ts_file):
    st = dict()
    ts = tskit.load(ts_file)
    for key, func in stats_functions.items():
        try:
            res = func(ts)
        except Exception:
            # Print the filename so it's easier to trace problems.
            warning(f"{ts_file} triggered exception")
            raise
        if res is not None:
            st[key] = res
    return st


def custom_violinplot(ax, data, labels):
    """
    Violin plot with a colour scheme shown in the matplotlib gallery.
    https://matplotlib.org/3.1.3/gallery/statistics/customized_violin.html
    """
    inds = list(range(1, len(labels)+1))
    quartile1, medians, quartile3 = np.percentile(data, [25, 50, 75], axis=1)
    parts = ax.violinplot(data, vert=False)
    collections = [parts[x] for x in parts.keys() if x != "bodies"] + parts["bodies"]
    for pc in collections:
        pc.set_facecolor("#D43F3A")
        pc.set_edgecolor("black")
        pc.set_alpha(1)
    ax.scatter(medians, inds, marker='o', fc="white", ec="black", s=30, zorder=3)
    ax.hlines(inds, quartile1, quartile3, color='k', linestyle='-', lw=5)
    ax.set_yticks(inds)
    ax.set_yticklabels(labels)


def do_plots(path, sim_key1, sim_key2, times, stats):
    plotdir = path / "plots"
    plotdir.mkdir(parents=True, exist_ok=True)

    cmap = plt.get_cmap("tab10")
    markers = "oXdPvp*"
    scale = 1.25
    fig_w, fig_h = plt.figaspect(9.0/16.0)
    figsize = (scale*fig_w, scale*fig_h)

    times1, times2 = times[sim_key1], times[sim_key2]
    stats1, stats2 = stats[sim_key1], stats[sim_key2]

    pdf = PdfPages(plotdir / f"{sim_key1}__{sim_key2}.pdf")

    # plot run times
    if len(times1) > 0 and len(times2) > 0:
        fig, ax = plt.subplots(figsize=figsize)
        label1 = sim_key1
        label2 = sim_key2
        f1 = simulation_functions.get(sim_key1)
        f2 = simulation_functions.get(sim_key2)
        if f1 is not None and f1.__doc__:
            label1 += "\n" + f1.__doc__
        if f2 is not None and f2.__doc__:
            label2 += "\n" + f2.__doc__
        custom_violinplot(ax, [times1, times2], [label1, label2])
        ax.set_title(f"Run time.")
        ax.set_xlabel("time (seconds)")
        ax.set_xlim(left=min(ax.get_xlim()[0], 0))
        fig.tight_layout()
        pdf.savefig(figure=fig)
        plt.close(fig)

    # QQ plots for each statistic
    quantiles = np.linspace(0, 1, 101)  # Use 101 to include 0.5.
    for stat_key in stats_functions.keys():
        if stat_key not in stats1[0]:
            continue
        inner_keys = stats1[0][stat_key].keys()
        assert inner_keys == stats2[0][stat_key].keys()
        ncols = int(np.ceil(np.sqrt(len(inner_keys))))
        nrows = int(np.ceil(len(inner_keys) / ncols))
        share = False
        if stat_key in ("linkage_disequilibrium", "allele_frequency_spectrum"):
            share = True
            shared_min = 1e9
            shared_max = -1e9
        fig, axs = plt.subplots(
                nrows=nrows, ncols=ncols, figsize=figsize,
                sharex="all" if share is True else "none",
                sharey="all" if share is True else "none",
                )
        axs = np.array(axs).reshape(-1)
        assert len(axs) >= len(inner_keys)
        imarker = itertools.cycle(markers)
        icolour = itertools.cycle(cmap.colors)
        save_fig = False
        for ax, inner_key in zip(axs, inner_keys):
            x = [d[stat_key][inner_key] for d in stats1]
            y = [d[stat_key][inner_key] for d in stats2]
            assert len(x) > 0 and len(y) > 0
            if np.all(np.isnan(x)) or np.all(np.isnan(y)):
                continue

            xq = np.nanquantile(x, quantiles)
            yq = np.nanquantile(y, quantiles)

            # Tails of the distribution are distinguished using open markers,
            # as opposed to solid/closed markers for the body. `hi` has +1 to
            # get equal numbers of points in each tail.
            lo, median, hi = 5, 50, 95+1
            colour = next(icolour)
            marker = next(imarker)
            ax.scatter(xq[:lo], yq[:lo], ec=colour, fc="none", marker=marker)
            ax.scatter(xq[hi:], yq[hi:], ec=colour, fc="none", marker=marker)
            ax.scatter(xq[lo:hi], yq[lo:hi], ec=colour, fc=colour, marker=marker)
            ax.scatter(xq[median], yq[median], ec="black", fc="black", marker=marker)
            ax.set_title(inner_key)

            # draw a diagonal line
            min_ = min(np.min(xq), np.min(yq))
            max_ = max(np.max(xq), np.max(yq))
            if share:
                shared_min = min(min_, shared_min)
                shared_max = max(max_, shared_max)
            else:
                ax.plot([min_, max_], [min_, max_],
                        c="lightgray", ls="--", lw=1, zorder=-10)

            save_fig = True

        if not save_fig:
            plt.close(fig)
            continue

        for i, ax in enumerate(axs):
            if not share and len(axs) > 15:
                # reduce clutter by hiding labels when we have lots of subplots
                ax.set_xticks([])
                ax.set_xticklabels([])
                ax.set_yticks([])
                ax.set_yticklabels([])
            if share and i < len(inner_keys):
                ax.plot([shared_min, shared_max], [shared_min, shared_max],
                        c="lightgray", ls="--", lw=1, zorder=-10)
            if i >= len(inner_keys):
                # hide axes that weren't drawn on
                ax.set_axis_off()

        # use a full-figure subplot for labels that span the other subplots
        ax = fig.add_subplot(111, frameon=False)
        ax.set_xticks([])
        ax.set_yticks([])
        stat_docs = stats_functions[stat_key].__doc__
        if stat_docs:
            title = f"{stat_key}: {stat_docs}"
        else:
            warning(f"No docstring for {stat_key}")
            title = f"{stat_key}"
        ax.set_title(title, pad=20)
        ax.set_xlabel(sim_key1, labelpad=30)
        ax.set_ylabel(sim_key2, labelpad=50)
        fig.tight_layout()
        pdf.savefig(figure=fig, bbox_inches='tight')
        plt.close(fig)

    pdf.close()


def parse_args():
    parser = argparse.ArgumentParser(
            description="Do validation simulations and make QQ plots.")
    parser.add_argument(
            "-o", "--output-folder", metavar="DIR",
            type=pathlib.Path, default=pathlib.Path("validation"),
            help="Folder to store validation plots and tree sequences "
                 "[%(default)s].")

    mutex_group = parser.add_mutually_exclusive_group()
    mutex_group.add_argument(
            "-n", "--no-plots", action="store_true", default=False,
            help="Don't make plots, just do the simulations [%(default)s].")
    mutex_group.add_argument(
            "-p", "--plot-only", action="store_true", default=False,
            help="Don't simulate, just make QQ plots from preexisting files "
                 "[%(default)s].")

    parser.add_argument(
            "-j", "--num-procs", metavar="NPROCS", type=int, default=1,
            help="Number of simulations to run simultaneously [%(default)s].")
    parser.add_argument(
            "-r", "--num-replicates", metavar="NREPS", type=int, default=100,
            help="Number of replicates for each simulation key [%(default)s].")
    parser.add_argument(
            "-s", "--seed", metavar="SEED", type=int, default=1234,
            help="Seed for the random number generator [%(default)s].")
    parser.add_argument(
            "keys", nargs="*",
            help="One or more scenarios to simulate and/or compare.")

    args = parser.parse_args()

    if len(args.keys) == 0:
        args.comparisons = default_comparisons
        args.keys = list(set(itertools.chain(*args.comparisons)))
    else:
        args.comparisons = itertools.combinations(args.keys, 2)

    # sort keys to get deterministic ordering from random number generator
    args.keys.sort()

    for key in args.keys:
        if key not in simulation_functions:
            if args.plot_only:
                # Might be a mistake, but continue anyway to allow validation
                # using arbitrary folders that are in the right place.
                warning(f"unknown scenario key ``{key}''")
            else:
                parser.error(f"unknown scenario key ``{key}''")

    return args


if __name__ == "__main__":
    args = parse_args()
    rng = random.Random(args.seed)

    files = dict()
    times = dict()
    stats = dict()

    with concurrent.futures.ProcessPoolExecutor(args.num_procs) as executor:
        for sim_keys in args.comparisons:
            j, k = sim_keys
            assert j != k
            print(f"{j} / {k}.", end="", flush=True)

            for key in sim_keys:
                if key in files:
                    assert key in times
                    assert key in stats
                    continue

                if not args.plot_only:
                    files[key], times[key] = do_simulations(
                            rng, args.output_folder, args.num_replicates,
                            executor, key)
                else:
                    files[key], times[key] = find_simulations(
                            args.output_folder, key)

                print(".", end="", flush=True)

                if not args.no_plots:
                    stats[key] = list(executor.map(compute_stats, files[key]))
                    print(".", end="", flush=True)

            if not args.no_plots:
                do_plots(args.output_folder, j, k, times, stats)

            print("done")