Randomness and replication

Simulations are used to generate random samples from complex distributions, which we usually don’t have analytical solutions for. In this section we describe how msprime generates and controls randomness, and also how to efficiently and idiomatically run many replicate simulations to summarise those distributions.

Random seeds

Stochastic simulations depend on a source of randomness, provided by a pseudorandom number generator. Msprime uses the GNU Scientific Library to generate high-quality random numbers. The particular trajectory produced by a pseudorandom number generator is controlled by the “seed” it is provided.

Note

By default, msprime generates random seeds using a private instance of random.Random, which should guarantee unique seeds are produced even if (e.g.) many simulations are started at the same time in different processes. In particular, simulations run concurrently in subprocesses using concurrent.futures or multiprocessing will be assigned unique seeds by default. See the Running in parallel section for more information and examples.

Thus, if we run two ancestry simulations with the same parameters, we will get different results:

from IPython.display import display, SVG

ts1 = msprime.sim_ancestry(2)
ts2 = msprime.sim_ancestry(2)
display(SVG(ts1.draw_svg(y_axis=True)), SVG(ts2.draw_svg(y_axis=True)))
_images/replication_3_0.svg_images/replication_3_1.svg

The random_seed argument to sim_ancestry() allows us specify seeds explicitly, making the output of the simulation fully deterministic:

ts1 = msprime.sim_ancestry(2, random_seed=42)
ts2 = msprime.sim_ancestry(2, random_seed=42)
display(SVG(ts1.draw_svg(y_axis=True)), SVG(ts2.draw_svg(y_axis=True)))
_images/replication_5_0.svg_images/replication_5_1.svg

Similarly, if we run two different mutation simulations on the same ancestry, we’ll get different sets of mutations (and, therefore, genome sequences):

ts = msprime.sim_ancestry(2, sequence_length=10, random_seed=42)
mts1 = msprime.sim_mutations(ts, rate=0.1)
mts2 = msprime.sim_mutations(ts, rate=0.1)
display(SVG(mts1.draw_svg(y_axis=True)), SVG(mts2.draw_svg(y_axis=True)))
_images/replication_7_0.svg_images/replication_7_1.svg

We can see that mutations occur at different positions along the genome, and also on different branches of the trees. If we provide the random_seed argument to sim_mutations(), we will get precisely the same result in both simulations:

ts = msprime.sim_ancestry(2, sequence_length=10, random_seed=42)
mts1 = msprime.sim_mutations(ts, rate=0.1, random_seed=666)
mts2 = msprime.sim_mutations(ts, rate=0.1, random_seed=666)
display(SVG(mts1.draw_svg(y_axis=True)), SVG(mts2.draw_svg(y_axis=True)))
_images/replication_9_0.svg_images/replication_9_1.svg

Running replicate simulations

Simulations are random, and we will therefore usually want to have many independent replicates for a particular set of parameters. In this section we provide some examples of how to efficiently run replicate simulations in msprime.

Ancestry simulations

When performing ancestry simulations with sim_ancestry(), the num_replicates parameter provides a convenient and efficient way to iterate over simulation replicates. For example, suppose we wish to examine the distribution of the time to the most recent common ancestor.

import numpy as np

num_replicates = 100
tmrca = np.zeros(num_replicates)
replicates = msprime.sim_ancestry(10, num_replicates=num_replicates, random_seed=42)
for replicate_index, ts in enumerate(replicates):
    tree = ts.first()
    tmrca[replicate_index] = tree.time(tree.root)
np.mean(tmrca), np.var(tmrca)
(3.799291711240622, 3.920289255957004)

Tip

Using the num_replicates argument like this can be substantially more efficient that running simulations independently for small simulations.

It’s important to note that the replicate simulations are generated lazily here on demand—the replicates variable is a Python iterator, and we use the built-in enumerate() function to easily get the index of each replicate in the list. Thus, no simulation is actually done when we call sim_ancestry here, it’s just setting up the underlying state. Each replicate simulation is done sequentially as we go through the loop. We therefore only use the memory that we need to represent the results of one simulation.

Warning

It’s a bad idea to store the result of sim_ancestry in a list like this:

replicates = list(msprime.sim_ancestry(10, num_replicates=100))

Unlike the efficient example above, this will run all 100 simulations before returning and store the resulting tree sequences in memory. This consumes approximately 100X more memory that the example above!

Note

The return type of sim_ancestry() changes when we use the num_replicates argument. If num_replicates is not specified or None, we return an instance of tskit.TreeSequence. If it is specified, we return an iterator over a set of tskit.TreeSequence instances.

Adding mutations

Things are a little more involved when we also want to simulate mutations on the simulated ancestries, because we cannot get a stream of replicate results directly from sim_ancestry(). In this section we provide idiomatic code to efficiently combine ancestry and mutation simulations.

See also

It is often assumed that mutations are a necessary part of simulating population genetic processes, but this is often not the case. See the tutorial on this topic for more information.

Using a generator

In this example we define a “generator” function that runs a set of replicate ancestry simulations, simulates mutations on them, and sequentially yields the resulting tree sequence to the calling code:

def sim_replicates(sample_size, num_replicates):
    ancestry_reps = msprime.sim_ancestry(
        samples=sample_size,
        population_size=100, sequence_length=1_000,
        num_replicates=num_replicates)
    for ts in ancestry_reps:
        mutated_ts = msprime.sim_mutations(ts, rate=1e-6)
        yield mutated_ts

In general it’s a good idea to encapsulate specific simulations in a function like this: any free parameters (like sample_size here) can be passed in as parameters to the function, and any fixed values (like e.g., sequence_length) can be specified in one place, avoiding repetition and potential for mistakes. Having defined the generator function, we can now run it for a given number of replicates and sample size. In this example, we compute the number of segregating sites in each replicate tree sequence, and store the values in a numpy array:

num_replicates = 100
S = np.zeros(num_replicates)
for replicate_index, ts in enumerate(sim_replicates(10, num_replicates)):
    S[replicate_index] = ts.segregating_sites(span_normalise=False, mode="site")
# Show the mean and variance of S
(np.mean(S), np.var(S))
(1.52, 1.5495999999999999)

Again, this approach means that we only have the tree sequence for one replicate simulation in memory at one time.

Using functional tools

Another way in which we can efficiently apply mutations to replicate simulations is using Python’s functional programming tools:

import functools

num_replicates = 100
S = np.zeros(num_replicates)
ancestry_reps = msprime.sim_ancestry(
    samples=10,
    population_size=100,
    sequence_length=1_000,
    num_replicates=num_replicates)
mutations_func = functools.partial(msprime.sim_mutations, rate=1e-6)
mutations_reps = map(mutations_func, ancestry_reps)

for replicate_index, ts in enumerate(mutations_reps):
    S[replicate_index] = ts.segregating_sites(span_normalise=False, mode="site")
(np.mean(S), np.var(S))
(1.47, 2.0691)

In this example we use Python’s map() function to iteratively apply the sim_mutations() function to the output of sim_ancestry(). Because map only expects a single argument and we also need to specify the rate value, we need to create a “partial” function which sets the mutation rate (and any other parameters we might like to provide) for sim_mutations(). This could equally be done using a lambda expression or indeed by defining a function in the usual way using def.

Warning

Do not set the random seed to a fixed value in a partial function like this! This would result in different random ancestry replicates being correctly generated but all mutations being thrown down using the same seed. This would lead to nasty bugs!

Specifying seeds explicitly

When we’re running large simulations it’s usually better to specify the seeds explicitly and run ancestry simulations individually. First we define a function that takes our free parameters (num_samples in this case) and random seed values for the ancestry and mutation simulations:

def run_sim(num_samples, ancestry_seed, mutation_seed):
    ancestry_ts = msprime.sim_ancestry(
        samples=num_samples,
        population_size=100,
        sequence_length=1_000,
        random_seed=ancestry_seed)
    mutated_ts = msprime.sim_mutations(
        ancestry_ts, rate=1e-6, random_seed=mutation_seed)
    return mutated_ts

Note

Note that this function is not a generator—it returns a single tree sequence, rather than an iterator over replicate tree sequences.

Having defined this function, we can then generate some random seeds using numpy (note that we seed the random generator to make the seeds themselves deterministic—otherwise, we might as well let msprime generate random seeds):

num_replicates = 100
rng = np.random.RandomState(42)
seeds = rng.randint(1, 2**31, size=(num_replicates, 2))

S = np.zeros(num_replicates)
for replicate_index in range(num_replicates):
    ts = run_sim(10, *seeds[replicate_index])
    S[replicate_index] = ts.segregating_sites(span_normalise=False, mode="site")

(np.mean(S), np.var(S))
(1.28, 1.1016)

Running in parallel

When we are running replicates of large simulations we will usually want to take advantage of the available CPU cores and run them in parallel. Since replicates are independent, this is quite easy to do using Python’s concurrent.futures module.

When running simulations in parallel it’s important to create a simple function that runs your simulation in the worker processes. First we define this function:

import concurrent.futures

def worker(seeds):
    ancestry_seed, mutation_seed = seeds
    ts = msprime.sim_ancestry(
        samples=10,
        population_size=100,
        sequence_length=1_000,
        random_seed=ancestry_seed)
    ts = msprime.sim_mutations(ts, rate=1e-6, random_seed=mutation_seed)
    return ts.segregating_sites(span_normalise=False, mode="site")

Note that this is almost the same as the function we defined in the Specifying seeds explicitly section, with a few differences:

  1. The worker function takes one parameter, seeds, which we unpack manually into the ancestry_seed and mutation_seed.

  2. We have no free parameters, as it is much simpler to fully specify the worker function than it is to use partial functions in the context of multiple processes.

  3. We return the final result that we are interested (the segregating sites value) in rather than the simulated tree sequence. There are two good reasons for this: (a) we want to do as much computation as we can in the worker process and as little as possible in the main process; and (b) we want to keep the values that we return from the worker process as small and simple as we can to avoid serialisation overhead.

Once we’ve defined the function, then we can generate some random seeds as before, and then use the map method of ProcessPoolExecutor to distribute the work among all available CPU cores:

num_replicates = 100
rng = np.random.RandomState(42)
seeds = rng.randint(1, 2**31, size=(num_replicates, 2))

with concurrent.futures.ProcessPoolExecutor() as executor:
    results = executor.map(worker, seeds)
    S = np.array(list(results))

(np.mean(S), np.var(S))
(1.28, 1.1016)

Note

This approach is a good pattern to use when running replicates of large simulations, requiring at least (say) 10 seconds to run. It will not be efficient as a way of running large numbers of small simulations, though.