# Tutorial¶

This is the tutorial for the Python interface to the `msprime`

library. Detailed API Documentation is also available for this
library. An **ms** compatible command line interface
is also available if you wish to use `msprime`

directly within
an existing work flow.

## Simulating trees¶

Running simulations is very straightforward in `msprime`

:

```
>>> import msprime
>>> tree_sequence = msprime.simulate(sample_size=5, Ne=1000)
>>> tree = tree_sequence.first()
>>> print(tree.draw(format="unicode"))
8
┏━━┻━┓
┃ 7
┃ ┏━┻┓
6 ┃ ┃
┏┻┓ ┃ ┃
┃ ┃ ┃ 5
┃ ┃ ┃ ┏┻┓
2 3 4 0 1
```

Here, we simulate the coalescent for a sample of size
5 with an effective population size of 1000,
and then print out a depiction of the resulting tree.
The `simulate()`

function returns a
`TreeSequence`

object, which provides a very
efficient way to access the correlated trees in simulations
involving recombination. In this example we know that
there can only be one tree because we have not provided
a value for `recombination_rate`

, and it
defaults to zero. Therefore, we access the only tree in the
sequence using the call `tree_sequence.first()`

. Finally,
we draw a simple depiction of the tree to the terminal
using the `draw()`

method.

Trees are represented within `msprime`

in a slightly unusual way. In
the majority of libraries dealing with trees, each node is
represented as an object in memory and the relationship
between nodes as pointers between these objects. In `msprime`

,
however, nodes are *integers*.
In the tree above, we can see that the leaves of the tree
are labelled with 0 to 4, and all the internal nodes of the tree
are also integers with the root of the tree being 8.

We can easily trace our path
back to the root for a particular sample using the
`parent()`

method:

```
>>> u = 0
>>> while u != msprime.NULL_NODE:
>>> print("node {}: time = {}".format(u, tree.time(u)))
>>> u = tree.parent(u)
node 0: time = 0.0
node 5: time = 93.8536165443679
node 7: time = 770.0314897144547
node 8: time = 1950.6090018042169
```

In this code chunk we iterate up the tree starting at node 0 and
stop when we get to the root. We know that a node is a root
if its parent is `msprime.NULL_NODE`

, which is a special
reserved node. (The value of the null node is -1, but we recommend
using the symbolic constant to make code more readable.) We also use
the `time()`

method to get the time
for each node, which corresponds to the time in generations
at which the coalescence event happened during the simulation.
We can also obtain the length of a branch joining a node to
its parent using the `branch_length()`

method:

```
>>> print(tree.branch_length(5))
676.1778731700867
```

The branch length for node 5 is about 676 generations as the time for node 5 is 94 and the time of its parent is around 770 generation. It is also often useful to obtain the total branch length of the tree, i.e., the sum of the lengths of all branches:

```
>>> print(tree.total_branch_length)
4926.506226955324
```

## Recombination¶

Simulating the history of a single locus is a very useful, but we are most
often interesting in simulating the history of our sample across large genomic
regions under the influence of recombination. The `msprime`

API is
specifically designed to make this common requirement both easy and efficient.
To model genomic sequences under the influence of recombination we have
two parameters to the `simulate()`

function.
The `length`

parameter specifies the length of the simulated sequence
in bases, and may be a floating point number. If `length`

is not
supplied, it is assumed to be 1. The `recombination_rate`

parameter specifies the rate of crossing over per base per generation,
and is zero by default. See the API Documentation for a discussion of the precise
recombination model used.

Here, we simulate the trees across over a 10kb region with a recombination rate of \(2 \times 10^{-8}\) per base per generation, with an effective population size of 1000:

```
>>> tree_sequence = msprime.simulate(
... sample_size=5, Ne=1000, length=1e4, recombination_rate=2e-8)
>>> for tree in tree_sequence.trees():
... print("-" * 20)
... print("tree {}: interval = {}".format(tree.index, tree.interval))
... print(tree.draw(format="unicode"))
--------------------
tree 0: interval = (0.0, 6016.224463474058)
9
┏━━┻━┓
┃ 8
┃ ┏━┻┓
7 ┃ ┃
┏┻┓ ┃ ┃
┃ ┃ ┃ 6
┃ ┃ ┃ ┏┻┓
1 2 0 3 4
--------------------
tree 1: interval = (6016.224463474058, 10000.0)
9
┏━━┻━━┓
8 ┃
┏━┻━┓ ┃
┃ 6 ┃
┃ ┏━┻┓ ┃
┃ ┃ 5 ┃
┃ ┃ ┏┻┓ ┃
0 4 1 3 2
```

In this example, we use the `trees()`

method to iterate over the trees in the sequence. For each tree
we print out its index (i.e., it’s position in the sequence) and
the interval the tree covers (i.e., the genomic
coordinates which all share precisely this tree) using the
`index`

and `interval`

attributes.
Thus, the first tree covers the
first 6kb of sequence and the second tree covers the remaining 4kb.
We can see
that these trees share a great deal of their structure, but that there are
also important differences between the trees.

Warning

Do not store the values returned from the
`trees()`

iterator in a list and operate
on them afterwards! For efficiency reasons `msprime`

uses the same
instance of `SparseTree`

for each tree in the sequence
and updates the internal state for each new tree. Therefore, if you store
the trees returned from the iterator in a list, they will all refer
to the same tree.

## Mutations¶

Mutations are generated in `msprime`

by throwing mutations down
on the branches of trees at a particular rate. The mutations are
generated under the infinite sites model, and so each mutation
occurs at a unique (floating point) point position along the
genomic interval occupied by a tree. The mutation rate for simulations
is specified using the `mutation_rate`

parameter of
`simulate()`

. For example, the following chunk simulates 50kb
of nonrecombining sequence with a mutation rate of \(2 \times 10^{-8}\)
per base per generation:

```
>>> tree_sequence = msprime.simulate(
... sample_size=5, Ne=1000, length=50e3, mutation_rate=2e-8, random_seed=30)
>>> tree = tree_sequence.first()
>>> for site in tree.sites():
... for mutation in site.mutations:
... print("Mutation @ position {:.2f} over node {}".format(
... site.position, mutation.node))
Mutation @ position 1179.94 over node 0
Mutation @ position 4485.17 over node 5
Mutation @ position 9788.56 over node 5
Mutation @ position 11949.32 over node 7
Mutation @ position 31454.99 over node 7
Mutation @ position 34832.55 over node 0
Mutation @ position 36747.63 over node 7
Mutation @ position 46692.51 over node 5
>>> print(tree.draw(format="unicode"))
8
┏━━┻━┓
┃ 7
┃ ┏━┻┓
┃ ┃ 6
┃ ┃ ┏┻┓
5 ┃ ┃ ┃
┏┻┓ ┃ ┃ ┃
0 3 1 2 4
```

## Variants¶

We are often interesting in accessing the sequence data that results from
simulations directly. The most efficient way to do this is by using
the `TreeSequence.variants()`

method, which returns an iterator
over all the `Variant`

objects arising from the trees and mutations.
Each variant contains a reference to the site object, as well as the
alleles and the observed sequences for each sample in the `genotypes`

field:

```
>>> tree_sequence = msprime.simulate(
... sample_size=20, Ne=1e4, length=5e3, recombination_rate=2e-8,
... mutation_rate=2e-8, random_seed=10)
>>> for variant in tree_sequence.variants():
... print(
... variant.site.id, variant.site.position,
... variant.alleles, variant.genotypes, sep="\t")
0 2432.768327416852 ('0', '1') [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
1 2577.6939414924095 ('0', '1') [1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1]
2 2844.682702049562 ('0', '1') [0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]
3 4784.266628557816 ('0', '1') [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
```

In this example we simulate some data and then print out the observed
sequences. We loop through each variant and print out the observed state of
each sample as an array of zeros and ones, along with the index and position
of the corresponding mutation. In this example, the
alleles are always `'0'`

(the ancestral state) and `'1'`

(the derived state), because we are simulating simple infinite sites mutations. More
complex models are possible, however.

This way of working with the sequence data is quite efficient because we do not need to keep the entire genotype matrix in memory at once. However, if we do want the full genotype matrix it is simple to obtain:

```
>>> A = tree_sequence.genotype_matrix()
>>> A
array([[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=uint8)
```

In this example, we run the same simulation but this time store entire variant matrix in a two-dimensional numpy array. This is useful for integrating with tools such as scikit allel.

## Historical samples¶

Simulating coalescent histories in which some of the samples are not
from the present time is straightforward in `msprime`

.
By using the `samples`

argument to `msprime.simulate()`

we can specify the location and time at which all samples are made.

```
def historical_samples_example():
samples = [
msprime.Sample(population=0, time=0),
msprime.Sample(0, 0), # Or, we can use positional arguments.
msprime.Sample(0, 1.0)
]
tree_seq = msprime.simulate(samples=samples)
tree = tree_seq.first()
for u in tree.nodes():
print(u, tree.parent(u), tree.time(u), sep="\t")
print(tree.draw(format="unicode"))
```

In this example we create three samples, two taken at the present time
and one taken 1.0 generations in the past. There are a number of
different ways in which we can describe the samples using the
`msprime.Sample`

object (samples can be provided as plain tuples also
if more convenient). Running this example, we get:

```
>>> historical_samples_example()
4 -1 1.3805592647990812
1 4 0.0
3 4 1.052473878566028
0 3 0.0
2 3 1.0
4
┏━┻┓
┃ 3
┃ ┏┻┓
┃ ┃ 2
┃ ┃
1 0
```

Because nodes `0`

and `1`

were sampled at time 0, their times in the tree
are both 0. Node `2`

was sampled at time 1.0, and so its time is recorded
as 1.0 in the tree.

## Replication¶

A common task for coalescent simulations is to check the accuracy of analytical
approximations to statistics of interest. To do this, we require many independent
replicates of a given simulation. `msprime`

provides a simple and efficient
API for replication: by providing the `num_replicates`

argument to the
`simulate()`

function, we can iterate over the replicates
in a straightforward manner. Here is an example where we compare the
analytical results for the number of segregating sites with simulations:

```
import msprime
import numpy as np
def segregating_sites_example(n, theta, num_replicates):
S = np.zeros(num_replicates)
replicates = msprime.simulate(
Ne=0.5,
sample_size=n,
mutation_rate=theta / 2,
num_replicates=num_replicates)
for j, tree_sequence in enumerate(replicates):
S[j] = tree_sequence.num_sites
# Now, calculate the analytical predictions
S_mean_a = np.sum(1 / np.arange(1, n)) * theta
S_var_a = (
theta * np.sum(1 / np.arange(1, n)) +
theta**2 * np.sum(1 / np.arange(1, n)**2))
print(" mean variance")
print("Observed {}\t\t{}".format(np.mean(S), np.var(S)))
print("Analytical {:.5f}\t\t{:.5f}".format(S_mean_a, S_var_a))
```

Running this code, we get:

```
>>> segregating_sites_example(10, 5, 100000)
mean variance
Observed 14.17893 53.0746740551
Analytical 14.14484 52.63903
```

Note that in this example we set \(N_e = 0.5\) and
the mutation rate to \(\theta / 2\) when calling `simulate()`

.
This is essential to convert the diploid per-generation time units
of msprime into the the haploid coalescent units required for many
theoretical results. However, it is important to note that conventions
vary widely, and great care is needed with such factor-of-two
rescalings.

## Population structure¶

Todo

Update this example to use Ne=0.5 as recommended elsewhere.

Population structure in `msprime`

closely follows the model used in the
`ms`

simulator: we have \(N\) demes with an \(N\times N\)
matrix describing the migration rates between these subpopulations. The
sample sizes, population sizes and growth rates of all demes
can be specified independently. Migration rates are specified using
a migration matrix. Unlike `ms`

however, all times and rates are specified
in generations and all populations sizes are absolute (that is, not
multiples of \(N_e\)).

In the following example, we calculate the mean coalescence time for a pair of lineages sampled in different demes in a symmetric island model, and compare this with the analytical expectation.

```
import msprime
import numpy as np
def migration_example(num_replicates=10**6):
# M is the overall symmetric migration rate, and d is the number
# of demes.
M = 0.2
d = 3
m = M / (4 * (d - 1))
# Allocate the initial sample. Because we are interested in the
# between deme coalescence times, we choose one sample each
# from the first two demes.
population_configurations = [
msprime.PopulationConfiguration(sample_size=1),
msprime.PopulationConfiguration(sample_size=1),
msprime.PopulationConfiguration(sample_size=0)]
# Now we set up the migration matrix. Since this is a symmetric
# island model, we have the same rate of migration between all
# pairs of demes. Diagonal elements must be zero.
migration_matrix = [
[0, m, m],
[m, 0, m],
[m, m, 0]]
# We pass these values to the simulate function, and ask it
# to run the required number of replicates.
replicates = msprime.simulate(
population_configurations=population_configurations,
migration_matrix=migration_matrix,
num_replicates=num_replicates)
# And then iterate over these replicates
T = np.zeros(num_replicates)
for i, tree_sequence in enumerate(replicates):
tree = tree_sequence.first()
T[i] = tree.time(tree.root) / 4
# Finally, calculate the analytical expectation and print
# out the results
analytical = d / 2 + (d - 1) / (2 * M)
print("Observed =", np.mean(T))
print("Predicted =", analytical)
```

Running this example we get:

```
>>> migration_example()
Observed = 6.50638181614
Predicted = 6.5
```

## Demography¶

Msprime provides a flexible and simple way to model past demographic events in arbitrary combinations. Here is an example describing the Gutenkunst et al. out-of-Africa model. See Figure 2B for a schematic of this model, and Table 1 for the values used.

Todo

Add a diagram of the model for convenience.

```
import math
def out_of_africa():
# First we set out the maximum likelihood values of the various parameters
# given in Table 1.
N_A = 7300
N_B = 2100
N_AF = 12300
N_EU0 = 1000
N_AS0 = 510
# Times are provided in years, so we convert into generations.
generation_time = 25
T_AF = 220e3 / generation_time
T_B = 140e3 / generation_time
T_EU_AS = 21.2e3 / generation_time
# We need to work out the starting (diploid) population sizes based on
# the growth rates provided for these two populations
r_EU = 0.004
r_AS = 0.0055
N_EU = N_EU0 / math.exp(-r_EU * T_EU_AS)
N_AS = N_AS0 / math.exp(-r_AS * T_EU_AS)
# Migration rates during the various epochs.
m_AF_B = 25e-5
m_AF_EU = 3e-5
m_AF_AS = 1.9e-5
m_EU_AS = 9.6e-5
# Population IDs correspond to their indexes in the population
# configuration array. Therefore, we have 0=YRI, 1=CEU and 2=CHB
# initially.
population_configurations = [
msprime.PopulationConfiguration(
sample_size=0, initial_size=N_AF),
msprime.PopulationConfiguration(
sample_size=1, initial_size=N_EU, growth_rate=r_EU),
msprime.PopulationConfiguration(
sample_size=1, initial_size=N_AS, growth_rate=r_AS)
]
migration_matrix = [
[ 0, m_AF_EU, m_AF_AS],
[m_AF_EU, 0, m_EU_AS],
[m_AF_AS, m_EU_AS, 0],
]
demographic_events = [
# CEU and CHB merge into B with rate changes at T_EU_AS
msprime.MassMigration(
time=T_EU_AS, source=2, destination=1, proportion=1.0),
msprime.MigrationRateChange(time=T_EU_AS, rate=0),
msprime.MigrationRateChange(
time=T_EU_AS, rate=m_AF_B, matrix_index=(0, 1)),
msprime.MigrationRateChange(
time=T_EU_AS, rate=m_AF_B, matrix_index=(1, 0)),
msprime.PopulationParametersChange(
time=T_EU_AS, initial_size=N_B, growth_rate=0, population_id=1),
# Population B merges into YRI at T_B
msprime.MassMigration(
time=T_B, source=1, destination=0, proportion=1.0),
# Size changes to N_A at T_AF
msprime.PopulationParametersChange(
time=T_AF, initial_size=N_A, population_id=0)
]
# Use the demography debugger to print out the demographic history
# that we have just described.
dd = msprime.DemographyDebugger(
population_configurations=population_configurations,
migration_matrix=migration_matrix,
demographic_events=demographic_events)
dd.print_history()
```

The `DemographyDebugger`

provides a method to debug the history that
you have described so that you can be sure that the migration rates, population
sizes and growth rates are all as you intend during each epoch:

```
=============================
Epoch: 0 -- 848.0 generations
=============================
start end growth_rate | 0 1 2
-------- -------- -------- | -------- -------- --------
0 |1.23e+04 1.23e+04 0 | 0 3e-05 1.9e-05
1 |2.97e+04 1e+03 0.004 | 3e-05 0 9.6e-05
2 |5.41e+04 510 0.0055 | 1.9e-05 9.6e-05 0
Events @ generation 848.0
- Mass migration: lineages move from 2 to 1 with probability 1.0
- Migration rate change to 0 everywhere
- Migration rate change for (0, 1) to 0.00025
- Migration rate change for (1, 0) to 0.00025
- Population parameter change for 1: initial_size -> 2100 growth_rate -> 0
==================================
Epoch: 848.0 -- 5600.0 generations
==================================
start end growth_rate | 0 1 2
-------- -------- -------- | -------- -------- --------
0 |1.23e+04 1.23e+04 0 | 0 0.00025 0
1 | 2.1e+03 2.1e+03 0 | 0.00025 0 0
2 | 510 2.27e-09 0.0055 | 0 0 0
Events @ generation 5600.0
- Mass migration: lineages move from 1 to 0 with probability 1.0
===================================
Epoch: 5600.0 -- 8800.0 generations
===================================
start end growth_rate | 0 1 2
-------- -------- -------- | -------- -------- --------
0 |1.23e+04 1.23e+04 0 | 0 0.00025 0
1 | 2.1e+03 2.1e+03 0 | 0.00025 0 0
2 |2.27e-09 5.17e-17 0.0055 | 0 0 0
Events @ generation 8800.0
- Population parameter change for 0: initial_size -> 7300
================================
Epoch: 8800.0 -- inf generations
================================
start end growth_rate | 0 1 2
-------- -------- -------- | -------- -------- --------
0 | 7.3e+03 7.3e+03 0 | 0 0.00025 0
1 | 2.1e+03 2.1e+03 0 | 0.00025 0 0
2 |5.17e-17 0 0.0055 | 0 0 0
```

Warning

The output of the `DemographyDebugger.print_history()`

method
is intended only for debugging purposes, and is not meant to be machine
readable. The format is also preliminary; if there is other information
that you think would be useful, please open an issue on GitHub

Once you are satisfied that the demographic history that you have built
is correct, it can then be simulated by calling the `simulate()`

function.

## Recombination maps¶

The `msprime`

API allows us to quickly and easily simulate data from an
arbitrary recombination map. In this example we read a recombination
map for human chromosome 22, and simulate a single replicate. After
the simulation is completed, we plot histograms of the recombination
rates and the simulated breakpoints. These show that density of
breakpoints follows the recombination rate closely.

```
import numpy as np
import scipy.stats
import matplotlib.pyplot as pyplot
def variable_recomb_example():
infile = "hapmap/genetic_map_GRCh37_chr22.txt"
# Read in the recombination map using the read_hapmap method,
recomb_map = msprime.RecombinationMap.read_hapmap(infile)
# Now we get the positions and rates from the recombination
# map and plot these using 500 bins.
positions = np.array(recomb_map.get_positions()[1:])
rates = np.array(recomb_map.get_rates()[1:])
num_bins = 500
v, bin_edges, _ = scipy.stats.binned_statistic(
positions, rates, bins=num_bins)
x = bin_edges[:-1][np.logical_not(np.isnan(v))]
y = v[np.logical_not(np.isnan(v))]
fig, ax1 = pyplot.subplots(figsize=(16, 6))
ax1.plot(x, y, color="blue")
ax1.set_ylabel("Recombination rate")
ax1.set_xlabel("Chromosome position")
# Now we run the simulation for this map. We assume Ne=10^4
# and have a sample of 100 individuals
tree_sequence = msprime.simulate(
sample_size=100,
Ne=10**4,
recombination_map=recomb_map)
# Now plot the density of breakpoints along the chromosome
breakpoints = np.array(list(tree_sequence.breakpoints()))
ax2 = ax1.twinx()
v, bin_edges = np.histogram(breakpoints, num_bins, density=True)
ax2.plot(bin_edges[:-1], v, color="green")
ax2.set_ylabel("Breakpoint density")
ax2.set_xlim(1.5e7, 5.3e7)
fig.savefig("hapmap_chr22.svg")
```

## Calculating LD¶

The `msprime`

API provides methods to efficiently calculate
population genetics statistics. For example, the `LdCalculator`

class allows us to compute pairwise linkage disequilibrium coefficients.
Here we use the `get_r2_matrix()`

method to easily make an
LD plot using matplotlib. (Thanks to
the excellent scikit-allel
for the basic plotting code
used here.)

```
import msprime
import matplotlib.pyplot as pyplot
def ld_matrix_example():
ts = msprime.simulate(100, recombination_rate=10, mutation_rate=20,
random_seed=1)
ld_calc = msprime.LdCalculator(ts)
A = ld_calc.r2_matrix()
# Now plot this matrix.
x = A.shape[0] / pyplot.rcParams['figure.dpi']
x = max(x, pyplot.rcParams['figure.figsize'][0])
fig, ax = pyplot.subplots(figsize=(x, x))
fig.tight_layout(pad=0)
im = ax.imshow(A, interpolation="none", vmin=0, vmax=1, cmap="Blues")
ax.set_xticks([])
ax.set_yticks([])
for s in 'top', 'bottom', 'left', 'right':
ax.spines[s].set_visible(False)
pyplot.gcf().colorbar(im, shrink=.5, pad=0)
pyplot.savefig("ld.svg")
```

## Working with threads¶

When performing large calculations it’s often useful to split the
work over multiple processes or threads. The msprime API can
be used without issues across multiple processes, and the Python
`multiprocessing`

module often provides a very effective way to
work with many replicate simulations in parallel.

When we wish to work with a single very large dataset, however, threads can
offer better resource usage because of the shared memory space. The Python
`threading`

library gives a very simple interface to lightweight CPU
threads and allows us to perform several CPU intensive tasks in parallel. The
`msprime`

API is designed to allow multiple threads to work in parallel when
CPU intensive tasks are being undertaken.

Note

In the CPython implementation the Global Interpreter Lock ensures that
only one thread executes Python bytecode at one time. This means that
Python code does not parallelise well across threads, but avoids a large
number of nasty pitfalls associated with multiple threads updating
data structures in parallel. Native C extensions like `numpy`

and `msprime`

release the GIL while expensive tasks are being performed, therefore
allowing these calculations to proceed in parallel.

In the following example we wish to find all mutations that are in approximate
LD (\(r^2 \geq 0.5\)) with a given set of mutations. We parallelise this
by splitting the input array between a number of threads, and use the
`LdCalculator.r2_array()`

method to compute the \(r^2\) value
both up and downstream of each focal mutation, filter out those that
exceed our threshold, and store the results in a dictionary. We also
use the very cool tqdm module to give us a
progress bar on this computation.

```
import threading
import numpy as np
import tqdm
import msprime
def find_ld_sites(
tree_sequence, focal_mutations, max_distance=1e6, r2_threshold=0.5,
num_threads=8):
results = {}
progress_bar = tqdm.tqdm(total=len(focal_mutations))
num_threads = min(num_threads, len(focal_mutations))
def thread_worker(thread_index):
ld_calc = msprime.LdCalculator(tree_sequence)
chunk_size = int(math.ceil(len(focal_mutations) / num_threads))
start = thread_index * chunk_size
for focal_mutation in focal_mutations[start: start + chunk_size]:
a = ld_calc.r2_array(
focal_mutation, max_distance=max_distance,
direction=msprime.REVERSE)
rev_indexes = focal_mutation - np.nonzero(a >= r2_threshold)[0] - 1
a = ld_calc.r2_array(
focal_mutation, max_distance=max_distance,
direction=msprime.FORWARD)
fwd_indexes = focal_mutation + np.nonzero(a >= r2_threshold)[0] + 1
indexes = np.concatenate((rev_indexes[::-1], fwd_indexes))
results[focal_mutation] = indexes
progress_bar.update()
threads = [
threading.Thread(target=thread_worker, args=(j,))
for j in range(num_threads)]
for t in threads:
t.start()
for t in threads:
t.join()
progress_bar.close()
return results
def threads_example():
ts = msprime.simulate(
sample_size=1000, Ne=1e4, length=1e7, recombination_rate=2e-8,
mutation_rate=2e-8)
counts = np.zeros(ts.num_sites)
for tree in ts.trees():
for site in tree.sites():
assert len(site.mutations) == 1
mutation = site.mutations[0]
counts[site.id] = tree.get_num_samples(mutation.node)
doubletons = np.nonzero(counts == 2)[0]
results = find_ld_sites(ts, doubletons, num_threads=8)
print(
"Found LD sites for", len(results), "doubleton sites out of",
ts.num_sites)
```

In this example, we first simulate 1000 samples of 10 megabases and find all
doubleton mutations in the resulting tree sequence. We then call the
`find_ld_sites()`

function to find all mutations that are within 1 megabase
of these doubletons and have an \(r^2\) statistic of greater than 0.5.

The `find_ld_sites()`

function performs these calculations in parallel using
8 threads. The real work is done in the nested `thread_worker()`

function,
which is called once by each thread. In the thread worker, we first allocate an
instance of the `LdCalculator`

class. (It is **critically important**
that each thread has its own instance of `LdCalculator`

, as the threads
will not work efficiently otherwise.) After this, each thread works out the
slice of the input array that it is responsible for, and then iterates over
each focal mutation in turn. After the \(r^2\) values have been calculated,
we then find the indexes of the mutations corresponding to values greater than
0.5 using `numpy.nonzero()`

. Finally, the thread stores the resulting array
of mutation indexes in the `results`

dictionary, and moves on to the next
focal mutation.

Running this example we get:

```
>>> threads_example()
100%|████████████████████████████████████████████████| 4045/4045 [00:09<00:00, 440.29it/s]
Found LD sites for 4045 doubleton mutations out of 60100
```

## Editing tree sequences¶

Sometimes we wish to make some minor modifications to a tree sequence that has
been generated by a simulation. However, tree sequence objects are **immutable**
and so we cannot edit a them in place. The answer is to use the
Tables API: we export the tree sequence to a set of
tables, edit these tables, and then create
a new tree sequence from them. In the following example, we use this approach
to remove all singleton sites from a given tree sequence.

```
def strip_singletons(ts):
tables = ts.dump_tables()
tables.sites.clear()
tables.mutations.clear()
for tree in ts.trees():
for site in tree.sites():
assert len(site.mutations) == 1 # Only supports infinite sites muts.
mut = site.mutations[0]
if tree.num_samples(mut.node) > 1:
site_id = tables.sites.add_row(
position=site.position,
ancestral_state=site.ancestral_state)
tables.mutations.add_row(
site=site_id, node=mut.node, derived_state=mut.derived_state)
return tables.tree_sequence()
```

This function takes a tree sequence containing some infinite sites mutations as
input, and returns a copy in which all singleton sites have been removed.
The approach is very simple: we get a copy of the underlying
table data in a `TableCollection`

object, and first clear the
site and mutation tables. We then consider each site in turn,
and if the allele frequency of
the mutation is greater than one, we add the site and mutation to our
output tables using `SiteTable.add_row()`

and `MutationTable.add_row()`

.
(In this case we consider only simple infinite sites mutations,
where we cannot have back or recurrent mutations. These would require a slightly
more involved approach where we keep a map of mutation IDs so that
mutation `parent`

values could be computed. We have also omitted the
site and mutation metadata in the interest of simplicity.)p

After considering each site, we then create a new tree sequence using
the `TableCollection.tree_sequence()`

method on our updated tables.
Using this function then, we get:

```
>>> ts = msprime.simulate(10, mutation_rate=10)
>>> ts.num_sites
50
>>> ts_new = strip_singletons(ts)
>>> ts_new.num_sites
44
>>>
```

Thus, we have removed 6 singleton sites from the tree sequence.

Todo

Add another example here where we use the array oriented API to edit the nodes and edges of a tree sequence. Perhaps decapitating would be a good example?

## Working with Tables¶

Todo

This section is a work in progress and needs to be revised.

Tables provide a convenient method for viewing, importing and exporting tree
sequences. `msprime`

provides direct access to the the columns of a table as
`numpy`

arrays: for instance, if `n`

is a `NodeTable`

, then `n.time`

will return an array containing the birth times of the individuals in the
table. *However*, it is important to note that this is *not* a shallow copy:
modifying `n.time`

will not change the node table `n`

. This may change in
the future, but currently there are two ways to modify tables: `.add_row()`

and `.set_columns()`

(and also `.clear()`

, which empties the table).

The example node table above would be constructed using `.add_row()`

as
follows:

```
n = msprime.NodeTable()
sv = [True, True, True, False, False, False, False]
tv = [0.0, 0.0, 0.0, 0.4, 0.5, 0.7, 1.0]
pv = [0, 0, 0, 0, 0, 0, 0]
for s, t, p in zip(sv, tv, pv):
n.add_row(flags=s, population=p, time=t)
print(n)
```

The `.add_row()`

method is natural (and should be reasonably efficient) if
new records appear one-by-one. In the example above it would have been more
natural to use `.set_columns()`

:

```
n = msprime.NodeTable()
n.set_columns(flags=sv, population=pv, time=tv)
```

Finally, here is an example where we add 1.4 to every `time`

except the first
in the NodeTable constructed above using `numpy`

indexing:

```
fn = n.flags
pn = n.population
tn = n.time
tn[1:] = tn[1:] + 1.4
n.set_columns(flags=fn, population=pn, time=tn)
```

Here is an example. Consider the following sequence of trees:

```
time
----
1.0 6
0.7 / \ 5
/ x / \
0.5 / 4 4 / 4
/ / \ / x / / \
0.4 / / \ / 3 / / \
/ / \ / / \ / / \
/ / \ / / x / / \
/ / \ / / \ / / \
0.0 0 1 2 1 0 2 0 1 2
position 0.0 0.2 0.8 1.0
```

First, we specify the nodes:

```
NodeTable:
id is_sample population time
0 1 0 0
1 1 0 0
2 1 0 0
3 0 0 0.4
4 0 0 0.5
5 0 0 0.7
6 0 0 1.0
```

Importantly, the first column, `id`

, is **not actually recorded**, and is
only shown when printing out node tables (as here) for convenience. This has
three samples: nodes 0, 1, and 2, and lists their birth times. Then, we
specify the edges:

```
EdgeTable:
left right parent children
0.2 0.8 3 0
0.2 0.8 3 2
0.0 0.2 4 1
0.0 0.2 4 2
0.2 0.8 4 1
0.2 0.8 4 3
0.8 1.0 4 1
0.8 1.0 4 2
0.8 1.0 5 0
0.8 1.0 5 4
0.0 0.2 6 0
0.0 0.2 6 4
```

Since node 3 is most recent, the edge that says that nodes 0 and 2 inherit from node 3 on the interval between 0.2 and 0.8 comes first. Next are the edges from node 4: there are three of these, for each of the three genomic intervals over which node 4 is ancestor to a distinct set of nodes. At this point, we know the full tree on the middle interval. Finally, edges specifying the common ancestor of 0 and 4 on the remaining intervals (parents 6 and 5 respectively) allow us to construct all trees across the entire interval.

In the depiction above, `x`

denotes mutations. Suppose that the first
mutation occurs at position 0.1 and the mutations in the second tree both
occurred at the same position, at 0.5 (with a back mutation). The positions
are recorded in the sites table:

```
SiteTable:
id position ancestral_state
0 0.1 0
1 0.5 0
```

As with node tables, the `id`

column is **not** actually recorded, but is
implied by the position in the table. The acutal mutations are then recorded:

```
MutationTable:
site node derived_state
0 4 1
1 3 1
1 2 0
```

This would then result in the following (two-locus) haplotypes for the three samples:

```
sample haplotype
------ ---------
0 01
1 10
2 11
```