def time_to_coalescence(n_genes,
pop_size=None,
haploid=True,
rng=None):
"""
A random draw from the "Kingman distribution" (continuous time version):
Time to go from n genes to n-1 genes; i.e. waiting time until two
lineages coalesce. This is a random number with an exponential
distribution with a rate of (n choose 2).
`pop_size` is the effective *haploid* population size; i.e., number of gene
in the population: 2 * N in a diploid population of N individuals,
or N in a haploid population of N individuals.
If `pop_size` is 1 or 0 or None, then time is in haploid population units;
i.e. where 1 unit of time equals 2N generations for a diploid population of
size N, or N generations for a haploid population of size N. Otherwise time
is in generations.
"""
if rng is None:
rng = GLOBAL_RNG
if not pop_size:
time_units = 1
else:
time_units = pop_size * 2
rate = probability.binomial_coefficient(n_genes, 2)
tmrca = rng.expovariate(rate)
return tmrca * pop_size
def time_to_coalescence(n_genes,
pop_size=None,
haploid=True,
rng=None):
"""
A random draw from the "Kingman distribution" (continuous time version):
Time to go from n genes to n-1 genes; i.e. waiting time until two
lineages coalesce. This is a random number with an exponential
distribution with a rate of (n choose 2).
`pop_size` is the effective *haploid* population size; i.e., number of gene
in the population: 2 * N in a diploid population of N individuals,
or N in a haploid population of N individuals.
If `pop_size` is 1 or 0 or None, then time is in haploid population units;
i.e. where 1 unit of time equals 2N generations for a diploid population of
size N, or N generations for a haploid population of size N. Otherwise time
is in generations.
"""
if rng is None:
rng = GLOBAL_RNG
if not pop_size:
time_units = 1
else:
time_units = pop_size * 2
rate = probability.binomial_coefficient(n_genes, 2)
tmrca = rng.expovariate(rate)
return tmrca * pop_size
def discrete_time_to_coalescence(n_genes,
pop_size=None,
rng=None):
"""
A random draw from the "Kingman distribution" (discrete time version):
Time to go from n genes to n-1 genes; i.e. waiting time until two
lineages coalesce.
`pop_size` is the effective *haploid* population size; i.e., number of
genes in the population: 2 * N in a diploid population of N individuals, or
N in a haploid population of N individuals.
If `pop_size` is 1 or 0 or None, then time is in haploid population units;
i.e. where 1 unit of time equals 2N generations for a diploid population of
size N, or N generations for a haploid population of size N. Otherwise time
is in generations.
"""
if not pop_size:
time_units = 1
else:
time_units = pop_size * 2
if rng is None:
rng = GLOBAL_RNG
p = float(probability.binomial_coefficient(n_genes, 2)) / time_units
tmrca = probability.geometric_rv(p)
return tmrca * time_units
def discrete_time_to_coalescence(n_genes,
pop_size=None,
rng=None):
"""
A random draw from the "Kingman distribution" (discrete time version):
Time to go from n genes to n-1 genes; i.e. waiting time until two
lineages coalesce.
`pop_size` is the effective *haploid* population size; i.e., number of
genes in the population: 2 * N in a diploid population of N individuals, or
N in a haploid population of N individuals.
If `pop_size` is 1 or 0 or None, then time is in haploid population units;
i.e. where 1 unit of time equals 2N generations for a diploid population of
size N, or N generations for a haploid population of size N. Otherwise time
is in generations.
"""
if not pop_size:
time_units = 1
else:
time_units = pop_size * 2
if rng is None:
rng = GLOBAL_RNG
p = float(probability.binomial_coefficient(n_genes, 2)) / time_units
tmrca = probability.geometric_rv(p)
return tmrca * time_units
def calc(self):
"""
Returns a summary of a set of sequences that can be partitioned into
the list of lists of taxa given by `taxon_groups`.
"""
diffs_x, mean_diffs_x, sq_diff_x = _count_differences(self.pop1_seqs, self.state_alphabet, self.ignore_uncertain)
diffs_y, mean_diffs_y, sq_diff_y = _count_differences(self.pop2_seqs, self.state_alphabet, self.ignore_uncertain)
d_x = diffs_x / probability.binomial_coefficient(len(self.pop1_seqs), 2)
d_y = diffs_y / probability.binomial_coefficient(len(self.pop2_seqs), 2)
d_xy = self._average_number_of_pairwise_differences_between_populations()
s2_x = (float(sq_diff_x) / probability.binomial_coefficient(len(self.pop1_seqs), 2) ) - (d_x ** 2)
s2_y = (float(sq_diff_y) / probability.binomial_coefficient(len(self.pop2_seqs), 2) ) - (d_y ** 2)
s2_xy = self._variance_of_pairwise_differences_between_populations(d_xy)
n = len(self.combined_seqs)
n_x = float(len(self.pop1_seqs))
n_y = float(len(self.pop2_seqs))
a = float(n * (n-1))
ax = float(n_x * (n_x - 1))
ay = float(n_y * (n_y - 1))
k = _average_number_of_pairwise_differences(self.combined_seqs, self.state_alphabet, self.ignore_uncertain)
n = len(self.combined_seqs)
# Hickerson 2006: pi #
self.average_number_of_pairwise_differences = k
# Hickerson 2006: pi_b #
self.average_number_of_pairwise_differences_between = d_xy
# Hickerson 2006: pi_w #
self.average_number_of_pairwise_differences_within = d_x + d_y
# Hickerson 2006: pi_net #
self.average_number_of_pairwise_differences_net = d_xy - (d_x + d_y)
# Hickerson 2006: S #
self.num_segregating_sites = _num_segregating_sites(self.combined_seqs, self.state_alphabet, self.ignore_uncertain)
# Hickerson 2006: theta #
a1 = sum([1.0/i for i in range(1, n)])
self.wattersons_theta = float(self.num_segregating_sites) / a1
# Wakeley 1996 #
self.wakeleys_psi = (float(1)/(a)) * ( ax * (math.sqrt(s2_x)/d_x) + ay * (math.sqrt(s2_y)/d_y) + (2 * n_x * n_y * math.sqrt(s2_xy)/k))
# Tajima's D #
self.tajimas_d = _tajimas_d(n, self.average_number_of_pairwise_differences, self.num_segregating_sites)
def _average_number_of_pairwise_differences(char_vectors, state_alphabet, ignore_uncertain=True):
"""
Returns $k$ (Tajima 1983; Wakely 1996), calculated for a set of sequences:
k = \frac{\right(\sum \sum \k_{ij}\left)}{n \choose 2}
where $k_{ij}$ is the number of pairwise differences between the
$i$th and $j$th sequence, and $n$ is the number of DNA sequences
sampled.
"""
sum_diff, mean_diff, sq_diff = _count_differences(char_vectors, state_alphabet, ignore_uncertain)
return sum_diff / probability.binomial_coefficient(len(char_vectors), 2)
def expected_tmrca(n_genes, pop_size=None):
"""
Expected (mean) value for the Time to the Most Recent Common Ancestor.
`n_genes` is the number of genes in the sample.
`pop_size` is the effective *haploid* population size; i.e., number of gene
in the population: 2 * N in a diploid population of N individuals,
or N in a haploid population of N individuals.
If `pop_size` is 1 or 0 or None, then time is in haploid population units;
i.e. where 1 unit of time equals 2N generations for a diploid population of
size N, or N generations for a haploid population of size N. Otherwise time
is in generations.
"""
nc2 = probability.binomial_coefficient(n_genes, 2)
tmrca = (float(1)/nc2)
return tmrca * pop_size
def _average_number_of_pairwise_differences(char_vectors,
state_alphabet,
ignore_uncertain=True):
"""
Returns $k$ (Tajima 1983; Wakely 1996), calculated for a set of sequences:
k = \frac{\right(\sum \sum \k_{ij}\left)}{n \choose 2}
where $k_{ij}$ is the number of pairwise differences between the
$i$th and $j$th sequence, and $n$ is the number of DNA sequences
sampled.
"""
sum_diff, mean_diff, sq_diff = _count_differences(char_vectors,
state_alphabet,
ignore_uncertain)
return sum_diff / probability.binomial_coefficient(len(char_vectors), 2)
def expected_tmrca(n_genes, pop_size=None, rng=None):
"""
Expected (mean) value for the Time to the Most Recent Common Ancestor.
`n_genes` is the number of genes in the sample.
`pop_size` is the effective *haploid* population size; i.e., number of gene
in the population: 2 * N in a diploid population of N individuals,
or N in a haploid population of N individuals.
If `pop_size` is 1 or 0 or None, then time is in haploid population units;
i.e. where 1 unit of time equals 2N generations for a diploid population of
size N, or N generations for a haploid population of size N. Otherwise time
is in generations.
"""
if rng is None:
rng = GLOBAL_RNG
nc2 = probability.binomial_coefficient(n_genes, 2)
tmrca = (float(1)/nc2)
return tmrca * pop_sze
def kl_divergence_coalescent_waiting_times(allele_waiting_time_dist, haploid_pop_size):
"""
`allele_branch_len_dist` is a dictionary with number of alleles as keys
and a list of waiting times associated with that number of alleles as
values. `haploid_pop_size` is the population size in terms of total numbers
of genes. This returns a the KL-divergence between the distribution of
waiting times and the Kingman coalescent distribution.
D_{\mathrm{KL}}(P\|Q) = \sum_i P(i) \log \frac{P(i)}{Q(i)}.
"""
d_kl = 0.0
for k, wts in allele_waiting_time_dist.items():
p = float(probability.binomial_coefficient(k, 2)) / haploid_pop_size
for t in wts:
# Kernel types:
#
# 'E' or 'Epanechnikov'
# Epanechnikov kernel (default)
#
# 'U' or 'Uniform'
# Uniform kernel
#
# 'T' or 'Triangle'
# Triangle kernel
#
# 'G' or 'Gaussian'
# Gaussian kernel
#
# 'B' or 'Biweight'
# Quartic/biweight kernel
#
# '3' or 'Triweight'
# Triweight kernel
#
# 'C' or 'Cosine'
# Cosine kernel
q = de_hoon_lib.pdf(wts, [k], kernel = 'Gaussian')
if q == 0:
q = 1e-100
d_kl += p * math.log(p/q)
return d_kl
def kl_divergence_coalescent_waiting_times(allele_waiting_time_dist, haploid_pop_size):
"""
`allele_branch_len_dist` is a dictionary with number of alleles as keys
and a list of waiting times associated with that number of alleles as
values. `haploid_pop_size` is the population size in terms of total numbers
of genes. This returns a the KL-divergence between the distribution of
waiting times and the Kingman coalescent distribution.
D_{\mathrm{KL}}(P\|Q) = \sum_i P(i) \log \frac{P(i)}{Q(i)}.
"""
d_kl = 0.0
for k, wts in allele_waiting_time_dist.items():
p = float(probability.binomial_coefficient(k, 2)) / haploid_pop_size
for t in wts:
# Kernel types:
#
# 'E' or 'Epanechnikov'
# Epanechnikov kernel (default)
#
# 'U' or 'Uniform'
# Uniform kernel
#
# 'T' or 'Triangle'
# Triangle kernel
#
# 'G' or 'Gaussian'
# Gaussian kernel
#
# 'B' or 'Biweight'
# Quartic/biweight kernel
#
# '3' or 'Triweight'
# Triweight kernel
#
# 'C' or 'Cosine'
# Cosine kernel
q = de_hoon_lib.pdf(wts, [k], kernel = 'Gaussian')
if q == 0:
q = 1e-100
d_kl += p * math.log(p/q)
return d_kl
def calc(self):
"""
Returns a summary of a set of sequences that can be partitioned into
the list of lists of taxa given by `taxon_groups`.
"""
diffs_x, mean_diffs_x, sq_diff_x = _count_differences(
self.pop1_seqs, self.state_alphabet, self.ignore_uncertain)
diffs_y, mean_diffs_y, sq_diff_y = _count_differences(
self.pop2_seqs, self.state_alphabet, self.ignore_uncertain)
d_x = diffs_x / probability.binomial_coefficient(
len(self.pop1_seqs), 2)
d_y = diffs_y / probability.binomial_coefficient(
len(self.pop2_seqs), 2)
d_xy = self._average_number_of_pairwise_differences_between_populations(
)
s2_x = (float(sq_diff_x) / probability.binomial_coefficient(
len(self.pop1_seqs), 2)) - (d_x**2)
s2_y = (float(sq_diff_y) / probability.binomial_coefficient(
len(self.pop2_seqs), 2)) - (d_y**2)
s2_xy = self._variance_of_pairwise_differences_between_populations(
d_xy)
n = len(self.combined_seqs)
n_x = float(len(self.pop1_seqs))
n_y = float(len(self.pop2_seqs))
a = float(n * (n - 1))
ax = float(n_x * (n_x - 1))
ay = float(n_y * (n_y - 1))
k = _average_number_of_pairwise_differences(self.combined_seqs,
self.state_alphabet,
self.ignore_uncertain)
n = len(self.combined_seqs)
# Hickerson 2006: pi #
self.average_number_of_pairwise_differences = k
# Hickerson 2006: pi_b #
self.average_number_of_pairwise_differences_between = d_xy
# Hickerson 2006: pi_w #
self.average_number_of_pairwise_differences_within = d_x + d_y
# Hickerson 2006: pi_net #
self.average_number_of_pairwise_differences_net = d_xy - (d_x + d_y)
# Hickerson 2006: S #
self.num_segregating_sites = _num_segregating_sites(
self.combined_seqs, self.state_alphabet, self.ignore_uncertain)
# Hickerson 2006: theta #
a1 = sum([1.0 / i for i in range(1, n)])
self.wattersons_theta = float(self.num_segregating_sites) / a1
# Wakeley 1996 #
self.wakeleys_psi = (float(1) /
(a)) * (ax * (math.sqrt(s2_x) / d_x) + ay *
(math.sqrt(s2_y) / d_y) +
(2 * n_x * n_y * math.sqrt(s2_xy) / k))
# Tajima's D #
self.tajimas_d = _tajimas_d(
n, self.average_number_of_pairwise_differences,
self.num_segregating_sites)