Source code for allel.stats.diversity

# -*- coding: utf-8 -*-
import logging


import numpy as np


from allel.model.ndarray import SortedIndex, AlleleCountsArray
from allel.model.util import locate_fixed_differences
from allel.util import asarray_ndim, ignore_invalid, check_dim0_aligned, \
    ensure_dim1_aligned
from allel.stats.window import windowed_statistic, per_base, moving_statistic


logger = logging.getLogger(__name__)
debug = logger.debug


[docs]def mean_pairwise_difference(ac, an=None, fill=np.nan): """Calculate for each variant the mean number of pairwise differences between chromosomes sampled from within a single population. Parameters ---------- ac : array_like, int, shape (n_variants, n_alleles) Allele counts array. an : array_like, int, shape (n_variants,), optional Allele numbers. If not provided, will be calculated from `ac`. fill : float Use this value where there are no pairs to compare (e.g., all allele calls are missing). Returns ------- mpd : ndarray, float, shape (n_variants,) Notes ----- The values returned by this function can be summed over a genome region and divided by the number of accessible bases to estimate nucleotide diversity, a.k.a. *pi*. Examples -------- >>> import allel >>> h = allel.HaplotypeArray([[0, 0, 0, 0], ... [0, 0, 0, 1], ... [0, 0, 1, 1], ... [0, 1, 1, 1], ... [1, 1, 1, 1], ... [0, 0, 1, 2], ... [0, 1, 1, 2], ... [0, 1, -1, -1]]) >>> ac = h.count_alleles() >>> allel.mean_pairwise_difference(ac) array([0. , 0.5 , 0.66666667, 0.5 , 0. , 0.83333333, 0.83333333, 1. ]) See Also -------- sequence_diversity, windowed_diversity """ # This function calculates the mean number of pairwise differences # between haplotypes within a single population, generalising to any number # of alleles. # check inputs ac = asarray_ndim(ac, 2) # total number of haplotypes if an is None: an = np.sum(ac, axis=1) else: an = asarray_ndim(an, 1) check_dim0_aligned(ac, an) # total number of pairwise comparisons for each variant: # (an choose 2) n_pairs = an * (an - 1) / 2 # number of pairwise comparisons where there is no difference: # sum of (ac choose 2) for each allele (i.e., number of ways to # choose the same allele twice) n_same = np.sum(ac * (ac - 1) / 2, axis=1) # number of pairwise differences n_diff = n_pairs - n_same # mean number of pairwise differences, accounting for cases where # there are no pairs with ignore_invalid(): mpd = np.where(n_pairs > 0, n_diff / n_pairs, fill) return mpd
[docs]def mean_pairwise_difference_between(ac1, ac2, an1=None, an2=None, fill=np.nan): """Calculate for each variant the mean number of pairwise differences between chromosomes sampled from two different populations. Parameters ---------- ac1 : array_like, int, shape (n_variants, n_alleles) Allele counts array from the first population. ac2 : array_like, int, shape (n_variants, n_alleles) Allele counts array from the second population. an1 : array_like, int, shape (n_variants,), optional Allele numbers for the first population. If not provided, will be calculated from `ac1`. an2 : array_like, int, shape (n_variants,), optional Allele numbers for the second population. If not provided, will be calculated from `ac2`. fill : float Use this value where there are no pairs to compare (e.g., all allele calls are missing). Returns ------- mpd : ndarray, float, shape (n_variants,) Notes ----- The values returned by this function can be summed over a genome region and divided by the number of accessible bases to estimate nucleotide divergence between two populations, a.k.a. *Dxy*. Examples -------- >>> import allel >>> h = allel.HaplotypeArray([[0, 0, 0, 0], ... [0, 0, 0, 1], ... [0, 0, 1, 1], ... [0, 1, 1, 1], ... [1, 1, 1, 1], ... [0, 0, 1, 2], ... [0, 1, 1, 2], ... [0, 1, -1, -1]]) >>> ac1 = h.count_alleles(subpop=[0, 1]) >>> ac2 = h.count_alleles(subpop=[2, 3]) >>> allel.mean_pairwise_difference_between(ac1, ac2) array([0. , 0.5 , 1. , 0.5 , 0. , 1. , 0.75, nan]) See Also -------- sequence_divergence, windowed_divergence """ # This function calculates the mean number of pairwise differences # between haplotypes from two different populations, generalising to any # number of alleles. # check inputs ac1 = asarray_ndim(ac1, 2) ac2 = asarray_ndim(ac2, 2) check_dim0_aligned(ac1, ac2) ac1, ac2 = ensure_dim1_aligned(ac1, ac2) # total number of haplotypes sampled from each population if an1 is None: an1 = np.sum(ac1, axis=1) else: an1 = asarray_ndim(an1, 1) check_dim0_aligned(ac1, an1) if an2 is None: an2 = np.sum(ac2, axis=1) else: an2 = asarray_ndim(an2, 1) check_dim0_aligned(ac2, an2) # total number of pairwise comparisons for each variant n_pairs = an1 * an2 # number of pairwise comparisons where there is no difference: # sum of (ac1 * ac2) for each allele (i.e., number of ways to # choose the same allele twice) n_same = np.sum(ac1 * ac2, axis=1) # number of pairwise differences n_diff = n_pairs - n_same # mean number of pairwise differences, accounting for cases where # there are no pairs with ignore_invalid(): mpd = np.where(n_pairs > 0, n_diff / n_pairs, fill) return mpd
[docs]def sequence_diversity(pos, ac, start=None, stop=None, is_accessible=None): """Estimate nucleotide diversity within a given region, which is the average proportion of sites (including monomorphic sites not present in the data) that differ between randomly chosen pairs of chromosomes. Parameters ---------- pos : array_like, int, shape (n_items,) Variant positions, using 1-based coordinates, in ascending order. ac : array_like, int, shape (n_variants, n_alleles) Allele counts array. start : int, optional The position at which to start (1-based). Defaults to the first position. stop : int, optional The position at which to stop (1-based). Defaults to the last position. is_accessible : array_like, bool, shape (len(contig),), optional Boolean array indicating accessibility status for all positions in the chromosome/contig. Returns ------- pi : ndarray, float, shape (n_windows,) Nucleotide diversity. Notes ----- If start and/or stop are not provided, uses the difference between the last and the first position as a proxy for the total number of sites, which can overestimate the sequence diversity. Examples -------- >>> import allel >>> g = allel.GenotypeArray([[[0, 0], [0, 0]], ... [[0, 0], [0, 1]], ... [[0, 0], [1, 1]], ... [[0, 1], [1, 1]], ... [[1, 1], [1, 1]], ... [[0, 0], [1, 2]], ... [[0, 1], [1, 2]], ... [[0, 1], [-1, -1]], ... [[-1, -1], [-1, -1]]]) >>> ac = g.count_alleles() >>> pos = [2, 4, 7, 14, 15, 18, 19, 25, 27] >>> pi = allel.sequence_diversity(pos, ac, start=1, stop=31) >>> pi 0.13978494623655915 """ # check inputs if not isinstance(pos, SortedIndex): pos = SortedIndex(pos, copy=False) ac = asarray_ndim(ac, 2) is_accessible = asarray_ndim(is_accessible, 1, allow_none=True) # deal with subregion if start is not None or stop is not None: loc = pos.locate_range(start, stop) pos = pos[loc] ac = ac[loc] if start is None: start = pos[0] if stop is None: stop = pos[-1] # calculate mean pairwise difference mpd = mean_pairwise_difference(ac, fill=0) # sum differences over variants mpd_sum = np.sum(mpd) # calculate value per base if is_accessible is None: n_bases = stop - start + 1 else: n_bases = np.count_nonzero(is_accessible[start-1:stop]) pi = mpd_sum / n_bases return pi
[docs]def sequence_divergence(pos, ac1, ac2, an1=None, an2=None, start=None, stop=None, is_accessible=None): """Estimate nucleotide divergence between two populations within a given region, which is the average proportion of sites (including monomorphic sites not present in the data) that differ between randomly chosen pairs of chromosomes, one from each population. Parameters ---------- pos : array_like, int, shape (n_items,) Variant positions, using 1-based coordinates, in ascending order. ac1 : array_like, int, shape (n_variants, n_alleles) Allele counts array for the first population. ac2 : array_like, int, shape (n_variants, n_alleles) Allele counts array for the second population. an1 : array_like, int, shape (n_variants,), optional Allele numbers for the first population. If not provided, will be calculated from `ac1`. an2 : array_like, int, shape (n_variants,), optional Allele numbers for the second population. If not provided, will be calculated from `ac2`. start : int, optional The position at which to start (1-based). Defaults to the first position. stop : int, optional The position at which to stop (1-based). Defaults to the last position. is_accessible : array_like, bool, shape (len(contig),), optional Boolean array indicating accessibility status for all positions in the chromosome/contig. Returns ------- Dxy : ndarray, float, shape (n_windows,) Nucleotide divergence. Examples -------- Simplest case, two haplotypes in each population:: >>> import allel >>> h = allel.HaplotypeArray([[0, 0, 0, 0], ... [0, 0, 0, 1], ... [0, 0, 1, 1], ... [0, 1, 1, 1], ... [1, 1, 1, 1], ... [0, 0, 1, 2], ... [0, 1, 1, 2], ... [0, 1, -1, -1], ... [-1, -1, -1, -1]]) >>> ac1 = h.count_alleles(subpop=[0, 1]) >>> ac2 = h.count_alleles(subpop=[2, 3]) >>> pos = [2, 4, 7, 14, 15, 18, 19, 25, 27] >>> dxy = sequence_divergence(pos, ac1, ac2, start=1, stop=31) >>> dxy 0.12096774193548387 """ # check inputs if not isinstance(pos, SortedIndex): pos = SortedIndex(pos, copy=False) ac1 = asarray_ndim(ac1, 2) ac2 = asarray_ndim(ac2, 2) if an1 is not None: an1 = asarray_ndim(an1, 1) if an2 is not None: an2 = asarray_ndim(an2, 1) is_accessible = asarray_ndim(is_accessible, 1, allow_none=True) # handle start/stop if start is not None or stop is not None: loc = pos.locate_range(start, stop) pos = pos[loc] ac1 = ac1[loc] ac2 = ac2[loc] if an1 is not None: an1 = an1[loc] if an2 is not None: an2 = an2[loc] if start is None: start = pos[0] if stop is None: stop = pos[-1] # calculate mean pairwise difference between the two populations mpd = mean_pairwise_difference_between(ac1, ac2, an1=an1, an2=an2, fill=0) # sum differences over variants mpd_sum = np.sum(mpd) # calculate value per base, N.B., expect pos is 1-based if is_accessible is None: n_bases = stop - start + 1 else: n_bases = np.count_nonzero(is_accessible[start-1:stop]) dxy = mpd_sum / n_bases return dxy
[docs]def windowed_diversity(pos, ac, size=None, start=None, stop=None, step=None, windows=None, is_accessible=None, fill=np.nan): """Estimate nucleotide diversity in windows over a single chromosome/contig. Parameters ---------- pos : array_like, int, shape (n_items,) Variant positions, using 1-based coordinates, in ascending order. ac : array_like, int, shape (n_variants, n_alleles) Allele counts array. size : int, optional The window size (number of bases). start : int, optional The position at which to start (1-based). stop : int, optional The position at which to stop (1-based). step : int, optional The distance between start positions of windows. If not given, defaults to the window size, i.e., non-overlapping windows. windows : array_like, int, shape (n_windows, 2), optional Manually specify the windows to use as a sequence of (window_start, window_stop) positions, using 1-based coordinates. Overrides the size/start/stop/step parameters. is_accessible : array_like, bool, shape (len(contig),), optional Boolean array indicating accessibility status for all positions in the chromosome/contig. fill : object, optional The value to use where a window is completely inaccessible. Returns ------- pi : ndarray, float, shape (n_windows,) Nucleotide diversity in each window. windows : ndarray, int, shape (n_windows, 2) The windows used, as an array of (window_start, window_stop) positions, using 1-based coordinates. n_bases : ndarray, int, shape (n_windows,) Number of (accessible) bases in each window. counts : ndarray, int, shape (n_windows,) Number of variants in each window. Examples -------- >>> import allel >>> g = allel.GenotypeArray([[[0, 0], [0, 0]], ... [[0, 0], [0, 1]], ... [[0, 0], [1, 1]], ... [[0, 1], [1, 1]], ... [[1, 1], [1, 1]], ... [[0, 0], [1, 2]], ... [[0, 1], [1, 2]], ... [[0, 1], [-1, -1]], ... [[-1, -1], [-1, -1]]]) >>> ac = g.count_alleles() >>> pos = [2, 4, 7, 14, 15, 18, 19, 25, 27] >>> pi, windows, n_bases, counts = allel.windowed_diversity( ... pos, ac, size=10, start=1, stop=31 ... ) >>> pi array([0.11666667, 0.21666667, 0.09090909]) >>> windows array([[ 1, 10], [11, 20], [21, 31]]) >>> n_bases array([10, 10, 11]) >>> counts array([3, 4, 2]) """ # check inputs if not isinstance(pos, SortedIndex): pos = SortedIndex(pos, copy=False) is_accessible = asarray_ndim(is_accessible, 1, allow_none=True) # calculate mean pairwise difference mpd = mean_pairwise_difference(ac, fill=0) # sum differences in windows mpd_sum, windows, counts = windowed_statistic( pos, values=mpd, statistic=np.sum, size=size, start=start, stop=stop, step=step, windows=windows, fill=0 ) # calculate value per base pi, n_bases = per_base(mpd_sum, windows, is_accessible=is_accessible, fill=fill) return pi, windows, n_bases, counts
[docs]def windowed_divergence(pos, ac1, ac2, size=None, start=None, stop=None, step=None, windows=None, is_accessible=None, fill=np.nan): """Estimate nucleotide divergence between two populations in windows over a single chromosome/contig. Parameters ---------- pos : array_like, int, shape (n_items,) Variant positions, using 1-based coordinates, in ascending order. ac1 : array_like, int, shape (n_variants, n_alleles) Allele counts array for the first population. ac2 : array_like, int, shape (n_variants, n_alleles) Allele counts array for the second population. size : int, optional The window size (number of bases). start : int, optional The position at which to start (1-based). stop : int, optional The position at which to stop (1-based). step : int, optional The distance between start positions of windows. If not given, defaults to the window size, i.e., non-overlapping windows. windows : array_like, int, shape (n_windows, 2), optional Manually specify the windows to use as a sequence of (window_start, window_stop) positions, using 1-based coordinates. Overrides the size/start/stop/step parameters. is_accessible : array_like, bool, shape (len(contig),), optional Boolean array indicating accessibility status for all positions in the chromosome/contig. fill : object, optional The value to use where a window is completely inaccessible. Returns ------- Dxy : ndarray, float, shape (n_windows,) Nucleotide divergence in each window. windows : ndarray, int, shape (n_windows, 2) The windows used, as an array of (window_start, window_stop) positions, using 1-based coordinates. n_bases : ndarray, int, shape (n_windows,) Number of (accessible) bases in each window. counts : ndarray, int, shape (n_windows,) Number of variants in each window. Examples -------- Simplest case, two haplotypes in each population:: >>> import allel >>> h = allel.HaplotypeArray([[0, 0, 0, 0], ... [0, 0, 0, 1], ... [0, 0, 1, 1], ... [0, 1, 1, 1], ... [1, 1, 1, 1], ... [0, 0, 1, 2], ... [0, 1, 1, 2], ... [0, 1, -1, -1], ... [-1, -1, -1, -1]]) >>> ac1 = h.count_alleles(subpop=[0, 1]) >>> ac2 = h.count_alleles(subpop=[2, 3]) >>> pos = [2, 4, 7, 14, 15, 18, 19, 25, 27] >>> dxy, windows, n_bases, counts = windowed_divergence( ... pos, ac1, ac2, size=10, start=1, stop=31 ... ) >>> dxy array([0.15 , 0.225, 0. ]) >>> windows array([[ 1, 10], [11, 20], [21, 31]]) >>> n_bases array([10, 10, 11]) >>> counts array([3, 4, 2]) """ # check inputs pos = SortedIndex(pos, copy=False) is_accessible = asarray_ndim(is_accessible, 1, allow_none=True) # calculate mean pairwise divergence mpd = mean_pairwise_difference_between(ac1, ac2, fill=0) # sum in windows mpd_sum, windows, counts = windowed_statistic( pos, values=mpd, statistic=np.sum, size=size, start=start, stop=stop, step=step, windows=windows, fill=0 ) # calculate value per base dxy, n_bases = per_base(mpd_sum, windows, is_accessible=is_accessible, fill=fill) return dxy, windows, n_bases, counts
[docs]def windowed_df(pos, ac1, ac2, size=None, start=None, stop=None, step=None, windows=None, is_accessible=None, fill=np.nan): """Calculate the density of fixed differences between two populations in windows over a single chromosome/contig. Parameters ---------- pos : array_like, int, shape (n_items,) Variant positions, using 1-based coordinates, in ascending order. ac1 : array_like, int, shape (n_variants, n_alleles) Allele counts array for the first population. ac2 : array_like, int, shape (n_variants, n_alleles) Allele counts array for the second population. size : int, optional The window size (number of bases). start : int, optional The position at which to start (1-based). stop : int, optional The position at which to stop (1-based). step : int, optional The distance between start positions of windows. If not given, defaults to the window size, i.e., non-overlapping windows. windows : array_like, int, shape (n_windows, 2), optional Manually specify the windows to use as a sequence of (window_start, window_stop) positions, using 1-based coordinates. Overrides the size/start/stop/step parameters. is_accessible : array_like, bool, shape (len(contig),), optional Boolean array indicating accessibility status for all positions in the chromosome/contig. fill : object, optional The value to use where a window is completely inaccessible. Returns ------- df : ndarray, float, shape (n_windows,) Per-base density of fixed differences in each window. windows : ndarray, int, shape (n_windows, 2) The windows used, as an array of (window_start, window_stop) positions, using 1-based coordinates. n_bases : ndarray, int, shape (n_windows,) Number of (accessible) bases in each window. counts : ndarray, int, shape (n_windows,) Number of variants in each window. See Also -------- allel.model.locate_fixed_differences """ # check inputs pos = SortedIndex(pos, copy=False) is_accessible = asarray_ndim(is_accessible, 1, allow_none=True) # locate fixed differences loc_df = locate_fixed_differences(ac1, ac2) # count number of fixed differences in windows n_df, windows, counts = windowed_statistic( pos, values=loc_df, statistic=np.count_nonzero, size=size, start=start, stop=stop, step=step, windows=windows, fill=0 ) # calculate value per base df, n_bases = per_base(n_df, windows, is_accessible=is_accessible, fill=fill) return df, windows, n_bases, counts
# noinspection PyPep8Naming
[docs]def watterson_theta(pos, ac, start=None, stop=None, is_accessible=None): """Calculate the value of Watterson's estimator over a given region. Parameters ---------- pos : array_like, int, shape (n_items,) Variant positions, using 1-based coordinates, in ascending order. ac : array_like, int, shape (n_variants, n_alleles) Allele counts array. start : int, optional The position at which to start (1-based). Defaults to the first position. stop : int, optional The position at which to stop (1-based). Defaults to the last position. is_accessible : array_like, bool, shape (len(contig),), optional Boolean array indicating accessibility status for all positions in the chromosome/contig. Returns ------- theta_hat_w : float Watterson's estimator (theta hat per base). Examples -------- >>> import allel >>> g = allel.GenotypeArray([[[0, 0], [0, 0]], ... [[0, 0], [0, 1]], ... [[0, 0], [1, 1]], ... [[0, 1], [1, 1]], ... [[1, 1], [1, 1]], ... [[0, 0], [1, 2]], ... [[0, 1], [1, 2]], ... [[0, 1], [-1, -1]], ... [[-1, -1], [-1, -1]]]) >>> ac = g.count_alleles() >>> pos = [2, 4, 7, 14, 15, 18, 19, 25, 27] >>> theta_hat_w = allel.watterson_theta(pos, ac, start=1, stop=31) >>> theta_hat_w 0.10557184750733138 """ # check inputs if not isinstance(pos, SortedIndex): pos = SortedIndex(pos, copy=False) is_accessible = asarray_ndim(is_accessible, 1, allow_none=True) if not hasattr(ac, 'count_segregating'): ac = AlleleCountsArray(ac, copy=False) # deal with subregion if start is not None or stop is not None: loc = pos.locate_range(start, stop) pos = pos[loc] ac = ac[loc] if start is None: start = pos[0] if stop is None: stop = pos[-1] # count segregating variants S = ac.count_segregating() # assume number of chromosomes sampled is constant for all variants n = ac.sum(axis=1).max() # (n-1)th harmonic number a1 = np.sum(1 / np.arange(1, n)) # calculate absolute value theta_hat_w_abs = S / a1 # calculate value per base if is_accessible is None: n_bases = stop - start + 1 else: n_bases = np.count_nonzero(is_accessible[start-1:stop]) theta_hat_w = theta_hat_w_abs / n_bases return theta_hat_w
# noinspection PyPep8Naming
[docs]def windowed_watterson_theta(pos, ac, size=None, start=None, stop=None, step=None, windows=None, is_accessible=None, fill=np.nan): """Calculate the value of Watterson's estimator in windows over a single chromosome/contig. Parameters ---------- pos : array_like, int, shape (n_items,) Variant positions, using 1-based coordinates, in ascending order. ac : array_like, int, shape (n_variants, n_alleles) Allele counts array. size : int, optional The window size (number of bases). start : int, optional The position at which to start (1-based). stop : int, optional The position at which to stop (1-based). step : int, optional The distance between start positions of windows. If not given, defaults to the window size, i.e., non-overlapping windows. windows : array_like, int, shape (n_windows, 2), optional Manually specify the windows to use as a sequence of (window_start, window_stop) positions, using 1-based coordinates. Overrides the size/start/stop/step parameters. is_accessible : array_like, bool, shape (len(contig),), optional Boolean array indicating accessibility status for all positions in the chromosome/contig. fill : object, optional The value to use where a window is completely inaccessible. Returns ------- theta_hat_w : ndarray, float, shape (n_windows,) Watterson's estimator (theta hat per base). windows : ndarray, int, shape (n_windows, 2) The windows used, as an array of (window_start, window_stop) positions, using 1-based coordinates. n_bases : ndarray, int, shape (n_windows,) Number of (accessible) bases in each window. counts : ndarray, int, shape (n_windows,) Number of variants in each window. Examples -------- >>> import allel >>> g = allel.GenotypeArray([[[0, 0], [0, 0]], ... [[0, 0], [0, 1]], ... [[0, 0], [1, 1]], ... [[0, 1], [1, 1]], ... [[1, 1], [1, 1]], ... [[0, 0], [1, 2]], ... [[0, 1], [1, 2]], ... [[0, 1], [-1, -1]], ... [[-1, -1], [-1, -1]]]) >>> ac = g.count_alleles() >>> pos = [2, 4, 7, 14, 15, 18, 19, 25, 27] >>> theta_hat_w, windows, n_bases, counts = allel.windowed_watterson_theta( ... pos, ac, size=10, start=1, stop=31 ... ) >>> theta_hat_w array([0.10909091, 0.16363636, 0.04958678]) >>> windows array([[ 1, 10], [11, 20], [21, 31]]) >>> n_bases array([10, 10, 11]) >>> counts array([3, 4, 2]) """ # flake8: noqa # check inputs if not isinstance(pos, SortedIndex): pos = SortedIndex(pos, copy=False) is_accessible = asarray_ndim(is_accessible, 1, allow_none=True) if not hasattr(ac, 'count_segregating'): ac = AlleleCountsArray(ac, copy=False) # locate segregating variants is_seg = ac.is_segregating() # count segregating variants in windows S, windows, counts = windowed_statistic(pos, is_seg, statistic=np.count_nonzero, size=size, start=start, stop=stop, step=step, windows=windows, fill=0) # assume number of chromosomes sampled is constant for all variants n = ac.sum(axis=1).max() # (n-1)th harmonic number a1 = np.sum(1 / np.arange(1, n)) # absolute value of Watterson's theta theta_hat_w_abs = S / a1 # theta per base theta_hat_w, n_bases = per_base(theta_hat_w_abs, windows=windows, is_accessible=is_accessible, fill=fill) return theta_hat_w, windows, n_bases, counts
# noinspection PyPep8Naming
[docs]def tajima_d(ac, pos=None, start=None, stop=None, min_sites=3): """Calculate the value of Tajima's D over a given region. Parameters ---------- ac : array_like, int, shape (n_variants, n_alleles) Allele counts array. pos : array_like, int, shape (n_items,), optional Variant positions, using 1-based coordinates, in ascending order. start : int, optional The position at which to start (1-based). Defaults to the first position. stop : int, optional The position at which to stop (1-based). Defaults to the last position. min_sites : int, optional Minimum number of segregating sites for which to calculate a value. If there are fewer, np.nan is returned. Defaults to 3. Returns ------- D : float Examples -------- >>> import allel >>> g = allel.GenotypeArray([[[0, 0], [0, 0]], ... [[0, 0], [0, 1]], ... [[0, 0], [1, 1]], ... [[0, 1], [1, 1]], ... [[1, 1], [1, 1]], ... [[0, 0], [1, 2]], ... [[0, 1], [1, 2]], ... [[0, 1], [-1, -1]], ... [[-1, -1], [-1, -1]]]) >>> ac = g.count_alleles() >>> allel.tajima_d(ac) 3.1445848780213814 >>> pos = [2, 4, 7, 14, 15, 18, 19, 25, 27] >>> allel.tajima_d(ac, pos=pos, start=7, stop=25) 3.8779735196179366 """ # check inputs if not hasattr(ac, 'count_segregating'): ac = AlleleCountsArray(ac, copy=False) # deal with subregion if pos is not None and (start is not None or stop is not None): if not isinstance(pos, SortedIndex): pos = SortedIndex(pos, copy=False) loc = pos.locate_range(start, stop) ac = ac[loc] # count segregating variants S = ac.count_segregating() if S < min_sites: return np.nan # assume number of chromosomes sampled is constant for all variants n = ac.sum(axis=1).max() # (n-1)th harmonic number a1 = np.sum(1 / np.arange(1, n)) # calculate Watterson's theta (absolute value) theta_hat_w_abs = S / a1 # calculate mean pairwise difference mpd = mean_pairwise_difference(ac, fill=0) # calculate theta_hat pi (sum differences over variants) theta_hat_pi_abs = np.sum(mpd) # N.B., both theta estimates are usually divided by the number of # (accessible) bases but here we want the absolute difference d = theta_hat_pi_abs - theta_hat_w_abs # calculate the denominator (standard deviation) a2 = np.sum(1 / (np.arange(1, n)**2)) b1 = (n + 1) / (3 * (n - 1)) b2 = 2 * (n**2 + n + 3) / (9 * n * (n - 1)) c1 = b1 - (1 / a1) c2 = b2 - ((n + 2) / (a1 * n)) + (a2 / (a1**2)) e1 = c1 / a1 e2 = c2 / (a1**2 + a2) d_stdev = np.sqrt((e1 * S) + (e2 * S * (S - 1))) # finally calculate Tajima's D D = d / d_stdev return D
# noinspection PyPep8Naming
[docs]def windowed_tajima_d(pos, ac, size=None, start=None, stop=None, step=None, windows=None, min_sites=3): """Calculate the value of Tajima's D in windows over a single chromosome/contig. Parameters ---------- pos : array_like, int, shape (n_items,) Variant positions, using 1-based coordinates, in ascending order. ac : array_like, int, shape (n_variants, n_alleles) Allele counts array. size : int, optional The window size (number of bases). start : int, optional The position at which to start (1-based). stop : int, optional The position at which to stop (1-based). step : int, optional The distance between start positions of windows. If not given, defaults to the window size, i.e., non-overlapping windows. windows : array_like, int, shape (n_windows, 2), optional Manually specify the windows to use as a sequence of (window_start, window_stop) positions, using 1-based coordinates. Overrides the size/start/stop/step parameters. min_sites : int, optional Minimum number of segregating sites for which to calculate a value. If there are fewer, np.nan is returned. Defaults to 3. Returns ------- D : ndarray, float, shape (n_windows,) Tajima's D. windows : ndarray, int, shape (n_windows, 2) The windows used, as an array of (window_start, window_stop) positions, using 1-based coordinates. counts : ndarray, int, shape (n_windows,) Number of variants in each window. Examples -------- >>> import allel >>> g = allel.GenotypeArray([[[0, 0], [0, 0]], ... [[0, 0], [0, 1]], ... [[0, 0], [1, 1]], ... [[0, 1], [1, 1]], ... [[1, 1], [1, 1]], ... [[0, 0], [1, 2]], ... [[0, 1], [1, 2]], ... [[0, 1], [-1, -1]], ... [[-1, -1], [-1, -1]]]) >>> ac = g.count_alleles() >>> pos = [2, 4, 7, 14, 15, 20, 22, 25, 27] >>> D, windows, counts = allel.windowed_tajima_d(pos, ac, size=20, step=10, start=1, stop=31) >>> D array([1.36521524, 4.22566622]) >>> windows array([[ 1, 20], [11, 31]]) >>> counts array([6, 6]) """ # check inputs if not isinstance(pos, SortedIndex): pos = SortedIndex(pos, copy=False) if not hasattr(ac, 'count_segregating'): ac = AlleleCountsArray(ac, copy=False) # assume number of chromosomes sampled is constant for all variants n = ac.sum(axis=1).max() # calculate constants a1 = np.sum(1 / np.arange(1, n)) a2 = np.sum(1 / (np.arange(1, n)**2)) b1 = (n + 1) / (3 * (n - 1)) b2 = 2 * (n**2 + n + 3) / (9 * n * (n - 1)) c1 = b1 - (1 / a1) c2 = b2 - ((n + 2) / (a1 * n)) + (a2 / (a1**2)) e1 = c1 / a1 e2 = c2 / (a1**2 + a2) # locate segregating variants is_seg = ac.is_segregating() # calculate mean pairwise difference mpd = mean_pairwise_difference(ac, fill=0) # define statistic to compute for each window # noinspection PyPep8Naming def statistic(w_is_seg, w_mpd): S = np.count_nonzero(w_is_seg) if S < min_sites: return np.nan pi = np.sum(w_mpd) d = pi - (S / a1) d_stdev = np.sqrt((e1 * S) + (e2 * S * (S - 1))) wD = d / d_stdev return wD D, windows, counts = windowed_statistic(pos, values=(is_seg, mpd), statistic=statistic, size=size, start=start, stop=stop, step=step, windows=windows, fill=np.nan) return D, windows, counts
[docs]def moving_tajima_d(ac, size, start=0, stop=None, step=None, min_sites=3): """Calculate the value of Tajima's D in moving windows of `size` variants. Parameters ---------- ac : array_like, int, shape (n_variants, n_alleles) Allele counts array. size : int The window size (number of variants). start : int, optional The index at which to start. stop : int, optional The index at which to stop. step : int, optional The number of variants between start positions of windows. If not given, defaults to the window size, i.e., non-overlapping windows. min_sites : int, optional Minimum number of segregating sites for which to calculate a value. If there are fewer, np.nan is returned. Defaults to 3. Returns ------- d : ndarray, float, shape (n_windows,) Tajima's D. Examples -------- >>> import allel >>> g = allel.GenotypeArray([[[0, 0], [0, 0]], ... [[0, 0], [0, 1]], ... [[0, 0], [1, 1]], ... [[0, 1], [1, 1]], ... [[1, 1], [1, 1]], ... [[0, 0], [1, 2]], ... [[0, 1], [1, 2]], ... [[0, 1], [-1, -1]], ... [[-1, -1], [-1, -1]]]) >>> ac = g.count_alleles() >>> D = allel.moving_tajima_d(ac, size=4, step=2) >>> D array([0.1676558 , 2.01186954, 5.70029703]) """ d = moving_statistic(values=ac, statistic=tajima_d, size=size, start=start, stop=stop, step=step, min_sites=min_sites) return d