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# ----------------------------------------------------------------------------
# Copyright (c) 2013--, scikit-bio development team.
#
# Distributed under the terms of the Modified BSD License.
#
# The full license is in the file COPYING.txt, distributed with this software.
# ----------------------------------------------------------------------------
import numpy as np
from ._base import osd
from skbio.diversity._util import _validate_counts_vector
from skbio.util._decorator import experimental
@experimental(as_of="0.4.0")
def chao1(counts, bias_corrected=True):
r"""Calculate chao1 richness estimator.
Uses the bias-corrected version unless `bias_corrected` is ``False`` *and*
there are both singletons and doubletons.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
bias_corrected : bool, optional
Indicates whether or not to use the bias-corrected version of the
equation. If ``False`` *and* there are both singletons and doubletons,
the uncorrected version will be used. The biased-corrected version will
be used otherwise.
Returns
-------
double
Computed chao1 richness estimator.
See Also
--------
chao1_ci
Notes
-----
The uncorrected version is based on Equation 6 in [1]_:
.. math::
chao1=S_{obs}+\frac{F_1^2}{2F_2}
where :math:`F_1` and :math:`F_2` are the count of singletons and
doubletons, respectively.
The bias-corrected version is defined as
.. math::
chao1=S_{obs}+\frac{F_1(F_1-1)}{2(F_2+1)}
References
----------
.. [1] Chao, A. 1984. Non-parametric estimation of the number of classes in
a population. Scandinavian Journal of Statistics 11, 265-270.
"""
counts = _validate_counts_vector(counts)
o, s, d = osd(counts)
if not bias_corrected and s and d:
return o + s ** 2 / (d * 2)
else:
return o + s * (s - 1) / (2 * (d + 1))
@experimental(as_of="0.4.0")
def chao1_ci(counts, bias_corrected=True, zscore=1.96):
"""Calculate chao1 confidence interval.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
bias_corrected : bool, optional
Indicates whether or not to use the bias-corrected version of the
equation. If ``False`` *and* there are both singletons and doubletons,
the uncorrected version will be used. The biased-corrected version will
be used otherwise.
zscore : scalar, optional
Score to use for confidence. Default of 1.96 is for a 95% confidence
interval.
Returns
-------
tuple
chao1 confidence interval as ``(lower_bound, upper_bound)``.
See Also
--------
chao1
Notes
-----
The implementation here is based on the equations in the EstimateS manual
[1]_. Different equations are employed to calculate the chao1 variance and
confidence interval depending on `bias_corrected` and the presence/absence
of singletons and/or doubletons.
Specifically, the following EstimateS equations are used:
1. No singletons, Equation 14.
2. Singletons but no doubletons, Equations 7, 13.
3. Singletons and doubletons, ``bias_corrected=True``, Equations 6, 13.
4. Singletons and doubletons, ``bias_corrected=False``, Equations 5, 13.
References
----------
.. [1] http://viceroy.eeb.uconn.edu/estimates/
"""
counts = _validate_counts_vector(counts)
o, s, d = osd(counts)
if s:
chao = chao1(counts, bias_corrected)
chaovar = _chao1_var(counts, bias_corrected)
return _chao_confidence_with_singletons(chao, o, chaovar, zscore)
else:
n = counts.sum()
return _chao_confidence_no_singletons(n, o, zscore)
def _chao1_var(counts, bias_corrected=True):
"""Calculates chao1 variance using decision rules in EstimateS."""
o, s, d = osd(counts)
if not d:
c = chao1(counts, bias_corrected)
return _chao1_var_no_doubletons(s, c)
if not s:
n = counts.sum()
return _chao1_var_no_singletons(n, o)
if bias_corrected:
return _chao1_var_bias_corrected(s, d)
else:
return _chao1_var_uncorrected(s, d)
def _chao1_var_uncorrected(singles, doubles):
"""Calculates chao1, uncorrected.
From EstimateS manual, equation 5.
"""
r = singles / doubles
return doubles * (.5 * r ** 2 + r ** 3 + .24 * r ** 4)
def _chao1_var_bias_corrected(s, d):
"""Calculates chao1 variance, bias-corrected.
`s` is the number of singletons and `d` is the number of doubletons.
From EstimateS manual, equation 6.
"""
return (s * (s - 1) / (2 * (d + 1)) + (s * (2 * s - 1) ** 2) /
(4 * (d + 1) ** 2) + (s ** 2 * d * (s - 1) ** 2) /
(4 * (d + 1) ** 4))
def _chao1_var_no_doubletons(s, chao1):
"""Calculates chao1 variance in absence of doubletons.
From EstimateS manual, equation 7.
`s` is the number of singletons, and `chao1` is the estimate of the mean of
Chao1 from the same dataset.
"""
return s * (s - 1) / 2 + s * (2 * s - 1) ** 2 / 4 - s ** 4 / (4 * chao1)
def _chao1_var_no_singletons(n, o):
"""Calculates chao1 variance in absence of singletons.
`n` is the number of individuals and `o` is the number of observed OTUs.
From EstimateS manual, equation 8.
"""
return o * np.exp(-n / o) * (1 - np.exp(-n / o))
def _chao_confidence_with_singletons(chao, observed, var_chao, zscore=1.96):
"""Calculates confidence bounds for chao1 or chao2.
Uses Eq. 13 of EstimateS manual.
`zscore` is the score to use for confidence. The default of 1.96 is for 95%
confidence.
"""
T = chao - observed
# if no diff betweeh chao and observed, CI is just point estimate of
# observed
if T == 0:
return observed, observed
K = np.exp(abs(zscore) * np.sqrt(np.log(1 + (var_chao / T ** 2))))
return observed + T / K, observed + T * K
def _chao_confidence_no_singletons(n, s, zscore=1.96):
"""Calculates confidence bounds for chao1/chao2 in absence of singletons.
Uses Eq. 14 of EstimateS manual.
`n` is the number of individuals and `s` is the number of OTUs.
"""
P = np.exp(-n / s)
return (max(s, s / (1 - P) - zscore * np.sqrt((s * P / (1 - P)))),
s / (1 - P) + zscore * np.sqrt(s * P / (1 - P)))
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