def osd(counts):
"""Calculate observed OTUs, singles, and doubles.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
osd : tuple
Observed OTUs, singles, and doubles.
See Also
--------
observed_otus
singles
doubles
Notes
-----
This is a convenience function used by many of the other measures that rely
on these three measures.
"""
counts = _validate_counts_vector(counts)
return observed_otus(counts), singles(counts), doubles(counts)
def equitability(counts, base=2):
"""Calculate equitability (Shannon index corrected for number of OTUs).
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
base : scalar, optional
Logarithm base to use in the calculations.
Returns
-------
double
Measure of equitability.
See Also
--------
shannon
Notes
-----
The implementation here is based on the description given in the SDR-IV
online manual [1]_.
References
----------
.. [1] http://www.pisces-conservation.com/sdrhelp/index.html
"""
counts = _validate_counts_vector(counts)
numerator = shannon(counts, base)
denominator = np.log(observed_otus(counts)) / np.log(base)
return numerator / denominator
def goods_coverage(counts):
r"""Calculate Good's coverage of counts.
Good's coverage estimator is defined as
.. math::
1-\frac{F_1}{N}
where :math:`F_1` is the number of singleton OTUs and :math:`N` is the
total number of individuals (sum of abundances for all OTUs).
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Good's coverage estimator.
"""
counts = _validate_counts_vector(counts)
f1 = singles(counts)
N = counts.sum()
return 1 - (f1 / N)
def robbins(counts):
r"""Calculate Robbins' estimator for the probability of unobserved outcomes.
Robbins' estimator is defined as:
.. math::
\frac{F_1}{n+1}
where :math:`F_1` is the number of singleton OTUs.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Robbins' estimate.
Notes
-----
Robbins' estimator is defined in [1]_. The estimate computed here is for
:math:`n-1` counts, i.e. the x-axis is off by 1.
References
----------
.. [1] Robbins, H. E (1968). Ann. of Stats. Vol 36, pp. 256-257.
"""
counts = _validate_counts_vector(counts)
return singles(counts) / counts.sum()
def fisher_alpha(counts):
r"""Calculate Fisher's alpha, a metric of diversity.
Fisher's alpha is estimated by solving the following equation for
:math:`\alpha`:
.. math::
S=\alpha\ln(1+\frac{N}{\alpha})
where :math:`S` is the number of OTUs and :math:`N` is the
total number of individuals in the sample.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Fisher's alpha.
Raises
------
RuntimeError
If the optimizer fails to converge (error > 1.0).
Notes
-----
The implementation here is based on the description given in the SDR-IV
online manual [1]_. Uses ``scipy.optimize.minimize_scalar`` to find
Fisher's alpha.
References
----------
.. [1] http://www.pisces-conservation.com/sdrhelp/index.html
"""
counts = _validate_counts_vector(counts)
n = counts.sum()
s = observed_otus(counts)
def f(alpha):
return (alpha * np.log(1 + (n / alpha)) - s) ** 2
# Temporarily silence RuntimeWarnings (invalid and division by zero) during
# optimization in case invalid input is provided to the objective function
# (e.g. alpha=0).
orig_settings = np.seterr(divide='ignore', invalid='ignore')
try:
alpha = minimize_scalar(f).x
finally:
np.seterr(**orig_settings)
if f(alpha) > 1.0:
raise RuntimeError("Optimizer failed to converge (error > 1.0), so "
"could not compute Fisher's alpha.")
return alpha
def test_validate_counts_vector(self):
# python list
obs = _validate_counts_vector([0, 2, 1, 3])
npt.assert_array_equal(obs, np.array([0, 2, 1, 3]))
self.assertEqual(obs.dtype, int)
# numpy array (no copy made)
data = np.array([0, 2, 1, 3])
obs = _validate_counts_vector(data)
npt.assert_array_equal(obs, data)
self.assertEqual(obs.dtype, int)
self.assertTrue(obs is data)
# single element
obs = _validate_counts_vector([42])
npt.assert_array_equal(obs, np.array([42]))
self.assertEqual(obs.dtype, int)
self.assertEqual(obs.shape, (1,))
# suppress casting to int
obs = _validate_counts_vector([42.2, 42.1, 0], suppress_cast=True)
npt.assert_array_equal(obs, np.array([42.2, 42.1, 0]))
self.assertEqual(obs.dtype, float)
# all zeros
obs = _validate_counts_vector([0, 0, 0])
npt.assert_array_equal(obs, np.array([0, 0, 0]))
self.assertEqual(obs.dtype, int)
# all zeros (single value)
obs = _validate_counts_vector([0])
npt.assert_array_equal(obs, np.array([0]))
self.assertEqual(obs.dtype, int)
def test_validate_counts_vector(self):
# python list
obs = _validate_counts_vector([0, 2, 1, 3])
npt.assert_array_equal(obs, np.array([0, 2, 1, 3]))
self.assertEqual(obs.dtype, int)
# numpy array (no copy made)
data = np.array([0, 2, 1, 3])
obs = _validate_counts_vector(data)
npt.assert_array_equal(obs, data)
self.assertEqual(obs.dtype, int)
self.assertTrue(obs is data)
# single element
obs = _validate_counts_vector([42])
npt.assert_array_equal(obs, np.array([42]))
self.assertEqual(obs.dtype, int)
self.assertEqual(obs.shape, (1, ))
# suppress casting to int
obs = _validate_counts_vector([42.2, 42.1, 0], suppress_cast=True)
npt.assert_array_equal(obs, np.array([42.2, 42.1, 0]))
self.assertEqual(obs.dtype, float)
# all zeros
obs = _validate_counts_vector([0, 0, 0])
npt.assert_array_equal(obs, np.array([0, 0, 0]))
self.assertEqual(obs.dtype, int)
# all zeros (single value)
obs = _validate_counts_vector([0])
npt.assert_array_equal(obs, np.array([0]))
self.assertEqual(obs.dtype, int)
def esty_ci(counts):
r"""Calculate Esty's CI.
Esty's CI is defined as
.. math::
F_1/N \pm z\sqrt{W}
where :math:`F_1` is the number of singleton OTUs, :math:`N` is the total
number of individuals (sum of abundances for all OTUs), and :math:`z` is a
constant that depends on the targeted confidence and based on the normal
distribution.
:math:`W` is defined as
.. math::
\frac{F_1(N-F_1)+2NF_2}{N^3}
where :math:`F_2` is the number of doubleton OTUs.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
tuple
Esty's confidence interval as ``(lower_bound, upper_bound)``.
Notes
-----
Esty's CI is defined in [1]_. :math:`z` is hardcoded for a 95% confidence
interval.
References
----------
.. [1] Esty, W. W. (1983). "A normal limit law for a nonparametric
estimator of the coverage of a random sample". Ann Statist 11: 905-912.
"""
counts = _validate_counts_vector(counts)
f1 = singles(counts)
f2 = doubles(counts)
n = counts.sum()
z = 1.959963985
W = (f1 * (n - f1) + 2 * n * f2) / (n ** 3)
return f1 / n - z * np.sqrt(W), f1 / n + z * np.sqrt(W)
def test_validate_counts_vector_invalid_input(self):
# wrong dtype
with self.assertRaises(TypeError):
_validate_counts_vector([0, 2, 1.2, 3])
# wrong number of dimensions (2-D)
with self.assertRaises(ValueError):
_validate_counts_vector([[0, 2, 1, 3], [4, 5, 6, 7]])
# wrong number of dimensions (scalar)
with self.assertRaises(ValueError):
_validate_counts_vector(1)
# negative values
with self.assertRaises(ValueError):
_validate_counts_vector([0, 0, 2, -1, 3])
def test_validate_counts_vector_invalid_input(self):
# wrong dtype
with self.assertRaises(TypeError):
_validate_counts_vector([0, 2, 1.2, 3])
# wrong number of dimensions (2-D)
with self.assertRaises(ValueError):
_validate_counts_vector([[0, 2, 1, 3], [4, 5, 6, 7]])
# wrong number of dimensions (scalar)
with self.assertRaises(ValueError):
_validate_counts_vector(1)
# negative values
with self.assertRaises(ValueError):
_validate_counts_vector([0, 0, 2, -1, 3])
def mcintosh_d(counts):
r"""Calculate McIntosh dominance index D.
McIntosh dominance index D is defined as:
.. math::
D = \frac{N - U}{N - \sqrt{N}}
where :math:`N` is the total number of individuals in the sample and
:math:`U` is defined as:
.. math::
U = \sqrt{\sum{{n_i}^2}}
where :math:`n_i` is the number of individuals in the :math:`i^{\text{th}}`
OTU.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
McIntosh dominance index D.
See Also
--------
mcintosh_e
Notes
-----
The index was proposed in [1]_. The implementation here is based on the
description given in the SDR-IV online manual [2]_.
References
----------
.. [1] McIntosh, R. P. 1967 An index of diversity and the relation of
certain concepts to diversity. Ecology 48, 1115-1126.
.. [2] http://www.pisces-conservation.com/sdrhelp/index.html
"""
counts = _validate_counts_vector(counts)
u = np.sqrt((counts * counts).sum())
n = counts.sum()
return (n - u) / (n - np.sqrt(n))
def kempton_taylor_q(counts, lower_quantile=0.25, upper_quantile=0.75):
"""Calculate Kempton-Taylor Q index of alpha diversity.
Estimates the slope of the cumulative abundance curve in the interquantile
range. By default, uses lower and upper quartiles, rounding inwards.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
lower_quantile : float, optional
Lower bound of the interquantile range. Defaults to lower quartile.
upper_quantile : float, optional
Upper bound of the interquantile range. Defaults to upper quartile.
Returns
-------
double
Kempton-Taylor Q index of alpha diversity.
Notes
-----
The index is defined in [1]_. The implementation here is based on the
description given in the SDR-IV online manual [2]_.
The implementation provided here differs slightly from the results given in
Magurran 1998. Specifically, we have 14 in the numerator rather than 15.
Magurran recommends counting half of the OTUs with the same # counts as the
point where the UQ falls and the point where the LQ falls, but the
justification for this is unclear (e.g. if there were a very large # OTUs
that just overlapped one of the quantiles, the results would be
considerably off). Leaving the calculation as-is for now, but consider
changing.
References
----------
.. [1] Kempton, R. A. and Taylor, L. R. (1976) Models and statistics for
species diversity. Nature, 262, 818-820.
.. [2] http://www.pisces-conservation.com/sdrhelp/index.html
"""
counts = _validate_counts_vector(counts)
n = len(counts)
lower = int(np.ceil(n * lower_quantile))
upper = int(n * upper_quantile)
sorted_counts = np.sort(counts)
return (upper - lower) / np.log(sorted_counts[upper] /
sorted_counts[lower])
def observed_otus(counts):
"""Calculate the number of distinct OTUs.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
int
Distinct OTU count.
"""
counts = _validate_counts_vector(counts)
return (counts != 0).sum()
def doubles(counts):
"""Calculate number of double occurrences (doubletons).
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
int
Doubleton count.
"""
counts = _validate_counts_vector(counts)
return (counts == 2).sum()
def mcintosh_e(counts):
r"""Calculate McIntosh's evenness measure E.
McIntosh evenness measure E is defined as:
.. math::
E = \frac{\sqrt{\sum{n_i^2}}}{\sqrt{((N-S+1)^2 + S -1}}
where :math:`n_i` is the number of individuals in the :math:`i^{\text{th}}`
OTU, :math:`N` is the total number of individuals, and :math:`S` is the
number of OTUs in the sample.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
McIntosh evenness measure E.
See Also
--------
mcintosh_d
Notes
-----
The implementation here is based on the description given in [1]_, **NOT**
the one in the SDR-IV online manual, which is wrong.
References
----------
.. [1] Heip & Engels (1974) Comparing Species Diversity and Evenness
Indices. p 560.
"""
counts = _validate_counts_vector(counts)
numerator = np.sqrt((counts * counts).sum())
n = counts.sum()
s = observed_otus(counts)
denominator = np.sqrt((n - s + 1) ** 2 + s - 1)
return numerator / denominator
def pielou_e(counts):
r"""Calculate Pielou's Evenness index J'.
Pielou's Evenness is defined as:
.. math::
J' = \frac{(H)}{\ln(S)}
where :math:`H` is the Shannon-Wiener entropy of counts and :math:`S` is
the number of OTUs in the sample.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Pielou's Evenness.
See Also
--------
shannon
heip_e
Notes
-----
The implementation here is based on the description in Wikipedia [1]_.
It was first proposed by E. C. Pielou [2]_ and is similar to Heip's
evenness [3]_.
References
----------
.. [1] https://en.wikipedia.org/wiki/Species_evenness
.. [2] Pielou, E. C., 1966. The measurement of diversity in different types
of biological collections. Journal of Theoretical Biology, 13, 131-44.
.. [3] Heip, C. 1974. A new index measuring evenness. J. Mar. Biol. Ass.
UK., 54, 555-557.
"""
counts = _validate_counts_vector(counts)
return shannon(counts, base=np.e) / np.log(observed_otus(counts))
def strong(counts):
r"""Calculate Strong's dominance index.
Strong's dominance index is defined as:
.. math::
D_w = max_i[(\frac{b_i}{N})-\frac{i}{S}]
where :math:`b_i` is the sequential cumulative totaling of the
:math:`i^{\text{th}}` OTU abundance values ranked from largest to smallest,
:math:`N` is the total number of individuals in the sample, and
:math:`S` is the number of OTUs in the sample. The expression in brackets
is computed for all OTUs, and :math:`max_i` denotes the maximum value in
brackets for any OTU.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Strong's dominance index (Dw).
Notes
-----
Strong's dominance index is defined in [1]_. The implementation here is
based on the description given in the SDR-IV online manual [2]_.
References
----------
.. [1] Strong, W. L., 2002 Assessing species abundance uneveness within and
between plant communities. Community Ecology, 3, 237-246.
.. [2] http://www.pisces-conservation.com/sdrhelp/index.html
"""
counts = _validate_counts_vector(counts)
n = counts.sum()
s = observed_otus(counts)
i = np.arange(1, len(counts) + 1)
sorted_sum = np.sort(counts)[::-1].cumsum()
return (sorted_sum / n - (i / s)).max()
def dominance(counts):
r"""Calculate dominance.
Dominance is defined as
.. math::
\sum{p_i^2}
where :math:`p_i` is the proportion of the entire community that OTU
:math:`i` represents.
Dominance can also be defined as 1 - Simpson's index. It ranges between
0 and 1.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Dominance.
See Also
--------
simpson
Notes
-----
The implementation here is based on the description given in [1]_.
References
----------
.. [1] http://folk.uio.no/ohammer/past/diversity.html
"""
counts = _validate_counts_vector(counts)
freqs = counts / counts.sum()
return (freqs * freqs).sum()
def heip_e(counts):
r"""Calculate Heip's evenness measure.
Heip's evenness is defined as:
.. math::
\frac{(e^H-1)}{(S-1)}
where :math:`H` is the Shannon-Wiener entropy of counts (using logarithm
base :math:`e`) and :math:`S` is the number of OTUs in the sample.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Heip's evenness measure.
See Also
--------
shannon
pielou_e
Notes
-----
The implementation here is based on the description in [1]_.
References
----------
.. [1] Heip, C. 1974. A new index measuring evenness. J. Mar. Biol. Ass.
UK., 54, 555-557.
"""
counts = _validate_counts_vector(counts)
return ((np.exp(shannon(counts, base=np.e)) - 1) /
(observed_otus(counts) - 1))
def simpson_e(counts):
r"""Calculate Simpson's evenness measure E.
Simpson's E is defined as
.. math::
E=\frac{1 / D}{S_{obs}}
where :math:`D` is dominance and :math:`S_{obs}` is the number of observed
OTUs.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Simpson's evenness measure E.
See Also
--------
dominance
enspie
simpson
Notes
-----
The implementation here is based on the description given in [1]_.
References
----------
.. [1] http://www.tiem.utk.edu/~gross/bioed/bealsmodules/simpsonDI.html
"""
counts = _validate_counts_vector(counts)
return enspie(counts) / observed_otus(counts)
def shannon(counts, base=2):
r"""Calculate Shannon entropy of counts, default in bits.
Shannon-Wiener diversity index is defined as:
.. math::
H = -\sum_{i=1}^s\left(p_i\log_2 p_i\right)
where :math:`s` is the number of OTUs and :math:`p_i` is the proportion of
the community represented by OTU :math:`i`.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
base : scalar, optional
Logarithm base to use in the calculations.
Returns
-------
double
Shannon diversity index H.
Notes
-----
The implementation here is based on the description given in the SDR-IV
online manual [1]_ except that the default logarithm base used here is 2
instead of :math:`e`.
References
----------
.. [1] http://www.pisces-conservation.com/sdrhelp/index.html
"""
counts = _validate_counts_vector(counts)
freqs = counts / counts.sum()
nonzero_freqs = freqs[freqs.nonzero()]
return -(nonzero_freqs * np.log(nonzero_freqs)).sum() / np.log(base)
def simpson(counts):
r"""Calculate Simpson's index.
Simpson's index is defined as ``1 - dominance``:
.. math::
1 - \sum{p_i^2}
where :math:`p_i` is the proportion of the community represented by OTU
:math:`i`.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Simpson's index.
See Also
--------
dominance
Notes
-----
The implementation here is ``1 - dominance`` as described in [1]_. Other
references (such as [2]_) define Simpson's index as ``1 / dominance``.
References
----------
.. [1] http://folk.uio.no/ohammer/past/diversity.html
.. [2] http://www.pisces-conservation.com/sdrhelp/index.html
"""
counts = _validate_counts_vector(counts)
return 1 - dominance(counts)
def enspie(counts):
r"""Calculate ENS_pie alpha diversity measure.
ENS_pie is equivalent to ``1 / dominance``:
.. math::
ENS_{pie} = \frac{1}{\sum_{i=1}^s{p_i^2}}
where :math:`s` is the number of OTUs and :math:`p_i` is the proportion of
the community represented by OTU :math:`i`.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
ENS_pie alpha diversity measure.
See Also
--------
dominance
Notes
-----
ENS_pie is defined in [1]_.
References
----------
.. [1] Chase and Knight (2013). "Scale-dependent effect sizes of ecological
drivers on biodiversity: why standardised sampling is not enough".
Ecology Letters, Volume 16, Issue Supplement s1, pgs 17-26.
"""
counts = _validate_counts_vector(counts)
return 1 / dominance(counts)
def berger_parker_d(counts):
r"""Calculate Berger-Parker dominance.
Berger-Parker dominance is defined as the fraction of the sample that
belongs to the most abundant OTU:
.. math::
d = \frac{N_{max}}{N}
where :math:`N_{max}` is defined as the number of individuals in the most
abundant OTU (or any of the most abundant OTUs in the case of ties), and
:math:`N` is defined as the total number of individuals in the sample.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Berger-Parker dominance.
Notes
-----
Berger-Parker dominance is defined in [1]_. The implementation here is
based on the description given in the SDR-IV online manual [2]_.
References
----------
.. [1] Berger & Parker (1970). SDR-IV online help.
.. [2] http://www.pisces-conservation.com/sdrhelp/index.html
"""
counts = _validate_counts_vector(counts)
return counts.max() / counts.sum()
def brillouin_d(counts):
r"""Calculate Brillouin index of alpha diversity.
This is calculated as follows:
.. math::
HB = \frac{\ln N!-\sum^s_{i=1}{\ln n_i!}}{N}
where :math:`N` is defined as the total number of individuals in the
sample, :math:`s` is the number of OTUs, and :math:`n_i` is defined as the
number of individuals in the :math:`i^{\text{th}}` OTU.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Brillouin index.
Notes
-----
The implementation here is based on the description given in the SDR-IV
online manual [1]_.
References
----------
.. [1] http://www.pisces-conservation.com/sdrhelp/index.html
"""
counts = _validate_counts_vector(counts)
nz = counts[counts.nonzero()]
n = nz.sum()
return (gammaln(n + 1) - gammaln(nz + 1).sum()) / n
def menhinick(counts):
r"""Calculate Menhinick's richness index.
Menhinick's richness index is defined as:
.. math::
D_{Mn} = \frac{S}{\sqrt{N}}
where :math:`S` is the number of OTUs and :math:`N` is the total number of
individuals in the sample.
Assumes square-root accumulation.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Menhinick's richness index.
Notes
-----
Based on the description in [1]_.
References
----------
.. [1] Magurran, A E 2004. Measuring biological diversity. Blackwell. pp.
76-77.
"""
counts = _validate_counts_vector(counts)
return observed_otus(counts) / np.sqrt(counts.sum())
def margalef(counts):
r"""Calculate Margalef's richness index.
Margalef's D is defined as:
.. math::
D = \frac{(S - 1)}{\ln N}
where :math:`S` is the number of OTUs and :math:`N` is the total number of
individuals in the sample.
Assumes log accumulation.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
Returns
-------
double
Margalef's richness index.
Notes
-----
Based on the description in [1]_.
References
----------
.. [1] Magurran, A E 2004. Measuring biological diversity. Blackwell. pp.
76-77.
"""
counts = _validate_counts_vector(counts)
return (observed_otus(counts) - 1) / np.log(counts.sum())
def faith_pd(counts, otu_ids, tree, validate=True):
""" Compute Faith's phylogenetic diversity metric (PD)
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
otu_ids: list, np.array
Vector of OTU ids corresponding to tip names in ``tree``. Must be the
same length as ``counts``.
tree: skbio.TreeNode
Tree relating the OTUs in otu_ids. The set of tip names in the tree can
be a superset of ``otu_ids``, but not a subset.
validate: bool, optional
If `False`, validation of the input won't be performed. This step can
be slow, so if validation is run elsewhere it can be disabled here.
However, invalid input data can lead to invalid results, so this step
should not be bypassed all together.
Returns
-------
float
The phylogenetic diversity (PD) of the samples.
Raises
------
ValueError
If ``counts`` and ``otu_ids`` are not equal in length.
MissingNodeError
If an OTU id is provided that does not correspond to a tip in the
tree.
Notes
-----
Faith's phylogenetic diversity, often referred to as PD, was originally
described in [1]_.
This implementation differs from that in PyCogent (and therefore QIIME
versions less than 2.0.0) by imposing a few additional restrictions on the
inputs. First, the input tree must be rooted. In PyCogent, if an unrooted
tree was provided that had a single trifurcating node (a newick convention
for unrooted trees) that node was considered the root of the tree. Next,
all OTU IDs must be tips in the tree. PyCogent would silently ignore OTU
IDs that were not present the tree. To reproduce Faith PD results from
PyCogent with scikit-bio, ensure that your PyCogent Faith PD calculations
are performed on a rooted tree and that all OTU IDs are present in the
tree.
References
----------
.. [1] Faith, D. P. Conservation evaluation and phylogenetic diversity.
Biol. Conserv. (1992).
"""
if validate:
counts = _validate_counts_vector(counts)
_validate_otu_ids_and_tree(counts, otu_ids, tree)
observed_otus = {o: c for o, c in zip(otu_ids, counts) if c >= 1}
observed_nodes = tree.observed_node_counts(observed_otus)
result = sum(o.length for o in observed_nodes if o.length is not None)
return result
def michaelis_menten_fit(counts, num_repeats=1, params_guess=None):
r"""Calculate Michaelis-Menten fit to rarefaction curve of observed OTUs.
The Michaelis-Menten equation is defined as:
.. math::
S=\frac{nS_{max}}{n+B}
where :math:`n` is the number of individuals and :math:`S` is the number of
OTUs. This function estimates the :math:`S_{max}` parameter.
The fit is made to datapoints for :math:`n=1,2,...,N`, where :math:`N` is
the total number of individuals (sum of abundances for all OTUs).
:math:`S` is the number of OTUs represented in a random sample of :math:`n`
individuals.
Parameters
----------
counts : 1-D array_like, int
Vector of counts.
num_repeats : int, optional
The number of times to perform rarefaction (subsampling without
replacement) at each value of :math:`n`.
params_guess : tuple, optional
Initial guess of :math:`S_{max}` and :math:`B`. If ``None``, default
guess for :math:`S_{max}` is :math:`S` (as :math:`S_{max}` should
be >= :math:`S`) and default guess for :math:`B` is ``round(N / 2)``.
Returns
-------
S_max : double
Estimate of the :math:`S_{max}` parameter in the Michaelis-Menten
equation.
See Also
--------
skbio.stats.subsample_counts
Notes
-----
There is some controversy about how to do the fitting. The ML model given
in [1]_ is based on the assumption that error is roughly proportional to
magnitude of observation, reasonable for enzyme kinetics but not reasonable
for rarefaction data. Here we just do a nonlinear curve fit for the
parameters using least-squares.
References
----------
.. [1] Raaijmakers, J. G. W. 1987 Statistical analysis of the
Michaelis-Menten equation. Biometrics 43, 793-803.
"""
counts = _validate_counts_vector(counts)
n_indiv = counts.sum()
if params_guess is None:
S_max_guess = observed_otus(counts)
B_guess = int(round(n_indiv / 2))
params_guess = (S_max_guess, B_guess)
# observed # of OTUs vs # of individuals sampled, S vs n
xvals = np.arange(1, n_indiv + 1)
ymtx = np.empty((num_repeats, len(xvals)), dtype=int)
for i in range(num_repeats):
ymtx[i] = np.asarray([observed_otus(subsample_counts(counts, n))
for n in xvals], dtype=int)
yvals = ymtx.mean(0)
# Vectors of actual vals y and number of individuals n.
def errfn(p, n, y):
return (((p[0] * n / (p[1] + n)) - y) ** 2).sum()
# Return S_max.
return fmin_powell(errfn, params_guess, ftol=1e-5, args=(xvals, yvals),
disp=False)[0]
