def create_fake_observation():
"""Create a subsample with defined property"""
# Create a subsample of a larger sample such that we can compute
# the expected probability of the unseen portion.
# This is used in the tests of lladser_pe and lladser_ci
counts = np.ones(1001, dtype='int64')
counts[0] = 9000
total = counts.sum()
fake_obs = subsample(counts, 1000)
exp_p = 1 - sum([x/total for (x, y) in zip(counts, fake_obs) if y > 0])
return fake_obs, exp_p
def create_fake_observation():
"""Create a subsample with defined property"""
# Create a subsample of a larger sample such that we can compute
# the expected probability of the unseen portion.
# This is used in the tests of lladser_pe and lladser_ci
counts = np.ones(1001, dtype='int64')
counts[0] = 9000
total = counts.sum()
fake_obs = subsample(counts, 1000)
exp_p = 1 - sum([x / total for (x, y) in zip(counts, fake_obs) if y > 0])
return fake_obs, exp_p
def michaelis_menten_fit(counts, num_repeats=1, params_guess=None):
"""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.math.subsample
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)
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, 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]