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,
return_b=False):
"""Michaelis-Menten fit to rarefaction curve of observed species
Note: there is some controversy about how to do the fitting. The ML model
givem by Raaijmakers 1987 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.
S = Smax*n/(B + n) . n: number of individuals, S: # of species
returns Smax
inputs:
num_repeats: will perform rarefaction (subsampling without replacement)
this many times at each value of n
params_guess: intial guess of Smax, B (None => default)
return_b: if True will return the estimate for Smax, B. Default is just Smax
the fit is made to datapoints where n = 1,2,...counts.sum(),
S = species represented in random sample of n individuals
"""
counts = asarray(counts)
if params_guess is None:
params_guess = array([100, 500])
# observed # of species vs # of individuals sampled, S vs n
xvals = arange(1, counts.sum() + 1)
ymtx = []
for i in range(num_repeats):
ymtx.append(array([observed_species(subsample(counts, n))
for n in xvals]))
ymtx = asarray(ymtx)
yvals = ymtx.mean(0)
# fit to obs_sp = max_sp * num_idiv / (num_indiv + B)
# return max_sp
def fitfn(p, n): # works with vectors of n, returns vector of S
return p[0] * n / (p[1] + n)
def errfn(p, n, y): # vectors of actual vals y and number of individuals n
return ((fitfn(p, n) - y) ** 2).sum()
p1 = fmin_powell(errfn, params_guess, args=(xvals, yvals), disp=0)
if return_b:
return p1
else:
return p1[0] # return only S_max, not the K_m (B) param
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
x = [9000]
x.extend([1] * 1000)
counts = np.array(x)
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
x = [9000]
x.extend([1] * 1000)
counts = np.array(x)
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]
def y(self):
try: return [self.div_fn(subsample(self.parent.sample.counts, k)) for k in self.x]
except ValueError: return [0 for k in self.x]
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]
