Module dadi.Godambe
Parameter uncertainties and likelihood ratio tests using Godambe information.
Expand source code
"""
Parameter uncertainties and likelihood ratio tests using Godambe information.
"""
import numpy
from dadi import Inference
from dadi.Spectrum_mod import Spectrum
two_pt_deriv_test = False
def hessian_elem(func, f0, p0, ii, jj, eps, args=(), one_sided=None):
"""
Calculate element [ii][jj] of the Hessian matrix, a matrix
of partial second derivatives w.r.t. to parameters ii and jj
func: Model function
f0: Evaluation of func at p0
p0: Parameters for func
eps: List of absolute step sizes to use for each parameter when taking
finite differences.
args: Additional arguments to func
one_sided: Optionally, pass in a sequence of length p0 that determines
whether a one-sided derivative will be used for each parameter.
"""
# Note that we need to specify dtype=float, to avoid this being an integer
# array which will silently fail when adding fractional eps.
if one_sided is None:
one_sided = [False]*len(p0)
pwork = numpy.array(p0, copy=True, dtype=float)
if ii == jj:
if pwork[ii] != 0 and not one_sided[ii]:
pwork[ii] = p0[ii] + eps[ii]
fp = func(pwork, *args)
pwork[ii] = p0[ii] - eps[ii]
fm = func(pwork, *args)
element = (fp - 2*f0 + fm)/eps[ii]**2
else:
pwork[ii] = p0[ii] + 2*eps[ii]
fpp = func(pwork, *args)
pwork[ii] = p0[ii] + eps[ii]
fp = func(pwork, *args)
element = (fpp - 2*fp + f0)/eps[ii]**2
else:
if pwork[ii] != 0 and pwork[jj] != 0 and not one_sided[ii] and not one_sided[jj]:
# f(xi + hi, xj + h)
pwork[ii] = p0[ii] + eps[ii]
pwork[jj] = p0[jj] + eps[jj]
fpp = func(pwork, *args)
# f(xi + hi, xj - hj)
pwork[ii] = p0[ii] + eps[ii]
pwork[jj] = p0[jj] - eps[jj]
fpm = func(pwork, *args)
# f(xi - hi, xj + hj)
pwork[ii] = p0[ii] - eps[ii]
pwork[jj] = p0[jj] + eps[jj]
fmp = func(pwork, *args)
# f(xi - hi, xj - hj)
pwork[ii] = p0[ii] - eps[ii]
pwork[jj] = p0[jj] - eps[jj]
fmm = func(pwork, *args)
element = (fpp - fpm - fmp + fmm)/(4 * eps[ii]*eps[jj])
else:
# f(xi + hi, xj + h)
pwork[ii] = p0[ii] + eps[ii]
pwork[jj] = p0[jj] + eps[jj]
fpp = func(pwork, *args)
# f(xi + hi, xj)
pwork[ii] = p0[ii] + eps[ii]
pwork[jj] = p0[jj]
fpm = func(pwork, *args)
# f(xi, xj + hj)
pwork[ii] = p0[ii]
pwork[jj] = p0[jj] + eps[jj]
fmp = func(pwork, *args)
element = (fpp - fpm - fmp + f0)/(eps[ii]*eps[jj])
return element
def get_hess(func, p0, eps, args=()):
"""
Calculate Hessian matrix of partial second derivatives.
Hij = dfunc/(dp_i dp_j)
func: Model function
p0: Parameter values to take derivative around
eps: Fractional stepsize to use when taking finite-difference derivatives
Note that if eps*param is < 1e-6, then the step size for that parameter
will simply be eps, to avoid numerical issues with small parameter
perturbations.
args: Additional arguments to func
"""
# Calculate step sizes for finite-differences.
eps_in = eps
eps = numpy.empty([len(p0)])
one_sided = [False]*len(p0)
for i, pval in enumerate(p0):
if pval != 0:
# Account for floating point arithmetic issues
if pval*eps_in < 1e-6:
eps[i] = eps_in
one_sided[i] = True
else:
eps[i] = eps_in*pval
else:
# Account for parameters equal to zero
eps[i] = eps_in
f0 = func(p0, *args)
hess = numpy.empty((len(p0), len(p0)))
for ii in range(len(p0)):
for jj in range(ii, len(p0)):
hess[ii][jj] = hessian_elem(func, f0, p0, ii, jj, eps, args=args, one_sided=one_sided)
hess[jj][ii] = hess[ii][jj]
return hess
def get_grad(func, p0, eps, args=()):
"""
Calculate gradient vector
func: Model function
p0: Parameters for func
eps: Fractional stepsize to use when taking finite-difference derivatives
Note that if eps*param is < 1e-6, then the step size for that parameter
will simply be eps, to avoid numerical issues with small parameter
perturbations.
args: Additional arguments to func
"""
# Calculate step sizes for finite-differences.
eps_in = eps
eps = numpy.empty([len(p0)])
one_sided = [False]*len(p0)
for i, pval in enumerate(p0):
if pval != 0:
# Account for floating point arithmetic issues
if pval*eps_in < 1e-6:
eps[i] = eps_in
one_sided[i] = True
else:
eps[i] = eps_in*pval
else:
# Account for parameters equal to zero
eps[i] = eps_in
grad = numpy.empty([len(p0), 1])
for ii in range(len(p0)):
pwork = numpy.array(p0, copy=True, dtype=float)
if p0[ii] != 0 and not one_sided[ii]:
pwork[ii] = p0[ii] + eps[ii]
fp = func(pwork, *args)
pwork[ii] = p0[ii] - eps[ii]
fm = func(pwork, *args)
grad[ii] = (fp - fm)/(2*eps[ii])
else:
# Do one-sided finite-difference
pwork[ii] = p0[ii] + eps[ii]
fp = func(pwork, *args)
pwork[ii] = p0[ii]
fm = func(pwork, *args)
grad[ii] = (fp - fm)/eps[ii]
if two_pt_deriv_test:
pwork[ii] = p0[ii] + 2*eps[ii]
fpp = func(pwork, *args)
grad[ii] = (4*fp - fpp - 3*fm)/(2*eps[ii])
return grad
cache = {}
def get_godambe(func_ex, grid_pts, all_boot, p0, data, eps, log=False,
just_hess=False, boot_theta_adjusts=[]):
"""
Godambe information and Hessian matrices
func_ex: Model function
grid_pts: Number of grid points to evaluate the model function
all_boot: List of bootstrap frequency spectra
p0: Best-fit parameters for func_ex.
data: Original data frequency spectrum
eps: Fractional stepsize to use when taking finite-difference derivatives
Note that if eps*param is < 1e-6, then the step size for that parameter
will simply be eps, to avoid numerical issues with small parameter
perturbations.
log: If True, calculate derivatives in terms of log-parameters
just_hess: If True, only evaluate and return the Hessian matrix
boot_theta_adjusts: Factors by which to adjust theta for each bootstrap
sample, relative to full data theta.
"""
ns = data.sample_sizes
if not boot_theta_adjusts:
boot_theta_adjusts = numpy.ones(len(all_boot))
# Cache evaluations of the frequency spectrum inside our hessian/J
# evaluation function
def func(params, data, theta_adjust=1):
key = (func_ex.__hash__(), tuple(params), tuple(ns), tuple(grid_pts))
if key not in cache:
cache[key] = func_ex(params, ns, grid_pts)
# theta_adjust deals with bootstraps that need different thetas
fs = theta_adjust*cache[key]
return Inference.ll(fs, data)
def log_func(logparams, data, theta_adjust=1):
return func(numpy.exp(logparams), data, theta_adjust)
# First calculate the observed hessian.
# theta_adjust defaults to 1.
if not log:
hess = -get_hess(func, p0, eps, args=[data])
else:
hess = -get_hess(log_func, numpy.log(p0), eps, args=[data])
if just_hess:
return hess
# Now the expectation of J over the bootstrap data
J = numpy.zeros((len(p0), len(p0)))
# cU is a column vector
cU = numpy.zeros((len(p0),1))
for ii, (boot,theta_adjust) in enumerate(zip(all_boot, boot_theta_adjusts)):
boot = Spectrum(boot)
if not log:
grad_temp = get_grad(func, p0, eps, args=[boot, theta_adjust])
else:
grad_temp = get_grad(log_func, numpy.log(p0), eps,
args=[boot, theta_adjust])
J_temp = numpy.outer(grad_temp, grad_temp)
J = J + J_temp
cU = cU + grad_temp
J = J/len(all_boot)
cU = cU/len(all_boot)
# G = H*J^-1*H
J_inv = numpy.linalg.inv(J)
godambe = numpy.dot(numpy.dot(hess, J_inv), hess)
return godambe, hess, J, cU
def GIM_uncert(func_ex, grid_pts, all_boot, p0, data, log=False,
multinom=True, eps=0.01, return_GIM=False,
boot_theta_adjusts=None):
"""
Parameter uncertainties from Godambe Information Matrix (GIM)
Returns standard deviations of parameter values.
func_ex: Model function
all_boot: List of bootstrap frequency spectra
p0: Best-fit parameters for func_ex
data: Original data frequency spectrum
eps: Fractional stepsize to use when taking finite-difference derivatives.
Note that if eps*param is < 1e-6, then the step size for that parameter
will simply be eps, to avoid numerical issues with small parameter
perturbations.
log: If True, assume log-normal distribution of parameters. Returned values
are then the standard deviations of the *logs* of the parameter values,
which can be interpreted as relative parameter uncertainties.
multinom: If True, assume model is defined without an explicit parameter for
theta. Because uncertainty in theta must be accounted for to get
correct uncertainties for other parameters, this function will
automatically consider theta if multinom=True. In that case, the
final entry of the returned uncertainties will correspond to
theta.
return_GIM: If true, also return the full GIM.
boot_theta_adjusts: Optionally, a sequence of *relative* values of theta
(compared to original data) to assume for bootstrap
data sets. Only valid when multinom=False.
"""
if multinom:
if boot_theta_adjusts:
raise ValueError('boot_thetas option can only be used with '
'multinom=False')
func_multi = func_ex
model = func_multi(p0, data.sample_sizes, grid_pts)
theta_opt = Inference.optimal_sfs_scaling(model, data)
p0 = list(p0) + [theta_opt]
func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts)
GIM, H, J, cU = get_godambe(func_ex, grid_pts, all_boot, p0, data, eps, log,
boot_theta_adjusts=boot_theta_adjusts)
uncerts = numpy.sqrt(numpy.diag(numpy.linalg.inv(GIM)))
if not return_GIM:
return uncerts
else:
return uncerts, GIM, H
def FIM_uncert(func_ex, grid_pts, p0, data, log=False, multinom=True, eps=0.01, return_FIM=False):
"""
Parameter uncertainties from Fisher Information Matrix
Returns standard deviations of parameter values.
func_ex: Model function
all_boot: List of bootstrap frequency spectra
p0: Best-fit parameters for func_ex
data: Original data frequency spectrum
eps: Fractional stepsize to use when taking finite-difference derivatives.
Note that if eps*param is < 1e-6, then the step size for that parameter
will simply be eps, to avoid numerical issues with small parameter
perturbations.
log: If True, assume log-normal distribution of parameters. Returned values
are then the standard deviations of the *logs* of the parameter values,
which can be interpreted as relative parameter uncertainties.
multinom: If True, assume model is defined without an explicit parameter for
theta. Because uncertainty in theta must be accounted for to get
correct uncertainties for other parameters, this function will
automatically consider theta if multinom=True. In that case, the
final entry of the returned uncertainties will correspond to
theta.
"""
if multinom:
func_multi = func_ex
model = func_multi(p0, data.sample_sizes, grid_pts)
theta_opt = Inference.optimal_sfs_scaling(model, data)
p0 = list(p0) + [theta_opt]
func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts)
H = get_godambe(func_ex, grid_pts, [], p0, data, eps, log, just_hess=True)
uncerts = numpy.sqrt(numpy.diag(numpy.linalg.inv(H)))
if not return_FIM:
return uncerts
else:
return uncerts, H
def LRT_adjust(func_ex, grid_pts, all_boot, p0, data, nested_indices,
multinom=True, eps=0.01, boot_theta_adjusts=None):
"""
First-order moment matching adjustment factor for likelihood ratio test
func_ex: Model function for complex model
grid_pts: Grid points at which to evaluate func_ex
all_boot: List of bootstrap frequency spectra
p0: Best-fit parameters for the simple model, with nested parameter
explicity defined. Although equal to values for simple model, should
be in a list form that can be taken in by the complex model you'd like
to evaluate.
data: Original data frequency spectrum
nested_indices: List of positions of nested parameters in complex model
parameter list
multinom: If True, assume model is defined without an explicit parameter for
theta. Because uncertainty in theta must be accounted for to get
correct uncertainties for other parameters, this function will
automatically consider theta if multinom=True.
eps: Fractional stepsize to use when taking finite-difference derivatives
Note that if eps*param is < 1e-6, then the step size for that parameter
will simply be eps, to avoid numerical issues with small parameter
perturbations.
boot_theta_adjusts: Optionally, a sequence of *relative* values of theta
(compared to original data) to assume for bootstrap
data sets. Only valid when multinom=False.
"""
if multinom and boot_theta_adjusts:
raise ValueError('Cannot use boot_theta_adjusts with multinom=True.')
if multinom:
func_multi = func_ex
model = func_multi(p0, data.sample_sizes, grid_pts)
theta_opt = Inference.optimal_sfs_scaling(model, data)
p0 = list(p0) + [theta_opt]
func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts)
# We only need to take derivatives with respect to the parameters in the
# complex model that have been set to specified values in the simple model
def diff_func(diff_params, ns, grid_pts):
# diff_params argument is only the nested parameters. All the rest
# should come from p0
full_params = numpy.array(p0, copy=True, dtype=float)
# Use numpy indexing to set relevant parameters
full_params[nested_indices] = diff_params
return func_ex(full_params, ns, grid_pts)
p_nested = numpy.asarray(p0)[nested_indices]
GIM, H, J, cU = get_godambe(diff_func, grid_pts, all_boot, p_nested, data,
eps, log=False, boot_theta_adjusts=boot_theta_adjusts)
adjust = len(nested_indices)/numpy.trace(numpy.dot(J, numpy.linalg.inv(H)))
return adjust
def sum_chi2_ppf(x, weights=(0,1)):
"""
Percent point function (inverse of cdf) of weighted sum of chi^2
distributions.
x: Value(s) at which to evaluate ppf
weights: Weights of chi^2 distributions, beginning with zero d.o.f.
For example, weights=(0,1) is the normal chi^2 distribution with 1
d.o.f. For single parameters on the boundary, the correct
distribution for the LRT is 0.5*chi^2_0 + 0.5*chi^2_1, which would
be weights=(0.5,0.5).
"""
import scipy.stats.distributions as ssd
# Ensure that weights are valid
if abs(numpy.sum(weights) - 1) > 1e-6:
raise ValueError('Weights must sum to 1.')
# A little clunky, but we want to handle x = 0.5, and x = [2, 3, 4]
# correctly. So if x is a scalar, we record that fact so we can return a
# scalar on output.
if numpy.isscalar(x):
scalar_input = True
# Convert x into an array, so we can index it easily.
x = numpy.atleast_1d(x)
# Calculate total cdf of all chi^2 dists with dof > 1.
# (ssd.chi2.cdf(x,0) is always nan, so we avoid that.)
cdf = numpy.sum([w*ssd.chi2.cdf(x, d+1) for (d, w)
in enumerate(weights[1:])], axis=0)
# Add in contribution from 0 d.o.f.
cdf[x > 0] += weights[0]
# Convert to ppf
ppf = 1-cdf
if scalar_input:
return ppf[0]
else:
return ppf
def Wald_stat(func_ex, grid_pts, all_boot, p0, data, nested_indices,
full_params, multinom=True, eps=0.01, adj_and_org=False):
# XXX: Implement boot_theta_adjusts
"""
Calculate test stastic from wald test
func_ex: Model function for complex model
all_boot: List of bootstrap frequency spectra
p0: Best-fit parameters for the simple model, with nested parameter
explicity defined. Although equal to values for simple model, should
be in a list form that can be taken in by the complex model you'd like
to evaluate.
data: Original data frequency spectrum
nested_indices: List of positions of nested parameters in complex model
parameter list
full_params: Parameter values for parameters found only in complex model,
Can either be array with just values found only in the compelx
model, or entire list of parameters from complex model.
multinom: If True, assume model is defined without an explicit parameter for
theta. Because uncertainty in theta must be accounted for to get
correct uncertainties for other parameters, this function will
automatically consider theta if multinom=True. In that case, the
final entry of the returned uncertainties will correspond to
theta.
eps: Fractional stepsize to use when taking finite-difference derivatives
Note that if eps*param is < 1e-6, then the step size for that parameter
will simply be eps, to avoid numerical issues with small parameter
perturbations.
adj_and_org: If False, return only adjusted Wald statistic. If True, also
return unadjusted statistic as second return value.
"""
if multinom:
func_multi = func_ex
model = func_multi(p0, data.sample_sizes, grid_pts)
theta_opt = Inference.optimal_sfs_scaling(model, data)
# Also need to extend full_params
if len(full_params) == len(p0):
full_params = numpy.concatenate((full_params, [theta_opt]))
p0 = list(p0) + [theta_opt]
func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts)
# We only need to take derivatives with respect to the parameters in the
# complex model that have been set to specified values in the simple model
def diff_func(diff_params, ns, grid_pts):
# diff_params argument is only the nested parameters. All the rest
# should come from p0
full_params = numpy.array(p0, copy=True, dtype=float)
# Use numpy indexing to set relevant parameters
full_params[nested_indices] = diff_params
return func_ex(full_params, ns, grid_pts)
# Reduce full params list to be same length as nested indices
if len(full_params) == len(p0):
full_params = numpy.asarray(full_params)[nested_indices]
if len(full_params) != len(nested_indices):
raise KeyError('Full parameters not equal in length to p0 or nested '
'indices')
p_nested = numpy.asarray(p0)[nested_indices]
GIM, H, J, cU = get_godambe(diff_func, grid_pts, all_boot, p_nested, data,
eps, log=False)
param_diff = full_params-p_nested
wald_adj = numpy.dot(numpy.dot(numpy.transpose(param_diff),GIM), param_diff)
wald_org = numpy.dot(numpy.dot(numpy.transpose(param_diff),H),param_diff)
if adj_and_org:
return wald_adj, wald_org
return wald_adj
def score_stat(func_ex, grid_pts, all_boot, p0, data, nested_indices,
multinom=True, eps=0.01, adj_and_org=False):
# XXX: Implement boot_theta_adjusts
"""
Calculate test stastic from score test
func_ex: Model function for complex model
grid_pts: Grid points to evaluate model function
all_boot: List of bootstrap frequency spectra
p0: Best-fit parameters for the simple model, with nested parameter
explicity defined. Although equal to values for simple model, should
be in a list form that can be taken in by the complex model you'd like
to evaluate.
data: Original data frequency spectrum
nested_indices: List of positions of nested parameters in complex model
parameter list
eps: Fractional stepsize to use when taking finite-difference derivatives
Note that if eps*param is < 1e-6, then the step size for that parameter
will simply be eps, to avoid numerical issues with small parameter
perturbations.
multinom: If True, assume model is defined without an explicit parameter for
theta. Because uncertainty in theta must be accounted for to get
correct uncertainties for other parameters, this function will
automatically consider theta if multinom=True.
adj_and_org: If False, return only adjusted score statistic. If True, also
return unadjusted statistic as second return value.
"""
if multinom:
func_multi = func_ex
model = func_multi(p0, data.sample_sizes, grid_pts)
theta_opt = Inference.optimal_sfs_scaling(model, data)
p0 = list(p0) + [theta_opt]
func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts)
# We only need to take derivatives with respect to the parameters in the
# complex model that have been set to specified values in the simple model
def diff_func(diff_params, ns, grid_pts):
# diff_params argument is only the nested parameters. All the rest
# should come from p0
full_params = numpy.array(p0, copy=True, dtype=float)
# Use numpy indexing to set relevant parameters
full_params[nested_indices] = diff_params
return func_ex(full_params, ns, grid_pts)
p_nested = numpy.asarray(p0)[nested_indices]
GIM, H, J, cU = get_godambe(diff_func, grid_pts, all_boot, p_nested, data,
eps, log=False)
score_org = numpy.dot(numpy.dot(numpy.transpose(cU),
numpy.linalg.inv(H)),cU)[0,0]
score_adj = numpy.dot(numpy.dot(numpy.transpose(cU),
numpy.linalg.inv(J)),cU)[0,0]
if adj_and_org:
return score_adj, score_org
return score_adjFunctions
def FIM_uncert(func_ex, grid_pts, p0, data, log=False, multinom=True, eps=0.01, return_FIM=False)-
Parameter uncertainties from Fisher Information Matrix
Returns standard deviations of parameter values.
func_ex: Model function all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for func_ex data: Original data frequency spectrum eps: Fractional stepsize to use when taking finite-difference derivatives. Note that if epsparam is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. log: If True, assume log-normal distribution of parameters. Returned values are then the standard deviations of the logs* of the parameter values, which can be interpreted as relative parameter uncertainties. multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. In that case, the final entry of the returned uncertainties will correspond to theta.
Expand source code
def FIM_uncert(func_ex, grid_pts, p0, data, log=False, multinom=True, eps=0.01, return_FIM=False): """ Parameter uncertainties from Fisher Information Matrix Returns standard deviations of parameter values. func_ex: Model function all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for func_ex data: Original data frequency spectrum eps: Fractional stepsize to use when taking finite-difference derivatives. Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. log: If True, assume log-normal distribution of parameters. Returned values are then the standard deviations of the *logs* of the parameter values, which can be interpreted as relative parameter uncertainties. multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. In that case, the final entry of the returned uncertainties will correspond to theta. """ if multinom: func_multi = func_ex model = func_multi(p0, data.sample_sizes, grid_pts) theta_opt = Inference.optimal_sfs_scaling(model, data) p0 = list(p0) + [theta_opt] func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts) H = get_godambe(func_ex, grid_pts, [], p0, data, eps, log, just_hess=True) uncerts = numpy.sqrt(numpy.diag(numpy.linalg.inv(H))) if not return_FIM: return uncerts else: return uncerts, H def GIM_uncert(func_ex, grid_pts, all_boot, p0, data, log=False, multinom=True, eps=0.01, return_GIM=False, boot_theta_adjusts=None)-
Parameter uncertainties from Godambe Information Matrix (GIM)
Returns standard deviations of parameter values.
func_ex: Model function all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for func_ex data: Original data frequency spectrum eps: Fractional stepsize to use when taking finite-difference derivatives. Note that if epsparam is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. log: If True, assume log-normal distribution of parameters. Returned values are then the standard deviations of the logs of the parameter values, which can be interpreted as relative parameter uncertainties. multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. In that case, the final entry of the returned uncertainties will correspond to theta. return_GIM: If true, also return the full GIM. boot_theta_adjusts: Optionally, a sequence of relative* values of theta (compared to original data) to assume for bootstrap data sets. Only valid when multinom=False.
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def GIM_uncert(func_ex, grid_pts, all_boot, p0, data, log=False, multinom=True, eps=0.01, return_GIM=False, boot_theta_adjusts=None): """ Parameter uncertainties from Godambe Information Matrix (GIM) Returns standard deviations of parameter values. func_ex: Model function all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for func_ex data: Original data frequency spectrum eps: Fractional stepsize to use when taking finite-difference derivatives. Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. log: If True, assume log-normal distribution of parameters. Returned values are then the standard deviations of the *logs* of the parameter values, which can be interpreted as relative parameter uncertainties. multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. In that case, the final entry of the returned uncertainties will correspond to theta. return_GIM: If true, also return the full GIM. boot_theta_adjusts: Optionally, a sequence of *relative* values of theta (compared to original data) to assume for bootstrap data sets. Only valid when multinom=False. """ if multinom: if boot_theta_adjusts: raise ValueError('boot_thetas option can only be used with ' 'multinom=False') func_multi = func_ex model = func_multi(p0, data.sample_sizes, grid_pts) theta_opt = Inference.optimal_sfs_scaling(model, data) p0 = list(p0) + [theta_opt] func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts) GIM, H, J, cU = get_godambe(func_ex, grid_pts, all_boot, p0, data, eps, log, boot_theta_adjusts=boot_theta_adjusts) uncerts = numpy.sqrt(numpy.diag(numpy.linalg.inv(GIM))) if not return_GIM: return uncerts else: return uncerts, GIM, H def LRT_adjust(func_ex, grid_pts, all_boot, p0, data, nested_indices, multinom=True, eps=0.01, boot_theta_adjusts=None)-
First-order moment matching adjustment factor for likelihood ratio test
func_ex: Model function for complex model grid_pts: Grid points at which to evaluate func_ex all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for the simple model, with nested parameter explicity defined. Although equal to values for simple model, should be in a list form that can be taken in by the complex model you'd like to evaluate. data: Original data frequency spectrum nested_indices: List of positions of nested parameters in complex model parameter list multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. eps: Fractional stepsize to use when taking finite-difference derivatives Note that if epsparam is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. boot_theta_adjusts: Optionally, a sequence of relative* values of theta (compared to original data) to assume for bootstrap data sets. Only valid when multinom=False.
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def LRT_adjust(func_ex, grid_pts, all_boot, p0, data, nested_indices, multinom=True, eps=0.01, boot_theta_adjusts=None): """ First-order moment matching adjustment factor for likelihood ratio test func_ex: Model function for complex model grid_pts: Grid points at which to evaluate func_ex all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for the simple model, with nested parameter explicity defined. Although equal to values for simple model, should be in a list form that can be taken in by the complex model you'd like to evaluate. data: Original data frequency spectrum nested_indices: List of positions of nested parameters in complex model parameter list multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. boot_theta_adjusts: Optionally, a sequence of *relative* values of theta (compared to original data) to assume for bootstrap data sets. Only valid when multinom=False. """ if multinom and boot_theta_adjusts: raise ValueError('Cannot use boot_theta_adjusts with multinom=True.') if multinom: func_multi = func_ex model = func_multi(p0, data.sample_sizes, grid_pts) theta_opt = Inference.optimal_sfs_scaling(model, data) p0 = list(p0) + [theta_opt] func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts) # We only need to take derivatives with respect to the parameters in the # complex model that have been set to specified values in the simple model def diff_func(diff_params, ns, grid_pts): # diff_params argument is only the nested parameters. All the rest # should come from p0 full_params = numpy.array(p0, copy=True, dtype=float) # Use numpy indexing to set relevant parameters full_params[nested_indices] = diff_params return func_ex(full_params, ns, grid_pts) p_nested = numpy.asarray(p0)[nested_indices] GIM, H, J, cU = get_godambe(diff_func, grid_pts, all_boot, p_nested, data, eps, log=False, boot_theta_adjusts=boot_theta_adjusts) adjust = len(nested_indices)/numpy.trace(numpy.dot(J, numpy.linalg.inv(H))) return adjust def Wald_stat(func_ex, grid_pts, all_boot, p0, data, nested_indices, full_params, multinom=True, eps=0.01, adj_and_org=False)-
Calculate test stastic from wald test
func_ex: Model function for complex model all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for the simple model, with nested parameter explicity defined. Although equal to values for simple model, should be in a list form that can be taken in by the complex model you'd like to evaluate. data: Original data frequency spectrum nested_indices: List of positions of nested parameters in complex model parameter list full_params: Parameter values for parameters found only in complex model, Can either be array with just values found only in the compelx model, or entire list of parameters from complex model. multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. In that case, the final entry of the returned uncertainties will correspond to theta. eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. adj_and_org: If False, return only adjusted Wald statistic. If True, also return unadjusted statistic as second return value.
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def Wald_stat(func_ex, grid_pts, all_boot, p0, data, nested_indices, full_params, multinom=True, eps=0.01, adj_and_org=False): # XXX: Implement boot_theta_adjusts """ Calculate test stastic from wald test func_ex: Model function for complex model all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for the simple model, with nested parameter explicity defined. Although equal to values for simple model, should be in a list form that can be taken in by the complex model you'd like to evaluate. data: Original data frequency spectrum nested_indices: List of positions of nested parameters in complex model parameter list full_params: Parameter values for parameters found only in complex model, Can either be array with just values found only in the compelx model, or entire list of parameters from complex model. multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. In that case, the final entry of the returned uncertainties will correspond to theta. eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. adj_and_org: If False, return only adjusted Wald statistic. If True, also return unadjusted statistic as second return value. """ if multinom: func_multi = func_ex model = func_multi(p0, data.sample_sizes, grid_pts) theta_opt = Inference.optimal_sfs_scaling(model, data) # Also need to extend full_params if len(full_params) == len(p0): full_params = numpy.concatenate((full_params, [theta_opt])) p0 = list(p0) + [theta_opt] func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts) # We only need to take derivatives with respect to the parameters in the # complex model that have been set to specified values in the simple model def diff_func(diff_params, ns, grid_pts): # diff_params argument is only the nested parameters. All the rest # should come from p0 full_params = numpy.array(p0, copy=True, dtype=float) # Use numpy indexing to set relevant parameters full_params[nested_indices] = diff_params return func_ex(full_params, ns, grid_pts) # Reduce full params list to be same length as nested indices if len(full_params) == len(p0): full_params = numpy.asarray(full_params)[nested_indices] if len(full_params) != len(nested_indices): raise KeyError('Full parameters not equal in length to p0 or nested ' 'indices') p_nested = numpy.asarray(p0)[nested_indices] GIM, H, J, cU = get_godambe(diff_func, grid_pts, all_boot, p_nested, data, eps, log=False) param_diff = full_params-p_nested wald_adj = numpy.dot(numpy.dot(numpy.transpose(param_diff),GIM), param_diff) wald_org = numpy.dot(numpy.dot(numpy.transpose(param_diff),H),param_diff) if adj_and_org: return wald_adj, wald_org return wald_adj def get_godambe(func_ex, grid_pts, all_boot, p0, data, eps, log=False, just_hess=False, boot_theta_adjusts=[])-
Godambe information and Hessian matrices
func_ex: Model function grid_pts: Number of grid points to evaluate the model function all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for func_ex. data: Original data frequency spectrum eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. log: If True, calculate derivatives in terms of log-parameters just_hess: If True, only evaluate and return the Hessian matrix boot_theta_adjusts: Factors by which to adjust theta for each bootstrap sample, relative to full data theta.
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def get_godambe(func_ex, grid_pts, all_boot, p0, data, eps, log=False, just_hess=False, boot_theta_adjusts=[]): """ Godambe information and Hessian matrices func_ex: Model function grid_pts: Number of grid points to evaluate the model function all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for func_ex. data: Original data frequency spectrum eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. log: If True, calculate derivatives in terms of log-parameters just_hess: If True, only evaluate and return the Hessian matrix boot_theta_adjusts: Factors by which to adjust theta for each bootstrap sample, relative to full data theta. """ ns = data.sample_sizes if not boot_theta_adjusts: boot_theta_adjusts = numpy.ones(len(all_boot)) # Cache evaluations of the frequency spectrum inside our hessian/J # evaluation function def func(params, data, theta_adjust=1): key = (func_ex.__hash__(), tuple(params), tuple(ns), tuple(grid_pts)) if key not in cache: cache[key] = func_ex(params, ns, grid_pts) # theta_adjust deals with bootstraps that need different thetas fs = theta_adjust*cache[key] return Inference.ll(fs, data) def log_func(logparams, data, theta_adjust=1): return func(numpy.exp(logparams), data, theta_adjust) # First calculate the observed hessian. # theta_adjust defaults to 1. if not log: hess = -get_hess(func, p0, eps, args=[data]) else: hess = -get_hess(log_func, numpy.log(p0), eps, args=[data]) if just_hess: return hess # Now the expectation of J over the bootstrap data J = numpy.zeros((len(p0), len(p0))) # cU is a column vector cU = numpy.zeros((len(p0),1)) for ii, (boot,theta_adjust) in enumerate(zip(all_boot, boot_theta_adjusts)): boot = Spectrum(boot) if not log: grad_temp = get_grad(func, p0, eps, args=[boot, theta_adjust]) else: grad_temp = get_grad(log_func, numpy.log(p0), eps, args=[boot, theta_adjust]) J_temp = numpy.outer(grad_temp, grad_temp) J = J + J_temp cU = cU + grad_temp J = J/len(all_boot) cU = cU/len(all_boot) # G = H*J^-1*H J_inv = numpy.linalg.inv(J) godambe = numpy.dot(numpy.dot(hess, J_inv), hess) return godambe, hess, J, cU def get_grad(func, p0, eps, args=())-
Calculate gradient vector
func: Model function p0: Parameters for func eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. args: Additional arguments to func
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def get_grad(func, p0, eps, args=()): """ Calculate gradient vector func: Model function p0: Parameters for func eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. args: Additional arguments to func """ # Calculate step sizes for finite-differences. eps_in = eps eps = numpy.empty([len(p0)]) one_sided = [False]*len(p0) for i, pval in enumerate(p0): if pval != 0: # Account for floating point arithmetic issues if pval*eps_in < 1e-6: eps[i] = eps_in one_sided[i] = True else: eps[i] = eps_in*pval else: # Account for parameters equal to zero eps[i] = eps_in grad = numpy.empty([len(p0), 1]) for ii in range(len(p0)): pwork = numpy.array(p0, copy=True, dtype=float) if p0[ii] != 0 and not one_sided[ii]: pwork[ii] = p0[ii] + eps[ii] fp = func(pwork, *args) pwork[ii] = p0[ii] - eps[ii] fm = func(pwork, *args) grad[ii] = (fp - fm)/(2*eps[ii]) else: # Do one-sided finite-difference pwork[ii] = p0[ii] + eps[ii] fp = func(pwork, *args) pwork[ii] = p0[ii] fm = func(pwork, *args) grad[ii] = (fp - fm)/eps[ii] if two_pt_deriv_test: pwork[ii] = p0[ii] + 2*eps[ii] fpp = func(pwork, *args) grad[ii] = (4*fp - fpp - 3*fm)/(2*eps[ii]) return grad def get_hess(func, p0, eps, args=())-
Calculate Hessian matrix of partial second derivatives. Hij = dfunc/(dp_i dp_j)
func: Model function p0: Parameter values to take derivative around eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. args: Additional arguments to func
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def get_hess(func, p0, eps, args=()): """ Calculate Hessian matrix of partial second derivatives. Hij = dfunc/(dp_i dp_j) func: Model function p0: Parameter values to take derivative around eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. args: Additional arguments to func """ # Calculate step sizes for finite-differences. eps_in = eps eps = numpy.empty([len(p0)]) one_sided = [False]*len(p0) for i, pval in enumerate(p0): if pval != 0: # Account for floating point arithmetic issues if pval*eps_in < 1e-6: eps[i] = eps_in one_sided[i] = True else: eps[i] = eps_in*pval else: # Account for parameters equal to zero eps[i] = eps_in f0 = func(p0, *args) hess = numpy.empty((len(p0), len(p0))) for ii in range(len(p0)): for jj in range(ii, len(p0)): hess[ii][jj] = hessian_elem(func, f0, p0, ii, jj, eps, args=args, one_sided=one_sided) hess[jj][ii] = hess[ii][jj] return hess def hessian_elem(func, f0, p0, ii, jj, eps, args=(), one_sided=None)-
Calculate element [ii][jj] of the Hessian matrix, a matrix of partial second derivatives w.r.t. to parameters ii and jj
func: Model function f0: Evaluation of func at p0 p0: Parameters for func eps: List of absolute step sizes to use for each parameter when taking finite differences. args: Additional arguments to func one_sided: Optionally, pass in a sequence of length p0 that determines whether a one-sided derivative will be used for each parameter.
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def hessian_elem(func, f0, p0, ii, jj, eps, args=(), one_sided=None): """ Calculate element [ii][jj] of the Hessian matrix, a matrix of partial second derivatives w.r.t. to parameters ii and jj func: Model function f0: Evaluation of func at p0 p0: Parameters for func eps: List of absolute step sizes to use for each parameter when taking finite differences. args: Additional arguments to func one_sided: Optionally, pass in a sequence of length p0 that determines whether a one-sided derivative will be used for each parameter. """ # Note that we need to specify dtype=float, to avoid this being an integer # array which will silently fail when adding fractional eps. if one_sided is None: one_sided = [False]*len(p0) pwork = numpy.array(p0, copy=True, dtype=float) if ii == jj: if pwork[ii] != 0 and not one_sided[ii]: pwork[ii] = p0[ii] + eps[ii] fp = func(pwork, *args) pwork[ii] = p0[ii] - eps[ii] fm = func(pwork, *args) element = (fp - 2*f0 + fm)/eps[ii]**2 else: pwork[ii] = p0[ii] + 2*eps[ii] fpp = func(pwork, *args) pwork[ii] = p0[ii] + eps[ii] fp = func(pwork, *args) element = (fpp - 2*fp + f0)/eps[ii]**2 else: if pwork[ii] != 0 and pwork[jj] != 0 and not one_sided[ii] and not one_sided[jj]: # f(xi + hi, xj + h) pwork[ii] = p0[ii] + eps[ii] pwork[jj] = p0[jj] + eps[jj] fpp = func(pwork, *args) # f(xi + hi, xj - hj) pwork[ii] = p0[ii] + eps[ii] pwork[jj] = p0[jj] - eps[jj] fpm = func(pwork, *args) # f(xi - hi, xj + hj) pwork[ii] = p0[ii] - eps[ii] pwork[jj] = p0[jj] + eps[jj] fmp = func(pwork, *args) # f(xi - hi, xj - hj) pwork[ii] = p0[ii] - eps[ii] pwork[jj] = p0[jj] - eps[jj] fmm = func(pwork, *args) element = (fpp - fpm - fmp + fmm)/(4 * eps[ii]*eps[jj]) else: # f(xi + hi, xj + h) pwork[ii] = p0[ii] + eps[ii] pwork[jj] = p0[jj] + eps[jj] fpp = func(pwork, *args) # f(xi + hi, xj) pwork[ii] = p0[ii] + eps[ii] pwork[jj] = p0[jj] fpm = func(pwork, *args) # f(xi, xj + hj) pwork[ii] = p0[ii] pwork[jj] = p0[jj] + eps[jj] fmp = func(pwork, *args) element = (fpp - fpm - fmp + f0)/(eps[ii]*eps[jj]) return element def score_stat(func_ex, grid_pts, all_boot, p0, data, nested_indices, multinom=True, eps=0.01, adj_and_org=False)-
Calculate test stastic from score test
func_ex: Model function for complex model grid_pts: Grid points to evaluate model function all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for the simple model, with nested parameter explicity defined. Although equal to values for simple model, should be in a list form that can be taken in by the complex model you'd like to evaluate. data: Original data frequency spectrum nested_indices: List of positions of nested parameters in complex model parameter list eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. adj_and_org: If False, return only adjusted score statistic. If True, also return unadjusted statistic as second return value.
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def score_stat(func_ex, grid_pts, all_boot, p0, data, nested_indices, multinom=True, eps=0.01, adj_and_org=False): # XXX: Implement boot_theta_adjusts """ Calculate test stastic from score test func_ex: Model function for complex model grid_pts: Grid points to evaluate model function all_boot: List of bootstrap frequency spectra p0: Best-fit parameters for the simple model, with nested parameter explicity defined. Although equal to values for simple model, should be in a list form that can be taken in by the complex model you'd like to evaluate. data: Original data frequency spectrum nested_indices: List of positions of nested parameters in complex model parameter list eps: Fractional stepsize to use when taking finite-difference derivatives Note that if eps*param is < 1e-6, then the step size for that parameter will simply be eps, to avoid numerical issues with small parameter perturbations. multinom: If True, assume model is defined without an explicit parameter for theta. Because uncertainty in theta must be accounted for to get correct uncertainties for other parameters, this function will automatically consider theta if multinom=True. adj_and_org: If False, return only adjusted score statistic. If True, also return unadjusted statistic as second return value. """ if multinom: func_multi = func_ex model = func_multi(p0, data.sample_sizes, grid_pts) theta_opt = Inference.optimal_sfs_scaling(model, data) p0 = list(p0) + [theta_opt] func_ex = lambda p, ns, pts: p[-1]*func_multi(p[:-1], ns, pts) # We only need to take derivatives with respect to the parameters in the # complex model that have been set to specified values in the simple model def diff_func(diff_params, ns, grid_pts): # diff_params argument is only the nested parameters. All the rest # should come from p0 full_params = numpy.array(p0, copy=True, dtype=float) # Use numpy indexing to set relevant parameters full_params[nested_indices] = diff_params return func_ex(full_params, ns, grid_pts) p_nested = numpy.asarray(p0)[nested_indices] GIM, H, J, cU = get_godambe(diff_func, grid_pts, all_boot, p_nested, data, eps, log=False) score_org = numpy.dot(numpy.dot(numpy.transpose(cU), numpy.linalg.inv(H)),cU)[0,0] score_adj = numpy.dot(numpy.dot(numpy.transpose(cU), numpy.linalg.inv(J)),cU)[0,0] if adj_and_org: return score_adj, score_org return score_adj def sum_chi2_ppf(x, weights=(0, 1))-
Percent point function (inverse of cdf) of weighted sum of chi^2 distributions.
x: Value(s) at which to evaluate ppf weights: Weights of chi^2 distributions, beginning with zero d.o.f. For example, weights=(0,1) is the normal chi^2 distribution with 1 d.o.f. For single parameters on the boundary, the correct distribution for the LRT is 0.5chi^2_0 + 0.5chi^2_1, which would be weights=(0.5,0.5).
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def sum_chi2_ppf(x, weights=(0,1)): """ Percent point function (inverse of cdf) of weighted sum of chi^2 distributions. x: Value(s) at which to evaluate ppf weights: Weights of chi^2 distributions, beginning with zero d.o.f. For example, weights=(0,1) is the normal chi^2 distribution with 1 d.o.f. For single parameters on the boundary, the correct distribution for the LRT is 0.5*chi^2_0 + 0.5*chi^2_1, which would be weights=(0.5,0.5). """ import scipy.stats.distributions as ssd # Ensure that weights are valid if abs(numpy.sum(weights) - 1) > 1e-6: raise ValueError('Weights must sum to 1.') # A little clunky, but we want to handle x = 0.5, and x = [2, 3, 4] # correctly. So if x is a scalar, we record that fact so we can return a # scalar on output. if numpy.isscalar(x): scalar_input = True # Convert x into an array, so we can index it easily. x = numpy.atleast_1d(x) # Calculate total cdf of all chi^2 dists with dof > 1. # (ssd.chi2.cdf(x,0) is always nan, so we avoid that.) cdf = numpy.sum([w*ssd.chi2.cdf(x, d+1) for (d, w) in enumerate(weights[1:])], axis=0) # Add in contribution from 0 d.o.f. cdf[x > 0] += weights[0] # Convert to ppf ppf = 1-cdf if scalar_input: return ppf[0] else: return ppf