def FIM_uncert(func_ex,
grid_pts,
p0,
data,
log=False,
multinom=True,
eps=0.01):
"""
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
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)
return numpy.sqrt(numpy.diag(numpy.linalg.inv(H)))
def LRT_adjust(func_ex,
grid_pts,
all_boot,
p0,
data,
nested_indices,
multinom=True,
eps=0.01):
# XXX: Need to implement boot_theta_adjusts
"""
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.
"""
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)
adjust = len(nested_indices) / numpy.trace(
numpy.dot(J, numpy.linalg.inv(H)))
return adjust
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
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):
"""
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
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