# This file is part of BurnMan - a thermoelastic and thermodynamic toolkit for
# the Earth and Planetary Sciences
# Copyright (C) 2012 - 2015 by the BurnMan team, released under the GNU
# GPL v2 or later.
from __future__ import absolute_import
from __future__ import print_function
import numpy as np
from . import nonlinear_fitting
from ..utils.misc import flatten
from ..utils.math import unit_normalize
from .nonlinear_fitting import nonlinear_least_squares_fit
[docs]
class MineralFit(object):
"""
Class for fitting mineral parameters to experimental data.
Instances of this class are passed to
:func:`burnman.nonlinear_least_squares_fit`.
For attributes added to this model when fitting is done,
please see the documentation for that function.
"""
def __init__(
self,
mineral,
data,
data_covariances,
flags,
fit_params,
mle_tolerances,
delta_params=None,
bounds=None,
):
self.m = mineral
self.data = data
self.data_covariances = data_covariances
self.flags = flags
self.fit_params = fit_params
self.mle_tolerances = mle_tolerances
if delta_params is None:
self.delta_params = self.get_params() * 1.0e-5 + 1.0e-10
else:
self.delta_params = delta_params
self.bounds = bounds
[docs]
def set_params(self, param_values):
i = 0
if self.bounds is not None:
param_values = np.clip(param_values, self.bounds[:, 0], self.bounds[:, 1])
for param in self.fit_params:
if isinstance(self.m.params[param], float):
self.m.params[param] = param_values[i]
i += 1
else:
for j in range(len(self.m.params[param])):
self.m.params[param][j] = param_values[i]
i += 1
[docs]
def get_params(self):
params = []
for i, param in enumerate(self.fit_params):
params.append(self.m.params[param])
return np.array(flatten([self.m.params[prm] for prm in self.fit_params]))
[docs]
def function(self, x, flag):
P, T, p = x
self.m.set_state(P, T)
return np.array([P, T, getattr(self.m, flag)])
[docs]
def normal(self, x, flag):
P, T, p = x
if flag == "V":
self.m.set_state(P, T)
dPdp = -self.m.isothermal_bulk_modulus_reuss / self.m.V
dpdT = self.m.alpha * self.m.V
elif flag == "H":
self.m.set_state(P, T)
dPdp = 1.0 / ((1.0 - T * self.m.alpha) * self.m.V)
dpdT = self.m.molar_heat_capacity_p
elif flag == "S":
self.m.set_state(P, T)
dPdp = -1.0 / (self.m.alpha * self.m.V)
dpdT = self.m.molar_heat_capacity_p / T
elif flag == "gibbs":
self.m.set_state(P, T)
dPdp = 1.0 / self.m.V
dpdT = -self.m.S
else:
dP = 1.0e5
dT = 1.0
dPdp = (2.0 * dP) / (
self.function([P + dP, T, 0.0], flag)[2]
- self.function([P - dP, T, 0.0], flag)[2]
)
dpdT = (
self.function([P, T + dT, 0.0], flag)[2]
- self.function([P, T - dT, 0.0], flag)[2]
) / (2.0 * dT)
dPdT = -dPdp * dpdT
n = np.array([-1.0, dPdT, dPdp])
return unit_normalize(n)
[docs]
def fit_PTp_data(
mineral,
fit_params,
flags,
data,
data_covariances=[],
mle_tolerances=[],
param_tolerance=1.0e-5,
delta_params=None,
bounds=None,
max_lm_iterations=50,
verbose=True,
):
"""
Given a mineral of any type, a list of fit parameters
and a set of P-T-property points and (optional) uncertainties,
this function returns a list of optimized parameters
and their associated covariances, fitted using the
scipy.optimize.curve_fit routine.
:param mineral: Mineral for which the parameters should be optimized.
:type mineral: :class:`burnman.Mineral`
:param fit_params: List of dictionary keys contained in mineral.params
corresponding to the variables to be optimized
during fitting. Initial guesses are taken from the existing
values for the parameters
:type fit_params: list of str
:param flags: Attribute names for the property to be fit for the whole
dataset or each datum individually (e.g. 'V')
:type flags: string or list of strings
:param data: Observed X-P-T-property values
:type data: 2D numpy.array
:param data_covariances: X-P-T-property covariances (optional)
If not given, all covariance matrices are chosen
such that all data points have equal weight,
with all error in the pressure.
:type data_covariances: 3D numpy.array
:param mle_tolerances: Tolerances for termination of the
maximum likelihood iterations (optional).
:type mle_tolerances: numpy.array
:param param_tolerance: Fractional tolerance for termination
of the nonlinear optimization (optional).
:type param_tolerance: float
:param delta_params: Initial values for the change in parameters (optional).
:type delta_params: numpy.array
:param bounds: Minimum and maximum bounds for the parameters (optional).
The shape must be (n_parameters, 2).
:type bounds: 2D numpy.array
:param max_lm_iterations: Maximum number of Levenberg-Marquardt iterations.
:type max_lm_iterations: int
:param verbose: Whether to print detailed information about the
optimization to screen.
:type verbose: bool
:returns: Model with optimized parameters.
:rtype: :class:`burnman.optimize.eos_fitting.MineralFit`
"""
# If only one property flag is given, assume it applies to all data
if type(flags) is str:
flags = np.array([flags] * len(data[:, 0]))
if len(flags) != len(data):
raise Exception(
f"The number of flags (n = {len(flags)}) must be equal "
f"to the number of data (n = {len(data)})."
)
# Apply mle tolerances if they dont exist
if len(mle_tolerances) == 0:
mineral.set_state(1.0e5, 300.0)
mle_tolerance_factor = 1.0e-5
mle_tolerances = np.empty(len(flags))
for i, flag in enumerate(flags):
if flag in ["gibbs", "enthalpy", "H", "helmholtz"]:
mle_tolerances[i] = 1.0 # 1 J
else:
mle_tolerances[i] = mle_tolerance_factor * getattr(mineral, flag)
# If covariance matrix is not given, apply unit weighting to all pressures
# (with zero errors on T and p)
covariances_defined = True
if len(data_covariances) == 0:
covariances_defined = False
data_covariances = np.zeros((len(data[:, 0]), len(data[0]), len(data[0])))
for i in range(len(data_covariances)):
data_covariances[i][0][0] = 1.0
model = MineralFit(
mineral=mineral,
data=data,
data_covariances=data_covariances,
flags=flags,
fit_params=fit_params,
delta_params=delta_params,
mle_tolerances=mle_tolerances,
bounds=bounds,
)
nonlinear_least_squares_fit(
model,
max_lm_iterations=max_lm_iterations,
param_tolerance=param_tolerance,
verbose=verbose,
)
if verbose is True and covariances_defined is True:
confidence_interval = 0.9
d = nonlinear_fitting.extreme_values(
model.weighted_residuals, confidence_interval
)
confidence_bound, indices, probabilities = d
if indices != []:
print(
"The function nonlinear_fitting.extreme_values"
"(model.weighted_residuals, confidence_interval) "
f"has determined that there are {len(indices):d} data points"
" which have residuals which are not expected at the "
f"{confidence_interval*100.:.1f}% confidence level "
f"(> {confidence_bound:.1f} s.d. away from the model fit).\n"
"Their indices and the probabilities of finding "
"such extreme values are:"
)
for i, idx in enumerate(indices):
print(
f"[{idx:d}]: {probabilities[i]:.4f} "
f"({np.abs(model.weighted_residuals[idx]):.1f} s.d. "
"from the model)"
)
print(
"You might consider removing them from your fit, "
"or increasing the uncertainties in their "
"measured values.\n"
)
return model
[docs]
def fit_PTV_data(
mineral,
fit_params,
data,
data_covariances=[],
delta_params=None,
bounds=None,
param_tolerance=1.0e-5,
max_lm_iterations=50,
verbose=True,
):
"""
A simple alias for the fit_PTp_data for when all the data is volume data
"""
return fit_PTp_data(
mineral=mineral,
flags="V",
data=data,
data_covariances=data_covariances,
fit_params=fit_params,
param_tolerance=param_tolerance,
delta_params=delta_params,
bounds=bounds,
max_lm_iterations=max_lm_iterations,
verbose=verbose,
)
[docs]
class SolutionFit(object):
"""
Class for fitting mineral parameters to experimental data.
Instances of this class are passed to
:func:`burnman.nonlinear_least_squares_fit`.
For attributes added to this model when fitting is done,
please see the documentation for that function.
"""
def __init__(
self,
solution,
data,
data_covariances,
flags,
fit_params,
mle_tolerances,
delta_params=None,
bounds=None,
):
self.m = solution
self.data = data
self.data_covariances = data_covariances
self.flags = flags
self.fit_params = fit_params
self.fit_params_strings = []
for p in fit_params:
if isinstance(p, list):
csv_list_mbrs = ",".join([str(i) for i in p[1:]])
self.fit_params_strings.append(f"{p[0]} ({csv_list_mbrs})")
else:
self.fit_params_strings.append(p)
self.mle_tolerances = mle_tolerances
if delta_params is None:
self.delta_params = self.get_params() * 1.0e-5 + 1.0e-10
else:
self.delta_params = delta_params
self.bounds = bounds
[docs]
def set_params(self, param_values):
# fit_params is a list of lists
# if the list has length 2, the first item should be an integer
# indicating the endmember number in the solution
# if the list has length 3, the first two items should be endmember
# numbers, and the third should be the interaction parameter type
# (E, S or V).
i = 0
if self.bounds is not None:
param_values = np.clip(param_values, self.bounds[:, 0], self.bounds[:, 1])
for param in self.fit_params:
value = param_values[i]
if len(param) == 2:
key, imbr = param
if isinstance(self.m.endmembers[imbr][0].params[key], float):
self.m.endmembers[imbr][0].params[key] = value
i += 1
else:
n_values = len(self.m.endmembers[imbr][0].params[key])
for j in range(n_values):
self.m.endmembers[imbr][0].params[key][j] = value
i += 1
elif len(param) == 3:
key, imbr, jmbr = param
ai = self.m.solution_model.alphas[imbr]
aj = self.m.solution_model.alphas[jmbr]
if key == "E":
self.m.solution_model.We[imbr, jmbr] = 2.0 * value / (ai * aj)
if key == "S":
self.m.solution_model.Ws[imbr, jmbr] = 2.0 * value / (ai * aj)
if key == "V":
self.m.solution_model.Wv[imbr, jmbr] = 2.0 * value / (ai * aj)
i += 1
else:
raise Exception("param length must be two or three")
[docs]
def get_params(self):
params = []
for param in self.fit_params:
if len(param) == 2:
key, imbr = param
value = self.m.endmembers[imbr][0].params[key]
if isinstance(value, float):
params.append(value)
else:
params.extend(list(value))
elif len(param) == 3:
key, imbr, jmbr = param
ai = self.m.solution_model.alphas[imbr]
aj = self.m.solution_model.alphas[jmbr]
if key == "E":
params.append(
self.m.solution_model.We[imbr, jmbr] * (ai * aj) / 2.0
)
if key == "S":
params.append(
self.m.solution_model.Ws[imbr, jmbr] * (ai * aj) / 2.0
)
if key == "V":
params.append(
self.m.solution_model.Wv[imbr, jmbr] * (ai * aj) / 2.0
)
else:
raise Exception("param length must be two or three")
return np.array(params)
[docs]
def function(self, x, flag):
self.m.set_composition(x[: self.m.n_endmembers])
P, T, p = x[self.m.n_endmembers :]
self.m.set_state(P, T)
f = np.copy(x)
f[-1] = getattr(self.m, flag)
return f
[docs]
def normal(self, x, flag):
self.m.set_composition(x[: self.m.n_endmembers])
P, T, p = x[self.m.n_endmembers :]
if flag == "V":
self.m.set_state(P, T)
dPdp = -self.m.isothermal_bulk_modulus_reuss / self.m.V
dpdT = self.m.alpha * self.m.V
elif flag == "H":
self.m.set_state(P, T)
dPdp = 1.0 / ((1.0 - T * self.m.alpha) * self.m.V)
dpdT = self.m.molar_heat_capacity_p
elif flag == "S":
self.m.set_state(P, T)
dPdp = -1.0 / (self.m.alpha * self.m.V)
dpdT = self.m.molar_heat_capacity_p / T
elif flag == "gibbs":
self.m.set_state(P, T)
dPdp = 1.0 / self.m.V
dpdT = -self.m.S
else:
dP = 1.0e5
dT = 1.0
xP0 = np.copy(x)
xP1 = np.copy(x)
xT0 = np.copy(x)
xT1 = np.copy(x)
xP0[-3] = xP1[-3] - dP
xP1[-3] = xP1[-3] + dP
xT0[-2] = xP1[-2] - dT
xT1[-2] = xP1[-2] + dT
dPdp = (2.0 * dP) / (
self.function(xP1, flag)[2] - self.function(xP0, flag)[2]
)
dpdT = (self.function(xT1, flag)[2] - self.function(xT0, flag)[2]) / (
2.0 * dT
)
dPdT = -dPdp * dpdT
n = np.zeros(len(x))
n[-3:] = np.array([-1.0, dPdT, dPdp])
return unit_normalize(n)
[docs]
def fit_XPTp_data(
solution,
fit_params,
flags,
data,
data_covariances=[],
mle_tolerances=[],
param_tolerance=1.0e-5,
delta_params=None,
bounds=None,
max_lm_iterations=50,
verbose=True,
):
"""
Given a symmetric solution, a list of fit parameters
and a set of P-T-property points and (optional) uncertainties,
this function returns a list of optimized parameters
and their associated covariances, fitted using the
scipy.optimize.curve_fit routine.
:param solution: Solution for which the parameters should be optimized.
:type solution: :class:`burnman.Solution`
:param fit_params: Variables to be optimized
during fitting. Each list is either of length two or three.
The first item of length-2 lists should be a
dictionary key contained in one of the endmember
mineral.params, and the second item should be the index of
the endmember in the solution (indexing starts from 0).
The first item of length-3 lists should be one of 'E', 'S' or
'V' (the excess energies, entropies or volumes in each binary).
The second two items should be the indices of the pair of
endmembers bounding the binary, in ascending order
(indexing starts from 0). Initial guesses are taken from the existing
values for the parameters.
:type fit_params: list of lists
:param flags: Attribute names for the property to be fit for the whole
dataset or each datum individually (e.g. 'V')
:type flags: string or list of strings
:param data: Observed X-P-T-property values
:type data: 2D numpy.array
:param data_covariances: X-P-T-property covariances (optional).
If not given, all covariance matrices are chosen
such that all data points have equal weight,
with all error in the pressure.
:type data_covariances: 3D numpy.array
:param mle_tolerances: Tolerances for termination of the
maximum likelihood iterations (optional).
:type mle_tolerances: numpy.array
:param param_tolerance: Fractional tolerance for termination
of the nonlinear optimization (optional).
:type param_tolerance: float
:param delta_params: Initial values for the change in parameters (optional).
:type delta_params: numpy.array
:param bounds: Minimum and maximum bounds for the parameters (optional).
The shape must be (n_parameters, 2).
:type bounds: 2D numpy.array
:param max_lm_iterations: Maximum number of Levenberg-Marquardt iterations.
:type max_lm_iterations: int
:param verbose: Whether to print detailed information about the
optimization to screen.
:type verbose: bool
:returns: Model with optimized parameters.
:rtype: :class:`burnman.optimize.eos_fitting.SolutionFit`
"""
# If only one property flag is given, assume it applies to all data
if type(flags) is str:
flags = np.array([flags] * len(data[:, 0]))
if len(flags) != len(data):
raise Exception(
f"The number of flags (n = {len(flags)}) must be equal "
f"to the number of data (n = {len(data)})."
)
# Apply mle tolerances if they dont exist
if len(mle_tolerances) == 0:
solution.set_state(1.0e5, 300.0)
mle_tolerance_factor = 1.0e-5
mle_tolerances = np.empty(len(flags))
for i, flag in enumerate(flags):
if flag in ["gibbs", "enthalpy", "H", "helmholtz"]:
mle_tolerances[i] = 1.0 # 1 J
else:
mle_tolerances[i] = mle_tolerance_factor * getattr(solution, flag)
# If covariance matrix is not given, apply unit weighting to all pressures
# (with zero errors on T and property)
covariances_defined = True
if len(data_covariances) == 0:
covariances_defined = False
nX = solution.n_endmembers
data_covariances = np.zeros((len(data[:, 0]), len(data[0]), len(data[0])))
for i in range(len(data_covariances)):
data_covariances[i][nX][nX] = 1.0
model = SolutionFit(
solution=solution,
data=data,
data_covariances=data_covariances,
flags=flags,
fit_params=fit_params,
mle_tolerances=mle_tolerances,
delta_params=delta_params,
bounds=bounds,
)
nonlinear_least_squares_fit(
model,
max_lm_iterations=max_lm_iterations,
param_tolerance=param_tolerance,
verbose=verbose,
)
if verbose is True and covariances_defined is True:
confidence_interval = 0.9
v_extreme = nonlinear_fitting.extreme_values(
model.weighted_residuals, confidence_interval
)
confidence_bound, indices, probabilities = v_extreme
if indices != []:
print(
"The function nonlinear_fitting.extreme_values"
"(model.weighted_residuals, confidence_interval) "
f"has determined that there are {len(indices):d} "
"data points which have residuals which are not "
f"expected at the {confidence_interval*100.:.1f}% "
"confidence level "
f"(> {confidence_bound:.1f} s.d. away from the model fit).\n"
"Their indices and the probabilities of "
"finding such extreme values are:"
)
for i, idx in enumerate(indices):
print(
f"[{idx:d}]: {probabilities[i]:.4f} "
f"({np.abs(model.weighted_residuals[idx]):.1f} s.d. "
"from the model)"
)
print(
"You might consider removing them from your fit, "
"or increasing the uncertainties in "
"their measured values.\n"
)
return model