# 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 scipy.stats import t, norm, genextreme
import copy
from ..utils.math import unit_normalize
import matplotlib.pyplot as plt
import matplotlib.colors as colors
from matplotlib.patches import Ellipse
[docs]
def nonlinear_least_squares_fit(
model, lm_damping=0.0, param_tolerance=1.0e-7, max_lm_iterations=100, verbose=False
):
"""
Function to compute the "best-fit" parameters for a model
by nonlinear least squares fitting.
The nonlinear least squares algorithm closely follows the logic in
Section 23.1 of Bayesian Probability Theory
(von der Linden et al., 2014; Cambridge University Press).
Parameters
----------
:param model: Model containing data to be fit, and functions to
aid in fitting.
:type model: object
:param lm_damping: Levenberg-Marquardt parameter for least squares minimization.
:type lm_damping: float
:param param_tolerance: Levenberg-Marquardt iterations are terminated when
the maximum fractional change in any of the parameters
during an iteration drops below this value
:type param_tolerance: float
:param max_lm_iterations: Maximum number of Levenberg-Marquardt iterations
:type max_lm_iterations: int
:param verbose: Print some information to standard output
:type verbose: bool
.. note:: The object passed as model must have the following attributes:
* data [2D numpy.array] - Elements of x[i][j] contain the
observed position of data point i.
* data_covariances [3D numpy.array] Elements of cov[i][j][k] contain
the covariance matrix of data point i.
* mle_tolerances [numpy.array] - The iterations to find the maximum likelihood
estimator for each observed data point will stop when mle_tolerances[i] <
np.linalg.norm(data_mle[i] - model.function(data_mle[i], flag))
* delta_params [numpy.array] - parameter perturbations used to compute the jacobian
Must also have the following methods:
* set_params(self, param_values) - Function to set parameters.
* get_params(self) - Function to get current model parameters.
* function(self, x) - Returns value of model function evaluated at x.
* normal(self, x) - Returns value of normal to the model function evaluated at x.
After this function has been performed, the following attributes are added to model:
* n_dof [int] - Degrees of freedom of the system.
* data_mle [2D numpy array] - Maximum likelihood estimates of the observed data points
on the best-fit curve.
* jacobian [2D numpy array] - d(weighted_residuals)/d(parameter).
* weighted_residuals [numpy array] - Weighted residuals.
* weights [numpy array] - 1/(data variances normal to the best fit curve).
* WSS [float] - Weighted sum of squares residuals.
* popt [numpy array] - Optimized parameters.
* pcov [2D numpy array] - Covariance matrix of optimized parameters.
* noise_variance [float] - Estimate of the variance of the data normal to the curve.
This function is available as ``burnman.nonlinear_least_squares_fit``.
"""
def _mle_estimate(x, x_m, cov, flag):
n = model.normal(x_m, flag)
var_n = abs_line_project(cov, n)
d = (x_m - x).dot(n)
x_mle = x + d * ((n.dot(cov)).T) / var_n
return x_mle, d, var_n
def _find_mle():
x_mle_arr = np.empty_like(model.data)
residual_arr = np.empty(n_data)
var_arr = np.empty(n_data)
for i, (x, cov, flag) in enumerate(
zip(*[model.data, model.data_covariances, model.flags])
):
x_mle_arr[i] = model.function(x, flag)
x_mle_est, residual_arr[i], var_arr[i] = _mle_estimate(
x, x_mle_arr[i], cov, flag
)
delta_x = x_mle_arr[i] - x
while np.linalg.norm(delta_x) > model.mle_tolerances[i]:
x_mle_est, residual_arr[i], var_arr[i] = _mle_estimate(
x, x_mle_arr[i], cov, flag
)
x_mle_arr[i] = model.function(x_mle_est, flag)
delta_x = x_mle_arr[i] - x_mle_est
return x_mle_arr, residual_arr / np.sqrt(var_arr), 1.0 / var_arr
def calculate_jacobian():
model.jacobian = np.empty((n_data, n_params))
diag_delta = np.diag(model.delta_params)
param_values = model.get_params()
for prm_i, value in enumerate(param_values):
model.set_params(param_values - diag_delta[prm_i])
x_mle_arr, residual_arr_0, weights_0 = _find_mle()
model.set_params(param_values + diag_delta[prm_i])
x_mle_arr, residual_arr_1, weights_1 = _find_mle()
model.jacobian[:, prm_i] = (residual_arr_1 - residual_arr_0) / (
2.0 * diag_delta[prm_i][prm_i]
)
model.set_params(param_values) # reset params
def _update_beta(lmbda):
# Performs a Levenberg-Marquardt iteration
# Note that if lambda = 0, this is a simple Gauss-Newton iteration
calculate_jacobian()
model.data_mle, model.weighted_residuals, model.weights = _find_mle()
J = model.jacobian # this the weighted Jacobian
JTJ = J.T.dot(J)
delta_beta = (
np.linalg.inv(JTJ + lmbda * np.diag(JTJ))
.dot(J.T)
.dot(model.weighted_residuals)
)
old_params = np.copy(model.get_params())
new_params = old_params - delta_beta
# f_delta_beta = delta_beta/new_params
model.set_params(new_params)
# set_params may modify the step to satisfy bounds on the problem
# We therefore need to get the params before
# calculating the fractional change.
new_params = model.get_params()
# In case the new_params object returns a very small value,
# modify to avoid a pointless comparison:
mod_params = np.where(
np.abs(new_params) < param_tolerance, param_tolerance, new_params
)
f_delta_beta = (old_params - new_params) / mod_params
return f_delta_beta
n_data = len(model.data)
params = model.get_params()
n_params = len(params)
model.dof = n_data - n_params
if not hasattr(model, "flags"):
model.flags = [None] * n_data
for n_it in range(max_lm_iterations):
# update the parameters with a LM iteration
f_delta_beta = _update_beta(lm_damping)
max_f = np.max(np.abs(f_delta_beta))
if verbose is True:
print(
"Iteration {0:d}: {1}. Max change in param: {2}".format(
n_it, model.get_params(), max_f
)
)
if max_f < param_tolerance:
break
J = model.jacobian
r = model.weighted_residuals
model.WSS = r.dot(r.T)
model.popt = model.get_params()
model.pcov = np.linalg.inv(J.T.dot(J)) * r.dot(r.T) / model.dof
# Estimate the noise variance normal to the curve
model.goodness_of_fit = model.WSS / model.dof
model.noise_variance = r.dot(np.diag(1.0 / model.weights)).dot(r.T) / model.dof
if verbose is True:
if n_it == max_lm_iterations - 1:
print(
f"Max iterations ({max_lm_iterations:d}) reached "
f"(param tolerance = {param_tolerance:1e})"
)
else:
print("Converged in {0:d} iterations".format(n_it))
print("\nOptimised parameter values:")
print(model.popt)
print("\nParameter covariance matrix:")
print(model.pcov)
print("")
[docs]
def confidence_prediction_bands(model, x_array, confidence_interval, f, flag=None):
"""
This function calculates the confidence and prediction bands of
the function f(x) from a best-fit model with uncertainties in its
parameters as calculated (for example) by
the function nonlinear_least_squares_fit().
The values are calculated via the delta method, which estimates
the variance of f evaluated at x as var(f(x)) = df(x)/dB var(B) df(x)/dB
where df(x)/dB is the vector of partial derivatives of f(x)
with respect to B.
:param model: As modified (for example) by the function
:func:`burnman.nonlinear_least_squares_fit`.
Should contain the following functions: get_params, set_params, function, normal
And attributes: delta_params, pcov, dof, noise_variance
:type model: object
:param x_array: Coordinates at which to evaluate the bounds.
:type x_array: 2D numpy.array
:param confidence_interval: Probability level of finding the true model
(confidence bound) or any new data point (probability bound).
For example, the 95% confidence bounds should be calculated using a
confidence interval of 0.95.
:type confidence_interval: float
:param f: The function defining the variable y=f(x) for which the
confidence and prediction bounds are desired.
:type f: function
:param flag: This (optional) flag is passed to model.function to control how the
modified position of x is calculated. This value is then used by f(x)
:type flag: type informed by model object
:returns: An element of bounds[i][j] gives the lower and upper confidence
(i=0, i=1) and prediction (i=2, i=3) bounds for the jth data point.
:rtype: 2D numpy.array
"""
# Check array dimensions
n_dimensions = len(model.data[0])
if len(x_array[0]) != n_dimensions:
raise Exception(
"Dimensions of each point must be the same as the "
"total number of dimensions"
)
param_values = model.get_params()
x_m_0s = np.empty_like(x_array)
f_m_0s = np.empty_like(x_array[:, 0])
for i, x in enumerate(x_array):
x_m_0s[i] = model.function(x, flag)
f_m_0s[i] = f(x)
diag_delta = np.diag(model.delta_params)
dxdbeta = np.empty([len(param_values), len(x_array)])
for i, value in enumerate(param_values):
model.set_params(param_values + diag_delta[i])
for j, x_m_0 in enumerate(x_m_0s):
x_m_1 = model.function(x_m_0, flag)
dxdbeta[i][j] = (f(x_m_1) - f_m_0s[j]) / diag_delta[i][i]
model.set_params(param_values) # reset params
variance = np.empty(len(x_array))
for i, Gprime in enumerate(dxdbeta.T):
variance[i] = Gprime.T.dot(model.pcov).dot(Gprime)
critical_value = t.isf(0.5 * (confidence_interval + 1.0), model.dof)
confidence_half_widths = critical_value * np.sqrt(variance)
prediction_half_widths = critical_value * np.sqrt(variance + model.noise_variance)
confidence_bound_0 = f_m_0s - confidence_half_widths
confidence_bound_1 = f_m_0s + confidence_half_widths
prediction_bound_0 = f_m_0s - prediction_half_widths
prediction_bound_1 = f_m_0s + prediction_half_widths
return np.array(
[confidence_bound_0, confidence_bound_1, prediction_bound_0, prediction_bound_1]
)
[docs]
def abs_line_project(M, n):
n = unit_normalize(n)
return n.dot(M).dot(n.T)
[docs]
def plot_cov_ellipse(cov, pos, nstd=2, ax=None, **kwargs):
"""
Plots an `nstd` sigma error ellipse based on the specified covariance
matrix (`cov`). Additional keyword arguments are passed on to the
ellipse patch artist.
:param cov: The 2x2 covariance matrix to base the ellipse on.
:type cov: numpy.array
:param pos: The location of the center of the ellipse. Expects a 2-element
sequence of [x0, y0].
:type pos: list or numpy.array
:param nstd: The radius of the ellipse in numbers of standard deviations.
Defaults to 2 standard deviations.
:type nstd: float
:param ax: The axis that the ellipse will be plotted on. Defaults to the
current axis.
:type ax: matplotlib.pyplot.axes
:param kwargs: Additional keyword arguments are passed on to the ellipse patch.
:returns: The covariance ellipse (already applied to the desired axes object).
:rtype: matplotlib.patches.Ellipse
"""
def eigsorted(cov):
vals, vecs = np.linalg.eigh(cov)
order = vals.argsort()[::-1]
return vals[order], vecs[:, order]
if ax is None:
ax = plt.gca()
vals, vecs = eigsorted(cov)
theta = np.degrees(np.arctan2(*vecs[:, 0][::-1]))
# Width and height are "full" widths, not radius
width, height = 2 * nstd * np.sqrt(vals)
ellip = Ellipse(xy=pos, width=width, height=height, angle=theta, **kwargs)
ax.add_artist(ellip)
return ellip
[docs]
def corner_plot(popt, pcov, param_names=[], n_std=1.0):
"""
Creates a corner plot of covariances
:param popt: Optimized parameters.
:type popt: numpy.array
:param pcov: Covariance matrix of the parameters.
:type pcov: 2D numpy.array
:param param_names: Parameter names.
:type param_names: list
:param n_std: Number of standard deviations for ellipse.
:type n_std: float
:returns: ``matplotlib.pyplot.figure`` and list of ``matplotlib.pyplot.Axes``
objects.
:rtype: tuple
"""
if len(pcov[0]) != len(pcov[:, 0]):
raise Exception("Covariance matrices must be square")
n_params = len(pcov[0])
if n_params < 2:
raise Exception(
"Covariance matrix must be at least 2x2 for " "a corner plot to be plotted"
)
# ellipse plotting is prone to rounding errors, so we scale the plots here
scaling = 1.0 / np.power(10.0, np.around(np.log10(np.abs(popt)) - 0.5))
scaling = np.outer(scaling, scaling)
fig, ax = plt.subplots(n_params - 1, n_params - 1)
fig.set_size_inches(3.0 * (n_params - 1), 3.0 * (n_params - 1))
for j in range(n_params - 1):
for i in range(j):
fig.delaxes(ax[i][j])
for i in range(j, n_params - 1):
ax[i][j].get_xaxis().get_major_formatter().set_useOffset(False)
ax[i][j].get_yaxis().get_major_formatter().set_useOffset(False)
ax[i][j].set_box_aspect(1)
if j > 0:
ax[i][j].get_yaxis().set_visible(False)
if i < n_params - 2:
ax[i][j].get_xaxis().set_visible(False)
indices = np.array([j, i + 1])
projected_cov = (pcov * scaling)[indices[:, None], indices]
scaled_pos = np.array(
[
popt[j] * np.sqrt(scaling[j][j]),
popt[i + 1] * np.sqrt(scaling[i + 1][i + 1]),
]
)
plot_cov_ellipse(
cov=projected_cov, pos=scaled_pos, nstd=n_std, ax=ax[i][j], color="grey"
)
maxx = 1.5 * n_std * np.sqrt(projected_cov[0][0])
maxy = 1.5 * n_std * np.sqrt(projected_cov[1][1])
ax[i][j].set_xlim(scaled_pos[0] - maxx, scaled_pos[0] + maxx)
ax[i][j].set_ylim(scaled_pos[1] - maxy, scaled_pos[1] + maxy)
if param_names != []:
for i in range(n_params - 1):
ax[n_params - 2][i].set_xlabel(
"{0:s} (x $10^{{{1:d}}}$)".format(
param_names[i], -int(np.log10(np.sqrt(scaling[i][i])))
)
)
for j in range(1, n_params):
ax[j - 1][0].set_ylabel(
"{0:s} (x $10^{{{1:d}}}$)".format(
param_names[j], -int(np.log10(np.sqrt(scaling[j][j])))
)
)
fig.set_tight_layout(True)
return fig, ax
[docs]
def weighted_residual_plot(
ax,
model,
flag=None,
sd_limit=3,
cmap=plt.cm.RdYlBu,
plot_axes=[0, 1],
scale_axes=[1.0, 1.0],
):
"""
Creates a plot of the weighted residuals
The user can choose the projection axes, and scaling to apply to those axes
The chosen color palette (cmap) is discretised by standard deviation up
to a cut off value of sd_limit.
:param ax: Plot.
:param type: ``matplotlib.pyplot.Axes``
:param model: A model as used by
:func:`burnman.nonlinear_least_squares_fit`.
Must contain the attributes model.data,
model.weighted_residuals and
model.flags (if flag is not None).
:type model: object
:param flag: String to determine which data to plot.
Finds matches with model.flags.
:type flag: str
:param sd_limit: Data with weighted residuals exceeding this
limit are plotted in black.
:type sd_limit: float
:param cmap: Color palette.
:type cmap: matplotlib color palette
:param plot_axes: Data axes to use as plot axes.
:type plot_axes: list of int
:param scale_axes: Plot axes are scaled by multiplication
of the data by these values.
:type scale_axes: list of float
:returns: Coloured scatter plot of the weighted residuals in data space.
:rtype: matplotlib Axes object
"""
if flag is None:
mask = range(len(model.data[:, 0]))
else:
mask = [i for i, flg in enumerate(model.flags) if flg == flag]
cmap_cp = copy.copy(cmap)
cmap_cp.set_under("k")
cmap_cp.set_over("k")
bounds = np.linspace(-sd_limit, sd_limit, sd_limit * 2 + 1)
norm = colors.BoundaryNorm(bounds, cmap_cp.N)
im = ax.scatter(
model.data[:, plot_axes[0]][mask] * scale_axes[0],
model.data[:, plot_axes[1]][mask] * scale_axes[1],
c=model.weighted_residuals[mask],
cmap=cmap_cp,
norm=norm,
s=50,
)
plt.colorbar(im, ax=ax, label="Misfit (standard deviations)")
[docs]
def extreme_values(weighted_residuals, confidence_interval):
"""
This function uses extreme value theory to calculate the number of
standard deviations away from the mean at which we should expect to bracket
*all* of our n data points at a certain confidence level.
It then uses that value to identify which (if any) of the data points
lie outside that region, and calculates the corresponding probabilities
of finding a data point at least that many standard deviations away.
:param weighted_residuals: Array of residuals weighted by the square root
of their variances wr_i = r_i/sqrt(var_i).
:type weighted_residuals: array of float
:param confidence_interval: Probability at which all the weighted residuals lie
within the confidence bounds.
:type confidence_interval: float
:returns: Number of standard deviations at which we should expect to encompass
all data at the user-defined confidence interval, the indices of weighted
residuals exceeding the confidence_interval defined by the user, and
the probabilities that the extreme data point of the distribution lies
further from the mean than the observed position wr_i for each i in
the "indices" output array.
:rtype: tuple of (float, numpy.array, numpy.array)
"""
n = len(weighted_residuals)
mean = norm.isf(1.0 / n)
# good approximation for > 10 data points
scale = 0.8 / np.power(np.log(n), 1.0 / 2.0)
# good approximation for > 10 data points
c = 0.33 / np.power(np.log(n), 3.0 / 4.0)
# We now need a 1-tailed probability from the given confidence_interval
# p_total = 1. - confidence_interval = p_upper + p_lower - p_upper*p_lower
# p_total = 1. - confidence_interval = 2p - p^2, therefore:
p = 1.0 - np.sqrt(confidence_interval)
confidence_bound = genextreme.isf(p, c, loc=mean, scale=scale)
indices = [
i for i, r in enumerate(weighted_residuals) if np.abs(r) > confidence_bound
]
# Convert back to 2-tailed probabilities
probabilities = 1.0 - np.power(
genextreme.sf(np.abs(weighted_residuals[indices]), c, loc=mean, scale=scale)
- 1.0,
2.0,
)
return confidence_bound, indices, probabilities
[docs]
def plot_residuals(ax, weighted_residuals, n_bins=None, flags=[]):
if flags is []:
flags = [""] * len(weighted_residuals)
list_flags = [""]
else:
list_flags = list(set(flags))
if n_bins is None:
try: # Only works for recent versions of numpy
bin_heights, bin_bounds = np.histogram(
weighted_residuals, bins="auto", density=True
)
n_bins = len(bin_heights)
except:
n_bins = 11.0
mask = [i for i, f in enumerate(flags)]
for flag in list_flags:
binwidth = np.ptp(weighted_residuals) / n_bins
dmin = min(weighted_residuals) - binwidth
dmax = max(weighted_residuals) + binwidth
bins = np.linspace(dmin, dmax, n_bins)
bin_heights, bin_bounds = np.histogram(
weighted_residuals[mask], bins=bins, density=True
)
normalisation = float(len(weighted_residuals[mask])) / float(
len(weighted_residuals)
)
bin_centers = (bin_bounds[:-1] + bin_bounds[1:]) / 2.0
bin_heights = bin_heights * normalisation
bin_widths = bin_bounds[1] - bin_bounds[0]
plt.bar(bin_centers, bin_heights, width=bin_widths, label=flag, alpha=0.2)
mask = [i for i, f in enumerate(flags) if f != flag and i in mask]
x = np.linspace(bin_bounds[0], bin_bounds[-1], 1001)
ax.plot(x, norm.pdf(x) * normalisation)
ax.set_title("Residual plot versus expected normal distribution")
ax.set_xlabel("Number of standard deviations from the mean")
ax.set_ylabel("Probability")
ax.legend(loc="upper right")