# Source code for burnman.utils.math

# This file is part of BurnMan - a thermoelastic and thermodynamic toolkit for
# the Earth and Planetary Sciences
# Copyright (C) 2012 - 2021 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.ndimage.filters import gaussian_filter
from scipy.interpolate import interp2d
from sympy import Matrix, Rational
import scipy.integrate as integrate
from collections import Counter
import itertools

from .reductions import row_reduce

[docs]def round_to_n(x, xerr, n):
return round(x, -int(np.floor(np.log10(np.abs(xerr)))) + (n - 1))

[docs]def unit_normalize(a, order=2, axis=-1):
"""
Calculates the L2 normalized array of numpy array a
of a given order and along a given axis.
"""
l2 = np.atleast_1d(np.apply_along_axis(np.linalg.norm, axis, a, order))

l2[l2 == 0] = 1
return a / np.expand_dims(l2, axis)[0][0]

[docs]def float_eq(a, b):
"""
Test if two floats are almost equal to each other
"""
return abs(a - b) < 1e-10 * max(1e-5, abs(a), abs(b))

[docs]def linear_interpol(x, x1, x2, y1, y2):
"""
Linearly interpolate to point x, between
the points (x1,y1), (x2,y2)
"""
assert(x1 <= x)
assert(x2 >= x)
assert(x1 <= x2)

alpha = (x - x1) / (x2 - x1)
return (1. - alpha) * y1 + alpha * y2

[docs]def bracket(fn, x0, dx, args=(), ratio=1.618, maxiter=100):
"""
Given a function and a starting guess, find two
inputs for the function that bracket a root.

Parameters
----------
fn : function
The function to bracket
x0 : float
The starting guess
dx : float
Small step for starting the search
args : parameter list
Additional arguments to give to fn
ratio :
The step size increases by this ratio
every step in the search. Defaults to
the golden ratio.
maxiter : int
The maximum number of steps before giving up.

Returns
-------
xa, xb, fa, fb: floats
xa and xb are the inputs which bracket a root of fn.
fa and fb are the values of the function at those points.
If the bracket function takes more than maxiter steps,
it raises a ValueError.
"""
niter = 0
dx = np.abs(dx)
assert(ratio > 1.0)

# Get the starting positions
f0 = fn(x0, *args)
x_left = x0 - dx
x_right = x0 + dx
f_left = fn(x_left, *args)
f_right = fn(x_right, *args)

# Overshot zero, try making dx smaller
if (f0 - f_left) * (f_right - f0) < 0.:
while (f0 - f_left) * (f_right - f0) < 0. and dx > np.finfo('float').eps and niter < maxiter:
dx /= ratio
x_left = x0 - dx
x_right = x0 + dx
f_left = fn(x_left, *args)
f_right = fn(x_right, *args)
niter += 1
if niter == maxiter:  # Couldn't find something with same slope in both directions
raise ValueError('Cannot find zero.')

niter = 0
slope = f_right - f0
if slope > 0. and f0 > 0.:  # Walk left
dx = -dx
x1 = x_left
f1 = f_left
elif slope > 0. and f0 < 0.:  # Walk right
x1 = x_right
f1 = f_right
elif slope < 0. and f0 > 0:  # Walk right
x1 = x_right
f1 = f_right
else:  # Walk left
dx = -dx
x1 = x_left
f1 = f_left

# Do the walking
while f0 * f1 > 0. and niter < maxiter:
dx *= ratio
xnew = x1 + dx
fnew = fn(xnew, *args)
x0 = x1
f0 = f1
x1 = xnew
f1 = fnew
niter += 1

if f0 * f1 > 0.:
raise ValueError('Cannot find zero.')
else:
return x0, x1, f0, f1

"""
Pads an ndarray according to an inverse mirror
scheme. For example, for a 1D array
[2, 4, 6, 7, 8] padded by 3 cells, we have:

-3 -2  0  |  2  4  6  7  8  |  9 10 12

Parameters
----------
array : numpy ndarray
The number of elements with which to pad the
array in each dimension.

Returns
-------

"""

for i, l in enumerate(array.shape)])

for i, n
in enumerate(array.shape)]))
for i, l
in enumerate(array.shape)]))

keys = list(counter.keys())
padded_indices = [keys[i] for i, value in enumerate(counter.values())
if value == 1]
for dimension, axis_idx in enumerate(idx)]) for idx in padded_indices])
for j in range(len(array.shape))])

[docs]def smooth_array(array, grid_spacing,
gaussian_rms_widths, truncate=4.0,
mode='inverse_mirror'):
"""
Creates a smoothed array by convolving it with a gaussian filter.
Grid resolutions and gaussian RMS widths are required for each of
the axes of the numpy array. The smoothing is truncated at a
user-defined number of standard deviations. The edges of the array
can be padded in a number of different ways given by the
'mode' parameter.

Parameters
----------
array : numpy ndarray
The array to smooth
grid_spacing : numpy array of floats
The spacing of points along each axis
gaussian_rms_widths : numpy array of floats
The Gaussian RMS widths/standard deviations for the
Gaussian convolution.
truncate : float (default=4.)
The number of standard deviations at which to truncate
the smoothing.
mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap', 'inverse_mirror'}
The mode parameter determines how the array borders are handled
either by scipy.ndimage.filters.gaussian_filter.
Default is 'inverse_mirror', which uses
:func:burnman.tools.math._pad_ndarray_inverse_mirror.

Returns
-------
smoothed_array: numpy ndarray
The smoothed array

"""

# gaussian_filter works with standard deviations normalised to
# the grid spacing.
sigma = tuple(np.array(gaussian_rms_widths)/np.array(grid_spacing))

if mode == 'inverse_mirror':
padding = tuple([int(np.ceil(truncate*s)) for s in sigma])
sigma=sigma)
for i, l in enumerate(array.shape)])
else:
smoothed_array = gaussian_filter(array, sigma=sigma, mode=mode)

return smoothed_array

[docs]def interp_smoothed_array_and_derivatives(array,
x_values, y_values,
x_stdev=0., y_stdev=0.,
truncate=4.,
mode='inverse_mirror',
indexing='xy'):
"""
Creates a smoothed array on a regular 2D grid. Smoothing
is achieved using :func:burnman.tools.math.smooth_array.
Outputs scipy.interpolate.interp2d() interpolators
which can be used to query the array, or its derivatives in the
x- and y- directions.

Parameters
----------
array : 2D numpy array
The array to smooth. Each element array[i][j]
corresponds to the position x_values[i], y_values[j]
x_values : 1D numpy array
The gridded x values over which to create the smoothed grid
y_values : 1D numpy array
The gridded y_values over which to create the smoothed grid
x_stdev : float
The standard deviation for the Gaussian filter along the x axis
y_stdev : float
The standard deviation for the Gaussian filter along the x axis
truncate : float (optional)
The number of standard deviations at which to truncate
the smoothing (default = 4.).
mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap', 'inverse_mirror'}
The mode parameter determines how the array borders are handled
either by scipy.ndimage.filters.gaussian_filter.
Default is 'inverse_mirror', which uses
:func:burnman.tools.math._pad_ndarray_inverse_mirror.
indexing : {'xy', 'ij'}, optional
Cartesian ('xy', default) or matrix ('ij') indexing of output.
See numpy.meshgrid for more details.

Returns
-------
interps: tuple of three interp2d functors
interpolation functions for the smoothed property and
the first derivatives with respect to x and y.

"""

dx = x_values[1] - x_values[0]
dy = y_values[1] - y_values[0]

if indexing == 'xy':
smoothed_array = smooth_array(array=array,
grid_spacing=np.array([dy, dx]),
gaussian_rms_widths=np.array([y_stdev,
x_stdev]),
truncate=truncate,
mode=mode)

elif indexing == 'ij':
smoothed_array = smooth_array(array=array,
grid_spacing=np.array([dx, dy]),
gaussian_rms_widths=np.array([x_stdev,
y_stdev]),
truncate=truncate,
mode=mode).T

else:
raise Exception('Indexing scheme not recognised. Should be ij or xy.')

interps = (interp2d(x_values, y_values, smoothed_array, kind='linear'),

return interps

[docs]def compare_l2(depth, calc, obs):
"""
Computes the L2 norm for N profiles at a time (assumed to be linear between points).

:type depths: array of float
:param depths: depths. :math:[m]
:type calc: list of arrays of float
:param calc: N arrays calculated values, e.g. [mat_vs,mat_vphi]
:type obs: list of arrays of float
:param obs: N arrays of values (observed or calculated) to compare to , e.g. [seis_vs, seis_vphi]

:returns: array of L2 norms of length N
:rtype: array of floats
"""
err = []
for i in range(len(calc)):
err.append(l2(depth, calc[i], obs[i]))

return err

[docs]def compare_chifactor(calc, obs):
"""
Computes the chi factor for N profiles at a time. Assumes a 1% a priori uncertainty on the seismic model.

:type calc: list of arrays of float
:param calc: N arrays calculated values, e.g. [mat_vs,mat_vphi]
:type obs: list of arrays of float
:param obs: N arrays of values (observed or calculated) to compare to , e.g. [seis_vs, seis_vphi]

:returns: error array of length N
:rtype: array of floats
"""
err = []
for i in range(len(calc)):
err.append(chi_factor(calc[i], obs[i]))

return err

[docs]def l2(x, funca, funcb):
"""
Computes the L2 norm for one profile(assumed to be linear between points).

:type x: array of float
:param x: depths :math:[m].
:type funca: list of arrays of float
:param funca: array calculated values
:type funcb: list of arrays of float
:param funcb: array of values (observed or calculated) to compare to

:returns: L2 norm
:rtype: array of floats
"""
diff = np.array(funca - funcb)
diff = diff * diff

return integrate.trapz(diff, x)

[docs]def nrmse(x, funca, funcb):
"""
Normalized root mean square error for one profile
:type x: array of float
:param x: depths in m.
:type funca: list of arrays of float
:param funca: array calculated values
:type funcb: list of arrays of float
:param funcb: array of values (observed or calculated) to compare to

:returns: RMS error
:rtype: array of floats

"""
diff = np.array(funca - funcb)
diff = diff * diff
rmse = np.sqrt(np.sum(diff) / x)
nrmse = rmse / (np.max(funca) - np.min(funca))

return nrmse

[docs]def chi_factor(calc, obs):
"""
:math:\\chi factor for one profile assuming 1% uncertainty on the reference model (obs)
:type calc: list of arrays of float
:param calc: array calculated values
:type obs: list of arrays of float
:param obs: array of reference values to compare to

:returns: :math:\\chi factor
:rtype: array of floats

"""

err = np.empty_like(calc)
for i in range(len(calc)):
err[i] = pow((calc[i] - obs[i]) / (0.01 * np.mean(obs)), 2.)

err_tot = np.sum(err) / len(err)

return err_tot

[docs]def independent_row_indices(array):
"""
Returns the indices corresponding to an independent set of rows
for a given array. The independent rows are determined from the pivots
used during row reduction/Gaussian elimination.

Parameters
----------
array : 2D numpy array of floats
The input array

Returns
-------
indices : 1D numpy array of integers
The indices corresponding to a set of independent rows
of the input array.
"""
m = Matrix(array.shape[0], array.shape[1],
lambda i, j: Rational(array[i, j]).limit_denominator(1000))
_, pivots, swaps = row_reduce(m, iszerofunc=lambda x: x.is_zero,
simpfunc=lambda x: Rational(x).limit_denominator(1000))
indices = np.array(range(len(array)))
for swap in np.array(swaps):
indices[swap] = indices[swap[::-1]]
return sorted(indices[:len(pivots)])

[docs]def array_to_rational_matrix(array):
"""
Converts a numpy array into a sympy matrix
filled with rationals
"""
return Matrix([[Rational(v).limit_denominator(1000)
for v in row]
for row in array])

[docs]def generate_complete_basis(incomplete_basis, array):
"""
Given a 2D array with independent rows and a second 2D array that spans a
larger space, creates a complete basis for the combined array using all
the rows of the first array, followed by any required rows of the
second array. So, for example, if the first array is:
[[1, 0, 0], [1, 1, 0]] and the second array is:
[[1, 0, 0], [0, 1, 0], [0, 0, 1]], the complete basis will be:
[[1, 0, 0], [1, 1, 0], [0, 0, 1]].

Parameters
----------
incomplete_basis : 2D numpy array
An array containing the basis to be completed.

array : 2D numpy array
An array spanning the full space for which a basis is required.

Returns
-------
complete_basis : 2D numpy array
An array containing the basis vectors spanning both of the
input arrays.
"""

incomplete_rank = array_to_rational_matrix(incomplete_basis).rank()
if incomplete_rank < len(incomplete_basis):
raise Exception('The incomplete basis is rank-deficient. '
'Remove one or more endmembers.')

a = np.concatenate((incomplete_basis, array))
complete_basis = np.array(a[independent_row_indices(a)],
dtype=float)

# Store the rank of the matrix for later comparison
len_basis = array_to_rational_matrix(complete_basis).rank()

# This next step ensures that all of the original
# rows are contained in the new basis in their original order
c = np.linalg.lstsq(np.array(complete_basis).astype(float).T,
np.array(incomplete_basis).astype(float).T,
rcond=None)[0].T

for row in np.eye(len(c[0])):
old_rank = array_to_rational_matrix(c).rank()
c2 = np.concatenate((c, [row]))
new_rank = array_to_rational_matrix(c2).rank()

if new_rank > old_rank:
c = c2

complete_basis = c.dot(complete_basis)

# Check that the matrix rank has not changed
if len_basis != array_to_rational_matrix(complete_basis).rank():
raise Exception('Basis length changed during conversion. '
'Report this bug to developers.')

return complete_basis.round(decimals=12) + 0.