"""
ANALYSIS CLASS FOR THE SC SAMPLER
"""
import numpy as np
import chaospy as cp
from itertools import product, chain, combinations
import pickle
import copy
from easyvvuq import OutputType
from .base import BaseAnalysisElement
from .results import AnalysisResults
import logging
# from scipy.special import comb
import pandas as pd
__author__ = "Wouter Edeling"
__copyright__ = """
Copyright 2018 Robin A. Richardson, David W. Wright
This file is part of EasyVVUQ
EasyVVUQ is free software: you can redistribute it and/or modify
it under the terms of the Lesser GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
EasyVVUQ is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
Lesser GNU General Public License for more details.
You should have received a copy of the Lesser GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
"""
__license__ = "LGPL"
[docs]class SCAnalysisResults(AnalysisResults):
def _get_sobols_first(self, qoi, input_):
raw_dict = AnalysisResults._keys_to_tuples(self.raw_data['sobols_first'])
result = raw_dict[AnalysisResults._to_tuple(qoi)][input_]
try:
return np.array([float(result)])
except TypeError:
return np.array(result)
[docs] def supported_stats(self):
"""Types of statistics supported by the describe method.
Returns
-------
list of str
"""
return ['mean', 'var', 'std']
def _describe(self, qoi, statistic):
if statistic in self.supported_stats():
return self.raw_data['statistical_moments'][qoi][statistic]
else:
raise NotImplementedError
[docs] def surrogate(self):
"""Return an SC surrogate model.
Returns
-------
A function that takes a dictionary of parameter - value pairs and returns
a dictionary with the results (same output as decoder).
"""
def surrogate_fn(inputs):
def swap(x):
if len(x) > 1:
return list(x)
else:
return x[0]
values = np.squeeze(np.array([inputs[key] for key in self.inputs])).T
results = dict([(qoi, swap(self.surrogate_(qoi, values))) for qoi in self.qois])
return results
return surrogate_fn
[docs]class SCAnalysis(BaseAnalysisElement):
def __init__(self, sampler=None, qoi_cols=None):
"""
Parameters
----------
sampler : SCSampler
Sampler used to initiate the SC analysis
qoi_cols : list or None
Column names for quantities of interest (for which analysis is
performed).
"""
if sampler is None:
msg = 'SC analysis requires a paired sampler to be passed'
raise RuntimeError(msg)
if qoi_cols is None:
raise RuntimeError("Analysis element requires a list of "
"quantities of interest (qoi)")
self.qoi_cols = qoi_cols
self.output_type = OutputType.SUMMARY
self.sampler = sampler
self.dimension_adaptive = sampler.dimension_adaptive
if self.dimension_adaptive:
self.adaptation_errors = []
self.mean_history = []
self.std_history = []
self.sparse = sampler.sparse
self.pce_coefs = {}
self.N_qoi = {}
for qoi_k in qoi_cols:
self.pce_coefs[qoi_k] = {}
self.N_qoi[qoi_k] = 0
[docs] def element_name(self):
"""Name for this element for logging purposes"""
return "SC_Analysis"
[docs] def element_version(self):
"""Version of this element for logging purposes"""
return "0.5"
[docs] def save_state(self, filename):
"""Saves the complete state of the analysis object to a pickle file,
except the sampler object (self.samples).
Parameters
----------
filename : string
name to the file to write the state to
"""
logging.debug("Saving analysis state to %s" % filename)
# make a copy of the state, and do not store the sampler as well
state = copy.copy(self.__dict__)
del state['sampler']
file = open(filename, 'wb')
pickle.dump(state, file)
file.close()
[docs] def load_state(self, filename):
"""Loads the complete state of the analysis object from a
pickle file, stored using save_state.
Parameters
----------
filename : string
name of the file to load
"""
logging.debug("Loading analysis state from %s" % filename)
file = open(filename, 'rb')
state = pickle.load(file)
for key in state.keys():
self.__dict__[key] = state[key]
file.close()
[docs] def analyse(self, data_frame=None, compute_moments=True, compute_Sobols=True):
"""Perform SC analysis on input `data_frame`.
Parameters
----------
data_frame : pandas.DataFrame
Input data for analysis.
Returns
-------
dict
Results dictionary with sub-dicts with keys:
['statistical_moments', 'sobol_indices'].
Each dict has an entry for each item in `qoi_cols`.
"""
if data_frame is None:
raise RuntimeError("Analysis element needs a data frame to "
"analyse")
elif isinstance(data_frame, pd.DataFrame) and data_frame.empty:
raise RuntimeError(
"No data in data frame passed to analyse element")
# the number of uncertain parameters
self.N = self.sampler.N
# tensor grid
self.xi_d = self.sampler.xi_d
# the maximum level (quad order) of the (sparse) grid
self.L = self.sampler.L
# if L < L_min: quadratures and interpolations are zero
# For full tensor grid: there is only one level: L_min = L
if not self.sparse:
self.L_min = self.L
self.l_norm = np.array([self.sampler.polynomial_order])
self.l_norm_min = self.l_norm
# For sparse grid: one or more levels
else:
self.L_min = 1
# multi indices (stored in l_norm) for isotropic sparse grid or
# dimension-adaptive grid before the 1st refinement.
# If dimension_adaptive and nadaptations > 0: l_norm
# is computed in self.adaptation_metric
if not self.dimension_adaptive or self.sampler.nadaptations == 0:
# the maximum level (quad order) of the (sparse) grid
self.l_norm = self.sampler.compute_sparse_multi_idx(self.L, self.N)
self.l_norm_min = np.ones(self.N, dtype=int)
# #compute generalized combination coefficients
self.comb_coef = self.compute_comb_coef()
# 1d weights and points per level
self.xi_1d = self.sampler.xi_1d
# self.wi_1d = self.compute_SC_weights(rule=self.sampler.quad_rule)
self.wi_1d = self.sampler.wi_1d
# Extract output values for each quantity of interest from Dataframe
logging.debug('Loading samples...')
qoi_cols = self.qoi_cols
samples = {k: [] for k in qoi_cols}
for run_id in data_frame[('run_id', 0)].unique():
for k in qoi_cols:
values = data_frame.loc[data_frame[('run_id', 0)] == run_id][k].values
samples[k].append(values.flatten())
self.samples = samples
logging.debug('done')
# assume that self.l_norm has changed, and that the interpolation
# must be initialised, see sc_expansion subroutine
self.init_interpolation = True
# same pce coefs must be computed for every qoi
if self.sparse:
for qoi_k in qoi_cols:
self.pce_coefs[qoi_k] = self.SC2PCE(samples[qoi_k], qoi_k)
# size of one code sample
self.N_qoi[qoi_k] = self.samples[qoi_k][0].size
# Compute descriptive statistics for each quantity of interest
results = {'statistical_moments': {},
'sobols_first': {k: {} for k in self.qoi_cols},
'sobols': {k: {} for k in self.qoi_cols}}
if compute_moments:
for qoi_k in qoi_cols:
if not self.sparse:
mean_k, var_k = self.get_moments(qoi_k)
std_k = np.sqrt(var_k)
else:
self.pce_coefs[qoi_k] = self.SC2PCE(self.samples[qoi_k], qoi_k)
mean_k, var_k, _ = self.get_pce_stats(self.l_norm, self.pce_coefs[qoi_k],
self.comb_coef)
std_k = np.sqrt(var_k)
# compute statistical moments
results['statistical_moments'][qoi_k] = {'mean': mean_k,
'var': var_k,
'std': std_k}
if compute_Sobols:
for qoi_k in qoi_cols:
if not self.sparse:
results['sobols'][qoi_k] = self.get_sobol_indices(qoi_k, 'first_order')
else:
_, _, _, results['sobols'][qoi_k] = self.get_pce_sobol_indices(
qoi_k, 'first_order')
for idx, param_name in enumerate(self.sampler.vary.get_keys()):
results['sobols_first'][qoi_k][param_name] = \
results['sobols'][qoi_k][(idx,)]
results = SCAnalysisResults(raw_data=results, samples=data_frame,
qois=qoi_cols, inputs=list(self.sampler.vary.get_keys()))
results.surrogate_ = self.surrogate
return results
[docs] def compute_comb_coef(self, **kwargs):
"""Compute general combination coefficients. These are the coefficients
multiplying the tensor products associated to each multi index l,
see page 12 Gerstner & Griebel, numerical integration using sparse grids
"""
if 'l_norm' in kwargs:
l_norm = kwargs['l_norm']
else:
l_norm = self.l_norm
comb_coef = {}
logging.debug('Computing combination coefficients...')
for k in l_norm:
coef = 0.0
# for every k, subtract all multi indices
for l in l_norm:
z = l - k
# if the results contains only 0's and 1's, then z is the
# vector that can be formed from a tensor product of unit vectors
# for which k+z is in self.l_norm
if np.array_equal(z, z.astype(bool)):
coef += (-1)**(np.sum(z))
comb_coef[tuple(k)] = coef
logging.debug('done')
return comb_coef
[docs] def adapt_dimension(self, qoi, data_frame, store_stats_history=True,
method='surplus', **kwargs):
"""Compute the adaptation metric and decide which of the admissible
level indices to include in next iteration of the sparse grid. The
adaptation metric is based on the hierarchical surplus, defined as the
difference between the new code values of the admissible level indices,
and the SC surrogate of the previous iteration. Alternatively, it can be
based on the difference between the output mean of the current level,
and the mean computed with one extra admissible index.
This subroutine must be called AFTER the code is evaluated at
the new points, but BEFORE the analysis is performed.
Parameters
----------
qoi : string
the name of the quantity of interest which is used
to base the adaptation metric on.
data_frame : pandas.DataFrame
store_stats_history : bool
store the mean and variance at each refinement in self.mean_history
and self.std_history. Used for checking convergence in the statistics
over the refinement iterations
method : string
name of the refinement error, default is 'surplus'. In this case the
error is based on the hierarchical surplus, which is an interpolation
based error. Another possibility is 'var',
in which case the error is based on the difference in the
variance between the current estimate and the estimate obtained
when a particular candidate direction is added.
"""
logging.debug('Refining sampling plan...')
# load the code samples
samples = []
if isinstance(data_frame, pd.DataFrame):
for run_id in data_frame[('run_id', 0)].unique():
values = data_frame.loc[data_frame[('run_id', 0)] == run_id][qoi].values
samples.append(values.flatten())
if method == 'var':
all_idx = np.concatenate((self.l_norm, self.sampler.admissible_idx))
self.xi_1d = self.sampler.xi_1d
self.wi_1d = self.sampler.wi_1d
self.pce_coefs[qoi] = self.SC2PCE(samples, qoi, verbose=True, l_norm=all_idx,
xi_d=self.sampler.xi_d)
_, var_l, _ = self.get_pce_stats(self.l_norm, self.pce_coefs[qoi], self.comb_coef)
# the currently accepted grid points
xi_d_accepted = self.sampler.generate_grid(self.l_norm)
# compute the hierarchical surplus based error for every admissible l
error = {}
for l in self.sampler.admissible_idx:
error[tuple(l)] = []
# compute the error based on the hierarchical surplus (interpolation based)
if method == 'surplus':
# collocation points of current level index l
X_l = [self.sampler.xi_1d[n][l[n]] for n in range(self.N)]
X_l = np.array(list(product(*X_l)))
# only consider new points, subtract the accepted points
X_l = setdiff2d(X_l, xi_d_accepted)
for xi in X_l:
# find the location of the current xi in the global grid
idx = np.where((xi == self.sampler.xi_d).all(axis=1))[0][0]
# hierarchical surplus error at xi
hier_surplus = samples[idx] - self.surrogate(qoi, xi)
if 'index' in kwargs:
hier_surplus = hier_surplus[kwargs['index']]
error[tuple(l)].append(np.abs(hier_surplus))
else:
error[tuple(l)].append(np.linalg.norm(hier_surplus, np.inf))
# compute mean error over all points in X_l
error[tuple(l)] = np.mean(error[tuple(l)])
# compute the error based on quadrature of the variance
elif method == 'var':
# create a candidate set of multi indices by adding the current
# admissible index to l_norm
candidate_l_norm = np.concatenate((self.l_norm, l.reshape([1, self.N])))
# now we must recompute the combination coefficients
c_l = self.compute_comb_coef(l_norm=candidate_l_norm)
_, var_candidate_l, _ = self.get_pce_stats(
candidate_l_norm, self.pce_coefs[qoi], c_l)
# error in var
error[tuple(l)] = np.linalg.norm(var_candidate_l - var_l, np.inf)
else:
logging.debug('Specified refinement method %s not recognized' % method)
logging.debug('Accepted are surplus, mean or var')
import sys
sys.exit()
for key in error.keys():
# logging.debug("Surplus error when l = %s is %s" % (key, error[key]))
logging.debug("Refinement error for l = %s is %s" % (key, error[key]))
# find the admissble index with the largest error
l_star = np.array(max(error, key=error.get)).reshape([1, self.N])
# logging.debug('Selecting %s for refinement.' % l_star)
logging.debug('Selecting %s for refinement.' % l_star)
# add max error to list
self.adaptation_errors.append(max(error.values()))
# add l_star to the current accepted level indices
self.l_norm = np.concatenate((self.l_norm, l_star))
# if someone executes this function twice for some reason,
# remove the duplicate l_star entry. Keep order unaltered
idx = np.unique(self.l_norm, axis=0, return_index=True)[1]
self.l_norm = np.array([self.l_norm[i] for i in sorted(idx)])
# peform the analyse step, but do not compute moments and Sobols
self.analyse(data_frame, compute_moments=False, compute_Sobols=False)
# if True store the mean and variance at eacht iteration of the adaptive
# algorithmn
if store_stats_history:
# mean_f, var_f = self.get_moments(qoi)
logging.debug('Storing moments of iteration %d' % self.sampler.nadaptations)
pce_coefs = self.SC2PCE(samples, qoi, verbose=True)
mean_f, var_f, _ = self.get_pce_stats(self.l_norm, pce_coefs, self.comb_coef)
self.mean_history.append(mean_f)
self.std_history.append(var_f)
logging.debug('done')
[docs] def merge_accepted_and_admissible(self, level=0, **kwargs):
"""In the case of the dimension-adaptive sampler, there are 2 sets of
quadrature multi indices. There are the accepted indices that are actually
used in the analysis, and the admissible indices, of which some might
move to the accepted set in subsequent iterations. This subroutine merges
the two sets of multi indices by moving all admissible to the set of
accepted indices.
Do this at the end, when no more refinements will be executed. The
samples related to the admissble indices are already computed, although
not used in the analysis. By executing this subroutine at very end, all
computed samples are used during the final postprocessing stage. Execute
campaign.apply_analysis to let the new set of indices take effect.
If further refinements are executed after all via sampler.look_ahead, the
number of new admissible samples to be computed can be very high,
especially in high dimensions. It is possible to undo the merge via
analysis.undo_merge before new refinements are made. Execute
campaign.apply_analysis again to let the old set of indices take effect.
"""
if 'include' in kwargs:
include = kwargs['include']
else:
include = np.arange(self.N)
if self.sampler.dimension_adaptive:
logging.debug('Moving admissible indices to the accepted set...')
# make a backup of l_norm, such that undo_merge can revert back
self.l_norm_backup = np.copy(self.l_norm)
# merge admissible and accepted multi indices
if level == 0:
merged_l = np.concatenate((self.l_norm, self.sampler.admissible_idx))
else:
admissible_idx = []
count = 0
for l in self.sampler.admissible_idx:
L = np.sum(l) - self.N + 1
tmp = np.where(l == L)[0]
if L <= level and np.in1d(tmp, include)[0]:
admissible_idx.append(l)
count += 1
admissible_idx = np.array(admissible_idx).reshape([count, self.N])
merged_l = np.concatenate((self.l_norm, admissible_idx))
# make sure final result contains only unique indices and store
# results in l_norm
idx = np.unique(merged_l, axis=0, return_index=True)[1]
# return np.array([merged_l[i] for i in sorted(idx)])
self.l_norm = np.array([merged_l[i] for i in sorted(idx)])
logging.debug('done')
[docs] def undo_merge(self):
"""This reverses the effect of the merge_accepted_and_admissble subroutine.
Execute if further refinement are required after all.
"""
if self.sampler.dimension_adaptive:
self.l_norm = self.l_norm_backup
logging.debug('Restored old multi indices.')
[docs] def get_adaptation_errors(self):
"""Returns self.adaptation_errors
"""
return self.adaptation_errors
[docs] def plot_stat_convergence(self):
"""Plots the convergence of the statistical mean and std dev over the different
refinements in a dimension-adaptive setting. Specifically the inf norm
of the difference between the stats of iteration i and iteration i-1
is plotted.
"""
if not self.dimension_adaptive:
logging.debug('Only works for the dimension adaptive sampler.')
return
K = len(self.mean_history)
if K < 2:
logging.debug('Means from at least two refinements are required')
return
else:
differ_mean = np.zeros(K - 1)
differ_std = np.zeros(K - 1)
for i in range(1, K):
differ_mean[i - 1] = np.linalg.norm(self.mean_history[i] -
self.mean_history[i - 1], np.inf)
# make relative
differ_mean[i - 1] = differ_mean[i - 1] / np.linalg.norm(self.mean_history[i - 1],
np.inf)
differ_std[i - 1] = np.linalg.norm(self.std_history[i] -
self.std_history[i - 1], np.inf)
# make relative
differ_std[i - 1] = differ_std[i - 1] / np.linalg.norm(self.std_history[i - 1],
np.inf)
import matplotlib.pyplot as plt
fig = plt.figure('stat_conv')
ax1 = fig.add_subplot(111, title='moment convergence')
ax1.set_xlabel('iteration', fontsize=12)
# ax1.set_ylabel(r'$ ||\mathrm{mean}_i - \mathrm{mean}_{i - 1}||_\infty$',
# color='r', fontsize=12)
ax1.set_ylabel(r'relative error mean', color='r', fontsize=12)
ax1.plot(range(2, K + 1), differ_mean, color='r', marker='+')
ax1.tick_params(axis='y', labelcolor='r')
ax2 = ax1.twinx() # instantiate a second axes that shares the same x-axis
# ax2.set_ylabel(r'$ ||\mathrm{var}_i - \mathrm{var}_{i - 1}||_\infty$',
# color='b', fontsize=12)
ax2.set_ylabel(r'relative error variance', fontsize=12, color='b')
ax2.plot(range(2, K + 1), differ_std, color='b', marker='*')
ax2.tick_params(axis='y', labelcolor='b')
plt.tight_layout()
plt.show()
[docs] def surrogate(self, qoi, x, L=None):
"""Use sc_expansion UQP as a surrogate
Parameters
----------
qoi : str
name of the qoi
x : array
location at which to evaluate the surrogate
L : int
level of the (sparse) grid, default = self.L
Returns
-------
the interpolated value of qoi at x (float, (N_qoi,))
"""
return self.sc_expansion(self.samples[qoi], x=x)
[docs] def quadrature(self, qoi, samples=None):
"""Computes a (Smolyak) quadrature
Parameters
----------
qoi : str
name of the qoi
samples: array
compute the mean by setting samples = self.samples.
To compute the variance, set samples = (self.samples - mean)**2
Returns
-------
the quadrature of qoi
"""
if samples is None:
samples = self.samples[qoi]
return self.combination_technique(qoi, samples)
[docs] def combination_technique(self, qoi, samples=None, **kwargs):
"""Efficient quadrature formulation for (sparse) grids. See:
Gerstner, Griebel, "Numerical integration using sparse grids"
Uses the general combination technique (page 12).
Parameters
----------
qoi : str
name of the qoi
samples : array
compute the mean by setting samples = self.samples.
To compute the variance, set samples = (self.samples - mean)**2
"""
if samples is None:
samples = self.samples[qoi]
# In the case of quadrature-based refinement, we need to specify
# l_norm, comb_coef and xi_d other than the current defualt values
if 'l_norm' in kwargs:
l_norm = kwargs['l_norm']
else:
l_norm = self.l_norm
if 'comb_coef' in kwargs:
comb_coef = kwargs['comb_coef']
else:
comb_coef = self.comb_coef
if 'xi_d' in kwargs:
xi_d = kwargs['xi_d']
else:
xi_d = self.xi_d
# quadrature Q
Q = 0.0
# loop over l
for l in l_norm:
# compute the tensor product of parameter and weight values
X_k = [self.xi_1d[n][l[n]] for n in range(self.N)]
W_k = [self.wi_1d[n][l[n]] for n in range(self.N)]
X_k = np.array(list(product(*X_k)))
W_k = np.array(list(product(*W_k)))
W_k = np.prod(W_k, axis=1)
W_k = W_k.reshape([W_k.shape[0], 1])
# scaling factor of combination technique
W_k = W_k * comb_coef[tuple(l)]
# find corresponding code values
f_k = np.array([samples[np.where((x == xi_d).all(axis=1))[0][0]] for x in X_k])
# quadrature of Q^1_{k1} X ... X Q^1_{kN} product
Q = Q + np.sum(f_k * W_k, axis=0).T
return Q
[docs] def get_moments(self, qoi):
"""
Parameters
----------
qoi : str
name of the qoi
Returns
-------
mean and variance of qoi (float (N_qoi,))
"""
logging.debug('Computing moments...')
# compute mean
mean_f = self.quadrature(qoi)
# compute variance
variance_samples = [(sample - mean_f)**2 for sample in self.samples[qoi]]
var_f = self.quadrature(qoi, samples=variance_samples)
logging.debug('done')
return mean_f, var_f
[docs] def sc_expansion(self, samples, x):
"""
Non recursive implementation of the SC expansion. Performs interpolation
of code output samples for both full and sparse grids.
Parameters
----------
samples : list
list of code output samples.
x : array
One or more locations in stochastic space at which to evaluate
the surrogate.
Returns
-------
surr : array
The interpolated values of the code output at input locations
specified by x.
"""
# Computing the tensor grid of each multiindex l (xi_d below)
# every time is slow. Instead store it globally, and only recompute when
# self.l_norm has changed, when the flag init_interpolation = True.
# This flag is set to True when self.analyse is executed
if self.init_interpolation:
self.xi_d_per_l = {}
for l in self.l_norm:
# all points corresponding to l
xi = [self.xi_1d[n][l[n]] for n in range(self.N)]
self.xi_d_per_l[tuple(l)] = np.array(list(product(*xi)))
self.init_interpolation = False
surr = 0.0
for l in self.l_norm:
# all points corresponding to l
# xi = [self.xi_1d[n][l[n]] for n in range(self.N)]
# xi_d = np.array(list(product(*xi)))
xi_d = self.xi_d_per_l[tuple(l)]
for xi in xi_d:
# indices of current collocation point
# in corresponding 1d colloc points (self.xi_1d[n][l[n]])
# These are the j of the 1D lagrange polynomials l_j(x), see
# lagrange_poly subroutine
idx = [(self.xi_1d[n][l[n]] == xi[n]).nonzero()[0][0] for n in range(self.N)]
# index of the code sample
sample_idx = np.where((xi == self.xi_d).all(axis=1))[0][0]
# values of Lagrange polynomials at x
if x.ndim == 1:
weight = [lagrange_poly(x[n], self.xi_1d[n][l[n]], idx[n])
for n in range(self.N)]
surr += self.comb_coef[tuple(l)] * samples[sample_idx] * np.prod(weight, axis=0)
# batch setting, if multiple x values are presribed
else:
weight = [lagrange_poly(x[:, n], self.xi_1d[n][l[n]], idx[n])
for n in range(self.N)]
surr += self.comb_coef[tuple(l)] * samples[sample_idx] * \
np.prod(weight, axis=0).reshape([-1, 1])
return surr
[docs] def get_sample_array(self, qoi):
"""
Parameters
----------
qoi : str
name of quantity of interest
Returns
-------
array of all samples of qoi
"""
return np.array([self.samples[qoi][k] for k in range(len(self.samples[qoi]))])
[docs] def adaptation_histogram(self):
"""Plots a bar chart of the maximum order of the quadrature rule
that is used in each dimension. Use in case of the dimension adaptive
sampler to get an idea of which parameters were more refined than others.
This gives only a first-order idea, as it only plots the max quad
order independently per input parameter, so higher-order refinements
that were made do not show up in the bar chart.
"""
import matplotlib.pyplot as plt
fig = plt.figure('adapt_hist', figsize=[4, 8])
ax = fig.add_subplot(111, ylabel='max quadrature order',
title='Number of refinements = %d'
% self.sampler.nadaptations)
# find max quad order for every parameter
adapt_measure = np.max(self.l_norm, axis=0)
ax.bar(range(adapt_measure.size), height=adapt_measure - 1)
params = list(self.sampler.vary.get_keys())
ax.set_xticks(range(adapt_measure.size))
ax.set_xticklabels(params)
plt.xticks(rotation=90)
plt.tight_layout()
plt.show()
[docs] def adaptation_table(self, **kwargs):
"""Plots a color-coded table of the quadrature-order refinement.
Shows in what order the parameters were refined, and unlike
adaptation_histogram, this also shows higher-order refinements.
Parameters
----------
**kwargs: can contain kwarg 'order' to specify the order in which
the variables on the x axis are plotted (e.g. in order of decreasing
1st order Sobol index).
Returns
-------
None.
"""
# if specified, plot the variables on the x axis in a given order
if 'order' in kwargs:
order = kwargs['order']
else:
order = range(self.N)
l = np.copy(self.l_norm)[:, order]
import matplotlib as mpl
import matplotlib.pyplot as plt
fig = plt.figure(figsize=[12, 6])
ax = fig.add_subplot(111)
# max quad order
M = np.max(l)
cmap = plt.get_cmap('Purples', M)
# plot 'heat map' of refinement
plt.imshow(l.T, cmap=cmap, aspect='auto')
norm = mpl.colors.Normalize(vmin=0, vmax=M - 1)
sm = plt.cm.ScalarMappable(cmap=cmap, norm=norm)
sm.set_array([])
cb = plt.colorbar(sm)
# plot the quad order in the middle of the colorbar intervals
p = np.linspace(0, M - 1, M + 1)
tick_p = 0.5 * (p[1:] + p[0:-1])
cb.set_ticks(tick_p)
cb.set_ticklabels(np.arange(M))
cb.set_label(r'quadrature order')
# plot the variables names on the x axis
ax.set_yticks(range(l.shape[1]))
params = np.array(list(self.sampler.vary.get_keys()))
ax.set_yticklabels(params[order], fontsize=12)
# ax.set_yticks(range(l.shape[0]))
ax.set_xlabel('iteration')
# plt.yticks(rotation=90)
plt.tight_layout()
plt.show()
[docs] def plot_grid(self):
"""Plots the collocation points for 2 and 3 dimensional problems
"""
import matplotlib.pyplot as plt
if self.N == 2:
fig = plt.figure()
ax = fig.add_subplot(111, xlabel=r'$x_1$', ylabel=r'$x_2$')
ax.plot(self.xi_d[:, 0], self.xi_d[:, 1], 'ro')
plt.tight_layout()
plt.show()
elif self.N == 3:
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d', xlabel=r'$x_1$',
ylabel=r'$x_2$', zlabel=r'$x_3$')
ax.scatter(self.xi_d[:, 0], self.xi_d[:, 1], self.xi_d[:, 2])
plt.tight_layout()
plt.show()
else:
logging.debug('Will only plot for N = 2 or N = 3.')
[docs] def SC2PCE(self, samples, qoi, verbose=True, **kwargs):
"""Computes the Polynomials Chaos Expansion coefficients from the SC
expansion via a transformation of basis (Lagrange polynomials basis -->
orthonomial basis).
Parameters
----------
samples : array
SC code samples from which to compute the PCE coefficients
qoi : string
Name of the QoI.
Returns
-------
pce_coefs : dict
PCE coefficients per multi index l
"""
pce_coefs = {}
if 'l_norm' in kwargs:
l_norm = kwargs['l_norm']
else:
l_norm = self.l_norm
if 'xi_d' in kwargs:
xi_d = kwargs['xi_d']
else:
xi_d = self.xi_d
# if not hasattr(self, 'pce_coefs'):
# self.pce_coefs = {}
count_l = 1
for l in l_norm:
if not tuple(l) in self.pce_coefs[qoi].keys():
# pce coefficients for current multi-index l
pce_coefs[tuple(l)] = {}
# 1d points generated by l
x_1d = [self.xi_1d[n][l[n]] for n in range(self.N)]
# 1d Lagrange polynomials generated by l
# EDIT: do not use chaospy for Lagrange, converts lagrange into monomial, requires
# Vandermonde matrix inversion to find coefficients, which becomes
# very ill conditioned rather quickly. Can no longer use cp.E to compute
# integrals, use GQ instead
# a_1d = [cp.lagrange_polynomial(sampler.xi_1d[n][l[n]]) for n in range(d)]
# N-dimensional grid generated by l
x_l = np.array(list(product(*x_1d)))
# all multi indices of the PCE expansion: k <= l
k_norm = list(product(*[np.arange(1, l[n] + 1) for n in range(self.N)]))
if verbose:
logging.debug('Computing PCE coefficients %d / %d' % (count_l, l_norm.shape[0]))
for k in k_norm:
# product of the PCE basis function or order k - 1 and all
# Lagrange basis functions in a_1d, per dimension
# [[phi_k[0]*a_1d[0]], ..., [phi_k[N-1]*a_1d[N-1]]]
# orthogonal polynomial generated by chaospy
phi_k = [cp.expansion.stieltjes(k[n] - 1,
dist=self.sampler.params_distribution[n],
normed=True)[-1] for n in range(self.N)]
# the polynomial order of each integrand phi_k*a_j = (k - 1) + (number of
# colloc. points - 1)
orders = [(k[n] - 1) + (self.xi_1d[n][l[n]].size - 1) for n in range(self.N)]
# will hold the products of PCE basis functions phi_k and lagrange
# interpolation polynomials a_1d
cross_prod = []
for n in range(self.N):
# GQ using n points can exactly integrate polynomials of order 2n-1:
# solve for required number of points n when given order
n_quad_points = int(np.ceil((orders[n] + 1) / 2))
# generate Gaussian quad rule
if isinstance(self.sampler.params_distribution[n], cp.DiscreteUniform):
xi = self.xi_1d[n][l[n]]
wi = self.wi_1d[n][l[n]]
else:
xi, wi = cp.generate_quadrature(
n_quad_points - 1, self.sampler.params_distribution[n], rule="G")
xi = xi[0]
# number of colloc points = number of Lagrange polynomials
n_lagrange_poly = int(self.xi_1d[n][l[n]].size)
# compute the v coefficients = coefficients of SC2PCE mapping
v_coefs_n = []
for j in range(n_lagrange_poly):
# compute values of Lagrange polys at quadrature points
l_j = np.array([lagrange_poly(xi[i], self.xi_1d[n][l[n]], j)
for i in range(xi.size)])
# each coef is the integral of the lagrange poly times the current
# orthogonal PCE poly
v_coefs_n.append(np.sum(l_j * phi_k[n](xi) * wi))
cross_prod.append(v_coefs_n)
# tensor product of all integrals
integrals = np.array(list(product(*cross_prod)))
# multiply over the number of parameters: v_prod = v_k1_j1 * ... * v_kd_jd
v_prod = np.prod(integrals, axis=1)
v_prod = v_prod.reshape([v_prod.size, 1])
# find corresponding code values
f_k = np.array([samples[np.where((x == xi_d).all(axis=1))[0][0]] for x in x_l])
# the sum of all code sample * v_{k,j_1} * ... * v_{k,j_N}
# equals the PCE coefficient
eta_k = np.sum(f_k * v_prod, axis=0).T
pce_coefs[tuple(l)][tuple(k)] = eta_k
else:
# pce coefs previously computed, just copy result
pce_coefs[tuple(l)] = self.pce_coefs[qoi][tuple(l)]
count_l += 1
logging.debug('done')
return pce_coefs
[docs] def generalized_pce_coefs(self, l_norm, pce_coefs, comb_coef):
"""
Computes the generalized PCE coefficients, defined as the linear combibation
of PCE coefficients which make it possible to write the dimension-adaptive
PCE expansion in standard form. See DOI: 10.13140/RG.2.2.18085.58083/1
Parameters
----------
l_norm : array
array of quadrature order multi indices
pce_coefs : tuple
tuple of PCE coefficients computed by SC2PCE subroutine
comb_coef : tuple
tuple of combination coefficients computed by compute_comb_coef
Returns
-------
gen_pce_coefs : tuple
The generalized PCE coefficients, indexed per multi index.
"""
assert self.sparse, "Generalized PCE coeffcients are computed only for sparse grids"
# the set of all forward neighbours of l: {k | k >= l}
F_l = {}
# the generalized PCE coefs, which turn the adaptive PCE into a standard PCE expansion
gen_pce_coefs = {}
for l in l_norm:
# {indices of k | k >= l}
idx = np.where((l <= l_norm).all(axis=1))[0]
F_l[tuple(l)] = l_norm[idx]
# the generalized PCE coefs are comb_coef[k] * pce_coefs[k][l], summed over k
# for a fixed l
gen_pce_coefs[tuple(l)] = 0.0
for k in F_l[tuple(l)]:
gen_pce_coefs[tuple(l)] += comb_coef[tuple(k)] * pce_coefs[tuple(k)][tuple(l)]
return gen_pce_coefs
[docs] def get_pce_stats(self, l_norm, pce_coefs, comb_coef):
"""Compute the mean and the variance based on the generalized PCE coefficients
See DOI: 10.13140/RG.2.2.18085.58083/1
Parameters
----------
l_norm : array
array of quadrature order multi indices
pce_coefs : tuple
tuple of PCE coefficients computed by SC2PCE subroutine
comb_coef : tuple
tuple of combination coefficients computed by compute_comb_coef
Returns
-------
tuple with mean and variance based on the PCE coefficients
"""
gen_pce_coefs = self.generalized_pce_coefs(l_norm, pce_coefs, comb_coef)
# with the generalized pce coefs, the standard PCE formulas for the mean and var
# can be used for the dimension-adaptive PCE
# the PCE mean is just the 1st generalized PCE coef
l1 = tuple(np.ones(self.N, dtype=int))
mean = gen_pce_coefs[l1]
# the variance is the sum of the squared generalized PCE coefs, excluding the 1st coef
D = 0.0
for l in l_norm[1:]:
D += gen_pce_coefs[tuple(l)] ** 2
return mean, D, gen_pce_coefs
[docs] def get_pce_sobol_indices(self, qoi, typ='first_order', **kwargs):
"""Computes Sobol indices using Polynomials Chaos coefficients. These
coefficients are computed from the SC expansion via a transformation
of basis (SC2PCE subroutine). This works better than computing the
Sobol indices directly from the SC expansion in the case of the
dimension-adaptive sampler. See DOI: 10.13140/RG.2.2.18085.58083/1
Method: J.D. Jakeman et al, "Adaptive multi-index collocation
for uncertainty quantification and sensitivity analysis", 2019.
(Page 18)
Parameters
----------
qoi : str
name of the Quantity of Interest for which to compute the indices
typ : str
Default = 'first_order'. 'all' is also possible
**kwargs : dict
if this contains 'samples', use these instead of the SC samples ]
in the database
Returns
-------
Tuple
Mean: PCE mean
Var: PCE variance
S_u: PCE Sobol indices, either the first order indices or all indices
"""
if 'samples' in kwargs:
samples = kwargs['samples']
N_qoi = samples[0].size
else:
samples = self.samples[qoi]
N_qoi = self.N_qoi[qoi]
# compute the (generalized) PCE coefficients and stats
self.pce_coefs[qoi] = self.SC2PCE(samples, qoi)
mean, D, gen_pce_coefs = self.get_pce_stats(
self.l_norm, self.pce_coefs[qoi], self.comb_coef)
logging.debug('Computing Sobol indices...')
# Universe = (0, 1, ..., N - 1)
U = np.arange(self.N)
# the powerset of U for either the first order or all Sobol indices
if typ == 'first_order':
P = [()]
for i in range(self.N):
P.append((i,))
else:
# all indices u
P = list(powerset(U))
# dict to hold the partial Sobol variances and Sobol indices
D_u = {}
S_u = {}
for u in P[1:]:
# complement of u
u_prime = np.delete(U, u)
k = []
D_u[u] = np.zeros(N_qoi)
S_u[u] = np.zeros(N_qoi)
# compute the set of multi indices corresponding to varying ONLY
# the inputs indexed by u
for l in self.l_norm:
# assume l_i = 1 for all i in u' until found otherwise
all_ones = True
for i_up in u_prime:
if l[i_up] != 1:
all_ones = False
break
# if l_i = 1 for all i in u'
if all_ones:
# assume all l_i for i in u are > 1
all_gt_one = True
for i_u in u:
if l[i_u] == 1:
all_gt_one = False
break
# if both conditions above are True, the current l varies
# only inputs indexed by u, add this l to k
if all_gt_one:
k.append(l)
logging.debug('Multi indices of dimension %s are %s' % (u, k))
# the partial variance of u is the sum of all variances index by k
for k_u in k:
D_u[u] = D_u[u] + gen_pce_coefs[tuple(k_u)] ** 2
# normalize D_u by total variance D to get the Sobol index
S_u[u] = D_u[u] / D
logging.debug('done')
return mean, D, D_u, S_u
# Start SC specific methods
[docs] @staticmethod
def compute_tensor_prod_u(xi, wi, u, u_prime):
"""
Calculate tensor products of weights and collocation points
with dimension of u and u'
Parameters
----------
xi (array of floats): 1D colloction points
wi (array of floats): 1D quadrature weights
u (array of int): dimensions
u_prime (array of int): remaining dimensions (u union u' = range(N))
Returns
dict of tensor products of weight and points for dimensions u and u'
-------
"""
# tensor products with dimension of u
xi_u = [xi[key] for key in u]
wi_u = [wi[key] for key in u]
xi_d_u = np.array(list(product(*xi_u)))
wi_d_u = np.array(list(product(*wi_u)))
# tensor products with dimension of u' (complement of u)
xi_u_prime = [xi[key] for key in u_prime]
wi_u_prime = [wi[key] for key in u_prime]
xi_d_u_prime = np.array(list(product(*xi_u_prime)))
wi_d_u_prime = np.array(list(product(*wi_u_prime)))
return xi_d_u, wi_d_u, xi_d_u_prime, wi_d_u_prime
[docs] def compute_marginal(self, qoi, u, u_prime, diff):
"""
Computes a marginal integral of the qoi(x) over the dimension defined
by u_prime, for every x value in dimensions u
Parameters
----------
- qoi (str): name of the quantity of interest
- u (array of int): dimensions which are not integrated
- u_prime (array of int): dimensions which are integrated
- diff (array of int): levels
Returns
- Values of the marginal integral
-------
"""
# 1d weights and points of the levels in diff
xi = [self.xi_1d[n][np.abs(diff)[n]] for n in range(self.N)]
wi = [self.wi_1d[n][np.abs(diff)[n]] for n in range(self.N)]
# compute tensor products and weights in dimension u and u'
xi_d_u, wi_d_u, xi_d_u_prime, wi_d_u_prime =\
self.compute_tensor_prod_u(xi, wi, u, u_prime)
idxs = np.argsort(np.concatenate((u, u_prime)))
# marginals h = f*w' integrated over u', so cardinality is that of u
h = [0.0] * xi_d_u.shape[0]
for i_u, xi_d_u_ in enumerate(xi_d_u):
for i_up, xi_d_u_prime_ in enumerate(xi_d_u_prime):
xi_s = np.concatenate((xi_d_u_, xi_d_u_prime_))[idxs]
# find the index of the corresponding code sample
idx = np.where(np.prod(self.xi_d == xi_s, axis=1))[0][0]
# perform quadrature
q_k = self.samples[qoi][idx]
h[i_u] += q_k * wi_d_u_prime[i_up].prod()
# return marginal and the weights of dimensions u
return h, wi_d_u
[docs] def get_sobol_indices(self, qoi, typ='first_order'):
"""
Computes Sobol indices using Stochastic Collocation. Method:
Tang (2009), GLOBAL SENSITIVITY ANALYSIS FOR STOCHASTIC COLLOCATION
EXPANSION.
Parameters
----------
qoi (str): name of the Quantity of Interest for which to compute the indices
typ (str): Default = 'first_order'. 'all' is also possible
Returns
-------
Either the first order or all Sobol indices of qoi
"""
logging.debug('Computing Sobol indices...')
# multi indices
U = np.arange(self.N)
if typ == 'first_order':
P = list(powerset(U))[0:self.N + 1]
elif typ == 'all':
# all indices u
P = list(powerset(U))
# get first two moments
mu, D = self.get_moments(qoi)
# partial variances
D_u = {P[0]: mu**2}
sobol = {}
for u in P[1:]:
# complement of u
u_prime = np.delete(U, u)
D_u[u] = 0.0
for l in self.l_norm:
# expand the multi-index indices of the tensor product
# (Q^1_{i1} - Q^1_{i1-1}) X ... X (Q^1_{id) - Q^1_{id-1})
diff_idx = np.array(list(product(*[[k, -(k - 1)] for k in l])))
# perform analysis on each Q^1_l1 X ... X Q^1_l_N tensor prod
for diff in diff_idx:
# if any Q^1_li is below the minimim level, Q^1_li is defined
# as zero: do not compute this Q^1_l1 X ... X Q^1_l_N tensor prod
if not (np.abs(diff) < self.l_norm_min).any():
# mariginal integral h, integrate over dimensions u'
h, wi_d_u = self.compute_marginal(qoi, u, u_prime, diff)
# square result and integrate over remaining dimensions u
for i_u in range(wi_d_u.shape[0]):
D_u[u] += np.sign(np.prod(diff)) * h[i_u]**2 * wi_d_u[i_u].prod()
# D_u[u] = D_u[u].flatten()
# all subsets of u
W = list(powerset(u))[0:-1]
# partial variance of u
for w in W:
D_u[u] -= D_u[w]
# compute Sobol index, only include points where D > 0
# sobol[u] = D_u[u][idx_gt0]/D[idx_gt0]
sobol[u] = D_u[u] / D
logging.debug('done.')
return sobol
[docs] def get_uncertainty_amplification(self, qoi):
"""
Computes a measure that signifies the ratio of output to input
uncertainty. It is computed as the (mean) Coefficient of Variation (V)
of the output divided by the (mean) CV of the input.
Parameters
----------
qoi (string): name of the Quantity of Interest
Returns
-------
blowup (float): the ratio output CV / input CV
"""
mean_f, var_f = self.get_moments(qoi)
std_f = np.sqrt(var_f)
mean_xi = []
std_xi = []
CV_xi = []
for param in self.sampler.params_distribution:
E = cp.E(param)
Std = cp.Std(param)
mean_xi.append(E)
std_xi.append(Std)
CV_xi.append(Std / E)
CV_in = np.mean(CV_xi)
CV_out = std_f / mean_f
idx = np.where(np.isnan(CV_out) == False)[0]
CV_out = np.mean(CV_out[idx])
blowup = CV_out / CV_in
print('-----------------')
print('Mean CV input = %.4f %%' % (100 * CV_in, ))
print('Mean CV output = %.4f %%' % (100 * CV_out, ))
print('Uncertainty amplification factor = %.4f/%.4f = %.4f' %
(CV_out, CV_in, blowup))
print('-----------------')
return blowup
[docs]def powerset(iterable):
"""powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)
Taken from: https://docs.python.org/3/library/itertools.html#recipes
Parameters
----------
iterable : iterable
Input sequence
Returns
-------
"""
s = list(iterable)
return chain.from_iterable(combinations(s, r) for r in range(len(s) + 1))
[docs]def lagrange_poly(x, x_i, j):
"""Lagrange polynomials used for interpolation
l_j(x) = product(x - x_m / x_j - x_m) with 0 <= m <= k
and m !=j
Parameters
----------
x : float
location at which to compute the polynomial
x_i : list or array of float
nodes of the Lagrange polynomials
j : int
index of node at which l_j(x_j) = 1
Returns
-------
float
l_j(x) calculated as shown above.
"""
l_j = 1.0
for m in range(len(x_i)):
if m != j:
denom = x_i[j] - x_i[m]
nom = x - x_i[m]
l_j *= nom / denom
return l_j
# implementation below is more beautiful, but slower
# x_i_ = np.delete(x_i, j)
# return np.prod((x - x_i_) / (x_i[j] - x_i_))
[docs]def setdiff2d(X, Y):
"""
Computes the difference of two 2D arrays X and Y
Parameters
----------
X : 2D numpy array
Y : 2D numpy array
Returns
-------
The difference X \\ Y as a 2D array
"""
diff = set(map(tuple, X)) - set(map(tuple, Y))
return np.array(list(diff))
