"""Analysis element for polynomial chaos expansion (PCE). We use ChaosPy
under the hood for this functionality.
"""
import logging
import chaospy as cp
import numpy as np
from easyvvuq import OutputType
from .base import BaseAnalysisElement
from .results import AnalysisResults
from .qmc_analysis import QMCAnalysisResults
__author__ = 'Jalal Lakhlili'
__license__ = "LGPL"
logger = logging.getLogger(__name__)
[docs]class PCEAnalysisResults(QMCAnalysisResults):
"""Analysis results for the PCEAnalysis class.
"""
def _get_sobols_first(self, qoi, input_):
"""Returns the first order sobol index for a given qoi wrt input variable.
Parameters
----------
qoi : str
Quantity of interest
input_ : str
Input variable
Returns
-------
float
First order sobol index.
"""
raw_dict = AnalysisResults._keys_to_tuples(self.raw_data['sobols_first'])
return raw_dict[AnalysisResults._to_tuple(qoi)][input_]
def _get_sobols_second(self, qoi, input_):
"""Returns the second order sobol index for a given qoi wrt input variable.
Parameters
----------
qoi : str
Quantity of interest
input_ : str
Input variable
Returns
-------
float
Second order sobol index.
"""
raw_dict = AnalysisResults._keys_to_tuples(self.raw_data['sobols_second'])
return dict([(in_, raw_dict[AnalysisResults._to_tuple(qoi)][input_][i])
for i, in_ in enumerate(self.inputs) if in_ != input_])
def _get_sobols_total(self, qoi, input_):
"""Returns the total order sobol index for a given qoi wrt input variable.
Parameters
----------
qoi : str
Quantity of interest
input_ : str
Input variable
Returns
-------
float
Total order sobol index.
"""
raw_dict = AnalysisResults._keys_to_tuples(self.raw_data['sobols_total'])
return raw_dict[AnalysisResults._to_tuple(qoi)][input_]
[docs] def supported_stats(self):
"""Types of statistics supported by the describe method.
Returns
-------
list of str
"""
return ['min', 'max', '10%', '90%', '1%', '99%', 'median',
'mean', 'var', 'std']
def _describe(self, qoi, statistic):
"""Returns descriptive statistics, similar to pandas describe.
Parameters
----------
qoi : str
Name of quantity of interest.
statistic : str
One of 'min', 'max', '10%', '90%', 'median', 'mean', 'var', 'std'
Returns
-------
float
Value of the requested statistic.
"""
if statistic not in self.supported_stats():
raise NotImplementedError
if statistic == 'min':
return np.array([v.lower[0] for _, v in enumerate(
self.raw_data['output_distributions'][qoi])])
elif statistic == 'max':
return np.array([v.upper[0] for _, v in enumerate(
self.raw_data['output_distributions'][qoi])])
elif statistic == '1%':
return self.raw_data['percentiles'][qoi]['p01']
elif statistic == '10%':
return self.raw_data['percentiles'][qoi]['p10']
elif statistic == '90%':
return self.raw_data['percentiles'][qoi]['p90']
elif statistic == '99%':
return self.raw_data['percentiles'][qoi]['p99']
elif statistic == 'median':
return self.raw_data['percentiles'][qoi]['p50']
else:
try:
return self.raw_data['statistical_moments'][qoi][statistic]
except KeyError:
raise NotImplementedError
[docs] def surrogate(self):
"""Return a PCE 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.array([inputs[key] for key in self.inputs])
results = dict([(qoi, swap((self.raw_data['fit'][qoi](*values)).T))
for qoi in self.qois])
return results
return surrogate_fn
[docs] def get_distribution(self, qoi):
"""Returns a distribution for the given qoi.
Parameters
----------
qoi: str
QoI name
Returns
-------
A ChaosPy PDF
"""
if qoi not in self.qois:
raise RuntimeError('no such quantity of interest - {}'.format(qoi))
return self.raw_data['output_distributions'][qoi]
[docs]class PCEAnalysis(BaseAnalysisElement):
def __init__(self, sampler=None, qoi_cols=None, sampling=False):
"""Analysis element for polynomial chaos expansion (PCE).
Parameters
----------
sampler : PCESampler
Sampler used to initiate the PCE analysis.
qoi_cols : list or None
Column names for quantities of interest (for which analysis is
performed).
sampling : True if chaospy sampling method to be used for calculating
statitical quantities; otherwise [default] the pce coefficients are used
"""
if sampler is None:
msg = 'PCE 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.sampling = sampling
self.output_type = OutputType.SUMMARY
self.sampler = sampler
[docs] def element_name(self):
"""Name for this element for logging purposes.
Returns
-------
str
"PCE_Analysis"
"""
return "PCE_Analysis"
[docs] def element_version(self):
"""Version of this element for logging purposes.
Returns
-------
str
Element version.
"""
return "0.6"
[docs] def analyse(self, data_frame=None):
"""Perform PCE analysis on input `data_frame`.
Parameters
----------
data_frame : pandas DataFrame
Input data for analysis.
Returns
-------
PCEAnalysisResults
Use it to get the sobol indices and other information.
"""
def sobols(P, coefficients):
""" Utility routine to calculate sobols based on coefficients
"""
A = np.array(P.coefficients) != 0
multi_indices = np.array([P.exponents[A[:, i]].sum(axis=0) for i in range(A.shape[1])])
sobol_mask = multi_indices != 0
_, index = np.unique(sobol_mask, axis=0, return_index=True)
index = np.sort(index)
sobol_idx_bool = sobol_mask[index]
sobol_idx_bool = np.delete(sobol_idx_bool, [0], axis=0)
n_sobol_available = sobol_idx_bool.shape[0]
if len(coefficients.shape) == 1:
n_out = 1
else:
n_out = coefficients.shape[1]
n_coeffs = coefficients.shape[0]
sobol_poly_idx = np.zeros([n_coeffs, n_sobol_available])
for i_sobol in range(n_sobol_available):
sobol_poly_idx[:, i_sobol] = np.all(sobol_mask == sobol_idx_bool[i_sobol], axis=1)
sobol = np.zeros([n_sobol_available, n_out])
for i_sobol in range(n_sobol_available):
sobol[i_sobol] = np.sum(
np.square(coefficients[sobol_poly_idx[:, i_sobol] == 1]), axis=0)
idx_sort_descend_1st = np.argsort(sobol[:, 0], axis=0)[::-1]
sobol = sobol[idx_sort_descend_1st, :]
sobol_idx_bool = sobol_idx_bool[idx_sort_descend_1st]
sobol_idx = [0 for _ in range(sobol_idx_bool.shape[0])]
for i_sobol in range(sobol_idx_bool.shape[0]):
sobol_idx[i_sobol] = np.array(
[i for i, x in enumerate(sobol_idx_bool[i_sobol, :]) if x])
var = ((coefficients[1:]**2).sum(axis=0))
sobol = sobol / (var + np.finfo(float).tiny)
return sobol, sobol_idx, sobol_idx_bool
if data_frame is None:
raise RuntimeError("Analysis element needs a data frame to "
"analyse")
elif data_frame.empty:
raise RuntimeError(
"No data in data frame passed to analyse element")
qoi_cols = self.qoi_cols
results = {'statistical_moments': {},
'percentiles': {},
'sobols_first': {k: {} for k in qoi_cols},
'sobols_second': {k: {} for k in qoi_cols},
'sobols_total': {k: {} for k in qoi_cols},
'correlation_matrices': {},
'output_distributions': {},
'fit': {},
'Fourier_coefficients': {},
}
# Get sampler informations
P = self.sampler.P
nodes = self.sampler._nodes
weights = self.sampler._weights
regression = self.sampler.regression
# Extract output values for each quantity of interest from Dataframe
# samples = {k: [] for k in qoi_cols}
# for run_id in data_frame[('run_id', 0)].unique():
# for k in qoi_cols:
# data = data_frame.loc[data_frame[('run_id', 0)] == run_id][k]
# samples[k].append(data.values.flatten())
samples = {k: [] for k in qoi_cols}
for k in qoi_cols:
samples[k] = data_frame[k].values
# Compute descriptive statistics for each quantity of interest
for k in qoi_cols:
# Approximation solver
if regression:
fit, fc = cp.fit_regression(P, nodes, samples[k], retall=1)
else:
fit, fc = cp.fit_quadrature(P, nodes, weights, samples[k], retall=1)
results['fit'][k] = fit
results['Fourier_coefficients'][k] = fc
# Percentiles: 1%, 10%, 50%, 90% and 99%
P01, P10, P50, P90, P99 = cp.Perc(
fit, [1, 10, 50, 90, 99], self.sampler.distribution).squeeze()
results['percentiles'][k] = {'p01': P01, 'p10': P10, 'p50': P50, 'p90': P90, 'p99': P99}
if self.sampling: # use chaospy's sampling method
# Statistical moments
mean = cp.E(fit, self.sampler.distribution)
var = cp.Var(fit, self.sampler.distribution)
std = cp.Std(fit, self.sampler.distribution)
results['statistical_moments'][k] = {'mean': mean,
'var': var,
'std': std}
# Sensitivity Analysis: First, Second and Total Sobol indices
sobols_first_narr = cp.Sens_m(fit, self.sampler.distribution)
sobols_second_narr = cp.Sens_m2(fit, self.sampler.distribution)
sobols_total_narr = cp.Sens_t(fit, self.sampler.distribution)
sobols_first_dict = {}
sobols_second_dict = {}
sobols_total_dict = {}
for i, param_name in enumerate(self.sampler.vary.vary_dict):
sobols_first_dict[param_name] = sobols_first_narr[i]
sobols_second_dict[param_name] = sobols_second_narr[i]
sobols_total_dict[param_name] = sobols_total_narr[i]
results['sobols_first'][k] = sobols_first_dict
results['sobols_second'][k] = sobols_second_dict
results['sobols_total'][k] = sobols_total_dict
else: # use PCE coefficients
# Statistical moments
mean = fc[0]
var = np.sum(fc[1:]**2, axis=0)
std = np.sqrt(var)
results['statistical_moments'][k] = {'mean': mean,
'var': var,
'std': std}
# Sensitivity Analysis: First, Second and Total Sobol indices
sobol, sobol_idx, _ = sobols(P, fc)
varied = [_ for _ in self.sampler.vary.get_keys()]
S1 = {_: np.zeros(sobol.shape[-1]) for _ in varied}
ST = {_: np.zeros(sobol.shape[-1]) for _ in varied}
# S2 = {_ : {__: np.zeros(sobol.shape[-1]) for __ in varied} for _ in varied}
# for v in varied: del S2[v][v]
S2 = {_: np.zeros((len(varied), sobol.shape[-1])) for _ in varied}
for n, si in enumerate(sobol_idx):
if len(si) == 1:
v = varied[si[0]]
S1[v] = sobol[n]
elif len(si) == 2:
v1 = varied[si[0]]
v2 = varied[si[1]]
# S2[v1][v2] = sobol[n]
# S2[v2][v1] = sobol[n]
S2[v1][si[1]] = sobol[n]
S2[v2][si[0]] = sobol[n]
for i in si:
ST[varied[i]] += sobol[n]
results['sobols_first'][k] = S1
results['sobols_second'][k] = S2
results['sobols_total'][k] = ST
# Correlation matrix
results['correlation_matrices'][k] = cp.Corr(
fit, self.sampler.distribution)
# Output distributions
try:
results['output_distributions'][k] = cp.QoI_Dist(
fit, self.sampler.distribution)
except Exception as e:
print('Error %s for %s' % (e.__class__.__name__, k))
# from traceback import print_exc
# print_exc()
results['output_distributions'][k] = None
return PCEAnalysisResults(raw_data=results, samples=data_frame,
qois=self.qoi_cols, inputs=list(self.sampler.vary.get_keys()))
