"""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
import numpoly
import warnings
from easyvvuq import OutputType
from .base import BaseAnalysisElement
from .results import AnalysisResults
from .qmc_analysis import QMCAnalysisResults
import traceback
__author__ = 'Jalal Lakhlili'
__license__ = "LGPL"
logger = logging.getLogger(__name__)
[docs]
class PCEAnalysisResults(QMCAnalysisResults):
"""Analysis results for the PCEAnalysis class.
"""
def _get_derivatives_first(self, qoi, input_):
"""Returns the first order derivative-based index for a given qoi wrt input variable.
Parameters
----------
qoi : str
Quantity of interest
input_ : str
Input variable
Returns
-------
float
First order derivative-based index.
"""
raw_dict = AnalysisResults._keys_to_tuples(self.raw_data['derivatives_first'])
return raw_dict[AnalysisResults._to_tuple(qoi)][input_]
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%':
if isinstance(self.raw_data['percentiles'][qoi]['p01'], np.ndarray):
return self.raw_data['percentiles'][qoi]['p01']
else:
return np.array([self.raw_data['percentiles'][qoi]['p01']])
elif statistic == '10%':
if isinstance(self.raw_data['percentiles'][qoi]['p10'], np.ndarray):
return self.raw_data['percentiles'][qoi]['p10']
else:
return np.array([self.raw_data['percentiles'][qoi]['p10']])
elif statistic == '90%':
if isinstance(self.raw_data['percentiles'][qoi]['p90'], np.ndarray):
return self.raw_data['percentiles'][qoi]['p90']
else:
return np.array([self.raw_data['percentiles'][qoi]['p90']])
elif statistic == '99%':
if isinstance(self.raw_data['percentiles'][qoi]['p99'], np.ndarray):
return self.raw_data['percentiles'][qoi]['p99']
else:
return np.array([self.raw_data['percentiles'][qoi]['p99']])
elif statistic == 'median':
if isinstance(self.raw_data['percentiles'][qoi]['p50'], np.ndarray):
return self.raw_data['percentiles'][qoi]['p50']
else:
return np.array([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, CorrelationMatrices=True, OutputDistributions=True):
"""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
statistical quantities; otherwise [default] the pce coefficients are used
CorrelationMatrices : boolean
if False then disable the calculation of the Correlation Matrices, otherwise
[default] calculate them
OutputDistributions : boolean
if False then disable the calculation of the Output Distributions, otherwise
[default] calculate them
"""
if sampler is None:
msg = 'PCE analysis requires a paired sampler to be passed'
raise RuntimeError(msg)
# Flag specifing if we should scale the runs with the nominal base run
self.relative_analysis = sampler.relative_analysis
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
self.CorrelationMatrices = CorrelationMatrices
self.OutputDistributions = OutputDistributions
[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
def build_surrogate_der(Y_hat, verbose=False):
'''Computes derivative of the polynomial Y_hat w.r.t. Vars
Parameter T specifies the time dimension
'''
# Build derivative with respect to all variables
dim = len(self.sampler.vary.vary_dict)
if dim < 1:
return 0
elif dim == 1:
Vars = [cp.variable(dim).names[0]]
else:
Vars = [v.names[0] for v in cp.variable(dim)]
T = len(Y_hat)
assert(len(Vars) == len(self.sampler.vary.vary_dict))
# derivative of the PCE expansion
# {dYhat_dx1: [t0, t1, ...],
# dYhat_dx2: [t0, t1, ...],
# ...,
# dYhat_dxN: [t0, t1, ...] }
dY_hat = {v:[cp.polynomial(0) for t in range(T)] for v in self.sampler.vary.vary_dict}
for t in range(T):
for n1, n2 in zip(Y_hat[t].names, Vars):
assert(n1 == n2)
for d_var_idx, (d_var, d_var_app) in enumerate(zip(Vars, self.sampler.vary.vary_dict)):
if verbose:
print(f'Computing derivative d(Y_hat)/d({d_var})')
print('='*40)
# Some variables are missing in the expression,
# then they must be constant terms only i.e. sum(exp==0)
if Y_hat[t].exponents.shape[1] < dim:
#exponents.shape: (n_summands, n_variables)
assert(sum(sum(np.array(Y_hat[t].exponents))) == 0)
continue
# Consider only polynomial components var^exp where exp > 0 (since the derivative decreases exp by -1)
components_mask = np.array(Y_hat[t].exponents[:,d_var_idx] > 0)
dY_hat_dvar_exp = Y_hat[t].exponents[components_mask]
dY_hat_dvar_coeff = np.array(Y_hat[t].coefficients)[components_mask]
# Iterate over all polynomial components (summands)
for i, (coeff, exp) in enumerate(zip(dY_hat_dvar_coeff, dY_hat_dvar_exp)):
assert(exp[d_var_idx] > 0)
# derivative = coeff*exp * var^(exp-1)
dY_hat_dvar_coeff[i] = coeff * exp[d_var_idx]
dY_hat_dvar_exp[i][d_var_idx] = exp[d_var_idx] - 1
dY_hat[d_var_app][t] = numpoly.construct.polynomial_from_attributes(
exponents=dY_hat_dvar_exp,
coefficients=dY_hat_dvar_coeff,
names=Y_hat[t].names,
retain_coefficients=True,
retain_names=True)
return dY_hat
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
T = len(data_frame[qoi_cols[0]].values[-1])
results = {'statistical_moments': {k: {'mean':np.zeros(T),
'var':np.zeros(T),
'std':np.zeros(T)} for k in qoi_cols},
'percentiles': {k: {'p01': np.zeros(T),
'p10': np.zeros(T),
'p50': np.zeros(T),
'p90': np.zeros(T),
'p99': np.zeros(T)} for k in qoi_cols},
'sobols_first': {k: {p: np.zeros(T) for p in self.sampler.vary.vary_dict} for k in qoi_cols},
'sobols_second': {k: {p: np.zeros(T) for p in self.sampler.vary.vary_dict} for k in qoi_cols},
'sobols_total': {k: {p: np.zeros(T) for p in self.sampler.vary.vary_dict} for k in qoi_cols},
'correlation_matrices': {k: {} for k in qoi_cols},
'output_distributions': {k: {} for k in qoi_cols},
'fit': {k: cp.polynomial(np.zeros(T)) for k in qoi_cols},
'Fourier_coefficients': {k: {p: np.zeros(T) for p in self.sampler.vary.vary_dict} for k in qoi_cols},
'derivatives_first': {k: {p: np.zeros(T) for p in self.sampler.vary.vary_dict} for k in qoi_cols},
}
# Get sampler informations
P = self.sampler.P
nodes = self.sampler._nodes
weights = self.sampler._weights
regression = self.sampler.regression
samples = {k: [] for k in qoi_cols}
for k in qoi_cols:
if self.relative_analysis:
base = data_frame[k].values[self.sampler.n_samples]
if np.all(np.array(base) == 0):
warnings.warn(f"Removing QoI {k} from the analysis, contains all zeros", RuntimeWarning)
continue
if np.any(np.array(base) == 0):
warnings.warn(f"Removing QoI {k} from the analysis, contains some zeros", RuntimeWarning)
continue
samples[k] = data_frame[k].values[:self.sampler.n_samples]
# Compute descriptive statistics for each quantity of interest
if regression:
fit, fc = cp.fit_regression(P, [n[:self.sampler.n_samples] for n in 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}
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
# Sensitivity Analysis: Derivative based
try:
dY_hat = build_surrogate_der(fit, verbose=False)
derivatives_first_dict = {}
Ndimensions = len(self.sampler.vary.vary_dict)
for i, param_name in enumerate(self.sampler.vary.vary_dict):
if self.sampler.nominal_value:
# Evaluate dY_hat['param'] at the nominal value of the parameters
values = self.sampler.nominal_value
logging.info(f"Using nominal value of the parameters to evaluate the derivative ")
derivatives_first_dict[param_name] = cp.polynomial(dY_hat[param_name])(*[v for v in values.values()])
elif all([type(v) == type(cp.Normal()) for v in self.sampler.vary.vary_dict.values()]):
# Evaluate dY_hat['param'] at the mean of the parameters
logging.info(f"Using mean value of the parameters to evaluate the derivative ")
derivatives_first_dict[param_name] = cp.polynomial(dY_hat[param_name])(*[v.get_mom_parameters()["shift"][0] for v in self.sampler.vary.vary_dict.values()])
elif all([type(v) == type(cp.Uniform()) for v in self.sampler.vary.vary_dict.values()]):
logging.info(f"Using mean value of the parameters to evaluate the derivative ")
# Evaluate dY_hat['param'] at the mean of the parameters
derivatives_first_dict[param_name] = cp.polynomial(dY_hat[param_name])(*[(v.lower + v.upper)/2.0 for v in self.sampler.vary.vary_dict.values()])
else:
# Evaluate dY_hat['param'] at the zero vector
logging.info(f"Using zero vector to evaluate the derivative ")
derivatives_first_dict[param_name] = cp.polynomial(dY_hat[param_name])(*np.zeros(Ndimensions))
results['derivatives_first'][k] = derivatives_first_dict
except Exception:
traceback.print_exc()
# Transform the relative numbers back to the absolute values
if self.relative_analysis:
base = data_frame[k].values[-1]
results['percentiles'][k]['p01'] = (1.0 + results['percentiles'][k]['p01']) * base
results['percentiles'][k]['p10'] = (1.0 + results['percentiles'][k]['p10']) * base
results['percentiles'][k]['p50'] = (1.0 + results['percentiles'][k]['p50']) * base
results['percentiles'][k]['p90'] = (1.0 + results['percentiles'][k]['p90']) * base
results['percentiles'][k]['p99'] = (1.0 + results['percentiles'][k]['p99']) * base
results['statistical_moments'][k]['mean'] = (1.0 + results['statistical_moments'][k]['mean']) * base
results['statistical_moments'][k]['var'] = (1.0 + results['statistical_moments'][k]['var']) * base
results['statistical_moments'][k]['std'] = (1.0 + results['statistical_moments'][k]['std']) * base
# Correlation matrix
try:
if self.sampler._is_dependent:
warnings.warn(f"Skipping computation of cp.Corr", RuntimeWarning)
results['correlation_matrices'][k] = None
else:
if self.CorrelationMatrices:
results['correlation_matrices'][k] = cp.Corr(fit, self.sampler.distribution)
else:
warnings.warn(f"Skipping computation of cp.Corr", RuntimeWarning)
results['correlation_matrices'][k] = None
except Exception as e:
print ('Error %s for %s when computing cp.Corr()'% (e.__class__.__name__, k))
results['correlation_matrices'][k] = None
# Output distributions
try:
if self.sampler._is_dependent:
warnings.warn(f"Skipping computation of cp.QoI_Dist", RuntimeWarning)
results['output_distributions'][k] = None
else:
if self.OutputDistributions:
results['output_distributions'][k] = cp.QoI_Dist( fit, self.sampler.distribution)
else:
warnings.warn(f"Skipping computation of cp.QoI_Dist", RuntimeWarning)
results['output_distributions'][k] = None
except Exception as e:
print ('Error %s for %s when computing cp.QoI_Dist()'% (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()))