def calculate_uqsa_measures(joint_dist, polynomial, alpha=5):
""" Use chaospy to calculate appropriate indices of uq and sa"""
dists = joint_dist
mean = cp.E(polynomial, dists)
var = cp.Var(polynomial, dists)
std = cp.Std(polynomial, dists)
conInt = cp.Perc(polynomial, [alpha / 2., 100 - alpha / 2.], joint_dist)
sens_m = cp.Sens_m(polynomial, dists)
sens_m2 = cp.Sens_m2(polynomial, dists)
sens_t = cp.Sens_t(polynomial, dists)
return dict(mean=mean,
var=var,
std=std,
conInt=conInt,
sens_m=sens_m,
sens_m2=sens_m2,
sens_t=sens_t)
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
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
results['output_distributions'][k] = cp.QoI_Dist(
fit, self.sampler.distribution)
return PCEAnalysisResults(raw_data=results,
samples=data_frame,
qois=self.qoi_cols,
inputs=list(self.sampler.vary.get_keys()))
Ns_pc = 200
polynomial_order = 3
# calculate sensitivity indices with gpc
Spc, Stpc, gpce_reg = polynomial_chaos_sens(Ns_pc, jpdf, polynomial_order,return_reg=True)
# compare the computations
import pandas as pd
row_labels = ['X_'+str(x) for x in range(1,N_terms*2+1)]
S=np.column_stack((Sa,Spc,Smc,Sta,Stpc,Stmc))
S_table = pd.DataFrame(S, columns=['Sa','Spc','Smc','Sta','Stpc','Stmc'], index=row_labels)
print(S_table.round(3))
# Second order indices with gpc
S2 = cp.Sens_m2(gpce_reg, jpdf) # second order indices with gpc
# print all second order indices
print(pd.DataFrame(S2,columns=row_labels,index=row_labels).round(3))
# sum all second order indices
SumS2=np.sum(np.triu(S2))
print('\nSum Sij = {:2.2f}'.format(SumS2))
# sum all first and second order indices
print('Sum Si + Sij = {:2.2f}\n'.format(np.sum(Spc)+SumS2))
# compare nonzero second order indices with analytical indices
Szw_pc=[S2[i,i+N_terms] for i in range(N_terms) ]
Szw_table=np.column_stack((Szw_pc,Szw,(Szw_pc-Szw)/Szw))
print(pd.DataFrame(Szw_table,columns=['Szw','Szw pc','Error%']).round(3))
def analyse(self, data_frame=None):
"""Perform PCE analysis on input `data_frame`.
Parameters
----------
data_frame : :obj:`pandas.DataFrame`
Input data for analysis.
Returns
-------
dict:
Contains analysis results in sub-dicts with keys -
['statistical_moments', 'percentiles', 'sobol_indices',
'correlation_matrices', 'output_distributions']
"""
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': {},
}
# Get the Polynomial
P = self.sampler.P
# Get the PCE variante to use (Regression or Projection)
regression = self.sampler.regression
# Compute nodes (and weights)
if regression:
nodes = cp.generate_samples(order=self.sampler.n_samples,
domain=self.sampler.distribution,
rule=self.sampler.rule)
else:
nodes, weights = cp.generate_quadrature(
order=self.sampler.quad_order,
dist=self.sampler.distribution,
rule=self.sampler.rule,
sparse=self.sampler.quad_sparse,
growth=self.sampler.quad_growth)
# 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.unique():
for k in qoi_cols:
data = data_frame.loc[data_frame['run_id'] == run_id][k]
samples[k].append(data.values)
# Compute descriptive statistics for each quantity of interest
for k in qoi_cols:
# Approximation solver
if regression:
if samples[k][0].dtype == object:
for i in range(self.sampler.count):
samples[k][i] = samples[k][i].astype("float64")
fit = cp.fit_regression(P, nodes, samples[k], "T")
else:
fit = cp.fit_quadrature(P, nodes, weights, samples[k])
# 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
}
# Percentiles (Pxx)
P10 = cp.Perc(fit, 10, self.sampler.distribution)
P90 = cp.Perc(fit, 90, self.sampler.distribution)
results['percentiles'][k] = {'p10': P10, 'p90': P90}
# 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 = {}
ipar = 0
i = 0
for param_name in self.sampler.vary.get_keys():
j = self.sampler.params_size[ipar]
sobols_first_dict[param_name] = sobols_first_narr[i:i + j]
sobols_second_dict[param_name] = sobols_second_narr[i:i + j]
sobols_total_dict[param_name] = sobols_total_narr[i:i + j]
i += j
ipar += 1
results['sobols_first'][k] = sobols_first_dict
results['sobols_second'][k] = sobols_second_dict
results['sobols_total'][k] = sobols_total_dict
# Correlation matrix
results['correlation_matrices'][k] = cp.Corr(
fit, self.sampler.distribution)
# Output distributions
results['output_distributions'][k] = cp.QoI_Dist(
fit, self.sampler.distribution)
return results
sai_file.write("\n")
sai_file.write("Total-effect index \n" )
#np.savetxt(sai_file, sait.T, fmt=['%.5f','%.5f', '%.5f','%.5f','%.5f','%.5f'], header = " b_ff, b_xx, b_fx, C, K, fr_epi")
#np.savetxt(sai_file, sait.T, fmt=['%.5f','%.5f', '%.5f','%.5f','%.5f','%.5f','%.5f','%.5f','%.5f'], header = " b_ff, b_xx, b_fx, C, K, fr_epi,fr_endo, sr_epi, sr_endo")
sai_file.write("\n")
sai_file.write("\n")
#output displacement/stress metrics
#sai1 = cp.Sens_m(qoi_hat[0],joint)
u_mean = cp.E(qoi2_hat, joint_KL)
u_var = cp.Var(qoi2_hat, joint_KL)
u_stdv = np.sqrt(u_var)
u_sai1 = cp.Sens_m(qoi2_hat,joint_KL)
u_sai2 = cp.Sens_m2(qoi2_hat,joint_KL)
u_sait = cp.Sens_t(qoi2_hat,joint_KL)
sff_mean = cp.E(qoi3_hat, joint_KL)
sff_var = cp.Var(qoi3_hat, joint_KL)
sff_stdv = np.sqrt(sff_var)
sff_sai1 = cp.Sens_m(qoi3_hat,joint_KL)
sff_sai2 = cp.Sens_m2(qoi3_hat,joint_KL)
sff_sait = cp.Sens_t(qoi3_hat,joint_KL)
# save displacement/stress results
umean.vector()[:] = u_mean
ustdv.vector()[:] = u_stdv
ums.vector()[:] = u_mean + u_stdv
umean_file << umean
holdout_file.write("Polynomial order: %i \n" % order)
holdout_file.write("Num. test samples: %i \n" % ntests)
holdout_file.write("Test sampling rule: %s \n" % test_rule)
holdout_file.write("\n")
# np.savetxt(holdout_file, mse, fmt=['%.3f','%.3f', '%.3f','%.3f','%.3f','%.3f','%.3f'], delimiter = " ", header = "dVol, dLen, dRen, dRep, dThi, dTwi, dWV")
holdout_file.write("Predictive error: \n")
holdout_file.writelines(["%s\n" % item for item in pe])
txt_file.write("\n")
holdout_file.write("Mean Square error: \n")
holdout_file.writelines(["%s\n" % item for item in mse])
# output sensitivity index
sai1 = cp.Sens_m(qoi1_hat, joint)
sait = cp.Sens_t(qoi1_hat, joint)
sai2 = cp.Sens_m2(qoi1_hat, joint)
# save sensitivity index
sai_file.write("Pressure (kPa) = %f \n" % pressure)
sai_file.write("First order Sensitivity index \n")
# np.savetxt(sai_file, sai1.T, fmt=['%.5f','%.5f', '%.5f','%.5f','%.5f','%.5f'], header = " b_ff, b_xx, b_fx, C, K, fr_epi")
np.savetxt(sai_file, sai1.T, fmt=['%.5f', '%.5f', '%.5f', '%.5f', '%.5f', '%.5f', '%.5f', '%.5f', '%.5f'],
header=" b_ff, b_xx, b_fx, C, K, fr_epi, fr_endo, sr_epi, sr_endo")
sai_file.write("\n")
sai_file.write("Total-effect index \n")
# np.savetxt(sai_file, sait.T, fmt=['%.5f','%.5f', '%.5f','%.5f','%.5f','%.5f'], header = " b_ff, b_xx, b_fx, C, K, fr_epi")
np.savetxt(sai_file, sait.T, fmt=['%.5f', '%.5f', '%.5f', '%.5f', '%.5f', '%.5f', '%.5f', '%.5f', '%.5f'],
header=" b_ff, b_xx, b_fx, C, K, fr_epi,fr_endo, sr_epi, sr_endo")
sai_file.write("\n")
def analyse(self, data_frame=None):
"""Perform PCE analysis on input `data_frame`.
Parameters
----------
data_frame : :obj:`pandas.DataFrame`
Input data for analysis.
Returns
-------
dict:
Contains analysis results in sub-dicts with keys -
['statistical_moments', 'percentiles', 'sobol_indices',
'correlation_matrices', 'output_distributions']
"""
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': {},
}
# 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())
# Compute descriptive statistics for each quantity of interest
for k in qoi_cols:
# Approximation solver
if regression:
fit = cp.fit_regression(P, nodes, samples[k])
else:
fit = cp.fit_quadrature(P, nodes, weights, samples[k])
# 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
}
# Percentiles: 10% and 90%
P10 = cp.Perc(fit, 10, self.sampler.distribution)
P90 = cp.Perc(fit, 90, self.sampler.distribution)
results['percentiles'][k] = {'p10': P10, 'p90': P90}
# 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
# Correlation matrix
results['correlation_matrices'][k] = cp.Corr(
fit, self.sampler.distribution)
# Output distributions
results['output_distributions'][k] = cp.QoI_Dist(
fit, self.sampler.distribution)
return PCEAnalysisResults(raw_data=results,
samples=data_frame,
qois=self.qoi_cols,
inputs=list(self.sampler.vary.get_keys()))