### Files

``````from __future__ import division
from __future__ import print_function

from scipy.stats import norm, gaussian_kde, rankdata

import numpy as np

from . import common_args

def analyze(problem, X, Y, num_resamples=10,
conf_level=0.95, print_to_console=False, seed=None):
"""Perform Delta Moment-Independent Analysis on model outputs.

Returns a dictionary with keys 'delta', 'delta_conf', 'S1', and 'S1_conf',
where each entry is a list of size D (the number of parameters) containing
the indices in the same order as the parameter file.

Parameters
----------
problem : dict
The problem definition
X: numpy.matrix
A NumPy matrix containing the model inputs
Y : numpy.array
A NumPy array containing the model outputs
num_resamples : int
The number of resamples when computing confidence intervals (default 10)
conf_level : float
The confidence interval level (default 0.95)
print_to_console : bool
Print results directly to console (default False)

References
----------
.. [1] Borgonovo, E. (2007). "A new uncertainty importance measure."
Reliability Engineering & System Safety, 92(6):771-784,
doi:10.1016/j.ress.2006.04.015.

.. [2] Plischke, E., E. Borgonovo, and C. L. Smith (2013). "Global
sensitivity measures from given data." European Journal of
Operational Research, 226(3):536-550, doi:10.1016/j.ejor.2012.11.047.

Examples
--------
>>> X = latin.sample(problem, 1000)
>>> Y = Ishigami.evaluate(X)
>>> Si = delta.analyze(problem, X, Y, print_to_console=True)
"""
if seed:
np.random.seed(seed)

D = problem['num_vars']
N = Y.size

if not 0 < conf_level < 1:
raise RuntimeError("Confidence level must be between 0-1.")

# equal frequency partition
exp = (2 / (7 + np.tanh((1500 - N) / 500)))
M = int(np.round( min(int(np.ceil(N**exp)), 48) ))
m = np.linspace(0, N, M + 1)
Ygrid = np.linspace(np.min(Y), np.max(Y), 100)

keys = ('delta', 'delta_conf', 'S1', 'S1_conf')
S = ResultDict((k, np.zeros(D)) for k in keys)
S['names'] = problem['names']

if print_to_console:
print("Parameter %s %s %s %s" % keys)

try:
for i in range(D):
X_i = X[:, i]
S['delta'][i], S['delta_conf'][i] = bias_reduced_delta(
Y, Ygrid, X_i, m, num_resamples, conf_level)
S['S1'][i] = sobol_first(Y, X_i, m)
S['S1_conf'][i] = sobol_first_conf(
Y, X_i, m, num_resamples, conf_level)
if print_to_console:
print("%s %f %f %f %f" % (S['names'][i], S['delta'][
i], S['delta_conf'][i], S['S1'][i], S['S1_conf'][i]))
except np.linalg.LinAlgError as e:
msg = "Singular matrix detected\n"
msg += "This may be due to the sample size ({}) being too small\n".format(Y.size)
msg += "If this is not the case, check Y values or raise an issue with the\n"
msg += "SALib team"

raise np.linalg.LinAlgError(msg)

return S

# Plischke et al. 2013 estimator (eqn 26) for d_hat

def calc_delta(Y, Ygrid, X, m):
N = len(Y)
fy = gaussian_kde(Y, bw_method='silverman')(Ygrid)
abs_fy = np.abs(fy)
xr = rankdata(X, method='ordinal')

d_hat = 0
for j in range(len(m) - 1):
ix = np.where((xr > m[j]) & (xr <= m[j + 1]))[0]
nm = len(ix)

Y_ix = Y[ix]
if not np.all(np.equal(Y_ix, Y_ix[0])):
fyc = gaussian_kde(Y_ix, bw_method='silverman')(Ygrid)
fy_ = np.abs(fy - fyc)
else:
fy_ = abs_fy

d_hat += (nm / (2 * N)) * np.trapz(fy_, Ygrid)

return d_hat

def bias_reduced_delta(Y, Ygrid, X, m, num_resamples, conf_level):
"""Plischke et al. 2013 bias reduction technique (eqn 30)"""
d = np.zeros(num_resamples)
d_hat = calc_delta(Y, Ygrid, X, m)

N = len(Y)
r = np.random.randint(N, size=(num_resamples, N))
for i in range(num_resamples):
r_i = r[i, :]
d[i] = calc_delta(Y[r_i], Ygrid, X[r_i], m)

d = 2 * d_hat - d
return (d.mean(), norm.ppf(0.5 + conf_level / 2) * d.std(ddof=1))

def sobol_first(Y, X, m):
xr = rankdata(X, method='ordinal')
Vi = 0
N = len(Y)
Y_mean = Y.mean()
for j in range(len(m) - 1):
ix = np.where((xr > m[j]) & (xr <= m[j + 1]))[0]
nm = len(ix)
Vi += (nm / N) * ((Y[ix].mean() - Y_mean)**2)
return Vi / np.var(Y)

def sobol_first_conf(Y, X, m, num_resamples, conf_level):
s = np.zeros(num_resamples)

N = len(Y)
r = np.random.randint(N, size=(num_resamples, N))
for i in range(num_resamples):
r_i = r[i, :]
s[i] = sobol_first(Y[r_i], X[r_i], m)

return norm.ppf(0.5 + conf_level / 2) * s.std(ddof=1)

def cli_parse(parser):
default=None,
help='Model input file')
default=10,
help='Number of bootstrap resamples for \
Sobol confidence intervals')
return parser

def cli_action(args):