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@tupui
Created December 7, 2018 16:11
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Visual explanation of Moment independent sensitivity analysis
r"""Visual explanation of Moment independent sensitivity analysis.
Moment-based method are based on the whole PDF to mitigate these
issues (Borgonovo2007). Based on the unconditional PDF, a conditional PDF per
parameter is computed. The more the conditional PDF deviates from the
unconditional PDF, the more the parameter has an impact on the quantity of
interest. The same procedure can be done using the Empirical Cumulative
Density Function (ECDF), respectively with the unconditional ECDF.
This visually shows this procedure. Bins of samples (red circles) are used to
compute a conditional PDF of the output. This PDF is compared to the
unconditional PDF (black).
Reference:
Borgonovo. A new uncertainty importance measure. RESS, 2007. DOI: 10.1016/j.ress.2006.04
---------------------------
MIT License
Copyright (c) 2018 Pamphile Tupui ROY
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
"""
import numpy as np
from scipy.stats import gaussian_kde
from batman.functions import Ishigami
from batman.space import Space
import matplotlib.pyplot as plt
p_labels = ['$x_1$', '$x_2$', '$x_3$']
sample = Space([[-np.pi, -np.pi, -np.pi], [np.pi, np.pi, np.pi]])
sample.sampling(1000)
n_dim = sample.shape[1]
func = Ishigami()
output = func(sample).flatten()
mini = np.min(output)
maxi = np.max(output)
n_bins = 10
bins = np.linspace(-np.pi, np.pi, num=n_bins, endpoint=False)
dx = bins[1] - bins[0]
# Moment explanation
# Unconditional PDF
xs = np.linspace(mini, maxi, 100)
pdf_u = gaussian_kde(output, bw_method="silverman")(xs)
fig, ax = plt.subplots(2, n_dim)
for i in range(n_dim):
xi = sample[:, i]
ax[0][i].scatter(xi, output, marker='+')
ax[0][i].set_xlabel(p_labels[i])
ax[1][i].plot(xs, pdf_u, c='k')
for bin_ in bins[:1]:
idx = np.where((bin_ <= xi) & (xi <= bin_ + dx))
xi_ = xi[idx]
y_ = output[idx]
ax[0][i].scatter(xi_, y_, c='r')
pdf_c = gaussian_kde(y_, bw_method="silverman")(xs)
ax[1][i].plot(xs, pdf_c, ls='--')
ax[0][0].set_ylabel('Y')
ax[1][0].set_ylabel('PDF')
ax[1][1].set_xlabel('Y')
plt.tight_layout()
plt.show()
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tupui commented Dec 7, 2018

capture d ecran 2018-12-07 a 17 11 06

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