Visual explanation of Moment independent sensitivity analysis

moment_independent_visually.py

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

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MIT License

Copyright (c) 2018 Pamphile Tupui ROY

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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()