Sobol' variance based sensitivity indices based on Saltelli2010 in python

sobol_saltelli.py

"""Sobol' indices.

Compute Sobol' variance based sensitivity indices.
Use formulations from Saltelli2010.

Reference:

Saltelli et al. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index,
Computer Physics Communications, 2010. DOI: 10.1016/j.cpc.2009.09.018

---------------------------

MIT License

Copyright (c) 2018 Pamphile Tupui ROY

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"""
import numpy as np
import itertools


def f_ishigami(x):
    """Ishigami function.
    
    :param array_like x: [n, [x1, x2, x3]].
    :return: Function evaluation.
    :rtype: array_like (n, 1)
    """
    x = np.atleast_2d(x)
    return np.array([np.sin(xi[0]) + 7 * np.sin(xi[1])**2 + \
                     0.1 * (xi[2]**4) * np.sin(xi[0]) for xi in x]).reshape(-1, 1)


def sobol(func, n_sample, dim, *, bounds=None, seed=None):
    """Sobol' indices.

    The total number of function call is N(p+2).
    Three matrices are required for the computation of
    the indices: A, B and a permutation matrix AB based
    on both A and B.

    :param callable func: Function to analyse.
    :param int n_sample: Number of samples.
    :param int dim: Number of dimensions.
    :param array_like bounds: Desired range of transformed data.
      The transformation apply the bounds on the sample and not
      the theoretical space, unit cube. Thus min and
      max values of the sample will coincide with the bounds.
      ([min, n_features], [max, n_features]).
    :param seed: Seed of the random number generator.
    :return: first orders and total orders indices.
    :rtype: list(float) (n_features), list(float) (n_features)
    """
    rng = np.random.default_rng(seed)
    A = rng.random((n_sample, dim))
    B = rng.random((n_sample, dim))

    if bounds is not None:
        bounds = np.asarray(bounds)
        min_ = bounds.min(axis=0)
        max_ = bounds.max(axis=0)

        A = (max_ - min_) * A + min_
        B = (max_ - min_) * B + min_

    f_A = func(A)
    f_B = func(B)

    var = np.var(np.vstack([f_A, f_B]), axis=0)

    # Total effect of pairs of factors: generalization of Saltenis
    # st2 = np.zeros((dim, dim))
    # for j, i in itertools.combinations(range(0, dim), 2):
    #     st2[i, j] = 1 / (2 * n_sample) * np.sum((f_AB[i] - f_AB[j]) ** 2, axis=0) / var

    f_AB = []
    for i in range(dim):
        f_AB.append(func(np.column_stack((A[:, 0:i], B[:, i], A[:, i+1:]))))

    f_AB = np.array(f_AB).reshape(dim, n_sample)

    s = 1 / n_sample * np.sum(f_B * (np.subtract(f_AB, f_A.flatten()).T), axis=0) / var
    st = 1 / (2 * n_sample) * np.sum(np.subtract(f_A.flatten(), f_AB).T ** 2, axis=0) / var

    return s, st


print(f_ishigami([[0, 1, 3], [2, 4, 2]]))

indices = sobol(f_ishigami, 2000, 3, [[-np.pi, -np.pi, -np.pi], [np.pi, np.pi, np.pi]])
print(indices)

Have you implemented the second and third order of Sobol Indices in another code?
Thank you in advance!

Hi @marcrocasalonso, thanks for checking this out. I did not, but you can find second order in SaLib. Personally I would not try to get third orders as their interpretation is quite complicated and they require a lot of computation. If you did not check it out, I have other gists that calculate SA using moment independent method instead (takes into account other moments, not just the variance).

Hi Pamphile, thank you. I will take a look into your gists :)

HI Pamphile, sorry for all my questions!
With your experience and background, do you think that the best approach is "Cosine transformation sensitivity indice" for the first order of sensitivity indices? I was using first, second and Total Sobol Indices to quantify the influence of the parameters. But maybe Cosine transformation could be a better approach for only the First order... What do you think?
Regards,
Marc

HI Pamphile, sorry for all my questions!

No problem @marcrocasalonso 😄 (not sure if I need to ping you so you know I commented...)

With your experience and background, do you think that the best approach is "Cosine transformation sensitivity indice" for the first order of sensitivity indices? I was using first, second and Total Sobol Indices to quantify the influence of the parameters. But maybe Cosine transformation could be a better approach for only the First order... What do you think?

Personally I would stick with first and total indices of Sobol' (don't forget the ', it's a diacritic part of the name 😉 ). COSI did not receive much validation, hence it might be harder to justify. If you need more information, moment independent methods are also a good alternative. And in case of correlations you can also use Shapley values.

thank you :) , I will try to go deeper into what you have told me.

@tupui Thanks for the notebook. However, I have one question. How can I use correlated variables in the code? Your response will be highly appreciated.

Hi @samdanicivilruet, thanks 😊 As I said above, Sobol’ is not well suited for correlated inputs. That’s a base assumption of the method. Instead I would suggest you to have a look at the moment independent method. I have a gist showing how this work and also SALib had an implementation.

@tupui thanks again for the clarification. I really appreciate it. Best regards