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)

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