s.
rng : `numpy.random.Generator`, optional
    Pseudorandom number generator state. When `rng` is None, a new
    `numpy.random.Generator` is created using entropy from the
    operating system. Types other than `numpy.random.Generator` are
    passed to `numpy.random.default_rng` to instantiate a ``Generator``.

    .. versionchanged:: 1.15.0

        As part of the `SPEC-007 <https://scientific-python.org/specs/spec-0007/>`_
        transition from use of `numpy.random.RandomState` to
        `numpy.random.Generator`, this keyword was changed from `random_state` to
        `rng`. For an interim period, both keywords will continue to work, although
        only one may be specified at a time. After the interim period, function
        calls using the `random_state` keyword will emit warnings. Following a
        deprecation period, the `random_state` keyword will be removed.

Returns
-------
res : SobolResult
    An object with attributes:

    first_order : ndarray of shape (s, d)
        First order Sobol' indices.
    total_order : ndarray of shape (s, d)
        Total order Sobol' indices.

    And method:

    bootstrap(confidence_level: float, n_resamples: int)
    -> BootstrapSobolResult

        A method providing confidence intervals on the indices.
        See `scipy.stats.bootstrap` for more details.

        The bootstrapping is done on both first and total order indices,
        and they are available in `BootstrapSobolResult` as attributes
        ``first_order`` and ``total_order``.

Notes
-----
The Sobol' method [1]_, [2]_ is a variance-based Sensitivity Analysis which
obtains the contribution of each parameter to the variance of the
quantities of interest (QoIs; i.e., the outputs of `func`).
Respective contributions can be used to rank the parameters and
also gauge the complexity of the model by computing the
model's effective (or mean) dimension.

.. note::

    Parameters are assumed to be independently distributed. Each
    parameter can still follow any distribution. In fact, the distribution
    is very important and should match the real distribution of the
    parameters.

It uses a functional decomposition of the variance of the function to
explore

.. math::

    \mathbb{V}(Y) = \sum_{i}^{d} \mathbb{V}_i (Y) + \sum_{i<j}^{d}
    \mathbb{V}_{ij}(Y) + ... + \mathbb{V}_{1,2,...,d}(Y),

introducing conditional variances:

.. math::

    \mathbb{V}_i(Y) = \mathbb{\mathbb{V}}[\mathbb{E}(Y|x_i)]
    \qquad
    \mathbb{V}_{ij}(Y) = \mathbb{\mathbb{V}}[\mathbb{E}(Y|x_i x_j)]
    - \mathbb{V}_i(Y) - \mathbb{V}_j(Y),

Sobol' indices are expressed as

.. math::

    S_i = \frac{\mathbb{V}_i(Y)}{\mathbb{V}[Y]}
    \qquad
    S_{ij} =\frac{\mathbb{V}_{ij}(Y)}{\mathbb{V}[Y]}.

:math:`S_{i}` corresponds to the first-order term which apprises the
contribution of the i-th parameter, while :math:`S_{ij}` corresponds to the
second-order term which informs about the contribution of interactions
between the i-th and the j-th parameters. These equations can be
generalized to compute higher order terms; however, they are expensive to
compute and their interpretation is complex.
This is why only first order indices are provided.

Total order indices represent the global contribution of the parameters
to the variance of the QoI and are defined as:

.. math::

    S_{T_i} = S_i + \sum_j S_{ij} + \sum_{j,k} S_{ijk} + ...
    = 1 - \frac{\mathbb{V}[\mathbb{E}(Y|x_{\sim i})]}{\mathbb{V}[Y]}.

First order indices sum to at most 1, while total order indices sum to at
least 1. If there are no interactions, then first and total order indices
are equal, and both first and total order indices sum to 1.

.. warning::

    Negative Sobol' values are due to numerical errors. Increasing the
    number of points `n` should help.

    The number of sample required to have a good analysis increases with
    the dimensionality of the problem. e.g. for a 3 dimension problem,
    consider at minima ``n >= 2**12``. The more complex the model is,
    the more samples will be needed.

    Even for a purely additive model, the indices may not sum to 1 due
    to numerical noise.

References
----------
.. [1] Sobol, I. M.. "Sensitivity analysis for nonlinear mathematical
   models." Mathematical Modeling and Computational Experiment, 1:407-414,
   1993.
.. [2] Sobol, I. M. (2001). "Global sensitivity indices for nonlinear
   mathematical models and their Monte Carlo estimates." Mathematics
   and Computers in Simulation, 55(1-3):271-280,
   :doi:`10.1016/S0378-4754(00)00270-6`, 2001.
.. [3] Saltelli, A. "Making best use of model evaluations to
   compute sensitivity indices."  Computer Physics Communications,
   145(2):280-297, :doi:`10.1016/S0010-4655(02)00280-1`, 2002.
.. [4] Saltelli, A., M. Ratto, T. Andres, F. Campolongo, J. Cariboni,
   D. Gatelli, M. Saisana, and S. Tarantola. "Global Sensitivity Analysis.
   The Primer." 2007.
.. [5] Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and
   S. Tarantola. "Variance based sensitivity analysis of model
   output. Design and estimator for the total sensitivity index."
   Computer Physics Communications, 181(2):259-270,
   :doi:`10.1016/j.cpc.2009.09.018`, 2010.
.. [6] Ishigami, T. and T. Homma. "An importance quantification technique
   in uncertainty analysis for computer models." IEEE,
   :doi:`10.1109/ISUMA.1990.151285`, 1990.

Examples
--------
The following is an example with the Ishigami function [6]_

.. math::

    Y(\mathbf{x}) = \sin x_1 + 7 \sin^2 x_2 + 0.1 x_3^4 \sin x_1,

with :math:`\mathbf{x} \in [-\pi, \pi]^3`. This function exhibits strong
non-linearity and non-monotonicity.

Remember, Sobol' indices assumes that samples are independently
distributed. In this case we use a uniform distribution on each marginals.

>>> import numpy as np
>>> from scipy.stats import sobol_indices, uniform
>>> rng = np.random.default_rng()
>>> def f_ishigami(x):
...     f_eval = (
...         np.sin(x[0])
...         + 7 * np.sin(x[1])**2
...         + 0.1 * (x[2]**4) * np.sin(x[0])
...     )
...     return f_eval
>>> indices = sobol_indices(
...     func=f_ishigami, n=1024,
...     dists=[
...         uniform(loc=-np.pi, scale=2*np.pi),
...         uniform(loc=-np.pi, scale=2*np.pi),
...         uniform(loc=-np.pi, scale=2*np.pi)
...     ],
...     rng=rng
... )
>>> indices.first_order
array([0.31637954, 0.43781162, 0.00318825])
>>> indices.total_order
array([0.56122127, 0.44287857, 0.24229595])

Confidence interval can be obtained using bootstrapping.

>>> boot = indices.bootstrap()

Then, this information can be easily visualized.

>>> import matplotlib.pyplot as plt
>>> fig, axs = plt.subplots(1, 2, figsize=(9, 4))
>>> _ = axs[0].errorbar(
...     [1, 2, 3], indices.first_order, fmt='o',
...     yerr=[
...         indices.first_order - boot.first_order.confidence_interval.low,
...         boot.first_order.confidence_interval.high - indices.first_order
...     ],
... )
>>> axs[0].set_ylabel("First order Sobol' indices")
>>> axs[0].set_xlabel('Input parameters')
>>> axs[0].set_xticks([1, 2, 3])
>>> _ = axs[1].errorbar(
...     [1, 2, 3], indices.total_order, fmt='o',
...     yerr=[
...         indices.total_order - boot.total_order.confidence_interval.low,
...         boot.total_order.confidence_interval.high - indices.total_order
...     ],
... )
>>> axs[1].set_ylabel("Total order Sobol' indices")
>>> axs[1].set_xlabel('Input parameters')
>>> axs[1].set_xticks([1, 2, 3])
>>> plt.tight_layout()
>>> plt.show()

.. note::

    By default, `scipy.stats.uniform` has support ``[0, 1]``.
    Using the parameters ``loc`` and ``scale``, one obtains the uniform
    distribution on ``[loc, loc + scale]``.

This result is particularly interesting because the first order index
:math:`S_{x_3} = 0` whereas its total order is :math:`S_{T_{x_3}} = 0.244`.
This means that higher order interactions with :math:`x_3` are responsible
for the difference. Almost 25% of the observed variance
on the QoI is due to the correlations between :math:`x_3` and :math:`x_1`,
although :math:`x_3` by itself has no impact on the QoI.

The following gives a visual explanation of Sobol' indices on this
function. Let's generate 1024 samples in :math:`[-\pi, \pi]^3` and
calculate the value of the output.

>>> from scipy.stats import qmc
>>> n_dim = 3
>>> p_labels = ['$x_1$', '$x_2$', '$x_3$']
>>> sample = qmc.Sobol(d=n_dim, seed=rng).random(1024)
>>> sample = qmc.scale(
...     sample=sample,
...     l_bounds=[-np.pi, -np.pi, -np.pi],
...     u_bounds=[np.pi, np.pi, np.pi]
... )
>>> output = f_ishigami(sample.T)

Now we can do scatter plots of the output with respect to each parameter.
This gives a visual way to understand how each parameter impacts the
output of the function.

>>> fig, ax = plt.subplots(1, n_dim, figsize=(12, 4))
>>> for i in range(n_dim):
...     xi = sample[:, i]
...     ax[i].scatter(xi, output, marker='+')
...     ax[i].set_xlabel(p_labels[i])
>>> ax[0].set_ylabel('Y')
>>> plt.tight_layout()
>>> plt.show()

Now Sobol' goes a step further:
by conditioning the output value by given values of the parameter
(black lines), the conditional output mean is computed. It corresponds to
the term :math:`\mathbb{E}(Y|x_i)`. Taking the variance of this term gives
the numerator of the Sobol' indices.

>>> 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]
>>> fig, ax = plt.subplots(1, n_dim, figsize=(12, 4))
>>> for i in range(n_dim):
...     xi = sample[:, i]
...     ax[i].scatter(xi, output, marker='+')
...     ax[i].set_xlabel(p_labels[i])
...     for bin_ in bins:
...         idx = np.where((bin_ <= xi) & (xi <= bin_ + dx))
...         xi_ = xi[idx]
...         y_ = output[idx]
...         ave_y_ = np.mean(y_)
...         ax[i].plot([bin_ + dx/2] * 2, [mini, maxi], c='k')
...         ax[i].scatter(bin_ + dx/2, ave_y_, c='r')
>>> ax[0].set_ylabel('Y')
>>> plt.tight_layout()
>>> plt.show()

Looking at :math:`x_3`, the variance
of the mean is zero leading to :math:`S_{x_3} = 0`. But we can further
observe that the variance of the output is not constant along the parameter
values of :math:`x_3`. This heteroscedasticity is explained by higher order
interactions. Moreover, an heteroscedasticity is also noticeable on
:math:`x_1` leading to an interaction between :math:`x_3` and :math:`x_1`.
On :math:`x_2`, the variance seems to be constant and thus null interaction
with this parameter can be supposed.

This case is fairly simple to analyse visually---although it is only a
qualitative analysis. Nevertheless, when the number of input parameters
increases such analysis becomes unrealistic as it would be difficult to
conclude on high-order terms. Hence the benefit of using Sobol' indices.

r