a BCCf`@sDddlmZddlZddlmZddlmZmZmZm Z ddl Z ddl m Z ddlmZddlmZddlmZmZe rddlmZdd lmZmZmZd gZd d d ddZd)dddd dddZd d d dddZd d d ddddZeGdddZ eGdd d Z!Gd!d"d"eZ"dddd#d$dd%d&dd d'd(d Z#dS)*) annotationsN) dataclass)CallableLiteralProtocol TYPE_CHECKING)ConfidenceInterval)check_random_state)BootstrapResult)qmc bootstrap) DecimalNumber IntNumberSeedType sobol_indicesz npt.ArrayLike np.ndarray)xreturncCsPt|}t|ddt|ddd|ddt|d}|S)aIshigami function. .. 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`. Parameters ---------- x : array_like ([x1, x2, x3], n) Returns ------- f : array_like (n,) Function evaluation. References ---------- .. [1] Ishigami, T. and T. Homma. "An importance quantification technique in uncertainty analysis for computer models." IEEE, :doi:`10.1109/ISUMA.1990.151285`, 1990. rg?)np atleast_2dsin)rZf_evalr]/var/www/html/django/DPS/env/lib/python3.9/site-packages/scipy/stats/_sensitivity_analysis.py f_ishigamis  rrz list[PPFDist]r)ndists random_staterc Cst|}tjd||dd|j}|d|d}z8t|D]*\}}||dd|f|dd|f<qsstrrr saltelli_2010ns%rBc@seZdZUded<ded<dS)BootstrapSobolResultr first_order total_orderN)__name__ __module__ __qualname____annotations__rrrrrCs rCc@seZdZUded<ded<ded<ded<ded<ded<d Zd ed <d Zd ed <d Zd ed <d Zded<dddddddZd S) SobolResultrrDrEr_indices_method_f_A_f_B_f_ABNznp.ndarray | None_A_B_ABzBootstrapResult | None_bootstrap_resultffffff?r rrC)confidence_level n_resamplesrcsfdd}jjd}tt|g|d||jd}|_tt|jj d|jj d|j d|j dd}tt|jj d|jj d|j d|j dd}t ||dS) a`Bootstrap Sobol' indices to provide confidence intervals. Parameters ---------- confidence_level : float, default: ``0.95`` The confidence level of the confidence intervals. n_resamples : int, default: ``999`` The number of resamples performed to form the bootstrap distribution of the indices. Returns ------- res : BootstrapSobolResult Bootstrap result containing the confidence intervals and the bootstrap distribution of the indices. An object with attributes: first_order : BootstrapResult Bootstrap result of the first order indices. total_order : BootstrapResult Bootstrap result of the total order indices. See `BootstrapResult` for more details. cs@jdd|f}jdd|f}jd|f}|||S)N.)rLrMrNrK)idxZf_A_Zf_B_Zf_AB_selfrr statisticsz(SobolResult.bootstrap..statisticrZBCa)rZmethodrVrUZbootstrap_resultr)confidence_intervalbootstrap_distributionstandard_error)rDrE)rLr4r rr5rRr rr\lowhighr]r^rC)rYrUrVrZrresrDrErrXrr s4  zSobolResult.bootstrap)rSrT) rFrGrHrIrOrPrQrRr rrrrrJs     rJc@seZdZeddddZdS)PPFDistzCallable[..., float])rcCsdSNrrXrrrr+sz PPFDist.ppfN)rFrGrHpropertyr+rrrrrbsrb)rr[r zWCallable[[np.ndarray], npt.ArrayLike] | dict[Literal['f_A', 'f_B', 'f_AB'], np.ndarray]zlist[PPFDist] | Nonez#Callable | Literal['saltelli_2010'])funcrrr[r rc st|}t|}||d@dkr(||kr0td|}t|sdti}z|}||Wqty}z*|dt|d}t||WYd}~qd}~00n8|t } t| j hdkrd t t}t|fd d } trd|durtd fd dt |||d\} } t | | d} | }|j d|krFtdfdd}| }|| }nd}z$tddd\}}}Wn0ty}zt||WYd}~n d}~00|j d|ks|j |j ks|j |j ks|j d|dkrt|tj||gdddd}||8}||8}||8}tjddd"| |||d\}}Wdn1sp0Yd|t|<d|t|<t||| |||d}tr|t| | | dtfi|S) a4Global sensitivity indices of Sobol'. Parameters ---------- func : callable or dict(str, array_like) If `func` is a callable, function to compute the Sobol' indices from. Its signature must be:: func(x: ArrayLike) -> ArrayLike with ``x`` of shape ``(d, n)`` and output of shape ``(s, n)`` where: - ``d`` is the input dimensionality of `func` (number of input variables), - ``s`` is the output dimensionality of `func` (number of output variables), and - ``n`` is the number of samples (see `n` below). Function evaluation values must be finite. If `func` is a dictionary, contains the function evaluations from three different arrays. Keys must be: ``f_A``, ``f_B`` and ``f_AB``. ``f_A`` and ``f_B`` should have a shape ``(s, n)`` and ``f_AB`` should have a shape ``(d, s, n)``. This is an advanced feature and misuse can lead to wrong analysis. n : int Number of samples used to generate the matrices ``A`` and ``B``. Must be a power of 2. The total number of points at which `func` is evaluated will be ``n*(d+2)``. dists : list(distributions), optional List of each parameter's distribution. The distribution of parameters depends on the application and should be carefully chosen. Parameters are assumed to be independently distributed, meaning there is no constraint nor relationship between their values. Distributions must be an instance of a class with a ``ppf`` method. Must be specified if `func` is a callable, and ignored otherwise. method : Callable or str, default: 'saltelli_2010' Method used to compute the first and total Sobol' indices. If a callable, its signature must be:: func(f_A: np.ndarray, f_B: np.ndarray, f_AB: np.ndarray) -> Tuple[np.ndarray, np.ndarray] with ``f_A, f_B`` of shape ``(s, n)`` and ``f_AB`` of shape ``(d, s, n)``. These arrays contain the function evaluations from three different sets of samples. The output is a tuple of the first and total indices with shape ``(s, d)``. This is an advanced feature and misuse can lead to wrong analysis. random_state : {None, int, `numpy.random.Generator`}, optional If `random_state` is an int or None, a new `numpy.random.Generator` is created using ``np.random.default_rng(random_state)``. If `random_state` is already a ``Generator`` instance, then the provided instance is used. 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= 2**12``. The more complex the model is, the more samples will be needed. Even for a purely addiditive 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) ... ], ... random_state=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. rrzGThe balance properties of Sobol' points require 'n' to be a power of 2.rBz, is not a valid 'method'. It must be one of z or a callable.N>r;r:r9zAIf 'method' is a callable, it must have the following signature: cst|||dS)zmWrap indices method to ensure proper output dimension. 1D when single output, 2D otherwise. r9r:r;)rZsqueezerf)indices_method_rrindices_methodosz%sobol_indices..indices_methodz2'dists' must be defined when 'func' is a callable.cst|Src)rr)r)rerr wrapped_func|sz#sobol_indices..wrapped_func)rrr )r2r3zN'func' output should have a shape ``(s, -1)`` with ``s`` the number of output.csH|j\}}}t|dd|||}|}t|d||fddS)Nrr%)r4rZmoveaxisr))r6r"rr;)rirrfuncABs zsobol_indices..funcABzWhen 'func' is a dictionary, it must contain the following keys: 'f_A', 'f_B' and 'f_AB'.'f_A' and 'f_B' should have a shape ``(s, n)`` and 'f_AB' should have a shape ``(d, s, n)``.r9r:r;r%r<r=ignore)divideinvalidrf)rDrErKrLrMrN)rOrPrQ)r intr-callablerBlowerKeyErrorsetinspect signature parametersr1r8r4rrr?r)ZerrstateisfinitedictupdaterJ)rerrr[r Zn_Zindices_methodsr/r0sigrhr2r3r6r9rjr:r;r?rDrErar)rergrirrsW "         2 )N)$ __future__rrs dataclassesrtypingrrrrnumpyrZscipy.stats._commonrZscipy.stats._qmcr Zscipy.stats._resamplingr Z scipy.statsr r Z numpy.typingZnptZscipy._lib._utilr rr__all__rr1r8rBrCrJrbrrrrrs6      $!/N