o Eb@sdZddlmZddlZddlZddlZddlZddlZddlm Z m Z ddl m Z m Z mZmZmZmZmZddlZerTddlm ZddlmZddlmZmZmZmZddlmZddlm Z dd l!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*dd l+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2gd Z3edTdUddZ4edVddZ4dWddZ4dddXdd Z5dd!d"d#dYd*d+Z6dZd/d0Z7d[d6d7Z8d\d9d:Z9d]dd^dddd"d?d_dDdEZ;GdFdGdGe ZGdLdMdMe<Z?GdNdOdOe<Z@GdPdQdQe<ZAd`dadRdSZBdS)bz&Quasi-Monte Carlo engines and helpers.) annotationsN)ABCabstractmethod)CallableClassVarDictListOptionaloverload TYPE_CHECKING)Literal) DecimalNumber GeneratorType IntNumberSeedType) rng_integers) initialize_v _cscramble_fill_p_cumulative_draw _fast_forward _categorizeinitialize_direction_numbers_MAXDIM_MAXBIT) _cy_wrapper_centered_discrepancy#_cy_wrapper_wrap_around_discrepancy_cy_wrapper_mixture_discrepancy_cy_wrapper_l2_star_discrepancy_cy_wrapper_update_discrepancy_cy_van_der_corput_scrambled_cy_van_der_corput) scale discrepancyupdate_discrepancy QMCEngineSobolHaltonLatinHypercubeMultinomialQMCMultivariateNormalQMC.seedOptional[IntNumber]returnnp.random.GeneratorcCdSNr+r1r12/usr/lib/python3/dist-packages/scipy/stats/_qmc.pycheck_random_state3r4rcCr/r0r1r2r1r1r3r47r5cCsN|dus t|tjtjfrtj|St|tjjtjjfr |St |d)aPTurn `seed` into a `numpy.random.Generator` instance. Parameters ---------- seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional If `seed` is None the `numpy.random.Generator` singleton is used. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Returns ------- seed : {`numpy.random.Generator`, `numpy.random.RandomState`} Random number generator. Nz9 cannot be used to seed a numpy.random.Generator instance) isinstancenumbersZIntegralnpintegerrandomZ default_rngZ RandomState Generator ValueErrorr2r1r1r3r4=s  F)reversesample npt.ArrayLikel_boundsu_boundsr=bool np.ndarraycCst|}t|}t|}|jdkstdt||\}}t||ks+tdt||jdkr8td|sTt|dkrHt|dksLtd||||St||krbt||ksftd||||S) aSample scaling from unit hypercube to different bounds. To convert a sample from :math:`[0, 1)` to :math:`[a, b), b>a`, with :math:`a` the lower bounds and :math:`b` the upper bounds. The following transformation is used: .. math:: (b - a) \cdot \text{sample} + a Parameters ---------- sample : array_like (n, d) Sample to scale. l_bounds, u_bounds : array_like (d,) Lower and upper bounds (resp. :math:`a`, :math:`b`) of transformed data. If `reverse` is True, range of the original data to transform to the unit hypercube. reverse : bool, optional Reverse the transformation from different bounds to the unit hypercube. Default is False. Returns ------- sample : array_like (n, d) Scaled sample. Examples -------- Transform 3 samples in the unit hypercube to bounds: >>> from scipy.stats import qmc >>> l_bounds = [-2, 0] >>> u_bounds = [6, 5] >>> sample = [[0.5 , 0.75], ... [0.5 , 0.5], ... [0.75, 0.25]] >>> sample_scaled = qmc.scale(sample, l_bounds, u_bounds) >>> sample_scaled array([[2. , 3.75], [2. , 2.5 ], [4. , 1.25]]) And convert back to the unit hypercube: >>> sample_ = qmc.scale(sample_scaled, l_bounds, u_bounds, reverse=True) >>> sample_ array([[0.5 , 0.75], [0.5 , 0.5 ], [0.75, 0.25]]) Sample is not a 2D arrayzBounds are not consistent a < bz3Sample dimension is different than bounds dimensionrSample is not in unit hypercubezSample is out of bounds) r8asarrayZ atleast_1dndimr<Zbroadcast_arraysalllenshape)r>r@rAr=lowerupperr1r1r3r"Zs" ;   r"CDrF) iterativemethodworkersrPrQ$Literal['CD', 'WD', 'MD', 'L2-star']rRrfloatcCstj|tjdd}|jdkstdt|dkr t|dks$tdt|}ttt t d}||vr<|||||d St|d t |) aDiscrepancy of a given sample. Parameters ---------- sample : array_like (n, d) The sample to compute the discrepancy from. iterative : bool, optional Must be False if not using it for updating the discrepancy. Default is False. Refer to the notes for more details. method : str, optional Type of discrepancy, can be ``CD``, ``WD``, ``MD`` or ``L2-star``. Refer to the notes for more details. Default is ``CD``. workers : int, optional Number of workers to use for parallel processing. If -1 is given all CPU threads are used. Default is 1. Returns ------- discrepancy : float Discrepancy. Notes ----- The discrepancy is a uniformity criterion used to assess the space filling of a number of samples in a hypercube. A discrepancy quantifies the distance between the continuous uniform distribution on a hypercube and the discrete uniform distribution on :math:`n` distinct sample points. The lower the value is, the better the coverage of the parameter space is. For a collection of subsets of the hypercube, the discrepancy is the difference between the fraction of sample points in one of those subsets and the volume of that subset. There are different definitions of discrepancy corresponding to different collections of subsets. Some versions take a root mean square difference over subsets instead of a maximum. A measure of uniformity is reasonable if it satisfies the following criteria [1]_: 1. It is invariant under permuting factors and/or runs. 2. It is invariant under rotation of the coordinates. 3. It can measure not only uniformity of the sample over the hypercube, but also the projection uniformity of the sample over non-empty subset of lower dimension hypercubes. 4. There is some reasonable geometric meaning. 5. It is easy to compute. 6. It satisfies the Koksma-Hlawka-like inequality. 7. It is consistent with other criteria in experimental design. Four methods are available: * ``CD``: Centered Discrepancy - subspace involves a corner of the hypercube * ``WD``: Wrap-around Discrepancy - subspace can wrap around bounds * ``MD``: Mixture Discrepancy - mix between CD/WD covering more criteria * ``L2-star``: L2-star discrepancy - like CD BUT variant to rotation See [2]_ for precise definitions of each method. Lastly, using ``iterative=True``, it is possible to compute the discrepancy as if we had :math:`n+1` samples. This is useful if we want to add a point to a sampling and check the candidate which would give the lowest discrepancy. Then you could just update the discrepancy with each candidate using `update_discrepancy`. This method is faster than computing the discrepancy for a large number of candidates. References ---------- .. [1] Fang et al. "Design and modeling for computer experiments". Computer Science and Data Analysis Series, 2006. .. [2] Zhou Y.-D. et al. Mixture discrepancy for quasi-random point sets. Journal of Complexity, 29 (3-4) , pp. 283-301, 2013. .. [3] T. T. Warnock. "Computational investigations of low discrepancy point sets". Applications of Number Theory to Numerical Analysis, Academic Press, pp. 319-343, 1972. Examples -------- Calculate the quality of the sample using the discrepancy: >>> from scipy.stats import qmc >>> space = np.array([[1, 3], [2, 6], [3, 2], [4, 5], [5, 1], [6, 4]]) >>> l_bounds = [0.5, 0.5] >>> u_bounds = [6.5, 6.5] >>> space = qmc.scale(space, l_bounds, u_bounds, reverse=True) >>> space array([[0.08333333, 0.41666667], [0.25 , 0.91666667], [0.41666667, 0.25 ], [0.58333333, 0.75 ], [0.75 , 0.08333333], [0.91666667, 0.58333333]]) >>> qmc.discrepancy(space) 0.008142039609053464 We can also compute iteratively the ``CD`` discrepancy by using ``iterative=True``. >>> disc_init = qmc.discrepancy(space[:-1], iterative=True) >>> disc_init 0.04769081147119336 >>> qmc.update_discrepancy(space[-1], space[:-1], disc_init) 0.008142039609053513 CdtypeZorderrDrErrFrG)rOZWDZMDzL2-starrRz* is not a valid method. It must be one of ) r8rHfloat64rIr<rJ_validate_workersrrrrset)r>rPrQrRmethodsr1r1r3r#s p r#x_new initial_discr cCstj|tjdd}tj|tjdd}|jdkstdt|dkr)t|dks-td|jdks6tdt|dkrDt|dksHtd |jd|jdkrVtd t|||S) aUpdate the centered discrepancy with a new sample. Parameters ---------- x_new : array_like (1, d) The new sample to add in `sample`. sample : array_like (n, d) The initial sample. initial_disc : float Centered discrepancy of the `sample`. Returns ------- discrepancy : float Centered discrepancy of the sample composed of `x_new` and `sample`. Examples -------- We can also compute iteratively the discrepancy by using ``iterative=True``. >>> from scipy.stats import qmc >>> space = np.array([[1, 3], [2, 6], [3, 2], [4, 5], [5, 1], [6, 4]]) >>> l_bounds = [0.5, 0.5] >>> u_bounds = [6.5, 6.5] >>> space = qmc.scale(space, l_bounds, u_bounds, reverse=True) >>> disc_init = qmc.discrepancy(space[:-1], iterative=True) >>> disc_init 0.04769081147119336 >>> qmc.update_discrepancy(space[-1], space[:-1], disc_init) 0.008142039609053513 rUrVrDrErrFrGzx_new is not a 1D arrayzx_new is not in unit hypercubez&x_new and sample must be broadcastable)r8rHrYrIr<rJrLr)r]r>r^r1r1r3r$<s%   r$i1inti2kdiscc Cs|jd}|d}d|dtjddt||ddft|t||ddf|dd}d|dtjddt||ddft|t||ddf|dd}d|dtdt||ddfd|tddt||ddfd||ddfd} d|dtdt||ddfd|tddt||ddfd||ddfd} dt|||ft|dd|ft|||f|dd|f} dt|||ft|dd|ft|||f|dd|f} | | } | |}|| }dt|||fdt|||f}dt|||fdt|||f}tdt||ddf}tdt||ddf}tddt||ddfd||ddfd}tddt||ddfd||ddfd}|||dd||||}||d|d||||}||||}tj|td }d |||g<t||}||| || d|}|S) aCentered discrepancy after an elementary perturbation of a LHS. An elementary perturbation consists of an exchange of coordinates between two points: ``sample[i1, k] <-> sample[i2, k]``. By construction, this operation conserves the LHS properties. Parameters ---------- sample : array_like (n, d) The sample (before permutation) to compute the discrepancy from. i1 : int The first line of the elementary permutation. i2 : int The second line of the elementary permutation. k : int The column of the elementary permutation. disc : float Centered discrepancy of the design before permutation. Returns ------- discrepancy : float Centered discrepancy of the design after permutation. References ---------- .. [1] Jin et al. "An efficient algorithm for constructing optimal design of computer experiments", Journal of Statistical Planning and Inference, 2005. r?g?g@NrFZaxisrDrWF)rLr8ZprodabsonesrBsum)r>r_rarbrcnZz_ijZc_i1jZc_i2jZc_i1i1Zc_i2i2ZnumZdenumZgammaZc_p_i1jZc_p_i2jZalphaZbetaZg_i1Zg_i2Zh_i1Zh_i2Zc_p_i1i1Zc_p_i2i2Zsum_maskZdisc_epr1r1r3_perturb_discrepancyxsj !  ($($&&((::$$  rlrjcCstj|d|ddktd}tdt|dddD],}d|ddB}d|||ddd|<d|||d|d@d ddd|<qtjdddt|d ddddBfS) aPrime numbers from 2 to *n*. Parameters ---------- n : int Sup bound with ``n >= 6``. Returns ------- primes : list(int) Primes in ``2 <= p < n``. Notes ----- Taken from [1]_ by P.T. Roy, written consent given on 23.04.2021 by the original author, Bruno Astrolino, for free use in SciPy under the 3-clause BSD. References ---------- .. [1] `StackOverflow `_. rDrfrFrdFNr)r8rhrBranger`Zr_Znonzero)rjZsieveirbr1r1r3primes_from_2_tos ,.rr List[int]cCsRgdd|}t||kr'd} t|d|}t||kr" |S|d7}q|S)zList of the n-first prime numbers. Parameters ---------- n : int Number of prime numbers wanted. Returns ------- primes : list(int) List of primes. )rDrm %)+/5;=CGIOSYaegkmqiii iiiii%i3i7i9i=iKiQi[i]iaigioiui{iiiiiiiiiiiiiiiiiiiiiii i ii#i-i3i9i;iAiKiQiWiYi_ieiiikiwiiiiiiiiiiiiiiiiiiiiiiiii)i+i5i7i;i=iGiUiYi[i_imiqisiwiiiiiiiiiiiiiiNizNot enough primesi)rKrr)rjprimesZ big_numberr1r1r3n_primess   rrD start_indexscrambler+rRbaserrrc Cs|dkrtd|r9t|}tdt|d}tjt|d|dd}|D]} || q)t |||||St ||||S)aDVan der Corput sequence. Pseudo-random number generator based on a b-adic expansion. Scrambling uses permutations of the remainders (see [1]_). Multiple permutations are applied to construct a point. The sequence of permutations has to be the same for all points of the sequence. Parameters ---------- n : int Number of element of the sequence. base : int, optional Base of the sequence. Default is 2. start_index : int, optional Index to start the sequence from. Default is 0. scramble : bool, optional If True, use Owen scrambling. Otherwise no scrambling is done. Default is True. seed : {None, int, `numpy.random.Generator`}, optional If `seed` is None the `numpy.random.Generator` singleton is used. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. workers : int, optional Number of workers to use for parallel processing. If -1 is given all CPU threads are used. Default is 1. Returns ------- sequence : list (n,) Sequence of Van der Corput. References ---------- .. [1] A. B. Owen. "A randomized Halton algorithm in R", arXiv:1706.02808, 2017. rDz'base' must be at least 26rFNrre) r<r4mathceilZlog2r8repeatarangeshuffler r!) rjrrrr+rRrngcount permutationsZpermr1r1r3van_der_corputs0 rc@sHeZdZdZedddd d ZeddddZdddZdddZdS)r%a A generic Quasi-Monte Carlo sampler class meant for subclassing. QMCEngine is a base class to construct a specific Quasi-Monte Carlo sampler. It cannot be used directly as a sampler. Parameters ---------- d : int Dimension of the parameter space. seed : {None, int, `numpy.random.Generator`}, optional If `seed` is None the `numpy.random.Generator` singleton is used. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Notes ----- By convention samples are distributed over the half-open interval ``[0, 1)``. Instances of the class can access the attributes: ``d`` for the dimension; and ``rng`` for the random number generator (used for the ``seed``). **Subclassing** When subclassing `QMCEngine` to create a new sampler, ``__init__`` and ``random`` must be redefined. * ``__init__(d, seed=None)``: at least fix the dimension. If the sampler does not take advantage of a ``seed`` (deterministic methods like Halton), this parameter can be omitted. * ``random(n)``: draw ``n`` from the engine and increase the counter ``num_generated`` by ``n``. Optionally, two other methods can be overwritten by subclasses: * ``reset``: Reset the engine to it's original state. * ``fast_forward``: If the sequence is deterministic (like Halton sequence), then ``fast_forward(n)`` is skipping the ``n`` first draw. Examples -------- To create a random sampler based on ``np.random.random``, we would do the following: >>> from scipy.stats import qmc >>> class RandomEngine(qmc.QMCEngine): ... def __init__(self, d, seed=None): ... super().__init__(d=d, seed=seed) ... ... ... def random(self, n=1): ... self.num_generated += n ... return self.rng.random((n, self.d)) ... ... ... def reset(self): ... super().__init__(d=self.d, seed=self.rng_seed) ... return self ... ... ... def fast_forward(self, n): ... self.random(n) ... return self After subclassing `QMCEngine` to define the sampling strategy we want to use, we can create an instance to sample from. >>> engine = RandomEngine(2) >>> engine.random(5) array([[0.22733602, 0.31675834], # random [0.79736546, 0.67625467], [0.39110955, 0.33281393], [0.59830875, 0.18673419], [0.67275604, 0.94180287]]) We can also reset the state of the generator and resample again. >>> _ = engine.reset() >>> engine.random(5) array([[0.22733602, 0.31675834], # random [0.79736546, 0.67625467], [0.39110955, 0.33281393], [0.59830875, 0.18673419], [0.67275604, 0.94180287]]) Nr2drr+rr-NonecCs@tt|tjs td||_t||_t ||_ d|_ dS)Nzd must be an integer valuer) r8Z issubdtypetyper9r<rr4rcopydeepcopyrng_seed num_generated)selfrr+r1r1r3__init__s    zQMCEngine.__init__rFrjrCcCsdS)a1Draw `n` in the half-open interval ``[0, 1)``. Parameters ---------- n : int, optional Number of samples to generate in the parameter space. Default is 1. Returns ------- sample : array_like (n, d) QMC sample. Nr1rrjr1r1r3r:szQMCEngine.randomcCs t|j}t||_d|_|S)zReset the engine to base state. Returns ------- engine : QMCEngine Engine reset to its base state. r)rrrr4rr)rr+r1r1r3resets zQMCEngine.resetcCs|j|d|S)a Fast-forward the sequence by `n` positions. Parameters ---------- n : int Number of points to skip in the sequence. Returns ------- engine : QMCEngine Engine reset to its base state. )rj)r:rr1r1r3 fast_forwards zQMCEngine.fast_forward)rrr+rr-rrFrjrr-rC)r-r%)rjrr-r%) __name__ __module__ __qualname____doc__rrr:rrr1r1r1r3r%`sX r%cs>eZdZdZddddfd dZ ddddddZZS)r'a Halton sequence. Pseudo-random number generator that generalize the Van der Corput sequence for multiple dimensions. The Halton sequence uses the base-two Van der Corput sequence for the first dimension, base-three for its second and base-:math:`n` for its n-dimension. Parameters ---------- d : int Dimension of the parameter space. scramble : bool, optional If True, use Owen scrambling. Otherwise no scrambling is done. Default is True. seed : {None, int, `numpy.random.Generator`}, optional If `seed` is None the `numpy.random.Generator` singleton is used. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Notes ----- The Halton sequence has severe striping artifacts for even modestly large dimensions. These can be ameliorated by scrambling. Scrambling also supports replication-based error estimates and extends applicabiltiy to unbounded integrands. References ---------- .. [1] Halton, "On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals", Numerische Mathematik, 1960. .. [2] A. B. Owen. "A randomized Halton algorithm in R", arXiv:1706.02808, 2017. Examples -------- Generate samples from a low discrepancy sequence of Halton. >>> from scipy.stats import qmc >>> sampler = qmc.Halton(d=2, scramble=False) >>> sample = sampler.random(n=5) >>> sample array([[0. , 0. ], [0.5 , 0.33333333], [0.25 , 0.66666667], [0.75 , 0.11111111], [0.125 , 0.44444444]]) Compute the quality of the sample using the discrepancy criterion. >>> qmc.discrepancy(sample) 0.088893711419753 If some wants to continue an existing design, extra points can be obtained by calling again `random`. Alternatively, you can skip some points like: >>> _ = sampler.fast_forward(5) >>> sample_continued = sampler.random(n=5) >>> sample_continued array([[0.3125 , 0.37037037], [0.8125 , 0.7037037 ], [0.1875 , 0.14814815], [0.6875 , 0.48148148], [0.4375 , 0.81481481]]) Finally, samples can be scaled to bounds. >>> l_bounds = [0, 2] >>> u_bounds = [10, 5] >>> qmc.scale(sample_continued, l_bounds, u_bounds) array([[3.125 , 3.11111111], [8.125 , 4.11111111], [1.875 , 2.44444444], [6.875 , 3.44444444], [4.375 , 4.44444444]]) TNrr+rrrrBr+rr-rcs*tj||d||_t||_||_dS)Nrr+)superrr+rrrrrrr+ __class__r1r3rKs  zHalton.__init__rFrXrjrRrCcsDtfddjD}j7_t|jjS)aDraw `n` in the half-open interval ``[0, 1)``. Parameters ---------- n : int, optional Number of samples to generate in the parameter space. Default is 1. workers : int, optional Number of workers to use for parallel processing. If -1 is given all CPU threads are used. Default is 1. It becomes faster than one worker for `n` greater than :math:`10^3`. Returns ------- sample : array_like (n, d) QMC sample. c s0g|]}tt|jjtjdqS)r)rr`rrrrr+).0ZbdimrjrrRr1r3 ks  z!Halton.random..)rZrrr8arrayTreshaper)rrjrRr>r1rr3r:Ts z Halton.randomrrrrBr+rr-rr)rjrrRrr-rC)rrrrrr: __classcell__r1r1rr3r'sQ r'cs\eZdZdZdddddd fddZd!d"ddZd!d"ddZd#d"ddZd$ddZZ S)%r(aLatin hypercube sampling (LHS). A Latin hypercube sample [1]_ generates :math:`n` points in :math:`[0,1)^{d}`. Each univariate marginal distribution is stratified, placing exactly one point in :math:`[j/n, (j+1)/n)` for :math:`j=0,1,...,n-1`. They are still applicable when :math:`n << d`. Parameters ---------- d : int Dimension of the parameter space. centered : bool, optional Center the point within the multi-dimensional grid. Default is False. optimization : {None, "random-cd"}, optional Whether to use an optimization scheme to construct a LHS. Default is None. * ``random-cd``: random permutations of coordinates to lower the centered discrepancy [5]_. The best design based on the centered discrepancy is constantly updated. Centered discrepancy-based design shows better space filling robustness toward 2D and 3D subprojections compared to using other discrepancy measures [6]_. .. versionadded:: 1.8.0 strength : {1, 2}, optional Strength of the LHS. ``strength=1`` produces a plain LHS while ``strength=2`` produces an orthogonal array based LHS of strength 2 [7]_, [8]_. In that case, only ``n=p**2`` points can be sampled, with ``p`` a prime number. It also constrains ``d <= p + 1``. Default is 1. .. versionadded:: 1.8.0 seed : {None, int, `numpy.random.Generator`}, optional If `seed` is None the `numpy.random.Generator` singleton is used. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Notes ----- When LHS is used for integrating a function :math:`f` over :math:`n`, LHS is extremely effective on integrands that are nearly additive [2]_. With a LHS of :math:`n` points, the variance of the integral is always lower than plain MC on :math:`n-1` points [3]_. There is a central limit theorem for LHS on the mean and variance of the integral [4]_, but not necessarily for optimized LHS due to the randomization. :math:`A` is called an orthogonal array of strength :math:`t` if in each n-row-by-t-column submatrix of :math:`A`: all :math:`p^t` possible distinct rows occur the same number of times. The elements of :math:`A` are in the set :math:`\{0, 1, ..., p-1\}`, also called symbols. The constraint that :math:`p` must be a prime number is to allow modular arithmetic. Strength 1 (plain LHS) brings an advantage over strength 0 (MC) and strength 2 is a useful increment over strength 1. Going to strength 3 is a smaller increment and scrambled QMC like Sobol', Halton are more performant [7]_. To create a LHS of strength 2, the orthogonal array :math:`A` is randomized by applying a random, bijective map of the set of symbols onto itself. For example, in column 0, all 0s might become 2; in column 1, all 0s might become 1, etc. Then, for each column :math:`i` and symbol :math:`j`, we add a plain, one-dimensional LHS of size :math:`p` to the subarray where :math:`A^i = j`. The resulting matrix is finally divided by :math:`p`. References ---------- .. [1] Mckay et al., "A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code." Technometrics, 1979. .. [2] M. Stein, "Large sample properties of simulations using Latin hypercube sampling." Technometrics 29, no. 2: 143-151, 1987. .. [3] A. B. Owen, "Monte Carlo variance of scrambled net quadrature." SIAM Journal on Numerical Analysis 34, no. 5: 1884-1910, 1997 .. [4] Loh, W.-L. "On Latin hypercube sampling." The annals of statistics 24, no. 5: 2058-2080, 1996. .. [5] Fang et al. "Design and modeling for computer experiments". Computer Science and Data Analysis Series, 2006. .. [6] Damblin et al., "Numerical studies of space filling designs: optimization of Latin Hypercube Samples and subprojection properties." Journal of Simulation, 2013. .. [7] A. B. Owen , "Orthogonal arrays for computer experiments, integration and visualization." Statistica Sinica, 1992. .. [8] B. Tang, "Orthogonal Array-Based Latin Hypercubes." Journal of the American Statistical Association, 1993. Examples -------- Generate samples from a Latin hypercube generator. >>> from scipy.stats import qmc >>> sampler = qmc.LatinHypercube(d=2) >>> sample = sampler.random(n=5) >>> sample array([[0.1545328 , 0.53664833], # random [0.84052691, 0.06474907], [0.52177809, 0.93343721], [0.68033825, 0.36265316], [0.26544879, 0.61163943]]) Compute the quality of the sample using the discrepancy criterion. >>> qmc.discrepancy(sample) 0.0196... # random Samples can be scaled to bounds. >>> l_bounds = [0, 2] >>> u_bounds = [10, 5] >>> qmc.scale(sample, l_bounds, u_bounds) array([[1.54532796, 3.609945 ], # random [8.40526909, 2.1942472 ], [5.2177809 , 4.80031164], [6.80338249, 3.08795949], [2.65448791, 3.83491828]]) Use the `optimization` keyword argument to produce a LHS with lower discrepancy at higher computational cost. >>> sampler = qmc.LatinHypercube(d=2, optimization="random-cd") >>> sample = sampler.random(n=5) >>> qmc.discrepancy(sample) 0.0176... # random Use the `strength` keyword argument to produce an orthogonal array based LHS of strength 2. In this case, the number of sample points must be the square of a prime number. >>> sampler = qmc.LatinHypercube(d=2, strength=2) >>> sample = sampler.random(n=9) >>> qmc.discrepancy(sample) 0.00526... # random Options could be combined to produce an optimized centered orthogonal array based LHS. After optimization, the result would not be guaranteed to be of strength 2. FrFN)centeredstrength optimizationr+rrrrBrr`rOptional[Literal['random-cd']]r+rr-rc stj||d||_|j|jd}z|||_Wnty3}z|dt|}t||d}~wwd|j i} ||durmz | }| ||_ Wntyd}z|dt| }t||d}~wwd|_ d|_ dSd|_ dS)Nr)rFrDz, is not a valid strength. It must be one of z random-cdz7 is not a valid optimization method. It must be one of di')rrr_random_random_oa_lhs lhs_methodKeyErrorr[r< _random_cdrMoptimization_method _n_nochange_n_iters) rrrrrr+Zlhs_method_strengthexcmessagerrr1r3rs>    zLatinHypercube.__init__rjrCcCs0||}|jdur||}|j|7_|S)a%Draw `n` in the half-open interval ``[0, 1)``. Parameters ---------- n : int, optional Number of samples to generate in the parameter space. Default is 1. Returns ------- sample : array_like (n, d) LHS sample. N)rrr)rrjlhsr1r1r3r:0s   zLatinHypercube.randomcCs||jrd}n |jj||jfd}ttd|d|jdf}t|jD]}|j||ddfq$|j }|||}|S)zBase LHS algorithm.rd)sizerFN) rrZuniformrr8tilerrprr)rrjsamplespermsrqr1r1r3rEs zLatinHypercube._randomrocCst|t}|d}|d}t|d}||vs||kr+td|ddd|j|dkr6tdtj||ftd}tt |d}tj tj |d d  d d|ddddf<t d|D]"}t|ddd f||dddf||ddd|df<qctj||ftd} t |D]} |j|} | |dd| f| dd| f<qtj||fd } td|jd|jd } t |D]2} t |D]+}|dd| f|k}| |}||dd| f|| dd| f|<| } qq| |} | ddd|jfS)z)Orthogonal array based LHS of strength 2.rDrFz8n is not the square of a prime number. Close values are Nz*n is too small for d. Must be n > (d-1)**2)rLrW)rDrFrer)rL)rrrr+)r8sqrtZastyper`rrr<rzerosrrstackZmeshgridrrpmodemptyrZ permutationr(rr:Zflattenr)rrjpZn_rowZn_colrZ oa_sampleZarraysZp_Z oa_sample_jrZ oa_lhs_sampleZ lhs_enginerbidxrr1r1r3rUsT    "   ( zLatinHypercube._random_oa_lhs best_samplec Cs4t|}|jdks |dkrt||jfSt|}|dkr|Sd|jdgd|dgd|dgf}d}d}||jkr||jkr|d7}t|jg|dR}t|jg|dR}t|jg|dR} t ||| ||} | |kr|| |f|||f|||f<|| |f<| }d}n|d7}||jkr||jks?|S)aOptimal LHS on CD. Create a base LHS and do random permutations of coordinates to lower the centered discrepancy. Because it starts with a normal LHS, it also works with the `centered` keyword argument. Two stopping criterion are used to stop the algorithm: at most, `_n_iters` iterations are performed; or if there is no improvement for `_n_nochange` consecutive iterations. rrFrD) rKrr8rr#rrrrrl) rrrjZ best_discZboundsZ n_nochangeZn_iterscolZrow_1Zrow_2rcr1r1r3rs:    zLatinHypercube._random_cd) rrrrBrr`rrr+rr-rrr)ro)rrCr-rC) rrrrrr:rrrrr1r1rr3r(us )  /r(cs|eZdZUdZeZded<eZded<dddd fddZ d!ddZ d"d#ddZ d$ddZ d%fdd Z d&ddZZS)'r&aEngine for generating (scrambled) Sobol' sequences. Sobol' sequences are low-discrepancy, quasi-random numbers. Points can be drawn using two methods: * `random_base2`: safely draw :math:`n=2^m` points. This method guarantees the balance properties of the sequence. * `random`: draw an arbitrary number of points from the sequence. See warning below. Parameters ---------- d : int Dimensionality of the sequence. Max dimensionality is 21201. scramble : bool, optional If True, use Owen scrambling. Otherwise no scrambling is done. Default is True. seed : {None, int, `numpy.random.Generator`}, optional If `seed` is None the `numpy.random.Generator` singleton is used. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Notes ----- Sobol' sequences [1]_ provide :math:`n=2^m` low discrepancy points in :math:`[0,1)^{d}`. Scrambling them [2]_ makes them suitable for singular integrands, provides a means of error estimation, and can improve their rate of convergence. There are many versions of Sobol' sequences depending on their 'direction numbers'. This code uses direction numbers from [3]_. Hence, the maximum number of dimension is 21201. The direction numbers have been precomputed with search criterion 6 and can be retrieved at https://web.maths.unsw.edu.au/~fkuo/sobol/. .. warning:: Sobol' sequences are a quadrature rule and they lose their balance properties if one uses a sample size that is not a power of 2, or skips the first point, or thins the sequence [4]_. If :math:`n=2^m` points are not enough then one should take :math:`2^M` points for :math:`M>m`. When scrambling, the number R of independent replicates does not have to be a power of 2. Sobol' sequences are generated to some number :math:`B` of bits. After :math:`2^B` points have been generated, the sequence will repeat. Currently :math:`B=30`. References ---------- .. [1] I. M. Sobol. The distribution of points in a cube and the accurate evaluation of integrals. Zh. Vychisl. Mat. i Mat. Phys., 7:784-802, 1967. .. [2] Art B. Owen. Scrambling Sobol and Niederreiter-Xing points. Journal of Complexity, 14(4):466-489, December 1998. .. [3] S. Joe and F. Y. Kuo. Constructing sobol sequences with better two-dimensional projections. SIAM Journal on Scientific Computing, 30(5):2635-2654, 2008. .. [4] Art B. Owen. On dropping the first Sobol' point. arXiv 2008.08051, 2020. Examples -------- Generate samples from a low discrepancy sequence of Sobol'. >>> from scipy.stats import qmc >>> sampler = qmc.Sobol(d=2, scramble=False) >>> sample = sampler.random_base2(m=3) >>> sample array([[0. , 0. ], [0.5 , 0.5 ], [0.75 , 0.25 ], [0.25 , 0.75 ], [0.375, 0.375], [0.875, 0.875], [0.625, 0.125], [0.125, 0.625]]) Compute the quality of the sample using the discrepancy criterion. >>> qmc.discrepancy(sample) 0.013882107204860938 To continue an existing design, extra points can be obtained by calling again `random_base2`. Alternatively, you can skip some points like: >>> _ = sampler.reset() >>> _ = sampler.fast_forward(4) >>> sample_continued = sampler.random_base2(m=2) >>> sample_continued array([[0.375, 0.375], [0.875, 0.875], [0.625, 0.125], [0.125, 0.625]]) Finally, samples can be scaled to bounds. >>> l_bounds = [0, 2] >>> u_bounds = [10, 5] >>> qmc.scale(sample_continued, l_bounds, u_bounds) array([[3.75 , 3.125], [8.75 , 4.625], [6.25 , 2.375], [1.25 , 3.875]]) z ClassVar[int]MAXDIMMAXBITTNrrrrrBr+rr-rcstj||d||jkrtd|jttj||jft d|_ t |j ||s4tj|t d|_ n| |j |_|jd|jdd|_dS)Nrz'Maximum supported dimensionality is {}.rfrDrFr)rrrr<formatrr8rrr`_svr_shift _scrambler_quasir _first_pointrrr1r3r+s    zSobol.__init__cCs~tt|jd|j|jftddtj|jtd|_|j |_ t t|jd|j|j|jftd}t |j||j d|_dS)zScramble the sequence.rD)rrWrfrN)r8dotrrrrr`rrrrZtrilrrr)rZltmr1r1r3rDs   zSobol._scramblerFrjrCcCstj||jftd}|jdkr@||d@dkstd|dkr$|j}n+t|d|j|j|j |j |t |j|gd|}nt||jd|j|j |j ||j|7_|S)a$Draw next point(s) in the Sobol' sequence. Parameters ---------- n : int, optional Number of samples to generate in the parameter space. Default is 1. Returns ------- sample : array_like (n, d) Sobol' sample. rfrrFzEThe balance properties of Sobol' points require n to be a power of 2.N) r8rrrTrwarningswarnrrrrZ concatenate)rrjr>r1r1r3r:Ss  z Sobol.randommcCs@d|}|j|}||d@dkstd|j||||S)aDraw point(s) from the Sobol' sequence. This function draws :math:`n=2^m` points in the parameter space ensuring the balance properties of the sequence. Parameters ---------- m : int Logarithm in base 2 of the number of samples; i.e., n = 2^m. Returns ------- sample : array_like (n, d) Sobol' sample. rDrFrzThe balance properties of Sobol' points require n to be a power of 2. {0} points have been previously generated, then: n={0}+2**{1}={2}. If you still want to do this, the function 'Sobol.random()' can be used.)rr<rr:)rrrjZtotal_nr1r1r3 random_base2vs   zSobol.random_base2cst|j|_|S)zReset the engine to base state. Returns ------- engine : Sobol Engine reset to its base state. )rrrrrrrr1r3rs z Sobol.resetcCsV|jdkrt|d|j|j|j|jnt||jd|j|j|j|j|7_|S)aFast-forward the sequence by `n` positions. Parameters ---------- n : int Number of points to skip in the sequence. Returns ------- engine : Sobol The fast-forwarded engine. rrF)rrrrrrr1r1r3rs zSobol.fast_forwardr)r-rrr)rrr-rC)r-r&)rjrr-r&)rrrrrr__annotations__rrrrr:rrrrr1r1rr3r&s  r    # r&csbeZdZdZ d!dddddd"fddZd#d$ddZd%fdd Zd&ddZd#d$dd ZZ S)'r*aiQMC sampling from a multivariate Normal :math:`N(\mu, \Sigma)`. Parameters ---------- mean : array_like (d,) The mean vector. Where ``d`` is the dimension. cov : array_like (d, d), optional The covariance matrix. If omitted, use `cov_root` instead. If both `cov` and `cov_root` are omitted, use the identity matrix. cov_root : array_like (d, d'), optional A root decomposition of the covariance matrix, where ``d'`` may be less than ``d`` if the covariance is not full rank. If omitted, use `cov`. inv_transform : bool, optional If True, use inverse transform instead of Box-Muller. Default is True. engine : QMCEngine, optional Quasi-Monte Carlo engine sampler. If None, `Sobol` is used. seed : {None, int, `numpy.random.Generator`}, optional If `seed` is None the `numpy.random.Generator` singleton is used. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Examples -------- >>> import matplotlib.pyplot as plt >>> from scipy.stats import qmc >>> engine = qmc.MultivariateNormalQMC(mean=[0, 5], cov=[[1, 0], [0, 1]]) >>> sample = engine.random(512) >>> _ = plt.scatter(sample[:, 0], sample[:, 1]) >>> plt.show() NT)cov_root inv_transformenginer+meanr?covOptional[npt.ArrayLike]rrrBr Optional[QMCEngine]r+rr-rc stj|ddd}|jd}|durltj|ddd}|jd|jdks'tdt||s3tdz tj|}WnHtjjyktj |\}} t |dksYtd t |d d}| t |}Ynw|durt |}|jd|jdkstdnd}tj||d ||_|sdt|d} n|} |durt| d |d |_nt|tr|j|krtd||_ntd||_||_dS)NFrFrZndminrrDz/Dimension mismatch between mean and covariance.z#Covariance matrix is not symmetric.g:0yEzCovariance matrix not PSD.grTrrr+zPDimension of `engine` must be consistent with dimensions of mean and covariance.F`engine` must be an instance of `scipy.stats.qmc.QMCEngine` or `None`.)r8rrLr<ZallcloseZ transposeZlinalgZcholeskyZ LinAlgErrorZeighrJZcliprZ atleast_2drr_inv_transformrrr&r r6r%r_mean _corr_matrix) rr r rrr r+rZeigvalZeigvecZ engine_dimrr1r3rsJ     zMultivariateNormalQMC.__init__rFrjrrCcCs"||}|j|7_||S)a%Draw `n` QMC samples from the multivariate Normal. Parameters ---------- n : int, optional Number of samples to generate in the parameter space. Default is 1. Returns ------- sample : array_like (n, d) Sample. )_standard_normal_samplesr _correlate)rrj base_samplesr1r1r3r:s  zMultivariateNormalQMC.randomct|j|S)zReset the engine to base state. Returns ------- engine : MultivariateNormalQMC Engine reset to its base state. rrr rrr1r3r* zMultivariateNormalQMC.resetrcCs$|jdur ||j|jS||jSr0)rr)rrr1r1r3r7s  z MultivariateNormalQMC._correlatec Cs|j|}|jrtjdd|dStd|jdd}t dt |dd|f}dt j |ddd|f}t |}t|}t||||gd|d}|ddd|jfS) a3Draw `n` QMC samples from the standard Normal :math:`N(0, I_d)`. Parameters ---------- n : int, optional Number of samples to generate in the parameter space. Default is 1. Returns ------- sample : array_like (n, d) Sample. rdgA?rrrDrNrF)r r:rstatsZnormZppfr8rrLrlogrZpicossinrrr) rrjrZevenZRsZthetasrrZtransf_samplesr1r1r3r>s    z.MultivariateNormalQMC._standard_normal_samplesr0)r r?r r rr rrBr r r+rr-rrr)r-r*)rrCr-rC) rrrrrr:rrrrr1r1rr3r*s# < r*csDeZdZdZddddfd d ZddddZdfdd ZZS)r)a:QMC sampling from a multinomial distribution. Parameters ---------- pvals : array_like (k,) Vector of probabilities of size ``k``, where ``k`` is the number of categories. Elements must be non-negative and sum to 1. engine : QMCEngine, optional Quasi-Monte Carlo engine sampler. If None, `Sobol` is used. seed : {None, int, `numpy.random.Generator`}, optional If `seed` is None the `numpy.random.Generator` singleton is used. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Examples -------- >>> from scipy.stats import qmc >>> engine = qmc.MultinomialQMC(pvals=[0.2, 0.4, 0.4]) >>> sample = engine.random(10) N)r r+pvalsr?r r r+rr-rcstj|ddd|_t|dkrtdtt|ds!td|dur.tdd|d|_nt |t r@|j dkrr1r1r3r:s zMultinomialQMC.randomcr)zReset the engine to base state. Returns ------- engine : MultinomialQMC Engine reset to its base state. rrrr1r3rrzMultinomialQMC.reset)rr?r r r+rr-rrr)r-r))rrrrrr:rrr1r1rr3r)_s r)cCsHt|}|dkrt}|durtd|S|dkr"td|d|S)a@Validate `workers` based on platform and value. Parameters ---------- workers : int, optional Number of workers to use for parallel processing. If -1 is given all CPU threads are used. Default is 1. Returns ------- Workers : int Number of CPU used by the algorithm rNzaCannot determine the number of cpus using os.cpu_count(), cannot use -1 for the number of workersrzInvalid number of workers: z, must be -1 or > 0)r`os cpu_countNotImplementedErrorr<rXr1r1r3rZsrZ).)r+r,r-r.)r+rr-rr0) r>r?r@r?rAr?r=rBr-rC) r>r?rPrBrQrSrRrr-rT)r]r?r>r?r^r r-rT) r>rCr_r`rar`rbr`rcrT)rjr`r-rC)rjrr-rs)rD)rjrrrrrrrBr+rrRrr-rCr)rRrr-r)CrZ __future__rrrr7r rabcrrtypingrrrrr r r Znumpyr8Z numpy.typingZnptZtyping_extensionsr Zscipy._lib._utilr rrrZ scipy.statsrrZscipy.stats._sobolrrrrrrrrrZscipy.stats._qmc_cyrrrrrr r!__all__r4r"r#r$rlrrrrr%r'r(r&r*r)rZr1r1r1r3sn $     ,$   "\ < Z )G{B'R