QMCPy Documentation¶

_images/discrete_distribution_overview.png

Discrete Distribution Class¶

_images/discrete_distribution_specific.png

Abstract Discrete Distribution Class¶

class qmcpy.discrete_distribution._discrete_distribution.DiscreteDistribution(dimension, seed)¶

Discrete Distribution abstract class. DO NOT INSTANTIATE.

__init__(dimension, seed)¶

Parameters

dimension (int or ndarray) – dimension of the generator. If an int is passed in, use sequence dimensions [0,…,dimensions-1]. If a ndarray is passed in, use these dimension indices in the sequence. Note that this is not relevant for IID generators.
seed (int or numpy.random.SeedSequence) – seed to create random number generator

gen_samples(*args)¶

ABSTRACT METHOD to generate samples from this discrete distribution.

Parameters: args (tuple) – tuple of positional argument. See implementations for details
Returns: n x d array of samples
Return type: ndarray

pdf(x)¶: ABSTRACT METHOD to evaluate pdf of distribution the samples mimic at locations of x.

spawn(s=1, dimensions=None)¶

Spawn new instances of the current discrete distribution but with new seeds and dimensions. Developed for multi-level and multi-replication (Q)MC algorithms.

Parameters

s (int) – number of spawn
dimensions (ndarray) – length s array of dimension for each spawn. Defaults to current dimension

Returns

list of DiscreteDistribution instances with new seeds and dimensions

Return type

list

Digital Net Base 2¶

class qmcpy.discrete_distribution.digital_net_b2.digital_net_b2.DigitalNetB2(dimension=1, randomize='LMS_DS', graycode=False, seed=None, generating_matrices='sobol_mat.21201.32.32.msb.npy', d_max=None, t_max=None, m_max=None, msb=None, t_lms=None, _verbose=False)¶

Quasi-Random digital nets in base 2.

>>> dnb2 = DigitalNetB2(2,seed=7)
>>> dnb2.gen_samples(4)
array([[0.56269008, 0.17377997],
       [0.346653  , 0.65070632],
       [0.82074548, 0.95490574],
       [0.10422261, 0.49458097]])
>>> dnb2.gen_samples(1)
array([[0.56269008, 0.17377997]])
>>> dnb2
DigitalNetB2 (DiscreteDistribution Object)
    d               2^(1)
    dvec            [0 1]
    randomize       LMS_DS
    graycode        0
    entropy         7
    spawn_key       ()
>>> DigitalNetB2(dimension=2,randomize=False,graycode=True).gen_samples(n_min=2,n_max=4)
array([[0.75, 0.25],
       [0.25, 0.75]])
>>> DigitalNetB2(dimension=2,randomize=False,graycode=False).gen_samples(n_min=2,n_max=4)
array([[0.25, 0.75],
       [0.75, 0.25]])

References

[1] Marius Hofert and Christiane Lemieux (2019). qrng: (Randomized) Quasi-Random Number Generators. R package version 0.0-7. https://CRAN.R-project.org/package=qrng.

[2] Faure, Henri, and Christiane Lemieux. “Implementation of Irreducible Sobol’ Sequences in Prime Power Bases.” Mathematics and Computers in Simulation 161 (2019): 13–22. Crossref. Web.

[3] F.Y. Kuo & D. Nuyens. Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients - a survey of analysis and implementation, Foundations of Computational Mathematics, 16(6):1631-1696, 2016. springer link: https://link.springer.com/article/10.1007/s10208-016-9329-5 arxiv link: https://arxiv.org/abs/1606.06613

[4] D. Nuyens, The Magic Point Shop of QMC point generators and generating vectors. MATLAB and Python software, 2018. Available from https://people.cs.kuleuven.be/~dirk.nuyens/ https://bitbucket.org/dnuyens/qmc-generators/src/cb0f2fb10fa9c9f2665e41419097781b611daa1e/cpp/digitalseq_b2g.hpp

[5] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., … Chintala, S. (2019). PyTorch: An Imperative Style, High-Performance Deep Learning Library. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d extquotesingle Alch'e-Buc, E. Fox, & R. Garnett (Eds.), Advances in Neural Information Processing Systems 32 (pp. 8024–8035). Curran Associates, Inc. Retrieved from http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf

[6] I.M. Sobol’, V.I. Turchaninov, Yu.L. Levitan, B.V. Shukhman: “Quasi-Random Sequence Generators” Keldysh Institute of Applied Mathematics, Russian Acamdey of Sciences, Moscow (1992).

[7] Sobol, Ilya & Asotsky, Danil & Kreinin, Alexander & Kucherenko, Sergei. (2011). Construction and Comparison of High-Dimensional Sobol’ Generators. Wilmott. 2011. 10.1002/wilm.10056.

[8] Paul Bratley and Bennett L. Fox. 1988. Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Softw. 14, 1 (March 1988), 88–100. DOI:https://doi.org/10.1145/42288.214372

__init__(dimension=1, randomize='LMS_DS', graycode=False, seed=None, generating_matrices='sobol_mat.21201.32.32.msb.npy', d_max=None, t_max=None, m_max=None, msb=None, t_lms=None, _verbose=False)¶

Parameters

dimension (int or ndarray) – dimension of the generator. If an int is passed in, use sequence dimensions [0,…,dimensions-1]. If a ndarray is passed in, use these dimension indices in the sequence.
randomize (bool) – apply randomization? True defaults to LMS_DS. Can also explicitly pass in ‘LMS_DS’: linear matrix scramble with digital shift ‘LMS’: linear matrix scramble only ‘DS’: digital shift only
graycode (bool) – indicator to use graycode ordering (True) or natural ordering (False)
seed (list) – int seed of list of seeds, one for each dimension.
generating_matrices (ndarray or str) – generating matricies or path to generating matrices. ndarray should have shape (d_max, m_max) where each int has t_max bits generating_matrices sould be formatted like gen_mat.21201.32.32.msb.npy with name.d_max.t_max.m_max.{msb,lsb}.npy
d_max (int) – max dimension
t_max (int) – number of bits in each int of each generating matrix. aka: number of rows in a generating matrix with ints expanded into columns
m_max (int) – 2^m_max is the number of samples supported. aka: number of columns in a generating matrix with ints expanded into columns
msb (bool) – bit storage as ints. e.g. if t_max=3, then 6 is [1 1 0] in MSB (True) and [0 1 1] in LSB (False)
t_lms (int) – LMS scrambling matrix will be t_lms x t_max for generating matrix of shape t_max x m_max
_verbose (bool) – print randomization details

gen_samples(n=None, n_min=0, n_max=8, warn=True, return_unrandomized=False)¶

Generate samples

Parameters

n (int) – if n is supplied, generate from n_min=0 to n_max=n samples. Otherwise use the n_min and n_max explicitly supplied as the following 2 arguments
n_min (int) – Starting index of sequence.
n_max (int) – Final index of sequence.
return_unrandomized (bool) – return unrandomized samples as well? If True, return randomized_samples,unrandomized_samples. Note that this only applies when randomize includes Digital Shift. Also note that unrandomized samples included linear matrix scrambling if applicable.

Returns

(n_max-n_min) x d (dimension) array of samples

Return type

ndarray

pdf(x)¶: pdf of a standard uniform

class qmcpy.discrete_distribution.digital_net_b2.digital_net_b2.Sobol(dimension=1, randomize='LMS_DS', graycode=False, seed=None, generating_matrices='sobol_mat.21201.32.32.msb.npy', d_max=None, t_max=None, m_max=None, msb=None, t_lms=None, _verbose=False)¶

Lattice¶

class qmcpy.discrete_distribution.lattice.lattice.Lattice(dimension=1, randomize=True, order='natural', seed=None, generating_vector='lattice_vec.3600.20.npy', d_max=None, m_max=None)¶

Quasi-Random Lattice nets in base 2.

>>> l = Lattice(2,seed=7)
>>> l.gen_samples(4)
array([[0.04386058, 0.58727432],
       [0.54386058, 0.08727432],
       [0.29386058, 0.33727432],
       [0.79386058, 0.83727432]])
>>> l.gen_samples(1)
array([[0.04386058, 0.58727432]])
>>> l
Lattice (DiscreteDistribution Object)
    d               2^(1)
    dvec            [0 1]
    randomize       1
    order           natural
    entropy         7
    spawn_key       ()
>>> Lattice(dimension=2,randomize=False,order='natural').gen_samples(4, warn=False)
array([[0.  , 0.  ],
       [0.5 , 0.5 ],
       [0.25, 0.75],
       [0.75, 0.25]])
>>> Lattice(dimension=2,randomize=False,order='linear').gen_samples(4, warn=False)
array([[0.  , 0.  ],
       [0.25, 0.75],
       [0.5 , 0.5 ],
       [0.75, 0.25]])
>>> Lattice(dimension=2,randomize=False,order='mps').gen_samples(4, warn=False)
array([[0.  , 0.  ],
       [0.5 , 0.5 ],
       [0.25, 0.75],
       [0.75, 0.25]])

References

[1] Sou-Cheng T. Choi, Yuhan Ding, Fred J. Hickernell, Lan Jiang, Lluis Antoni Jimenez Rugama, Da Li, Jagadeeswaran Rathinavel, Xin Tong, Kan Zhang, Yizhi Zhang, and Xuan Zhou, GAIL: Guaranteed Automatic Integration Library (Version 2.3) [MATLAB Software], 2019. Available from http://gailgithub.github.io/GAIL_Dev/

[2] F.Y. Kuo & D. Nuyens. Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients - a survey of analysis and implementation, Foundations of Computational Mathematics, 16(6):1631-1696, 2016. springer link: https://link.springer.com/article/10.1007/s10208-016-9329-5 arxiv link: https://arxiv.org/abs/1606.06613

[3] D. Nuyens, The Magic Point Shop of QMC point generators and generating vectors. MATLAB and Python software, 2018. Available from https://people.cs.kuleuven.be/~dirk.nuyens/

[4] Constructing embedded lattice rules for multivariate integration R Cools, FY Kuo, D Nuyens - SIAM J. Sci. Comput., 28(6), 2162-2188.

[5] L’Ecuyer, Pierre & Munger, David. (2015). LatticeBuilder: A General Software Tool for Constructing Rank-1 Lattice Rules. ACM Transactions on Mathematical Software. 42. 10.1145/2754929.

__init__(dimension=1, randomize=True, order='natural', seed=None, generating_vector='lattice_vec.3600.20.npy', d_max=None, m_max=None)¶

Parameters

dimension (int or ndarray) – dimension of the generator. If an int is passed in, use sequence dimensions [0,…,dimensions-1]. If a ndarray is passed in, use these dimension indices in the sequence.
randomize (bool) – If True, apply shift to generated samples. Note: Non-randomized lattice sequence includes the origin.
order (str) – ‘linear’, ‘natural’, or ‘mps’ ordering.
seed (None or int or numpy.random.SeedSeq) – seed the random number generator for reproducibility
generating_vector (ndarray or str) – generating matrix or path to generating matrices. ndarray should have shape (d_max). a string generating_vector should be formatted like ‘lattice_vec.3600.20.npy’ where ‘name.d_max.m_max.npy’
d_max (int) – maximum dimension
m_max (int) – 2^m_max is the max number of supported samples

Note

d_max and m_max are required if generating_vector is a ndarray. If generating_vector is an string (path), d_max and m_max can be taken from the file name if None

gen_samples(n=None, n_min=0, n_max=8, warn=True, return_unrandomized=False)¶

Generate lattice samples

Parameters

n (int) – if n is supplied, generate from n_min=0 to n_max=n samples. Otherwise use the n_min and n_max explicitly supplied as the following 2 arguments
n_min (int) – Starting index of sequence.
n_max (int) – Final index of sequence.
return_unrandomized (bool) – return samples without randomization as 2nd return value. Will not be returned if randomize=False.

Returns

(n_max-n_min) x d (dimension) array of samples

Return type

ndarray

Note

Lattice generates in blocks from 2**m to 2**(m+1) so generating n_min=3 to n_max=9 requires necessarily produces samples from n_min=2 to n_max=16 and automatically subsets. May be inefficient for non-powers-of-2 samples sizes.

pdf(x)¶: pdf of a standard uniform

Halton¶

class qmcpy.discrete_distribution.halton.Halton(dimension=1, randomize=True, generalize=True, seed=None)¶

Quasi-Random Halton nets.

>>> h_qrng = Halton(2,randomize='QRNG',generalize=True,seed=7)
>>> h_qrng.gen_samples(4)
array([[0.35362988, 0.38733489],
       [0.85362988, 0.72066823],
       [0.10362988, 0.05400156],
       [0.60362988, 0.498446  ]])
>>> h_qrng.gen_samples(1)
array([[0.35362988, 0.38733489]])
>>> h_qrng
Halton (DiscreteDistribution Object)
    d               2^(1)
    dvec            [0 1]
    randomize       QRNG
    generalize      1
    entropy         7
    spawn_key       ()
>>> h_owen = Halton(2,randomize='OWEN',generalize=False,seed=7)
>>> h_owen.gen_samples(4)
array([[0.64637012, 0.48226667],
       [0.14637012, 0.81560001],
       [0.89637012, 0.14893334],
       [0.39637012, 0.59337779]])
>>> h_owen
Halton (DiscreteDistribution Object)
    d               2^(1)
    dvec            [0 1]
    randomize       OWEN
    generalize      0
    entropy         7
    spawn_key       ()

References

[1] Marius Hofert and Christiane Lemieux (2019). qrng: (Randomized) Quasi-Random Number Generators. R package version 0.0-7. https://CRAN.R-project.org/package=qrng.

[2] Owen, A. B. “A randomized Halton algorithm in R,” 2017. arXiv:1706.02808 [stat.CO]

__init__(dimension=1, randomize=True, generalize=True, seed=None)¶

Parameters

dimension (int or ndarray) – dimension of the generator. If an int is passed in, use sequence dimensions [0,…,dimensions-1]. If a ndarray is passed in, use these dimension indices in the sequence.
randomize (str/bool) – select randomization method from ‘QRNG” [1], (max dimension = 360, supports generalize=True, default if randomize=True) or ‘OWEN’ [2], (max dimension = 1000),
generalize (bool) – generalize flag, only applicable to the QRNG generator
seed (None or int or numpy.random.SeedSeq) – seed the random number generator for reproducibility

gen_samples(n=None, n_min=0, n_max=8, warn=True)¶

Generate samples

Parameters

n (int) – if n is supplied, generate from n_min=0 to n_max=n samples. Otherwise use the n_min and n_max explicitly supplied as the following 2 arguments
n_min (int) – Starting index of sequence.
n_max (int) – Final index of sequence.

Returns

(n_max-n_min) x d (dimension) array of samples

Return type

ndarray

halton_owen(n, n0, d0=0)¶: see gen_samples method and [1] Owen, A. B.A randomized Halton algorithm in R2017. arXiv:1706.02808 [stat.CO].

pdf(x)¶: ABSTRACT METHOD to evaluate pdf of distribution the samples mimic at locations of x.

IID Standard Uniform¶

class qmcpy.discrete_distribution.iid_std_uniform.IIDStdUniform(dimension=1, seed=None)¶

A wrapper around NumPy’s IID Standard Uniform generator numpy.random.rand.

>>> dd = IIDStdUniform(dimension=2,seed=7)
>>> dd.gen_samples(4)
array([[0.04386058, 0.58727432],
       [0.3691824 , 0.65212985],
       [0.69669968, 0.10605352],
       [0.63025643, 0.13630282]])
>>> dd
IIDStdUniform (DiscreteDistribution Object)
    d               2^(1)
    entropy         7
    spawn_key       ()

__init__(dimension=1, seed=None)¶

Parameters

dimension (int) – dimension of samples
seed (None or int or numpy.random.SeedSeq) – seed the random number generator for reproducibility

gen_samples(n)¶

Generate samples

Parameters: n (int) – Number of observations to generate
Returns: n x self.d array of samples
Return type: ndarray

pdf(x)¶: ABSTRACT METHOD to evaluate pdf of distribution the samples mimic at locations of x.

True Measure Class¶

Abstract Measure Class¶

class qmcpy.true_measure._true_measure.TrueMeasure¶

True Measure abstract class. DO NOT INSTANTIATE.

gen_samples(*args, **kwargs)¶

Generate samples from the discrete distribution and transform them via the transform method.

Parameters

args (tuple) – positional arguments to the discrete distributions gen_samples method
kwargs (dict) – keyword arguments to the discrete distributions gen_samples method

Returns

n x d matrix of transformed samples

Return type

ndarray

spawn(s=1, dimensions=None)¶

Spawn new instances of the current discrete distribution but with new seeds and dimensions. Developed for multi-level and multi-replication (Q)MC algorithms.

Parameters

s (int) – number of spawn
dimensions (ndarray) – length s array of dimension for each spawn. Defaults to current dimension

Returns

list of TrueMeasures linked to newly spawned DiscreteDistributions

Return type

list

Uniform¶

class qmcpy.true_measure.uniform.Uniform(sampler, lower_bound=0.0, upper_bound=1.0)¶

>>> u = Uniform(DigitalNetB2(2,seed=7),lower_bound=[0,.5],upper_bound=[2,3])
>>> u.gen_samples(4)
array([[1.12538017, 0.93444992],
       [0.693306  , 2.12676579],
       [1.64149095, 2.88726434],
       [0.20844522, 1.73645241]])
>>> u
Uniform (TrueMeasure Object)
    lower_bound     [0.  0.5]
    upper_bound     [2 3]

__init__(sampler, lower_bound=0.0, upper_bound=1.0)¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
lower_bound (float) – a for Uniform(a,b)
upper_bound (float) – b for Uniform(a,b)

Gaussian¶

class qmcpy.true_measure.gaussian.Gaussian(sampler, mean=0.0, covariance=1.0, decomp_type='PCA')¶

Normal Measure.

>>> g = Gaussian(DigitalNetB2(2,seed=7),mean=[1,2],covariance=[[9,4],[4,5]])
>>> g.gen_samples(4)
array([[-0.23979685,  2.98944192],
       [ 2.45994002,  2.17853622],
       [-0.22923897, -1.92667105],
       [ 4.6127697 ,  4.25820377]])
>>> g
Gaussian (TrueMeasure Object)
    mean            [1 2]
    covariance      [[9 4]
                    [4 5]]
    decomp_type     PCA

__init__(sampler, mean=0.0, covariance=1.0, decomp_type='PCA')¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
mean (float) – mu for Normal(mu,sigma^2)
covariance (ndarray) – sigma^2 for Normal(mu,sigma^2). A float or d (dimension) vector input will be extended to covariance*eye(d)
decomp_type (str) – method of decomposition either “PCA” for principal component analysis or “Cholesky” for cholesky decomposition.

class qmcpy.true_measure.gaussian.Normal(sampler, mean=0.0, covariance=1.0, decomp_type='PCA')¶

Brownian Motion¶

class qmcpy.true_measure.brownian_motion.BrownianMotion(sampler, t_final=1, drift=0, decomp_type='PCA')¶

Geometric Brownian Motion.

>>> bm = BrownianMotion(DigitalNetB2(4,seed=7),t_final=2,drift=2)
>>> bm.gen_samples(2)
array([[0.44018403, 1.62690477, 2.69418273, 4.21753174],
       [1.97549563, 2.27002956, 2.92802765, 4.77126959]])
>>> bm
BrownianMotion (TrueMeasure Object)
    time_vec        [0.5 1.  1.5 2. ]
    drift           2^(1)
    mean            [1. 2. 3. 4.]
    covariance      [[0.5 0.5 0.5 0.5]
                    [0.5 1.  1.  1. ]
                    [0.5 1.  1.5 1.5]
                    [0.5 1.  1.5 2. ]]
    decomp_type     PCA

__init__(sampler, t_final=1, drift=0, decomp_type='PCA')¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
t_final (float) – end time for the Brownian Motion.
drift (int) – Gaussian mean is time_vec*drift
decomp_type (str) – method of decomposition either “PCA” for principal component analysis or “Cholesky” for cholesky decomposition.

Lebesgue¶

class qmcpy.true_measure.lebesgue.Lebesgue(sampler)¶

>>> Lebesgue(Gaussian(DigitalNetB2(2,seed=7)))
Lebesgue (TrueMeasure Object)
    transform       Gaussian (TrueMeasure Object)
                       mean            0
                       covariance      1
                       decomp_type     PCA
>>> Lebesgue(Uniform(DigitalNetB2(2,seed=7)))
Lebesgue (TrueMeasure Object)
    transform       Uniform (TrueMeasure Object)
                       lower_bound     0
                       upper_bound     1

__init__(sampler)¶

Parameters: sampler (TrueMeasure) – A true measure by which to compose a transform.

Continuous Bernoulli¶

class qmcpy.true_measure.bernoulli_cont.BernoulliCont(sampler, lam=0.5)¶

>>> bc = BernoulliCont(DigitalNetB2(2,seed=7),lam=.2)
>>> bc.gen_samples(4)
array([[0.39545122, 0.10073414],
       [0.21719142, 0.48293404],
       [0.68958314, 0.90847415],
       [0.05871131, 0.33436033]])
>>> bc
BernoulliCont (TrueMeasure Object)
    lam             0.200

See https://en.wikipedia.org/wiki/Continuous_Bernoulli_distribution

__init__(sampler, lam=0.5)¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
lam (ndarray) – 0 < lambda < 1, a shape parameter, independent for each dimension

Johnson’s SU¶

class qmcpy.true_measure.johnsons_su.JohnsonsSU(sampler, gamma=1, xi=1, delta=2, lam=2)¶

>>> jsu = JohnsonsSU(DigitalNetB2(2,seed=7),gamma=1,xi=2,delta=3,lam=4)
>>> jsu.gen_samples(4)
array([[ 0.86224892, -0.76967276],
       [ 0.07317047,  1.17727769],
       [ 1.89093286,  2.9341619 ],
       [-1.30283298,  0.62269632]])
>>> jsu
JohnsonsSU (TrueMeasure Object)
    gamma           1
    xi              2^(1)
    delta           3
    lam             2^(2)

See https://en.wikipedia.org/wiki/Johnson%27s_SU-distribution

__init__(sampler, gamma=1, xi=1, delta=2, lam=2)¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
gamma (ndarray) – gamma
xi (ndarray) – xi
delta (ndarray) – delta > 0
lam (ndarray) – lambda > 0

Kumaraswamy¶

class qmcpy.true_measure.kumaraswamy.Kumaraswamy(sampler, a=2, b=2)¶

>>> k = Kumaraswamy(DigitalNetB2(2,seed=7),a=[1,2],b=[3,4])
>>> k.gen_samples(4)
array([[0.24096272, 0.21587652],
       [0.13227662, 0.4808615 ],
       [0.43615893, 0.73428949],
       [0.03602294, 0.39602319]])
>>> k
Kumaraswamy (TrueMeasure Object)
    a               [1 2]
    b               [3 4]

See https://en.wikipedia.org/wiki/Kumaraswamy_distribution

__init__(sampler, a=2, b=2)¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
a (ndarray) – alpha > 0
b (ndarray) – beta > 0

SciPy Wrapper¶

class qmcpy.true_measure.scipy_wrapper.SciPyWrapper(sampler, scipy_distrib, **scipy_distrib_kwargs)¶

>>> triangular = SciPyWrapper(
...     sampler = DigitalNetB2(2,seed=7),
...     scipy_distrib = scipy.stats.triang,
...     c = [0.1,.2],
...     loc = [1,2],
...     scale = [3,4])
>>> triangular.gen_samples(4)
array([[2.11792393, 2.74571838],
       [1.6995412 , 3.88553573],
       [2.79502629, 5.24025887],
       [1.30634136, 3.45650562]])
>>> triangular
SciPyWrapper (TrueMeasure Object)
    scipy_distrib   triang
    c               [0.1 0.2]
    loc             [1 2]
    scale           [3 4]

__init__(sampler, scipy_distrib, **scipy_distrib_kwargs)¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
scipy_stat_distrib (float) – a CONTINUOUS UNIVARIATE scipy.stats distribution e.g. scipy.stats.norm, see https://docs.scipy.org/doc/scipy/reference/stats.html#continuous-distributions
**scipy_distrib_kwargs (float) – keyword arguments for scipy_stat_distrib.{pdf,ppf or pmf}. Note that you may pass in vectors of keyword arguments and they will be distributed appropriotly across the dimensions. Also note that positional aguments to scipy_stat_distrib.{pdf,ppf or pmf} must still be supplied as keyword arguments to QMCPy’s SciPyWrapper, see e.g. the above doctest and https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.triang.html#scipy.stats.triang

Integrand Class¶

Abstract Integrand Class¶

class qmcpy.integrand._integrand.Integrand(dprime, parallel)¶

Integrand abstract class. DO NOT INSTANTIATE.

__init__(dprime, parallel)¶

Parameters

dprime (tuple) – function output dimension shape.
parallel (int) – If parallel is False, 0, or 1: function evaluation is done in serial fashion. Otherwise, parallel specifies the number of CPUs used by multiprocessing.Pool. Passing parallel=True sets the number of CPUs equal to os.cpu_count().

bound_fun(bound_low, bound_high)¶

Compute the bounds on the combined function based on bounds for the individual functions. Defaults to the identity where we essentially do not combine integrands, but instead integrate each function individually.

Parameters

bound_low (ndarray) – length Integrand.dprime lower error bound
bound_high (ndarray) – length Integrand.dprime upper error bound

Returns

(tuple) containing

(ndarray): lower bound on function combining estimates
(ndarray): upper bound on function combining estimates
(ndarray): bool flags to override sufficient combined integrand estimation, e.g., when approximating a ratio of integrals, if the denominator’s bounds straddle 0, then returning True here forces ratio to be flagged as insufficiently approximated.

dependency(flags_comb)¶

takes a vector of indicators of weather of not the error bound is satisfied for combined integrands and which returns flags for individual integrands. For example, if we are taking the ratio of 2 individual integrands, then getting flag_comb=True means the ratio has not been approximated to within the tolerance, so the dependency function should return [True,True] indicating that both the numerator and denominator integrands need to be better approximated. :param flags_comb: flags indicating weather the combined integrals are insufficiently approximated :type flags_comb: bool ndarray

Returns: length (Integrand.dprime) flags for individual integrands
Return type: (bool ndarray)

f(x, periodization_transform='NONE', compute_flags=None, *args, **kwargs)¶

Evaluate transformed integrand based on true measures and discrete distribution

Parameters

x (ndarray) – n x d array of samples from a discrete distribution
periodization_transform (str) – periodization transform
compute_flags (ndarray) – TODO
*args – other ordered args to g
**kwargs (dict) – other keyword args to g

Returns

length n vector of function evaluations

Return type

ndarray

g(t, *args, **kwargs)¶

ABSTRACT METHOD for original integrand to be integrated.

Parameters: t (ndarray) – n x d array of samples to be input into original integrand.
Returns: n vector of function evaluations
Return type: ndarray

spawn(levels)¶

Spawn new instances of the current integrand at the specified levels.

Parameters: levels (ndarray) – array of levels at which to spawn new integrands
Returns: list of Integrands linked to newly spawned TrueMeasures and DiscreteDistributions
Return type: list

Custom Function¶

class qmcpy.integrand.custom_fun.CustomFun(true_measure, g, dprime=1, parallel=False)¶

Integrand wrapper for a user’s function

>>> cf = CustomFun(
...     true_measure = Gaussian(DigitalNetB2(2,seed=7),mean=[1,2]),
...     g = lambda x: x[:,0]**2*x[:,1],
...     dprime = 1)
>>> x = cf.discrete_distrib.gen_samples(2**10)
>>> y = cf.f(x)
>>> y.shape
(1024, 1)
>>> y.mean()
3.995...
>>> cf = CustomFun(
...     true_measure = Uniform(DigitalNetB2(3,seed=7),lower_bound=[2,3,4],upper_bound=[4,5,6]),
...     g = lambda x,compute_flags=None: x,
...     dprime = 3)
>>> x = cf.discrete_distrib.gen_samples(2**10)
>>> y = cf.f(x)
>>> y.shape
(1024, 3)
>>> y.mean(0)
array([3., 4., 5.])

__init__(true_measure, g, dprime=1, parallel=False)¶

Parameters

true_measure (TrueMeasure) – a TrueMeasure instance.
g (function) – a function handle.
dprime (tuple) – function output dimension shape.
parallel (int) – If parallel is False, 0, or 1: function evaluation is done in serial fashion. Otherwise, parallel specifies the number of CPUs used by multiprocessing.Pool. Passing parallel=True sets the number of CPUs equal to os.cpu_count(). Do NOT set g to a lambda function when doing parallel computation

g(t, *args, **kwargs)¶

ABSTRACT METHOD for original integrand to be integrated.

Parameters: t (ndarray) – n x d array of samples to be input into original integrand.
Returns: n vector of function evaluations
Return type: ndarray

Keister Function¶

class qmcpy.integrand.keister.Keister(sampler)¶

$f(\boldsymbol{t}) = \pi^{d/2} \cos(\| \boldsymbol{t} \|)$ .

The standard example integrates the Keister integrand with respect to an IID Gaussian distribution with variance 1./2.

>>> k = Keister(DigitalNetB2(2,seed=7))
>>> x = k.discrete_distrib.gen_samples(2**10)
>>> y = k.f(x)
>>> y.mean()
1.808...
>>> k.true_measure
Gaussian (TrueMeasure Object)
    mean            0
    covariance      2^(-1)
    decomp_type     PCA
>>> k = Keister(Gaussian(DigitalNetB2(2,seed=7),mean=0,covariance=2))
>>> x = k.discrete_distrib.gen_samples(2**12)
>>> y = k.f(x)
>>> y.mean()
1.808...
>>> yp = k.f(x,periodization_transform='c2sin')
>>> yp.mean()
1.807...

References

[1] B. D. Keister, Multidimensional Quadrature Algorithms, Computers in Physics, 10, pp. 119-122, 1996.

__init__(sampler)¶

Parameters: sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform

exact_integ(d)¶: computes the true value of the Keister integral in dimension d. Accuracy might degrade as d increases due to round-off error. :param d: :return: true_integral

g(t)¶

ABSTRACT METHOD for original integrand to be integrated.

Parameters: t (ndarray) – n x d array of samples to be input into original integrand.
Returns: n vector of function evaluations
Return type: ndarray

Box Integral¶

class qmcpy.integrand.box_integral.BoxIntegral(sampler, s=array([1, 2]))¶

$B_s(x) = \left(\sum_{j=1}^d x_j^2 \right)^{s/2}$

>>> l1 = BoxIntegral(DigitalNetB2(2,seed=7), s=[7])
>>> x1 = l1.discrete_distrib.gen_samples(2**10)
>>> y1 = l1.f(x1)
>>> y1.shape
(1024, 1)
>>> y1.mean(0)
array([0.75156724])
>>> l2 = BoxIntegral(DigitalNetB2(5,seed=7), s=[-7,7])
>>> x2 = l2.discrete_distrib.gen_samples(2**10)
>>> y2 = l2.f(x2,compute_flags=[1,1])
>>> y2.shape
(1024, 2)
>>> y2.mean(0)
array([ 6.67548708, 10.52267786])

References:

[1] D.H. Bailey, J.M. Borwein, R.E. Crandall,Box integrals, Journal of Computational and Applied Mathematics, Volume 206, Issue 1, 2007, Pages 196-208, ISSN 0377-0427, https://doi.org/10.1016/j.cam.2006.06.010. (https://www.sciencedirect.com/science/article/pii/S0377042706004250)

[2] https://www.davidhbailey.com/dhbpapers/boxintegrals.pdf

__init__(sampler, s=array([1, 2]))¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
s (list or ndarray) – vectorized s parameter, len(s) is the number of vectorized integrals to evalute.

g(t, **kwargs)¶

ABSTRACT METHOD for original integrand to be integrated.

Parameters: t (ndarray) – n x d array of samples to be input into original integrand.
Returns: n vector of function evaluations
Return type: ndarray

European Option¶

class qmcpy.integrand.european_option.EuropeanOption(sampler, volatility=0.5, start_price=30, strike_price=35, interest_rate=0, t_final=1, call_put='call')¶

European financial option.

>>> eo = EuropeanOption(DigitalNetB2(4,seed=7),call_put='put')
>>> eo
EuropeanOption (Integrand Object)
    volatility      2^(-1)
    call_put        put
    start_price     30
    strike_price    35
    interest_rate   0
>>> x = eo.discrete_distrib.gen_samples(2**12)
>>> y = eo.f(x)
>>> y.mean()
9.209...
>>> eo = EuropeanOption(BrownianMotion(DigitalNetB2(4,seed=7),drift=1),call_put='put')
>>> x = eo.discrete_distrib.gen_samples(2**12)
>>> y = eo.f(x)
>>> y.mean()
9.162...
>>> eo.get_exact_value()
9.211452976234058

__init__(sampler, volatility=0.5, start_price=30, strike_price=35, interest_rate=0, t_final=1, call_put='call')¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
volatility (float) – sigma, the volatility of the asset
start_price (float) – S(0), the asset value at t=0
strike_price (float) – strike_price, the call/put offer
interest_rate (float) – r, the annual interest rate
t_final (float) – exercise time
call_put (str) – ‘call’ or ‘put’ option

g(t)¶: See abstract method.

get_exact_value()¶

Get the fair price of a European call/put option.

Returns: fair price
Return type: float

Asian Option¶

class qmcpy.integrand.asian_option.AsianOption(sampler, volatility=0.5, start_price=30.0, strike_price=35.0, interest_rate=0.0, t_final=1, call_put='call', mean_type='arithmetic', multilevel_dims=None, _dim_frac=0)¶

Asian financial option.

>>> ac = AsianOption(DigitalNetB2(4,seed=7))
>>> ac
AsianOption (Integrand Object)
    volatility      2^(-1)
    call_put        call
    start_price     30
    strike_price    35
    interest_rate   0
    mean_type       arithmetic
    dim_frac        0
>>> x = ac.discrete_distrib.gen_samples(2**12)
>>> y = ac.f(x)
>>> y.mean()
1.768...
>>> level_dims = [2,4,8]
>>> ac2_multilevel = AsianOption(DigitalNetB2(seed=7),multilevel_dims=level_dims)
>>> levels_to_spawn = arange(ac2_multilevel.max_level+1)
>>> ac2_single_levels = ac2_multilevel.spawn(levels_to_spawn)
>>> yml = 0
>>> for ac2_single_level in ac2_single_levels:
...     x = ac2_single_level.discrete_distrib.gen_samples(2**12)
...     level_est = ac2_single_level.f(x).mean()
...     yml += level_est
>>> yml
1.779...

__init__(sampler, volatility=0.5, start_price=30.0, strike_price=35.0, interest_rate=0.0, t_final=1, call_put='call', mean_type='arithmetic', multilevel_dims=None, _dim_frac=0)¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
volatility (float) – sigma, the volatility of the asset
start_price (float) – S(0), the asset value at t=0
strike_price (float) – strike_price, the call/put offer
interest_rate (float) – r, the annual interest rate
t_final (float) – exercise time
mean_type (string) – ‘arithmetic’ or ‘geometric’ mean
multilevel_dims (list of ints) – list of dimensions at each level. Leave as None for single-level problems
_dim_frac (float) – for internal use only, users should not set this parameter.

g(t)¶

ABSTRACT METHOD for original integrand to be integrated.

Parameters: t (ndarray) – n x d array of samples to be input into original integrand.
Returns: n vector of function evaluations
Return type: ndarray

Multilevel Call Options with Milstein Discretization¶

class qmcpy.integrand.ml_call_options.MLCallOptions(sampler, option='european', volatility=0.2, start_strike_price=100.0, interest_rate=0.05, t_final=1.0, _level=0)¶

Various call options from finance using Milstein discretization with $2^l$ timesteps on level $l$ .

>>> mlco_original = MLCallOptions(DigitalNetB2(seed=7))
>>> mlco_original
MLCallOptions (Integrand Object)
    option          european
    sigma           0.200
    k               100
    r               0.050
    t               1
    b               85
    level           0
>>> mlco_ml_dims = mlco_original.spawn(levels=arange(4))
>>> yml = 0
>>> for mlco in mlco_ml_dims:
...     x = mlco.discrete_distrib.gen_samples(2**10)
...     yml += mlco.f(x).mean()
>>> yml
10.393...

References:

[1] M.B. Giles. Improved multilevel Monte Carlo convergence using the Milstein scheme. 343-358, in Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, 2008. http://people.maths.ox.ac.uk/~gilesm/files/mcqmc06.pdf.

__init__(sampler, option='european', volatility=0.2, start_strike_price=100.0, interest_rate=0.05, t_final=1.0, _level=0)¶

Parameters

sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform
option_type (str) – type of option in [“European”,”Asian”]
volatility (float) – sigma, the volatility of the asset
start_strike_price (float) – S(0), the asset value at t=0, and K, the strike price. Assume start_price = strike_price
interest_rate (float) – r, the annual interest rate
t_final (float) – exercise time
_level (int) – for internal use only, users should not set this parameter.

g(t)¶

Parameters: t (ndarray) – Gaussian(0,1^2) samples
Returns: First, an ndarray of length 6 vector of summary statistic sums. Second, a float of cost on this level.
Return type: tuple

get_exact_value()¶: Print exact analytic value, based on s0=k.

Linear Function¶

class qmcpy.integrand.linear0.Linear0(sampler)¶

>>> l = Linear0(DigitalNetB2(100,seed=7))
>>> x = l.discrete_distrib.gen_samples(2**10)
>>> y = l.f(x)
>>> y.mean()
-1.175...e-08
>>> ytf = l.f(x,periodization_transform='C1SIN')
>>> ytf.mean()
-4.050...e-12

__init__(sampler)¶

Parameters: sampler (DiscreteDistribution/TrueMeasure) – A discrete distribution from which to transform samples or a true measure by which to compose a transform

g(t)¶

ABSTRACT METHOD for original integrand to be integrated.

Parameters: t (ndarray) – n x d array of samples to be input into original integrand.
Returns: n vector of function evaluations
Return type: ndarray

Sobol’ Indices¶

class qmcpy.integrand.sobol_indices.SensitivityIndices(integrand, indices='singletons')¶

class qmcpy.integrand.sobol_indices.SobolIndices(integrand, indices='singletons')¶

Sobol’ Indicies in QMCPy.

>>> dnb2 = DigitalNetB2(dimension=3,seed=7)
>>> keister_d = Keister(dnb2)
>>> keister_indices = SobolIndices(keister_d,indices='singletons')
>>> sc = CubQMCNetG(keister_indices,abs_tol=1e-3)
>>> solution,data = sc.integrate()
>>> solution.squeeze()
array([[0.32803639, 0.32795358, 0.32807359],
       [0.33884667, 0.33857811, 0.33884115]])
>>> data
LDTransformData (AccumulateData Object)
    solution        [[0.328 0.328 0.328]
                    [0.339 0.339 0.339]]
    indv_error      [[0.002 0.002 0.002]
                    [0.002 0.002 0.002]
                    [0.    0.    0.   ]
                    [0.    0.    0.   ]
                    [0.001 0.001 0.001]
                    [0.003 0.003 0.003]]
    ci_low          [[1.67  1.67  1.671]
                    [1.725 1.724 1.725]
                    [2.168 2.168 2.168]
                    [2.168 2.168 2.168]
                    [9.799 9.799 9.799]
                    [9.797 9.797 9.797]]
    ci_high         [[1.675 1.674 1.675]
                    [1.73  1.729 1.73 ]
                    [2.168 2.168 2.168]
                    [2.169 2.169 2.169]
                    [9.802 9.802 9.802]
                    [9.803 9.803 9.803]]
    ci_comb_low     [[0.327 0.327 0.328]
                    [0.338 0.338 0.338]]
    ci_comb_high    [[0.329 0.329 0.329]
                    [0.34  0.339 0.34 ]]
    flags_comb      [[False False False]
                    [False False False]]
    flags_indv      [[False False False]
                    [False False False]
                    [False False False]
                    [False False False]
                    [False False False]
                    [False False False]]
    n_total         2^(16)
    n               [[65536. 65536. 65536.]
                    [32768. 32768. 32768.]
                    [65536. 65536. 65536.]
                    [32768. 32768. 32768.]
                    [65536. 65536. 65536.]
                    [32768. 32768. 32768.]]
    time_integrate  ...
CubQMCNetG (StoppingCriterion Object)
    abs_tol         0.001
    rel_tol         0
    n_init          2^(10)
    n_max           2^(35)
SobolIndices (Integrand Object)
    indices         [[0]
                    [1]
                    [2]]
    n_multiplier    3
Gaussian (TrueMeasure Object)
    mean            0
    covariance      2^(-1)
    decomp_type     PCA
DigitalNetB2 (DiscreteDistribution Object)
    d               6
    dvec            [0 1 2 3 4 5]
    randomize       LMS_DS
    graycode        0
    entropy         7
    spawn_key       (0,)

References

[1] Art B. Owen.Monte Carlo theory, methods and examples. 2013. Appendix A.

__init__(integrand, indices='singletons')¶

Parameters

integrand (Integrand) – integrand to find Sobol’ indices of
indices (list of lists) – each element of indices should be a list of indices, u, at which to compute the Sobol’ indices. The default indices=’singletons’ sets indices=[[0],[1],…[d-1]]. Should not include [], the null set

bound_fun(bound_low, bound_high)¶

Compute the bounds on the combined function based on bounds for the individual functions. Defaults to the identity where we essentially do not combine integrands, but instead integrate each function individually.

Parameters

bound_low (ndarray) – length Integrand.dprime lower error bound
bound_high (ndarray) – length Integrand.dprime upper error bound

Returns

(tuple) containing

(ndarray): lower bound on function combining estimates
(ndarray): upper bound on function combining estimates
(ndarray): bool flags to override sufficient combined integrand estimation, e.g., when approximating a ratio of integrals, if the denominator’s bounds straddle 0, then returning True here forces ratio to be flagged as insufficiently approximated.

dependency(flags_comb)¶

takes a vector of indicators of weather of not the error bound is satisfied for combined integrands and which returns flags for individual integrands. For example, if we are taking the ratio of 2 individual integrands, then getting flag_comb=True means the ratio has not been approximated to within the tolerance, so the dependency function should return [True,True] indicating that both the numerator and denominator integrands need to be better approximated. :param flags_comb: flags indicating weather the combined integrals are insufficiently approximated :type flags_comb: bool ndarray

Returns: length (Integrand.dprime) flags for individual integrands
Return type: (bool ndarray)

f(x, *args, **kwargs)¶

Evaluate transformed integrand based on true measures and discrete distribution

Parameters

x (ndarray) – n x d array of samples from a discrete distribution
periodization_transform (str) – periodization transform
compute_flags (ndarray) – TODO
*args – other ordered args to g
**kwargs (dict) – other keyword args to g

Returns

length n vector of function evaluations

Return type

ndarray

Stopping Criterion Algorithms¶

Abstract Stopping Criterion Class¶

class qmcpy.stopping_criterion._stopping_criterion.StoppingCriterion(allowed_levels, allowed_distribs, allow_vectorized_integrals)¶

Stopping Criterion abstract class. DO NOT INSTANTIATE.

__init__(allowed_levels, allowed_distribs, allow_vectorized_integrals)¶

Parameters

distribution (DiscreteDistribution) – a DiscreteDistribution
allowed_levels (list) – which integrand types are supported: ‘single’, ‘fixed-multi’, ‘adaptive-multi’
allowed_distribs (list) – list of compatible DiscreteDistribution classes

integrate()¶

ABSTRACT METHOD to determine the number of samples needed to satisfy the tolerance.

Returns

tuple containing:

solution (float): approximation to the integral
data (AccumulateData): an AccumulateData object

Return type

tuple

set_tolerance(*args, **kwargs)¶: ABSTRACT METHOD to reset the absolute tolerance.

Guaranteed Digital Net Cubature (QMC)¶

class qmcpy.stopping_criterion.cub_qmc_net_g.CubQMCNetG(integrand, abs_tol=0.01, rel_tol=0.0, n_init=1024.0, n_max=34359738368.0, fudge=<function CubQMCNetG.<lambda>>, check_cone=False, control_variates=[], control_variate_means=[], update_beta=False, error_fun=<function CubQMCNetG.<lambda>>)¶

Quasi-Monte Carlo method using Sobol’ cubature over the d-dimensional region to integrate within a specified generalized error tolerance with guarantees under Walsh-Fourier coefficients cone decay assumptions.

>>> k = Keister(DigitalNetB2(2,seed=7))
>>> sc = CubQMCNetG(k,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
LDTransformData (AccumulateData Object)
    solution        1.809
    indv_error      0.005
    ci_low          1.804
    ci_high         1.814
    ci_comb_low     1.804
    ci_comb_high    1.814
    flags_comb      0
    flags_indv      0
    n_total         2^(10)
    n               2^(10)
    time_integrate  ...
CubQMCNetG (StoppingCriterion Object)
    abs_tol         0.050
    rel_tol         0
    n_init          2^(10)
    n_max           2^(35)
Keister (Integrand Object)
Gaussian (TrueMeasure Object)
    mean            0
    covariance      2^(-1)
    decomp_type     PCA
DigitalNetB2 (DiscreteDistribution Object)
    d               2^(1)
    dvec            [0 1]
    randomize       LMS_DS
    graycode        0
    entropy         7
    spawn_key       ()
>>> dd = DigitalNetB2(3,seed=7)
>>> g1 = CustomFun(Uniform(dd,0,2),lambda t: 10*t[:,0]-5*t[:,1]**2+t[:,2]**3)
>>> cv1 = CustomFun(Uniform(dd,0,2),lambda t: t[:,0])
>>> cv2 = CustomFun(Uniform(dd,0,2),lambda t: t[:,1]**2)
>>> sc = CubQMCNetG(g1,abs_tol=1e-6,check_cone=True,
...     control_variates = [cv1,cv2],
...     control_variate_means = [1,4/3])
>>> sol,data = sc.integrate()
>>> sol
array([5.33333333])
>>> exactsol = 16/3
>>> abs(sol-exactsol)<1e-6
array([ True])
>>> dnb2 = DigitalNetB2(3,seed=7)
>>> f = BoxIntegral(dnb2, s=[-1,1])
>>> abs_tol = 1e-3
>>> sc = CubQMCNetG(f, abs_tol=abs_tol)
>>> solution,data = sc.integrate()
>>> solution
array([1.18944142, 0.96064165])
>>> sol3neg1 = -pi/4-1/2*log(2)+log(5+3*sqrt(3))
>>> sol31 = sqrt(3)/4+1/2*log(2+sqrt(3))-pi/24
>>> true_value = array([sol3neg1,sol31])
>>> (abs(true_value-solution)<abs_tol).all()
True
>>> f2 = BoxIntegral(dnb2,s=[3,4])
>>> sc = CubQMCNetG(f2,control_variates=f,control_variate_means=true_value,update_beta=True)
>>> solution,data = sc.integrate()
>>> solution
array([1.10168119, 1.26661293])
>>> data
LDTransformData (AccumulateData Object)
    solution        [1.102 1.267]
    indv_error      [0.002 0.005]
    ci_low          [1.099 1.262]
    ci_high         [1.104 1.271]
    ci_comb_low     [1.099 1.262]
    ci_comb_high    [1.104 1.271]
    flags_comb      [False False]
    flags_indv      [False False]
    n_total         2^(10)
    n               [1024. 1024.]
    time_integrate  ...
CubQMCNetG (StoppingCriterion Object)
    abs_tol         0.010
    rel_tol         0
    n_init          2^(10)
    n_max           2^(35)
    cv              BoxIntegral (Integrand Object)
                       s               [-1  1]
    cv_mu           [1.19  0.961]
    update_beta     1
BoxIntegral (Integrand Object)
    s               [3 4]
Uniform (TrueMeasure Object)
    lower_bound     0
    upper_bound     1
DigitalNetB2 (DiscreteDistribution Object)
    d               3
    dvec            [0 1 2]
    randomize       LMS_DS
    graycode        0
    entropy         7
    spawn_key       ()
>>> cf = CustomFun(
...     true_measure = Uniform(DigitalNetB2(6,seed=7)),
...     g = lambda x,compute_flags=None: (2*arange(1,7)*x).reshape(-1,2,3),
...     dprime = (2,3))
>>> sol,data = CubQMCNetG(cf,abs_tol=1e-6).integrate()
>>> data
LDTransformData (AccumulateData Object)
    solution        [[1. 2. 3.]
                    [4. 5. 6.]]
    indv_error      [[2.825e-08 6.101e-07 2.456e-10]
                    [4.547e-12 3.725e-07 3.499e-09]]
    ci_low          [[1. 2. 3.]
                    [4. 5. 6.]]
    ci_high         [[1. 2. 3.]
                    [4. 5. 6.]]
    ci_comb_low     [[1. 2. 3.]
                    [4. 5. 6.]]
    ci_comb_high    [[1. 2. 3.]
                    [4. 5. 6.]]
    flags_comb      [[False False False]
                    [False False False]]
    flags_indv      [[False False False]
                    [False False False]]
    n_total         2^(13)
    n               [[2048. 1024. 1024.]
                    [8192. 4096. 2048.]]
    time_integrate  ...
CubQMCNetG (StoppingCriterion Object)
    abs_tol         1.00e-06
    rel_tol         0
    n_init          2^(10)
    n_max           2^(35)
CustomFun (Integrand Object)
Uniform (TrueMeasure Object)
    lower_bound     0
    upper_bound     1
DigitalNetB2 (DiscreteDistribution Object)
    d               6
    dvec            [0 1 2 3 4 5]
    randomize       LMS_DS
    graycode        0
    entropy         7
    spawn_key       ()

Original Implementation:

https://github.com/GailGithub/GAIL_Dev/blob/master/Algorithms/IntegrationExpectation/cubSobol_g.m

References

[1] Fred J. Hickernell and Lluis Antoni Jimenez Rugama, Reliable adaptive cubature using digital sequences, 2014. Submitted for publication: arXiv:1410.8615.

[2] Sou-Cheng T. Choi, Yuhan Ding, Fred J. Hickernell, Lan Jiang, Lluis Antoni Jimenez Rugama, Da Li, Jagadeeswaran Rathinavel, Xin Tong, Kan Zhang, Yizhi Zhang, and Xuan Zhou, GAIL: Guaranteed Automatic Integration Library (Version 2.3) [MATLAB Software], 2019. Available from http://gailgithub.github.io/GAIL_Dev/

Guarantee:: This algorithm computes the integral of real valued functions in $[0,1]^d$ with a prescribed generalized error tolerance. The Fourier coefficients of the integrand are assumed to be absolutely convergent. If the algorithm terminates without warning messages, the output is given with guarantees under the assumption that the integrand lies inside a cone of functions. The guarantee is based on the decay rate of the Fourier coefficients. For integration over domains other than $[0,1]^d$ , this cone condition applies to $f \circ \psi$ (the composition of the functions) where $\psi$ is the transformation function for $[0,1]^d$ to the desired region. For more details on how the cone is defined, please refer to the references below.

__init__(integrand, abs_tol=0.01, rel_tol=0.0, n_init=1024.0, n_max=34359738368.0, fudge=<function CubQMCNetG.<lambda>>, check_cone=False, control_variates=[], control_variate_means=[], update_beta=False, error_fun=<function CubQMCNetG.<lambda>>)¶

Parameters

integrand (Integrand) – an instance of Integrand
abs_tol (float) – absolute error tolerance
rel_tol (float) – relative error tolerance
n_init (int) – initial number of samples
n_max (int) – maximum number of samples
fudge (function) – positive function multiplying the finite sum of Fast Fourier coefficients specified in the cone of functions
check_cone (boolean) – check if the function falls in the cone
control_variates (list) – list of integrand objects to be used as control variates. Control variates are currently only compatible with single level problems. The same discrete distribution instance must be used for the integrand and each of the control variates.
control_variate_means (list) – list of means for each control variate
update_beta (bool) – update control variate beta coefficients at each iteration?
error_fun – function taking in the approximate solution vector, absolute tolerance, and relative tolerance which returns the approximate error. Default indicates integration until either absolute OR relative tolerance is satisfied.

class qmcpy.stopping_criterion.cub_qmc_net_g.CubQMCSobolG(integrand, abs_tol=0.01, rel_tol=0.0, n_init=1024.0, n_max=34359738368.0, fudge=<function CubQMCNetG.<lambda>>, check_cone=False, control_variates=[], control_variate_means=[], update_beta=False, error_fun=<function CubQMCNetG.<lambda>>)¶

Guaranteed Lattice Cubature (QMC)¶

class qmcpy.stopping_criterion.cub_qmc_lattice_g.CubQMCLatticeG(integrand, abs_tol=0.01, rel_tol=0.0, n_init=1024.0, n_max=34359738368.0, fudge=<function CubQMCLatticeG.<lambda>>, check_cone=False, ptransform='Baker', error_fun=<function CubQMCLatticeG.<lambda>>)¶

Stopping Criterion quasi-Monte Carlo method using rank-1 Lattices cubature over a d-dimensional region to integrate within a specified generalized error tolerance with guarantees under Fourier coefficients cone decay assumptions.

>>> k = Keister(Lattice(2,seed=7))
>>> sc = CubQMCLatticeG(k,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
LDTransformData (AccumulateData Object)
    solution        1.810
    indv_error      0.005
    ci_low          1.806
    ci_high         1.815
    ci_comb_low     1.806
    ci_comb_high    1.815
    flags_comb      0
    flags_indv      0
    n_total         2^(10)
    n               2^(10)
    time_integrate  ...
CubQMCLatticeG (StoppingCriterion Object)
    abs_tol         0.050
    rel_tol         0
    n_init          2^(10)
    n_max           2^(35)
Keister (Integrand Object)
Gaussian (TrueMeasure Object)
    mean            0
    covariance      2^(-1)
    decomp_type     PCA
Lattice (DiscreteDistribution Object)
    d               2^(1)
    dvec            [0 1]
    randomize       1
    order           natural
    entropy         7
    spawn_key       ()
>>> f = BoxIntegral(Lattice(3,seed=7), s=[-1,1])
>>> abs_tol = 1e-3
>>> sc = CubQMCLatticeG(f, abs_tol=abs_tol)
>>> solution,data = sc.integrate()
>>> solution
array([1.18954582, 0.96056304])
>>> sol3neg1 = -pi/4-1/2*log(2)+log(5+3*sqrt(3))
>>> sol31 = sqrt(3)/4+1/2*log(2+sqrt(3))-pi/24
>>> true_value = array([sol3neg1,sol31])
>>> (abs(true_value-solution)<abs_tol).all()
True
>>> cf = CustomFun(
...     true_measure = Uniform(Lattice(6,seed=7)),
...     g = lambda x,compute_flags=None: (2*arange(1,7)*x).reshape(-1,2,3),
...     dprime = (2,3))
>>> sol,data = CubQMCLatticeG(cf,abs_tol=1e-6).integrate()
>>> data
LDTransformData (AccumulateData Object)
    solution        [[1. 2. 3.]
                    [4. 5. 6.]]
    indv_error      [[9.658e-07 4.835e-07 7.244e-07]
                    [9.655e-07 3.017e-07 3.625e-07]]
    ci_low          [[1. 2. 3.]
                    [4. 5. 6.]]
    ci_high         [[1. 2. 3.]
                    [4. 5. 6.]]
    ci_comb_low     [[1. 2. 3.]
                    [4. 5. 6.]]
    ci_comb_high    [[1. 2. 3.]
                    [4. 5. 6.]]
    flags_comb      [[False False False]
                    [False False False]]
    flags_indv      [[False False False]
                    [False False False]]
    n_total         2^(15)
    n               [[ 8192. 16384. 16384.]
                    [16384. 32768. 32768.]]
    time_integrate  ...
CubQMCLatticeG (StoppingCriterion Object)
    abs_tol         1.00e-06
    rel_tol         0
    n_init          2^(10)
    n_max           2^(35)
CustomFun (Integrand Object)
Uniform (TrueMeasure Object)
    lower_bound     0
    upper_bound     1
Lattice (DiscreteDistribution Object)
    d               6
    dvec            [0 1 2 3 4 5]
    randomize       1
    order           natural
    entropy         7
    spawn_key       ()

Original Implementation:

https://github.com/GailGithub/GAIL_Dev/blob/master/Algorithms/IntegrationExpectation/cubLattice_g.m

References

[1] Lluis Antoni Jimenez Rugama and Fred J. Hickernell, “Adaptive multidimensional integration based on rank-1 lattices,” Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, Belgium, April 2014 (R. Cools and D. Nuyens, eds.), Springer Proceedings in Mathematics and Statistics, vol. 163, Springer-Verlag, Berlin, 2016, arXiv:1411.1966, pp. 407-422.

[2] Sou-Cheng T. Choi, Yuhan Ding, Fred J. Hickernell, Lan Jiang, Lluis Antoni Jimenez Rugama, Da Li, Jagadeeswaran Rathinavel, Xin Tong, Kan Zhang, Yizhi Zhang, and Xuan Zhou, GAIL: Guaranteed Automatic Integration Library (Version 2.3) [MATLAB Software], 2019. Available from http://gailgithub.github.io/GAIL_Dev/

Guarantee: This algorithm computes the integral of real valued functions in $[0,1]^d$ with a prescribed generalized error tolerance. The Fourier coefficients of the integrand are assumed to be absolutely convergent. If the algorithm terminates without warning messages, the output is given with guarantees under the assumption that the integrand lies inside a cone of functions. The guarantee is based on the decay rate of the Fourier coefficients. For integration over domains other than $[0,1]^d$ , this cone condition applies to $f \circ \psi$ (the composition of the functions) where $\psi$ is the transformation function for $[0,1]^d$ to the desired region. For more details on how the cone is defined, please refer to the references below.

__init__(integrand, abs_tol=0.01, rel_tol=0.0, n_init=1024.0, n_max=34359738368.0, fudge=<function CubQMCLatticeG.<lambda>>, check_cone=False, ptransform='Baker', error_fun=<function CubQMCLatticeG.<lambda>>)¶

Parameters

integrand (Integrand) – an instance of Integrand
abs_tol (float) – absolute error tolerance
rel_tol (float) – relative error tolerance
n_init (int) – initial number of samples
n_max (int) – maximum number of samples
fudge (function) – positive function multiplying the finite sum of Fast Fourier coefficients specified in the cone of functions
check_cone (boolean) – check if the function falls in the cone
error_fun – function taking in the approximate solution vector, absolute tolerance, and relative tolerance which returns the approximate error. Default indicates integration until either absolute OR relative tolerance is satisfied.

Bayesian Lattice Cubature (QMC)¶

class qmcpy.stopping_criterion.cub_qmc_bayes_lattice_g.CubBayesLatticeG(integrand, abs_tol=0.01, rel_tol=0, n_init=256, n_max=4194304, order=2, alpha=0.01, ptransform='C1sin', error_fun=<function CubBayesLatticeG.<lambda>>)¶

Stopping criterion for Bayesian Cubature using rank-1 Lattice sequence with guaranteed accuracy over a d-dimensional region to integrate within a specified generalized error tolerance with guarantees under Bayesian assumptions.

>>> k = Keister(Lattice(2, order='linear', seed=123456789))
>>> sc = CubBayesLatticeG(k,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
LDTransformBayesData (AccumulateData Object)
    solution        1.808
    indv_error      6.41e-04
    ci_low          1.808
    ci_high         1.809
    ci_comb_low     1.808
    ci_comb_high    1.809
    flags_comb      0
    flags_indv      0
    n_total         2^(8)
    n               2^(8)
    time_integrate  ...
CubBayesLatticeG (StoppingCriterion Object)
    abs_tol         0.050
    rel_tol         0
    n_init          2^(8)
    n_max           2^(22)
    order           2^(1)
Keister (Integrand Object)
Gaussian (TrueMeasure Object)
    mean            0
    covariance      2^(-1)
    decomp_type     PCA
Lattice (DiscreteDistribution Object)
    d               2^(1)
    dvec            [0 1]
    randomize       1
    order           linear
    entropy         123456789
    spawn_key       ()

Adapted from

GAIL cubBayesLattice_g.

Reference

[1] Sou-Cheng T. Choi, Yuhan Ding, Fred J. Hickernell, Lan Jiang, Lluis Antoni Jimenez Rugama, Da Li, Jagadeeswaran Rathinavel, Xin Tong, Kan Zhang, Yizhi Zhang, and Xuan Zhou, GAIL: Guaranteed Automatic Integration Library (Version 2.3) [MATLAB Software], 2019. Available from GAIL.

Guarantee

This algorithm attempts to calculate the integral of function f over the hyperbox [0,1]^d to a prescribed error tolerance tolfun:= max(abstol, reltol*| I |) with guaranteed confidence level, e.g., 99% when alpha=0.5%. If the algorithm terminates without showing any warning messages and provides an answer Q, then the following inequality would be satisfied:

Pr(| Q - I | <= tolfun) = 99%.

This Bayesian cubature algorithm guarantees for integrands that are considered to be an instance of a gaussian process that fall in the middle of samples space spanned. Where The sample space is spanned by the covariance kernel parametrized by the scale and shape parameter inferred from the sampled values of the integrand. For more details on how the covariance kernels are defined and the parameters are obtained, please refer to the references below.

Bayesian Digital Net Cubature (QMC)¶

class qmcpy.stopping_criterion.cub_qmc_bayes_net_g.CubBayesNetG(integrand, abs_tol=0.01, rel_tol=0, n_init=256, n_max=4194304, alpha=0.01, error_fun=<function CubBayesNetG.<lambda>>)¶

Stopping criterion for Bayesian Cubature using digital net sequence with guaranteed accuracy over a d-dimensional region to integrate within a specified generalized error tolerance with guarantees under Bayesian assumptions.

>>> k = Keister(DigitalNetB2(2, seed=123456789))
>>> sc = CubBayesNetG(k,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
LDTransformBayesData (AccumulateData Object)
    solution        1.812
    indv_error      0.015
    ci_low          1.796
    ci_high         1.827
    ci_comb_low     1.796
    ci_comb_high    1.827
    flags_comb      0
    flags_indv      0
    n_total         2^(8)
    n               2^(8)
    time_integrate  ...
CubBayesNetG (StoppingCriterion Object)
    abs_tol         0.050
    rel_tol         0
    n_init          2^(8)
    n_max           2^(22)
Keister (Integrand Object)
Gaussian (TrueMeasure Object)
    mean            0
    covariance      2^(-1)
    decomp_type     PCA
DigitalNetB2 (DiscreteDistribution Object)
    d               2^(1)
    dvec            [0 1]
    randomize       LMS_DS
    graycode        0
    entropy         123456789
    spawn_key       ()

Adapted from

GAIL cubBayesNet_g.

Reference

[1] Sou-Cheng T. Choi, Yuhan Ding, Fred J. Hickernell, Lan Jiang, Lluis Antoni Jimenez Rugama, Da Li, Jagadeeswaran Rathinavel, Xin Tong, Kan Zhang, Yizhi Zhang, and Xuan Zhou, GAIL: Guaranteed Automatic Integration Library (Version 2.3) [MATLAB Software], 2019. Available from GAIL.

Guarantee

This algorithm attempts to calculate the integral of function f over the hyperbox [0,1]^d to a prescribed error tolerance

tolfun:= max(abstol, reltol*| I |)

with guaranteed confidence level, e.g., 99% when alpha=0.5%. If the algorithm terminates without showing any warning messages and provides an answer Q, then the following inequality would be satisfied:

Pr(| Q - I | <= tolfun) = 99%.

This Bayesian cubature algorithm guarantees for integrands that are considered to be an instance of a gaussian process that fall in the middle of samples space spanned. Where The sample space is spanned by the covariance kernel parametrized by the scale and shape parameter inferred from the sampled values of the integrand. For more details on how the covariance kernels are defined and the parameters are obtained, please refer to the references below.

class qmcpy.stopping_criterion.cub_qmc_bayes_net_g.CubBayesSobolG(integrand, abs_tol=0.01, rel_tol=0, n_init=256, n_max=4194304, alpha=0.01, error_fun=<function CubBayesNetG.<lambda>>)¶

CLT QMC Cubature (with Replications)¶

class qmcpy.stopping_criterion.cub_qmc_clt.CubQMCCLT(integrand, abs_tol=0.01, rel_tol=0.0, n_init=256.0, n_max=1073741824, inflate=1.2, alpha=0.01, replications=16.0, error_fun=<function CubQMCCLT.<lambda>>)¶

Stopping criterion based on Central Limit Theorem for multiple replications.

>>> k = Keister(Lattice(seed=7))
>>> sc = CubQMCCLT(k,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> solution
array([1.38030146])
>>> data
MeanVarDataRep (AccumulateData Object)
    solution        1.380
    indv_error      6.92e-04
    ci_low          1.380
    ci_high         1.381
    ci_comb_low     1.380
    ci_comb_high    1.381
    flags_comb      0
    flags_indv      0
    n_total         2^(12)
    n               2^(12)
    n_rep           2^(8)
    time_integrate  ...
CubQMCCLT (StoppingCriterion Object)
    inflate         1.200
    alpha           0.010
    abs_tol         0.050
    rel_tol         0
    n_init          2^(8)
    n_max           2^(30)
    replications    2^(4)
Keister (Integrand Object)
Gaussian (TrueMeasure Object)
    mean            0
    covariance      2^(-1)
    decomp_type     PCA
Lattice (DiscreteDistribution Object)
    d               1
    dvec            0
    randomize       1
    order           natural
    entropy         7
    spawn_key       ()
>>> f = BoxIntegral(Lattice(3,seed=7), s=[-1,1])
>>> abs_tol = 1e-3
>>> sc = CubQMCCLT(f, abs_tol=abs_tol)
>>> solution,data = sc.integrate()
>>> solution
array([1.19023153, 0.96068581])
>>> data
MeanVarDataRep (AccumulateData Object)
    solution        [1.19  0.961]
    indv_error      [0.001 0.001]
    ci_low          [1.19 0.96]
    ci_high         [1.191 0.961]
    ci_comb_low     [1.19 0.96]
    ci_comb_high    [1.191 0.961]
    flags_comb      [False False]
    flags_indv      [False False]
    n_total         2^(21)
    n               [2097152.    8192.]
    n_rep           [131072.    512.]
    time_integrate  ...
CubQMCCLT (StoppingCriterion Object)
    inflate         1.200
    alpha           0.010
    abs_tol         0.001
    rel_tol         0
    n_init          2^(8)
    n_max           2^(30)
    replications    2^(4)
BoxIntegral (Integrand Object)
    s               [-1  1]
Uniform (TrueMeasure Object)
    lower_bound     0
    upper_bound     1
Lattice (DiscreteDistribution Object)
    d               3
    dvec            [0 1 2]
    randomize       1
    order           natural
    entropy         7
    spawn_key       ()
>>> sol3neg1 = -pi/4-1/2*log(2)+log(5+3*sqrt(3))
>>> sol31 = sqrt(3)/4+1/2*log(2+sqrt(3))-pi/24
>>> true_value = array([sol3neg1,sol31])
>>> (abs(true_value-solution)<abs_tol).all()
True
>>> cf = CustomFun(
...     true_measure = Uniform(DigitalNetB2(6,seed=7)),
...     g = lambda x,compute_flags=None: (2*arange(1,7)*x).reshape(-1,2,3),
...     dprime = (2,3))
>>> sol,data = CubQMCCLT(cf,abs_tol=1e-4).integrate()
>>> data
MeanVarDataRep (AccumulateData Object)
    solution        [[1. 2. 3.]
                    [4. 5. 6.]]
    indv_error      [[2.484e-05 0.000e+00 0.000e+00]
                    [5.708e-06 2.178e-10 0.000e+00]]
    ci_low          [[1. 2. 3.]
                    [4. 5. 6.]]
    ci_high         [[1. 2. 3.]
                    [4. 5. 6.]]
    ci_comb_low     [[1. 2. 3.]
                    [4. 5. 6.]]
    ci_comb_high    [[1. 2. 3.]
                    [4. 5. 6.]]
    flags_comb      [[False False False]
                    [False False False]]
    flags_indv      [[False False False]
                    [False False False]]
    n_total         2^(14)
    n               [[ 4096.  4096.  4096.]
                    [16384.  4096.  4096.]]
    n_rep           [[ 256.  256.  256.]
                    [1024.  256.  256.]]
    time_integrate  ...
CubQMCCLT (StoppingCriterion Object)
    inflate         1.200
    alpha           0.010
    abs_tol         1.00e-04
    rel_tol         0
    n_init          2^(8)
    n_max           2^(30)
    replications    2^(4)
CustomFun (Integrand Object)
Uniform (TrueMeasure Object)
    lower_bound     0
    upper_bound     1
DigitalNetB2 (DiscreteDistribution Object)
    d               6
    dvec            [0 1 2 3 4 5]
    randomize       LMS_DS
    graycode        0
    entropy         7
    spawn_key       ()

__init__(integrand, abs_tol=0.01, rel_tol=0.0, n_init=256.0, n_max=1073741824, inflate=1.2, alpha=0.01, replications=16.0, error_fun=<function CubQMCCLT.<lambda>>)¶

Parameters

integrand (Integrand) – an instance of Integrand
inflate (float) – inflation factor when estimating variance
alpha (float) – significance level for confidence interval
abs_tol (float) – absolute error tolerance
rel_tol (float) – relative error tolerance
n_max (int) – maximum number of samples
replications (int) – number of replications
error_fun – function taking in the approximate solution vector, absolute tolerance, and relative tolerance which returns the approximate error. Default indicates integration until either absolute OR relative tolerance is satisfied.

integrate()¶: See abstract method.

set_tolerance(abs_tol=None, rel_tol=None)¶

See abstract method.

Parameters

abs_tol (float) – absolute tolerance. Reset if supplied, ignored if not.
rel_tol (float) – relative tolerance. Reset if supplied, ignored if not.

Guaranteed MC Cubature¶

class qmcpy.stopping_criterion.cub_mc_g.CubMCG(integrand, abs_tol=0.01, rel_tol=0.0, n_init=1024.0, n_max=10000000000.0, inflate=1.2, alpha=0.01, control_variates=[], control_variate_means=[])¶

Stopping criterion with guaranteed accuracy.

>>> k = Keister(IIDStdUniform(2,seed=7))
>>> sc = CubMCG(k,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
MeanVarData (AccumulateData Object)
    solution        1.807
    error_bound     0.050
    n_total         15256
    n               14232
    levels          1
    time_integrate  ...
CubMCG (StoppingCriterion Object)
    abs_tol         0.050
    rel_tol         0
    n_init          2^(10)
    n_max           10000000000
    inflate         1.200
    alpha           0.010
Keister (Integrand Object)
Gaussian (TrueMeasure Object)
    mean            0
    covariance      2^(-1)
    decomp_type     PCA
IIDStdUniform (DiscreteDistribution Object)
    d               2^(1)
    entropy         7
    spawn_key       ()
>>> dd = IIDStdUniform(1,seed=7)
>>> k = Keister(dd)
>>> cv1 = CustomFun(Uniform(dd),lambda x: sin(pi*x).sum(1))
>>> cv1mean = 2/pi
>>> cv2 = CustomFun(Uniform(dd),lambda x: (-3*(x-.5)**2+1).sum(1))
>>> cv2mean = 3/4
>>> sc1 = CubMCG(k,abs_tol=.05,control_variates=[cv1,cv2],control_variate_means=[cv1mean,cv2mean])
>>> sol,data = sc1.integrate()
>>> sol
1.384...

Original Implementation:

https://github.com/GailGithub/GAIL_Dev/blob/master/Algorithms/IntegrationExpectation/meanMC_g.m

References

[1] Fred J. Hickernell, Lan Jiang, Yuewei Liu, and Art B. Owen, “Guaranteed conservative fixed width confidence intervals via Monte Carlo sampling,” Monte Carlo and Quasi-Monte Carlo Methods 2012 (J. Dick, F. Y. Kuo, G. W. Peters, and I. H. Sloan, eds.), pp. 105-128, Springer-Verlag, Berlin, 2014. DOI: 10.1007/978-3-642-41095-6_5

[2] Sou-Cheng T. Choi, Yuhan Ding, Fred J. Hickernell, Lan Jiang, Lluis Antoni Jimenez Rugama, Da Li, Jagadeeswaran Rathinavel, Xin Tong, Kan Zhang, Yizhi Zhang, and Xuan Zhou, GAIL: Guaranteed Automatic Integration Library (Version 2.3) [MATLAB Software], 2019. Available from http://gailgithub.github.io/GAIL_Dev/

Guarantee:: This algorithm attempts to calculate the mean, mu, of a random variable to a prescribed error tolerance, _tol_fun:= max(abstol,reltol*|mu|), with guaranteed confidence level 1-alpha. If the algorithm terminates without showing any warning messages and provides an answer tmu, then the follow inequality would be satisfied: $\P(| mu - tmu | \\le tol\_fun) \\ge 1 - \\alpha$ .

__init__(integrand, abs_tol=0.01, rel_tol=0.0, n_init=1024.0, n_max=10000000000.0, inflate=1.2, alpha=0.01, control_variates=[], control_variate_means=[])¶

Parameters

integrand (Integrand) – an instance of Integrand
inflate – inflation factor when estimating variance
alpha – significance level for confidence interval
abs_tol – absolute error tolerance
rel_tol – relative error tolerance
n_init – initial number of samples
n_max – maximum number of samples
control_variates (list) – list of integrand objects to be used as control variates. Control variates are currently only compatible with single level problems. The same discrete distribution instance must be used for the integrand and each of the control variates.
control_variate_means (list) – list of means for each control variate

integrate()¶: See abstract method.

set_tolerance(abs_tol=None, rel_tol=None)¶

See abstract method.

Parameters

abs_tol (float) – absolute tolerance. Reset if supplied, ignored if not.
rel_tol (float) – relative tolerance. Reset if supplied, ignored if not.

CLT MC Cubature¶

class qmcpy.stopping_criterion.cub_mc_clt.CubMCCLT(integrand, abs_tol=0.01, rel_tol=0.0, n_init=1024.0, n_max=10000000000.0, inflate=1.2, alpha=0.01, control_variates=[], control_variate_means=[])¶

Stopping criterion based on the Central Limit Theorem.

>>> ao = AsianOption(IIDStdUniform(seed=7))
>>> sc = CubMCCLT(ao,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
MeanVarData (AccumulateData Object)
    solution        1.519
    error_bound     0.046
    n_total         96028
    n               95004
    levels          1
    time_integrate  ...
CubMCCLT (StoppingCriterion Object)
    abs_tol         0.050
    rel_tol         0
    n_init          2^(10)
    n_max           10000000000
    inflate         1.200
    alpha           0.010
AsianOption (Integrand Object)
    volatility      2^(-1)
    call_put        call
    start_price     30
    strike_price    35
    interest_rate   0
    mean_type       arithmetic
    dim_frac        0
BrownianMotion (TrueMeasure Object)
    time_vec        1
    drift           0
    mean            0
    covariance      1
    decomp_type     PCA
IIDStdUniform (DiscreteDistribution Object)
    d               1
    entropy         7
    spawn_key       ()
>>> ao = AsianOption(IIDStdUniform(seed=7),multilevel_dims=[2,4,8])
>>> sc = CubMCCLT(ao,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> dd = IIDStdUniform(1,seed=7)
>>> k = Keister(dd)
>>> cv1 = CustomFun(Uniform(dd),lambda x: sin(pi*x).sum(1))
>>> cv1mean = 2/pi
>>> cv2 = CustomFun(Uniform(dd),lambda x: (-3*(x-.5)**2+1).sum(1))
>>> cv2mean = 3/4
>>> sc1 = CubMCCLT(k,abs_tol=.05,control_variates=[cv1,cv2],control_variate_means=[cv1mean,cv2mean])
>>> sol,data = sc1.integrate()
>>> sol
1.381...

__init__(integrand, abs_tol=0.01, rel_tol=0.0, n_init=1024.0, n_max=10000000000.0, inflate=1.2, alpha=0.01, control_variates=[], control_variate_means=[])¶

Parameters

integrand (Integrand) – an instance of Integrand
inflate (float) – inflation factor when estimating variance
alpha (float) – significance level for confidence interval
abs_tol (float) – absolute error tolerance
rel_tol (float) – relative error tolerance
n_max (int) – maximum number of samples
control_variates (list) – list of integrand objects to be used as control variates. Control variates are currently only compatible with single level problems. The same discrete distribution instance must be used for the integrand and each of the control variates.
control_variate_means (list) – list of means for each control variate

integrate()¶: See abstract method.

set_tolerance(abs_tol=None, rel_tol=None)¶

See abstract method.

Parameters

abs_tol (float) – absolute tolerance. Reset if supplied, ignored if not.
rel_tol (float) – relative tolerance. Reset if supplied, ignored if not.

Continuation Multilevel QMC Cubature¶

class qmcpy.stopping_criterion.cub_qmc_ml_cont.CubQMCMLCont(integrand, abs_tol=0.05, alpha=0.01, rmse_tol=None, n_init=256.0, n_max=10000000000.0, replications=32.0, levels_min=2, levels_max=10, n_tols=10, tol_mult=1.6681005372000588, theta_init=0.5)¶

Stopping criterion based on continuation multi-level quasi-Monte Carlo.

>>> mlco = MLCallOptions(Lattice(seed=7))
>>> sc = CubQMCMLCont(mlco,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
MLQMCData (AccumulateData Object)
    solution        10.421
    n_total         98304
    n_level         [2048.  256.  256.  256.  256.]
    levels          5
    mean_level      [10.054  0.183  0.102  0.054  0.028]
    var_level       [2.027e-04 5.129e-05 2.656e-05 1.064e-05 3.466e-06]
    bias_estimate   0.016
    time_integrate  ...
CubQMCMLCont (StoppingCriterion Object)
    rmse_tol        0.019
    n_init          2^(8)
    n_max           10000000000
    replications    2^(5)
    levels_min      2^(1)
    levels_max      10
    n_tols          10
    tol_mult        1.668
    theta_init      2^(-1)
    theta           2^(-3)
MLCallOptions (Integrand Object)
    option          european
    sigma           0.200
    k               100
    r               0.050
    t               1
    b               85
    level           0
Gaussian (TrueMeasure Object)
    mean            0
    covariance      1
    decomp_type     PCA
Lattice (DiscreteDistribution Object)
    d               1
    dvec            0
    randomize       1
    order           natural
    entropy         7
    spawn_key       ()

References

[1] https://github.com/PieterjanRobbe/MultilevelEstimators.jl

__init__(integrand, abs_tol=0.05, alpha=0.01, rmse_tol=None, n_init=256.0, n_max=10000000000.0, replications=32.0, levels_min=2, levels_max=10, n_tols=10, tol_mult=1.6681005372000588, theta_init=0.5)¶

Parameters

integrand (Integrand) – integrand with multi-level g method
abs_tol (float) – absolute tolerance
alpha (float) – uncertaintly level. If rmse_tol not supplied, then rmse_tol = abs_tol/norm.ppf(1-alpha/2)
rmse_tol (float) – root mean squared error If supplied (not None), then absolute tolerance and alpha are ignored in favor of the rmse tolerance
n_max (int) – maximum number of samples
replications (int) – number of replications on each level
levels_min (int) – minimum level of refinement >= 2
levels_max (int) – maximum level of refinement >= Lmin
n_tols (int) – number of coarser tolerances to run
tol_mult (float) – coarser tolerance multiplication factor
theta_init (float) – initial error splitting constant

integrate()¶

ABSTRACT METHOD to determine the number of samples needed to satisfy the tolerance.

Returns

tuple containing:

solution (float): approximation to the integral
data (AccumulateData): an AccumulateData object

Return type

tuple

set_tolerance(abs_tol=None, alpha=0.01, rmse_tol=None)¶

See abstract method.

Parameters

integrand (Integrand) – integrand with multi-level g method
abs_tol (float) – absolute tolerance. Reset if supplied, ignored if not.
alpha (float) – uncertaintly level. If rmse_tol not supplied, then rmse_tol = abs_tol/norm.ppf(1-alpha/2)
rel_tol (float) – relative tolerance. Reset if supplied, ignored if not. Takes priority over aboluste tolerance and alpha if supplied.

Multilevel QMC Cubature¶

class qmcpy.stopping_criterion.cub_qmc_ml.CubQMCML(integrand, abs_tol=0.05, alpha=0.01, rmse_tol=None, n_init=256.0, n_max=10000000000.0, replications=32.0, levels_min=2, levels_max=10)¶

Stopping criterion based on multi-level quasi-Monte Carlo.

>>> mlco = MLCallOptions(Lattice(seed=7))
>>> sc = CubQMCML(mlco,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
MLQMCData (AccumulateData Object)
    solution        10.434
    n_total         172032
    n_level         [4096.  256.  256.  256.  256.  256.]
    levels          6
    mean_level      [10.053  0.183  0.102  0.054  0.028  0.014]
    var_level       [5.699e-05 5.129e-05 2.656e-05 1.064e-05 3.466e-06 1.113e-06]
    bias_estimate   0.007
    time_integrate  ...
CubQMCML (StoppingCriterion Object)
    rmse_tol        0.019
    n_init          2^(8)
    n_max           10000000000
    replications    2^(5)
MLCallOptions (Integrand Object)
    option          european
    sigma           0.200
    k               100
    r               0.050
    t               1
    b               85
    level           0
Gaussian (TrueMeasure Object)
    mean            0
    covariance      1
    decomp_type     PCA
Lattice (DiscreteDistribution Object)
    d               1
    dvec            0
    randomize       1
    order           natural
    entropy         7
    spawn_key       ()

References

[1] M.B. Giles and B.J. Waterhouse. ‘Multilevel quasi-Monte Carlo path simulation’. pp.165-181 in Advanced Financial Modelling, in Radon Series on Computational and Applied Mathematics, de Gruyter, 2009. http://people.maths.ox.ac.uk/~gilesm/files/radon.pdf

__init__(integrand, abs_tol=0.05, alpha=0.01, rmse_tol=None, n_init=256.0, n_max=10000000000.0, replications=32.0, levels_min=2, levels_max=10)¶

Parameters

integrand (Integrand) – integrand with multi-level g method
abs_tol (float) – absolute tolerance
alpha (float) – uncertaintly level. If rmse_tol not supplied, then rmse_tol = abs_tol/norm.ppf(1-alpha/2)
rmse_tol (float) – root mean squared error If supplied (not None), then absolute tolerance and alpha are ignored in favor of the rmse tolerance
n_max (int) – maximum number of samples
replications (int) – number of replications on each level
levels_min (int) – minimum level of refinement >= 2
levels_max (int) – maximum level of refinement >= Lmin

integrate()¶: See abstract method.

set_tolerance(abs_tol=None, alpha=0.01, rmse_tol=None)¶

See abstract method.

Parameters

integrand (Integrand) – integrand with multi-level g method
abs_tol (float) – absolute tolerance. Reset if supplied, ignored if not.
alpha (float) – uncertaintly level. If rmse_tol not supplied, then rmse_tol = abs_tol/norm.ppf(1-alpha/2)
rel_tol (float) – relative tolerance. Reset if supplied, ignored if not. Takes priority over aboluste tolerance and alpha if supplied.

Continuation Multilevel MC Cubature¶

class qmcpy.stopping_criterion.cub_mc_ml_cont.CubMCMLCont(integrand, abs_tol=0.05, alpha=0.01, rmse_tol=None, n_init=256.0, n_max=10000000000.0, levels_min=2, levels_max=10, n_tols=10, tol_mult=1.6681005372000588, theta_init=0.5)¶

Stopping criterion based on continuation multi-level monte carlo.

>>> mlco = MLCallOptions(IIDStdUniform(seed=7))
>>> sc = CubMCMLCont(mlco,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
MLMCData (AccumulateData Object)
    solution        10.400
    n_total         1193331
    levels          2^(2)
    n_level         [1133772.   22940.    8676.    2850.]
    mean_level      [10.059  0.186  0.105  0.05 ]
    var_level       [1.959e+02 1.603e-01 4.567e-02 1.013e-02]
    cost_per_sample [1. 2. 4. 8.]
    alpha           0.942
    beta            1.992
    gamma           1.000
    time_integrate  ...
CubMCMLCont (StoppingCriterion Object)
    rmse_tol        0.019
    n_init          2^(8)
    levels_min      2^(1)
    levels_max      10
    n_tols          10
    tol_mult        1.668
    theta_init      2^(-1)
    theta           2^(-1)
MLCallOptions (Integrand Object)
    option          european
    sigma           0.200
    k               100
    r               0.050
    t               1
    b               85
    level           0
Gaussian (TrueMeasure Object)
    mean            0
    covariance      1
    decomp_type     PCA
IIDStdUniform (DiscreteDistribution Object)
    d               1
    entropy         7
    spawn_key       ()

References

[1] https://github.com/PieterjanRobbe/MultilevelEstimators.jl

__init__(integrand, abs_tol=0.05, alpha=0.01, rmse_tol=None, n_init=256.0, n_max=10000000000.0, levels_min=2, levels_max=10, n_tols=10, tol_mult=1.6681005372000588, theta_init=0.5)¶

Parameters

integrand (Integrand) – integrand with multi-level g method
abs_tol (float) – absolute tolerance. Reset if supplied, ignored if not.
alpha (float) – uncertaintly level. If rmse_tol not supplied, then rmse_tol = abs_tol/norm.ppf(1-alpha/2)
rel_tol (float) – relative tolerance. Reset if supplied, ignored if not. Takes priority over absolute tolerance and alpha if supplied.
n_init (int) – initial number of samples
n_max (int) – maximum number of samples
levels_min (int) – minimum level of refinement >= 2
levels_max (int) – maximum level of refinement >= Lmin
n_tols (int) – number of coarser tolerances to run
tol_mult (float) – coarser tolerance multiplication factor
theta_init (float) – initial error splitting constant

integrate()¶

ABSTRACT METHOD to determine the number of samples needed to satisfy the tolerance.

Returns

tuple containing:

solution (float): approximation to the integral
data (AccumulateData): an AccumulateData object

Return type

tuple

set_tolerance(abs_tol=None, alpha=0.01, rmse_tol=None)¶

See abstract method.

Parameters

abs_tol (float) – absolute tolerance. Reset if supplied, ignored if not.
alpha (float) – uncertaintly level. If rmse_tol not supplied, then rmse_tol = abs_tol/norm.ppf(1-alpha/2)
rel_tol (float) – relative tolerance. Reset if supplied, ignored if not. Takes priority over aboluste tolerance and alpha if supplied.

Multilevel MC Cubature¶

class qmcpy.stopping_criterion.cub_mc_ml.CubMCML(integrand, abs_tol=0.05, alpha=0.01, rmse_tol=None, n_init=256.0, n_max=10000000000.0, levels_min=2, levels_max=10, alpha0=- 1.0, beta0=- 1.0, gamma0=- 1.0)¶

Stopping criterion based on multi-level monte carlo.

>>> mlco = MLCallOptions(IIDStdUniform(seed=7))
>>> sc = CubMCML(mlco,abs_tol=.05)
>>> solution,data = sc.integrate()
>>> data
MLMCData (AccumulateData Object)
    solution        10.450
    n_total         1213658
    levels          7
    n_level         [1.173e+06 2.369e+04 1.174e+04 3.314e+03 1.144e+03 4.380e+02 1.690e+02]
    mean_level      [1.006e+01 1.856e-01 1.053e-01 5.127e-02 2.699e-02 1.558e-02 7.068e-03]
    var_level       [1.958e+02 1.596e-01 4.603e-02 1.057e-02 2.978e-03 8.701e-04 2.552e-04]
    cost_per_sample [ 1.  2.  4.  8. 16. 32. 64.]
    alpha           0.936
    beta            1.870
    gamma           1.000
    time_integrate  ...
CubMCML (StoppingCriterion Object)
    rmse_tol        0.019
    n_init          2^(8)
    levels_min      2^(1)
    levels_max      10
    theta           2^(-1)
MLCallOptions (Integrand Object)
    option          european
    sigma           0.200
    k               100
    r               0.050
    t               1
    b               85
    level           0
Gaussian (TrueMeasure Object)
    mean            0
    covariance      1
    decomp_type     PCA
IIDStdUniform (DiscreteDistribution Object)
    d               1
    entropy         7
    spawn_key       ()

Original Implementation:

http://people.maths.ox.ac.uk/~gilesm/mlmc/#MATLAB

References

[1] M.B. Giles. ‘Multi-level Monte Carlo path simulation’. Operations Research, 56(3):607-617, 2008. http://people.maths.ox.ac.uk/~gilesm/files/OPRE_2008.pdf.

__init__(integrand, abs_tol=0.05, alpha=0.01, rmse_tol=None, n_init=256.0, n_max=10000000000.0, levels_min=2, levels_max=10, alpha0=- 1.0, beta0=- 1.0, gamma0=- 1.0)¶

Parameters

integrand (Integrand) – integrand with multi-level g method
abs_tol (float) – absolute tolerance. Reset if supplied, ignored if not.
alpha (float) – uncertaintly level. If rmse_tol not supplied, then rmse_tol = abs_tol/norm.ppf(1-alpha/2)
rel_tol (float) – relative tolerance. Reset if supplied, ignored if not. Takes priority over absolute tolerance and alpha if supplied.
n_init (int) – initial number of samples
n_max (int) – maximum number of samples
levels_min (int) – minimum level of refinement >= 2
levels_max (int) – maximum level of refinement >= Lmin
alpha0 (float) – weak error is O(2^{-alpha0*level})
beta0 (float) – variance is O(2^{-bet0a*level})
gamma0 (float) – sample cost is O(2^{gamma0*level})

Note

if alpha, beta, gamma are not positive, then they will be estimated

integrate()¶: See abstract method.

set_tolerance(abs_tol=None, alpha=0.01, rmse_tol=None)¶

See abstract method.

Parameters

abs_tol (float) – absolute tolerance. Reset if supplied, ignored if not.
alpha (float) – uncertaintly level. If rmse_tol not supplied, then rmse_tol = abs_tol/norm.ppf(1-alpha/2)
rel_tol (float) – relative tolerance. Reset if supplied, ignored if not. Takes priority over aboluste tolerance and alpha if supplied.

Utilities¶

qmcpy.util.latnetbuilder_linker.latnetbuilder_linker(lnb_dir='./', out_dir='./', fout_prefix='lnb4qmcpy')¶

Parameters

lnb_dir (str) – relative path to directory where outputMachine.txt is stored e.g. ‘my_lnb/poly_lat/’
out_dir (str) – relative path to directory where output should be stored e.g. ‘my_lnb/poly_lat_qmcpy/’
fout_prefix (str) – start of output file name. e.g. ‘my_poly_lat_vec’

Returns

path to file which can be passed into QMCPy’s Lattice or Sobol’ in order to use: the linked latnetbuilder generating vector/matrix e.g. ‘my_poly_lat_vec.10.16.npy’

Return type

str

Adapted from latnetbuilder parser:: https://github.com/umontreal-simul/latnetbuilder/blob/master/python-wrapper/latnetbuilder/parse_output.py#L74