""" Generation of various types of draws.
:author: Michel Bierlaire
:date: Tue Jun 18 19:05:13 2019
"""
# Too constraining
# pylint: disable=invalid-name, too-many-arguments, too-many-locals, too-many-statements
import numpy as np
import biogeme.exceptions as excep
[docs]def getLatinHypercubeDraws(
sample_size, number_of_draws, symmetric=False, uniformNumbers=None
):
"""Implementation of the Modified Latin Hypercube Sampling proposed
by `Hess et al., (2006)`_.
.. _`Hess et al., (2006)`:
https://doi.org/10.1016/j.trb.2004.10.005
:param sample_size: number of observations for which draws must be
generated. If None, a one dimensional array
will be generated. If it has a values k, then k
series of draws will be generated
:type sample_size: int
:param number_of_draws: number of draws to generate.
:type number_of_draws: int
:param symmetric: if True, draws from [-1: 1] are generated.
If False, draws from [0: 1] are generated. Default: False
:type symmetric: bool
:param uniformNumbers: numpy with uniformly distributed numbers.
If None, the numpy uniform number generator is used.
:type uniformNumbers: numpy.array
:return: numpy array with the draws
:rtype: numpy.array
Example::
latinHypercube = dr.getLatinHypercubeDraws(sample_size=3,
number_of_draws=10)
array([[0.43362897, 0.5275741 , 0.09215663, 0.94056236, 0.34376868,
0.87195551, 0.41495219, 0.71736691, 0.23198736, 0.145561 ],
[0.30520544, 0.78082964, 0.83591146, 0.2733167 , 0.53890906,
0.61607469, 0.00699715, 0.17179441, 0.7557228 , 0.39733102],
[0.49676864, 0.67073483, 0.9788854 , 0.5726069 , 0.11894558,
0.05515471, 0.2640275 , 0.82093696, 0.92034628, 0.64866597]])
:raise BiogemeError: if the number of draws is not positive.
:raise BiogemeError: if the sample size is not positive.
:raise BiogemeError: if the number of uniform draws is inconsistent.
"""
if number_of_draws <= 0:
raise excep.BiogemeError(f'Invalid number of draws: {number_of_draws}.')
if sample_size <= 0:
raise excep.BiogemeError(
f'Invalid sample size: {sample_size} when generating draws.'
)
totalSize = number_of_draws * sample_size
if uniformNumbers is None:
uniformNumbers = np.random.uniform(size=totalSize)
else:
if uniformNumbers.size != totalSize:
errorMsg = (
f'A total of {totalSize} uniform draws '
f'must be provided, and not {uniformNumbers.size}.'
)
raise excep.BiogemeError(errorMsg)
uniformNumbers.shape = (totalSize,)
numbers = np.array(
[(float(i) + uniformNumbers[i]) / float(totalSize) for i in range(totalSize)]
)
if symmetric:
numbers = 2.0 * numbers - 1.0
np.random.shuffle(numbers)
numbers.shape = (sample_size, number_of_draws)
return numbers
[docs]def getHaltonDraws(
sample_size, number_of_draws, symmetric=False, base=2, skip=0, shuffled=False
):
"""Generate Halton draws.
Implementation by Cristian Arteaga, University of Nevada Las Vegas,
:param sample_size: number of observations for which draws must be
generated. If None, a one dimensional array
will be generated. If it has a values k, then k
series of draws will be generated
:type sample_size: int
:param number_of_draws: number of draws to generate.
:type number_of_draws: int
:param symmetric: if True, draws from [-1: 1] are generated.
If False, draws from [0: 1] are generated. Default: False
:type symmetric: bool
:param base: generate Halton draws for a given basis.
Ideally, it should be a prime number. Default: 2.
:type base: int
:param skip: the number of elements of the sequence to be discarded.
:type skip: int
:param shuffled: if True, each series is shuffled
:type shuffled: bool
:return: numpy array with the draws
:rtype: numpy.array
Example::
halton = dr.getHaltonDraws(sample_size=2, number_of_draws=10, base=3)
array([[0.33333333, 0.66666667, 0.11111111, 0.44444444, 0.77777778,
0.22222222, 0.55555556, 0.88888889, 0.03703704, 0.37037037],
[0.7037037 , 0.14814815, 0.48148148, 0.81481481, 0.25925926,
0.59259259, 0.92592593, 0.07407407, 0.40740741, 0.74074074]])
:raise BiogemeError: if the number of draws is not positive.
:raise BiogemeError: if the sample size is not positive.
"""
if number_of_draws <= 0:
raise excep.BiogemeError(f'Invalid number of draws: {number_of_draws}.')
if sample_size <= 0:
raise excep.BiogemeError(
f'Invalid sample size: {sample_size} when generating draws.'
)
length = number_of_draws * sample_size
req_length = length + skip + 1
numbers = np.empty(req_length)
numbers[0] = 0
numbers_idx = 1
t = 1
while numbers_idx < req_length:
d = 1 / base**t
numbers_size = numbers_idx
i = 1
while i < base and numbers_idx < req_length:
max_numbers = min(req_length - numbers_idx, numbers_size)
numbers[numbers_idx : numbers_idx + max_numbers] = (
numbers[:max_numbers] + d * i
)
numbers_idx += max_numbers
i += 1
t += 1
numbers = numbers[skip + 1 : length + skip + 1]
if shuffled:
np.random.shuffle(numbers)
if symmetric:
numbers = 2.0 * numbers - 1.0
numbers.shape = (sample_size, number_of_draws)
return numbers
[docs]def getAntithetic(unif, sample_size, number_of_draws):
"""Returns antithetic uniform draws
:param unif: function taking two arguments (sample_size, number_of_draws)
and returning U[0, 1] draws
:type unif: function
:param sample_size: number of observations for which draws must be
generated. If None, a one dimensional array
will be generated. If it has a values k, then k
series of draws will be generated
:type sample_size: int
:param number_of_draws: number of draws to generate.
:type number_of_draws: int
:return: numpy array with the antithetic draws
:rtype: numpy.array
Example::
draws = dr.getAntithetic(dr.getUniform,
sample_size=3,
number_of_draws=10)
array([[0.48592363, 0.13648133, 0.35925946, 0.32431338, 0.32997936,
0.51407637, 0.86351867, 0.64074054, 0.67568662, 0.67002064],
[0.89261997, 0.0331808 , 0.30767182, 0.93433648, 0.17196124,
0.10738003, 0.9668192 , 0.69232818, 0.06566352, 0.82803876],
[0.81095587, 0.96171364, 0.40984817, 0.72177258, 0.16481096,
0.18904413, 0.03828636, 0.59015183, 0.27822742, 0.83518904]])
"""
R = int(number_of_draws / 2.0)
draws = unif(sample_size, R)
return np.concatenate((draws, 1 - draws), axis=1)
[docs]def getNormalWichuraDraws(
sample_size, number_of_draws, uniformNumbers=None, antithetic=False
):
"""Generate pseudo-random numbers from a normal distribution N(0, 1)
It uses the Algorithm AS241 by `Wichura (1988)`_
which produces the normal deviate z corresponding to a given lower
tail area of p; z is accurate to about 1 part in :math:`10^{16}`.
.. _`Wichura (1988)`:
http://www.jstor.org/stable/2347330
:param sample_size: number of observations for which draws must be
generated. If None, a one dimensional array
will be generated. If it has a values k, then k
series of draws will be generated
:type sample_size: int
:param number_of_draws: number of draws to generate.
:type number_of_draws: int
:param uniformNumbers: numpy with uniformly distributed numbers.
If None, the numpy uniform number generator is used.
:type uniformNumbers: numpy.array
:param antithetic: if True, only half of the draws are
actually generated, and the series are completed
with their antithetic version.
:type antithetic: bool
:return: numpy array with the draws
:rtype: numpy.array
Example::
draws = dr.getNormalWichuraDraws(sample_size=3, number_of_draws=10)
array(
[[ 0.52418458, -1.04344204, -2.11642482, 0.48257162, -2.67188279,
-1.89993283, 0.28251041, -0.38424425, 1.53182226, 0.30651874],
[-0.7937038 , -0.07884121, -0.91005616, -0.98855175, 1.09405753,
-0.5997651 , -1.70785113, 1.57571384, -0.33208723, -1.03510102],
[-0.13853654, 0.92595498, -0.80136586, 1.68454196, 0.9955927 ,
-0.28615154, 2.10635541, 0.0436191 , -0.25417774, 0.01026933]]
)
:raise BiogemeError: if the number of draws is not positive.
:raise BiogemeError: if the sample size is not positive.
"""
if number_of_draws <= 0:
raise excep.BiogemeError(f'Invalid number of draws: {number_of_draws}.')
if antithetic:
if number_of_draws % 2 != 0:
errorMsg = (
f'Please specify an even number of draws for '
f'antithetic draws. Requested number of '
f'{number_of_draws}.'
)
raise excep.BiogemeError(errorMsg)
number_of_draws = int(number_of_draws / 2.0)
if sample_size <= 0:
raise excep.BiogemeError(
f'Invalid sample size: {sample_size} when generating draws.'
)
totalSize = number_of_draws * sample_size
split2 = 5.0e00
const1 = 0.180625e00
const2 = 1.6e00
a0 = 3.3871328727963666080e00
a1 = 1.3314166789178437745e02
a2 = 1.9715909503065514427e03
a3 = 1.3731693765509461125e04
a4 = 4.5921953931549871457e04
a5 = 6.7265770927008700853e04
a6 = 3.3430575583588128105e04
a7 = 2.5090809287301226727e03
b1 = 4.2313330701600911252e01
b2 = 6.8718700749205790830e02
b3 = 5.3941960214247511077e03
b4 = 2.1213794301586595867e04
b5 = 3.9307895800092710610e04
b6 = 2.8729085735721942674e04
b7 = 5.2264952788528545610e03
c0 = 1.42343711074968357734e00
c1 = 4.63033784615654529590e00
c2 = 5.76949722146069140550e00
c3 = 3.64784832476320460504e00
c4 = 1.27045825245236838258e00
c5 = 2.41780725177450611770e-01
c6 = 2.27238449892691845833e-02
c7 = 7.74545014278341407640e-04
d1 = 2.05319162663775882187e00
d2 = 1.67638483018380384940e00
d3 = 6.89767334985100004550e-01
d4 = 1.48103976427480074590e-01
d5 = 1.51986665636164571966e-02
d6 = 5.47593808499534494600e-04
d7 = 1.05075007164441684324e-09
e0 = 6.65790464350110377720e00
e1 = 5.46378491116411436990e00
e2 = 1.78482653991729133580e00
e3 = 2.96560571828504891230e-01
e4 = 2.65321895265761230930e-02
e5 = 1.24266094738807843860e-03
e6 = 2.71155556874348757815e-05
e7 = 2.01033439929228813265e-07
f1 = 5.99832206555887937690e-01
f2 = 1.36929880922735805310e-01
f3 = 1.48753612908506148525e-02
f4 = 7.86869131145613259100e-04
f5 = 1.84631831751005468180e-05
f6 = 1.42151175831644588870e-07
f7 = 2.04426310338993978564e-15
if uniformNumbers is None:
uniformNumbers = np.random.uniform(size=totalSize)
elif uniformNumbers.size != totalSize:
errorMsg = (
f'A total of {totalSize} uniform draws must be '
f'provided, and not {uniformNumbers.size}.'
)
raise excep.BiogemeError(errorMsg)
uniformNumbers.shape = (totalSize,)
q = uniformNumbers - 0.5
draws = np.zeros(uniformNumbers.shape)
r = np.zeros(uniformNumbers.shape)
cond1 = np.abs(uniformNumbers) <= 0.45
r[cond1] = const1 - q[cond1] * q[cond1]
draws[cond1] = (
q[cond1]
* (
(
(
(
(((a7 * r[cond1] + a6) * r[cond1] + a5) * r[cond1] + a4)
* r[cond1]
+ a3
)
* r[cond1]
+ a2
)
* r[cond1]
+ a1
)
* r[cond1]
+ a0
)
/ (
(
(
(
(((b7 * r[cond1] + b6) * r[cond1] + b5) * r[cond1] + b4)
* r[cond1]
+ b3
)
* r[cond1]
+ b2
)
* r[cond1]
+ b1
)
* r[cond1]
+ 1
)
)
cond2 = np.abs(uniformNumbers) > 0.45
cond2a = np.logical_and(cond2, q < 0.0)
cond2b = np.logical_and(cond2, q >= 0.0)
r[cond2a] = uniformNumbers[cond2a]
r[cond2b] = 1 - uniformNumbers[cond2b]
cond2c = np.logical_and(cond2, r <= 0)
cond2d = np.logical_and(cond2, r > 0)
draws[cond2c] = 0.0
r[cond2d] = np.sqrt(-np.log(r[cond2d]))
cond2d_a = np.logical_and(cond2d, r <= split2)
cond2d_b = np.logical_and(cond2d, r > split2)
r[cond2d_a] = r[cond2d_a] - const2
draws[cond2d_a] = (
(
(
(
(((c7 * r[cond2d_a] + c6) * r[cond2d_a] + c5) * r[cond2d_a] + c4)
* r[cond2d_a]
+ c3
)
* r[cond2d_a]
+ c2
)
* r[cond2d_a]
+ c1
)
* r[cond2d_a]
+ c0
) / (
(
(
(
(((d7 * r[cond2d_a] + d6) * r[cond2d_a] + d5) * r[cond2d_a] + d4)
* r[cond2d_a]
+ d3
)
* r[cond2d_a]
+ d2
)
* r[cond2d_a]
+ d1
)
* r[cond2d_a]
+ 1
)
r[cond2d_b] = r[cond2d_b] - split2
draws[cond2d_b] = (
(
(
(
(((e7 * r[cond2d_b] + e6) * r[cond2d_b] + e5) * r[cond2d_b] + e4)
* r[cond2d_b]
+ e3
)
* r[cond2d_b]
+ e2
)
* r[cond2d_b]
+ e1
)
* r[cond2d_b]
+ e0
) / (
(
(
(
(((f7 * r[cond2d_b] + f6) * r[cond2d_b] + f5) * r[cond2d_b] + f4)
* r[cond2d_b]
+ f3
)
* r[cond2d_b]
+ f2
)
* r[cond2d_b]
+ f1
)
* r[cond2d_b]
+ 1
)
draws[cond2a] = -draws[cond2a]
draws.shape = (sample_size, number_of_draws)
if antithetic:
draws = np.concatenate((draws, -draws), axis=1)
return draws