"""
Authors:
Original FORTRAN77 version of i4_sobol by Bennett Fox.
MATLAB version by John Burkardt.
PYTHON version by Corrado Chisari
Original Python version of is_prime by Corrado Chisari
Original MATLAB versions of other functions by John Burkardt.
PYTHON versions by Corrado Chisari
Modified Python version by Holger Nahrstaedt
Original code is available from
http://people.sc.fsu.edu/~jburkardt/py_src/sobol/sobol.html
"""
from __future__ import division
import warnings
import numpy as np
from .base import InitialPointGenerator
from ..space import Space
from sklearn.utils import check_random_state
[docs]class Sobol(InitialPointGenerator):
"""Generates a new quasirandom Sobol' vector with each call.
Parameters
----------
skip : int
Skipped seed number.
randomize : bool, default=False
When set to True, random shift is applied.
Notes
-----
Sobol' sequences [1]_ provide :math:`n=2^m` low discrepancy points in
:math:`[0,1)^{dim}`. Scrambling them makes them suitable for singular
integrands, provides a means of error estimation, and can improve their
rate of convergence.
There are many versions of Sobol' sequences depending on their
'direction numbers'. Here, the maximum number of dimension is 40.
The routine adapts the ideas of Antonov and Saleev [2]_.
.. warning::
Sobol' sequences are a quadrature rule and they lose their balance
properties if one uses a sample size that is not a power of 2, or skips
the first point, or thins the sequence [5]_.
If :math:`n=2^m` points are not enough then one should take :math:`2^M`
points for :math:`M>m`. When scrambling, the number R of independent
replicates does not have to be a power of 2.
Sobol' sequences are generated to some number :math:`B` of bits. Then
after :math:`2^B` points have been generated, the sequence will repeat.
Currently :math:`B=30`.
References
----------
.. [1] I. M. Sobol. The distribution of points in a cube and the accurate
evaluation of integrals. Zh. Vychisl. Mat. i Mat. Phys., 7:784-802,
1967.
.. [2] Antonov, Saleev,
USSR Computational Mathematics and Mathematical Physics,
Volume 19, 1980, pages 252 - 256.
.. [3] Paul Bratley, Bennett Fox,
Algorithm 659:
Implementing Sobol's Quasirandom Sequence Generator,
ACM Transactions on Mathematical Software,
Volume 14, Number 1, pages 88-100, 1988.
.. [4] Bennett Fox,
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,
.. [5] Art B. Owen. On dropping the first Sobol' point. arXiv 2008.08051,
2020.
"""
[docs] def __init__(self, skip=0, randomize=True):
if not (skip & (skip - 1) == 0):
raise ValueError("The balance properties of Sobol' points require"
" skipping a power of 2.")
if skip != 0:
warnings.warn(f"{skip} points have been skipped: "
f"{skip} points can be generated before the "
f"sequence repeats.")
self.skip = skip
self.num_generated = 0
self.randomize = randomize
self.dim_max = 40
self.log_max = 30
self.atmost = 2 ** self.log_max - 1
self.lastq = None
self.maxcol = None
self.poly = None
self.recipd = None
self.seed_save = -1
self.v = np.zeros((self.dim_max, self.log_max))
self.dim_num_save = -1
def init(self, dim_num):
self.dim_num_save = dim_num
self.v = np.zeros((self.dim_max, self.log_max))
self.v[0:40, 0] = np.transpose([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1])
self.v[2:40, 1] = np.transpose([1, 3, 1, 3, 1, 3, 3, 1, 3, 1, 3, 1,
3, 1, 1, 3, 1, 3, 1, 3, 1, 3, 3, 1,
3, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 3,
1, 3])
self.v[3:40, 2] = np.transpose([7, 5, 1, 3, 3, 7, 5, 5, 7, 7, 1,
3, 3, 7, 5, 1, 1, 5, 3, 3, 1, 7, 5,
1, 3, 3, 7, 5, 1, 1, 5, 7, 7, 5, 1,
3, 3])
self.v[5:40, 3] = np.transpose([1, 7, 9, 13, 11, 1, 3, 7, 9,
5, 13, 13, 11, 3, 15, 5, 3, 15,
7, 9, 13, 9, 1, 11, 7, 5, 15,
1, 15, 11, 5, 3, 1, 7, 9])
self.v[7:40, 4] = np.transpose([9, 3, 27, 15, 29, 21, 23, 19,
11, 25, 7, 13, 17, 1, 25, 29,
3, 31, 11, 5, 23, 27, 19, 21,
5, 1, 17, 13, 7, 15, 9, 31, 9])
self.v[13:40, 5] = np.transpose([37, 33, 7, 5, 11, 39, 63, 27,
17, 15, 23, 29, 3, 21, 13, 31,
25, 9, 49, 33, 19, 29, 11, 19,
27, 15, 25])
self.v[19:40, 6] = np.transpose([13, 33, 115, 41, 79, 17, 29, 119,
75, 73, 105, 7, 59, 65, 21, 3,
113, 61, 89, 45, 107])
self.v[37:40, 7] = np.transpose([7, 23, 39])
# Set POLY.
self.poly = [1, 3, 7, 11, 13, 19, 25, 37, 59, 47, 61, 55, 41, 67, 97,
91, 109, 103, 115, 131, 193, 137, 145, 143, 241, 157,
185, 167, 229, 171, 213, 191, 253, 203, 211, 239, 247,
285, 369, 299]
# Find the number of bits in ATMOST.
self.maxcol = _bit_hi1(self.atmost)
# Initialize row 1 of V.
self.v[0, 0:self.maxcol] = 1
# Check parameters.
if dim_num < 1 or self.dim_max < dim_num:
raise ValueError(f'I4_SOBOL - Fatal error!\n'
f' The spatial dimension DIM_NUM should '
f'satisfy:\n'
f' 1 <= DIM_NUM <= {self.dim_max}\n'
f' But this input value is DIM_NUM = {dim_num}')
# Initialize the remaining rows of V.
for i in range(2, dim_num + 1):
# The bits of the integer POLY(I) gives the form of polynomial I.
# Find the degree of polynomial I from binary encoding.
j = self.poly[i - 1]
m = 0
j //= 2
while j > 0:
j //= 2
m += 1
# Expand this bit pattern to separate components
# of the logical array INCLUD.
j = self.poly[i - 1]
includ = np.zeros(m)
for k in range(m, 0, -1):
j2 = j // 2
includ[k - 1] = (j != 2 * j2)
j = j2
# Calculate the remaining elements of row I as explained
# in Bratley and Fox, section 2.
for j in range(m + 1, self.maxcol + 1):
newv = self.v[i - 1, j - m - 1]
p2 = 1
for k in range(1, m + 1):
p2 *= 2
if includ[k - 1]:
newv = np.bitwise_xor(
int(newv), int(p2 * self.v[i - 1, j - k - 1]))
self.v[i - 1, j - 1] = newv
# Multiply columns of V by appropriate power of 2.
p2 = 1
for j in range(self.maxcol - 1, 0, -1):
p2 *= 2
self.v[0:dim_num, j - 1] = self.v[0:dim_num, j - 1] * p2
# RECIPD is 1/(common denominator of the elements in V).
self.recipd = 1.0 / (2 * p2)
self.lastq = np.zeros(dim_num)
[docs] def generate(self, dimensions, n_samples, random_state=None):
"""Creates samples from Sobol' set.
Parameters
----------
dimensions : list, shape (n_dims,)
List of search space dimensions.
Each search dimension can be defined either as
- a `(lower_bound, upper_bound)` tuple (for `Real` or `Integer`
dimensions),
- a `(lower_bound, upper_bound, "prior")` tuple (for `Real`
dimensions),
- as a list of categories (for `Categorical` dimensions), or
- an instance of a `Dimension` object (`Real`, `Integer` or
`Categorical`).
n_samples : int
The order of the Sobol' sequence. Defines the number of samples.
random_state : int, RandomState instance, or None (default)
Set random state to something other than None for reproducible
results.
Returns
-------
sample : array_like (n_samples, dim)
Sobol' set.
"""
total_n_samples = self.num_generated + n_samples
if not (total_n_samples & (total_n_samples - 1) == 0):
warnings.warn("The balance properties of Sobol' points require "
"n to be a power of 2. {0} points have been "
"previously generated, then: n={0}+{1}={2}. "
.format(self.num_generated, n_samples,
total_n_samples))
if self.skip != 0 and total_n_samples > self.skip:
raise ValueError(f"{self.skip} points have been skipped: "
f"generating "
f"{n_samples} more points would cause the "
f"sequence to repeat.")
rng = check_random_state(random_state)
space = Space(dimensions)
n_dim = space.n_dims
transformer = space.get_transformer()
space.set_transformer("normalize")
r = np.full((n_samples, n_dim), np.nan)
seed = self.skip
for j in range(n_samples):
r[j, 0:n_dim], seed = self._sobol(n_dim, seed)
if self.randomize:
r = _random_shift(r, rng)
r = space.inverse_transform(r)
space.set_transformer(transformer)
self.num_generated += n_samples
return r
def _sobol(self, dim_num, seed):
"""Generates a new quasirandom Sobol' vector with each call.
Parameters
----------
dim_num : int
Number of spatial dimensions.
`dim_num` must satisfy 1 <= DIM_NUM <= 40.
seed : int
the `seed` for the sequence.
This is essentially the index in the sequence of the quasirandom
value to be generated. On output, `seed` has been set to the
appropriate next value, usually simply `seed`+1.
If `seed` is less than 0 on input, it is treated as though it were 0.
An input value of 0 requests the first (0-th) element of
the sequence.
Returns
-------
vector, seed : np.array (n_dim,), int
The next quasirandom vector and the seed of its next vector.
"""
# Things to do only if the dimension changed.
if dim_num != self.dim_num_save:
self.init(dim_num)
seed = int(np.floor(seed))
if seed < 0:
seed = 0
pos_lo0 = 1
if seed == 0:
self.lastq = np.zeros(dim_num)
elif seed == self.seed_save + 1:
# Find the position of the right-hand zero in SEED.
pos_lo0 = _bit_lo0(seed)
elif seed <= self.seed_save:
self.seed_save = 0
self.lastq = np.zeros(dim_num)
for seed_temp in range(int(self.seed_save), int(seed)):
pos_lo0 = _bit_lo0(seed_temp)
for i in range(1, dim_num + 1):
self.lastq[i - 1] = np.bitwise_xor(
int(self.lastq[i - 1]),
int(self.v[i - 1, pos_lo0 - 1]))
pos_lo0 = _bit_lo0(seed)
elif self.seed_save + 1 < seed:
for seed_temp in range(int(self.seed_save + 1), int(seed)):
pos_lo0 = _bit_lo0(seed_temp)
for i in range(1, dim_num + 1):
self.lastq[i - 1] = np.bitwise_xor(
int(self.lastq[i - 1]),
int(self.v[i - 1, pos_lo0 - 1]))
pos_lo0 = _bit_lo0(seed)
# Check that the user is not calling too many times!
if self.maxcol < pos_lo0:
raise ValueError(f'I4_SOBOL - Fatal error!\n'
f' Too many calls!\n'
f' MAXCOL = {self.maxcol}\n'
f' L = {pos_lo0}\n')
# Calculate the new components of QUASI.
quasi = np.zeros(dim_num)
for i in range(1, dim_num + 1):
quasi[i - 1] = self.lastq[i - 1] * self.recipd
self.lastq[i - 1] = np.bitwise_xor(
int(self.lastq[i - 1]), int(self.v[i - 1, pos_lo0 - 1]))
self.seed_save = seed
seed += 1
return [quasi, seed]
def _bit_hi1(n):
"""Returns the position of the high 1 bit base 2 in an integer.
Parameters
----------
n : int
Input, should be positive.
"""
bin_repr = np.binary_repr(n)
most_left_one = bin_repr.find('1')
if most_left_one == -1:
return 0
else:
return len(bin_repr) - most_left_one
def _bit_lo0(n):
"""Returns the position of the low 0 bit base 2 in an integer.
Parameters
----------
n : int
Input, should be positive.
"""
bin_repr = np.binary_repr(n)
most_right_zero = bin_repr[::-1].find('0')
if most_right_zero == -1:
most_right_zero = len(bin_repr)
return most_right_zero + 1
def _random_shift(dm, random_state=None):
"""Random shifting of a vector.
Randomization of the quasi-MC samples can be achieved in the easiest manner
by random shift (or the Cranley-Patterson rotation).
References
-----------
.. [1] C. Lemieux, "Monte Carlo and Quasi-Monte Carlo Sampling," Springer
Series in Statistics 692, Springer Science+Business Media, New York,
2009
Parameters
----------
dm : array, shape(n, d)
Input matrix.
random_state : int, RandomState instance, or None (default)
Set random state to something other than None for reproducible
results.
Returns
-------
dm : array, shape(n, d)
Randomized Sobol' design matrix.
"""
rng = check_random_state(random_state)
# Generate random shift matrix from uniform distribution
shift = np.repeat(rng.rand(1, dm.shape[1]), dm.shape[0], axis=0)
# Return the shifted Sobol' design
return (dm + shift) % 1
