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"""
Inspired by https://github.com/jonathf/chaospy/blob/master/chaospy/
distributions/sampler/sequences/halton.py
"""
import numpy as np
from sklearn.utils import check_random_state
from ..space import Space
from .base import InitialPointGenerator
class Halton(InitialPointGenerator):
"""Creates `Halton` sequence samples.
In statistics, Halton sequences are sequences used to generate
points in space for numerical methods such as Monte Carlo simulations.
Although these sequences are deterministic, they are of low discrepancy,
that is, appear to be random
for many purposes. They were first introduced in 1960 and are an example
of a quasi-random number sequence. They generalise the one-dimensional
van der Corput sequences.
For ``dim == 1`` the sequence falls back to Van Der Corput sequence.
Parameters
----------
min_skip : int
Minimum skipped seed number. When `min_skip != max_skip`
a random number is picked.
max_skip : int
Maximum skipped seed number. When `min_skip != max_skip`
a random number is picked.
primes : tuple, default=None
The (non-)prime base to calculate values along each axis. If
empty or None, growing prime values starting from 2 will be used.
"""
def __init__(self, min_skip=0, max_skip=0, primes=None):
self.primes = primes
self.min_skip = min_skip
self.max_skip = max_skip
def generate(self, dimensions, n_samples, random_state=None):
"""Creates samples from Halton 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 Halton 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
-------
np.array, shape=(n_dim, n_samples)
Halton set.
"""
rng = check_random_state(random_state)
if self.primes is None:
primes = []
else:
primes = list(self.primes)
space = Space(dimensions)
n_dim = space.n_dims
transformer = space.get_transformer()
space.set_transformer("normalize")
if len(primes) < n_dim:
prime_order = 10 * n_dim
while len(primes) < n_dim:
primes = _create_primes(prime_order)
prime_order *= 2
primes = primes[:n_dim]
assert len(primes) == n_dim, "not enough primes"
if self.min_skip == self.max_skip:
skip = self.min_skip
elif self.min_skip < 0 and self.max_skip < 0:
skip = max(primes)
elif self.min_skip < 0 or self.max_skip < 0:
skip = np.max(self.min_skip, self.max_skip)
else:
skip = rng.randint(self.min_skip, self.max_skip)
out = np.empty((n_dim, n_samples))
indices = [idx + skip for idx in range(n_samples)]
for dim_ in range(n_dim):
out[dim_] = _van_der_corput_samples(indices, number_base=primes[dim_])
out = space.inverse_transform(np.transpose(out))
space.set_transformer(transformer)
return out
def _van_der_corput_samples(idx, number_base=2):
"""Create `Van Der Corput` low discrepancy sequence samples.
A van der Corput sequence is an example of the simplest one-dimensional
low-discrepancy sequence over the unit interval; it was first described in
1935 by the Dutch mathematician J. G. van der Corput. It is constructed by
reversing the base-n representation of the sequence of natural numbers
(1, 2, 3, ...).
In practice, use Halton sequence instead of Van Der Corput, as it is the
same, but generalized to work in multiple dimensions.
Parameters
----------
idx (int, numpy.ndarray):
The index of the sequence. If array is provided, all values in
array is returned.
number_base : int
The numerical base from where to create the samples from.
Returns
-------
float, numpy.ndarray
Van der Corput samples.
"""
assert number_base > 1
idx = np.asarray(idx).flatten()
out = np.zeros(len(idx), dtype=float)
base = float(number_base)
active = np.ones(len(idx), dtype=bool)
while np.any(active):
out[active] += (idx[active] % number_base) / base
idx //= number_base
base *= number_base
active = idx > 0
return out
def _create_primes(threshold):
"""Generate prime values using sieve of Eratosthenes method.
Parameters
----------
threshold : int
The upper bound for the size of the prime values.
Returns
------
List
All primes from 2 and up to ``threshold``.
"""
if threshold == 2:
return [2]
elif threshold < 2:
return []
numbers = list(range(3, threshold + 1, 2))
root_of_threshold = threshold**0.5
half = int((threshold + 1) / 2 - 1)
idx = 0
counter = 3
while counter <= root_of_threshold:
if numbers[idx]:
idy = int((counter * counter - 3) / 2)
numbers[idy] = 0
while idy < half:
numbers[idy] = 0
idy += counter
idx += 1
counter = 2 * idx + 3
return [2] + [number for number in numbers if number]
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