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@tupui
Last active August 9, 2021 15:50
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Orthogonal Latin Hypercube Sampling
"""Orthogonal Latin Hypercube Sampling.
---------------------------
MIT License
Copyright (c) 2021 Pamphile Tupui ROY
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
"""
import functools
import numpy as np
from scipy import spatial
from scipy.stats import qmc
from scipy import optimize
import matplotlib.pyplot as plt
import seaborn as sns
def oa_lhs(p, d, seed):
oa_sample = np.zeros(shape=(p**2, p+1))
arrays = np.tile(np.arange(p), (2, 1))
oa_sample[:, :2] = np.stack(np.meshgrid(*arrays), axis=-1).reshape(-1, 2)
for p_ in range(1, p):
oa_sample[:, 2+p_-1] = np.mod(oa_sample[:, 0] + p_*oa_sample[:, 1], p)
# scramble the OA
oa_sample_ = np.empty(shape=(p**2, p+1))
for j in range(p+1):
perms = np.random.permutation(p)
for k in range(p):
idx = np.where(oa_sample[:, j] == k)[0]
oa_sample_[idx, j] = perms[k]
oa_sample = oa_sample_
# sample is a randomized OA from now
# and the following is making it an OA-LHS
oa_lhs_sample = np.zeros(shape=(p**2, p+1))
for j in range(p+1):
for k in range(p):
idx = np.where(oa_sample[:, j] == k)[0]
lhs = LatinHypercube(d=1, centered=True, seed=seed).random(p).flatten()
oa_lhs_sample[:, j][idx] = lhs + oa_sample[:, j][idx]
oa_lhs_sample /= p
if d is not None:
oa_lhs_sample = oa_lhs_sample[:, :d]
return oa_lhs_sample
rng = np.random.default_rng()
p, d = 7, 3
oa_lhs_sample = oa_lhs(p, d, seed=rng)
sns.pairplot(pd.DataFrame(oa_lhs_sample), diag_kind="hist", corner=True, diag_kws={"bins": p})
@kstoneriv3
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Nice visualization!

How about using below in the scrambling instead of using for loop? This fancy indexing runs OK in my local python environment.

    # scramble the OA
    perms = np.random.permutation(p)
    oa_sample_ = perms[oa_sample]

@kstoneriv3
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A few more questions:

  • Are we assuming that d<=p+1?

  • In line 71,

    if d is not None:
        idx = seed.permutation(p+1)[:d]
        oa_lhs_sample = oa_lhs_sample[:, idx]

feels more natural to me as all the columns have equal probability to be selected.

  • The code for generating an orthogonal array might be broken, as there does not seem to be exactly one point in each cell of 7x7 grid. There should be test for making sure that the generated samples are actually OAs of strength 2.
for i in range(p):
    for j in range(i+1, p):
        samples_2d = oa_samples[:, [i, j]]
        cart_prod = ((samples_2d * p) // 1).astype(int)  # should be an cartesian product of [0, 1, ..., p-1] x [0, 1, ..., p-1] 
        count_cells = np.zeros([p, p])
        count_cells[cart_prod] += 1  # all pairs of 0 to p-1 must be counted once
        assert np.all(count_cells == 1)

@tupui
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tupui commented Aug 5, 2021

@kstoneriv3 thanks for the feedbacks!

How about using below in the scrambling instead of using for loop? This fancy indexing runs OK in my local python environment.

    # scramble the OA
    perms = np.random.permutation(p)
    oa_sample_ = perms[oa_sample]

I was not sure about this because it would mean that the permutations are not independent. So we change all the 0 for a 1 in all columns. Whereas the current version changes only one column at a time. If we go with your version, I don't see how this is doing any scrambling. In the end we just renamed the symbols if we do this.

  • Are we assuming that d<=p+1?

Yes.

  • In line 71,
    if d is not None:
        idx = seed.permutation(p+1)[:d]
        oa_lhs_sample = oa_lhs_sample[:, idx]

feels more natural to me as all the columns have equal probability to be selected.

Totally, it's a mistake of mine. Thanks.

  • The code for generating an orthogonal array might be broken, as there does not seem to be exactly one point in each cell of 7x7 grid. There should be test for making sure that the generated samples are actually OAs of strength 2.

I could not run your snippet.

@kstoneriv3
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I was not sure about this because it would mean that the permutations are not independent. So we change all the 0 for a 1 in all columns. Whereas the current version changes only one column at a time. If we go with your version, I don't see how this is doing any scrambling. In the end, we just renamed the symbols if we do this.

Ah, you are right! I now have an intuition that permutations should be independently sampled for each column. Then the code near line 71 does not need any fix.

@kstoneriv3
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kstoneriv3 commented Aug 7, 2021

I could not run your snippet.

If runs once you define oa_lhs_sample and p.

import numpy as np
# define OA
p = 2
oa_lhs_sample = np.array([[0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 1, 0]]).T / 2 + 0.25

for i in range(p):
    for j in range(i+1, p):
        samples_2d = oa_lhs_sample[:, [i, j]]
        cart_prod = ((samples_2d * p) // 1).astype(int)  # should be an cartesian product of [0, 1, ..., p-1] x [0, 1, ..., p-1] 
        count_cells = np.zeros([p, p])
        count_cells[cart_prod] += 1  # all pairs of 0 to p-1 must be counted once
        assert np.all(count_cells == 1)

@tupui
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tupui commented Aug 7, 2021

Thanks for your inputs @kstoneriv3. I opened a PR if you did not see scipy/scipy#14546. Your test to check that we have an OA would be a nice addition. I will see how I can make this work. Or you're free to propose something over there.

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