"""
---
Copyright (c) 2018 Baskar Ganapathysubramanian, Balaji Sesha Sarath Pokuri
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.
---
"""
## --- end license text --- ##
"""
Utility functions that will be used in kernels, acquisition functions
and bayes optimization
"""
from typing import List
import numpy as np
[docs]def erf(x):
"""
error function of x: used in calculating cumulative distribution.
Unable to import from scipy, so writing our own function
:param x: float
:return: error function of x
"""
# save the sign of x
sign = 1 if x >= 0 else -1
x = abs(x)
# constants
a1 = 0.254829592
a2 = -0.284496736
a3 = 1.421413741
a4 = -1.453152027
a5 = 1.061405429
p = 0.3275911
# A&S formula 7.1.26
t = 1.0 / (1.0 + p * x)
y = 1.0 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * np.exp(-x * x)
return sign * y # erf(-x) = -erf(x)
[docs]def pdf_normal(x: np.array, mean: np.array = 0., sigma_sq: float = 1.) -> float:
"""
Probability distribution function of standard normal distribution,
returns the pdf of a location from given mean with variance sigma_sq
:param x: scalar/location
:param mean: mean of distribution
:param sigma_sq: variance of distribution (sigma^2)
:return: pdf of normal distribution
"""
assert type(x) == type(mean), "PDF: x({}), mean({}) are of different types".format(type(x), type(mean))
dist = distance(x, mean)
return 1.0/np.sqrt(2 * np.pi * sigma_sq) * np.exp(-dist*dist/(2*sigma_sq))
[docs]def cdf_normal(x: np.array, mean: np.array = 0., sigma_sq: np.array = 1.) -> float:
"""
Cumulative distribution function of standard normal distribution
:param x: scalar / location
:param mean: mean of distribution
:param sigma_sq: variance of distribution (sigma^2)
:return: cdf of normal distribution
"""
return 0.5 * (1 + erf(distance(x, mean)/np.sqrt(2.0 * sigma_sq)))
[docs]def distance(x1: np.array, x2: np.array) -> float:
"""
returns the distance between two query points
:param x1: point 1
:param x2: point 2
:return: euclidean distance between the two points
"""
assert x1.shape == x2.shape
dist = np.linalg.norm(x1 - x2)
if dist < 1e-16:
return 0.0
else:
return dist
################################################################################
"""
This code was originally published by the following individuals for use with
Scilab:
Copyright (C) 2012 - 2013 - Michael Baudin
Copyright (C) 2012 - Maria Christopoulou
Copyright (C) 2010 - 2011 - INRIA - Michael Baudin
Copyright (C) 2009 - Yann Collette
Copyright (C) 2009 - CEA - Jean-Marc Martinez
website: forge.scilab.org/index.php/p/scidoe/sourcetree/master/macros
Much thanks goes to these individuals. It has been converted to Python by
Abraham Lee.
"""
[docs]def lhs(n: int, samples: int = None, criterion: str = None, iterations: int = None):
"""
Generate a latin-hypercube design
:param n: The number of factors to generate samples for
:param samples: The number of samples to generate for each factor (Default: n)
:param criterion: Allowable values are "center" or "c", "maximin" or "m", "centermaximin" or "cm", and \
"correlation" or "corr". If no value given, the design is centermaximin.
:param iterations: The number of iterations in the maximin and correlations algorithms (Default: 5).
:return H: An n-by-samples design matrix that has been normalized so factor values are uniformly \
spaced between zero and one.
:Example:
A 3-factor design (defaults to 3 samples)::
>>> lhs(3)
array([[ 0.40069325, 0.08118402, 0.69763298],
[ 0.19524568, 0.41383587, 0.29947106],
[ 0.85341601, 0.75460699, 0.360024 ]])
A 4-factor design with 6 samples::
>>> lhs(4, samples=6)
array([[ 0.27226812, 0.02811327, 0.62792445, 0.91988196],
[ 0.76945538, 0.43501682, 0.01107457, 0.09583358],
[ 0.45702981, 0.76073773, 0.90245401, 0.18773015],
[ 0.99342115, 0.85814198, 0.16996665, 0.65069309],
[ 0.63092013, 0.22148567, 0.33616859, 0.36332478],
[ 0.05276917, 0.5819198 , 0.67194243, 0.78703262]])
A 2-factor design with 5 centered samples::
>>> lhs(2, samples=5, criterion='center')
array([[ 0.3, 0.5],
[ 0.7, 0.9],
[ 0.1, 0.3],
[ 0.9, 0.1],
[ 0.5, 0.7]])
A 3-factor design with 4 samples where the minimum distance between
all samples has been maximized::
>>> lhs(3, samples=4, criterion='maximin')
array([[ 0.02642564, 0.55576963, 0.50261649],
[ 0.51606589, 0.88933259, 0.34040838],
[ 0.98431735, 0.0380364 , 0.01621717],
[ 0.40414671, 0.33339132, 0.84845707]])
A 4-factor design with 5 samples where the samples are as uncorrelated
as possible (within 10 iterations)::
>>> lhs(4, samples=5, criterion='correlate', iterations=10)
"""
H = None
if samples is None:
samples = n
if criterion is not None:
assert criterion.lower() in ('center', 'c', 'maximin', 'm',
'centermaximin', 'cm', 'correlation',
'corr'), 'Invalid value for "criterion": {}'.format(criterion)
else:
H = _lhsclassic(n, samples)
if criterion is None:
criterion = 'centermaximin'
if iterations is None:
iterations = 5
if H is None:
if criterion.lower() in ('center', 'c'):
H = _lhscentered(n, samples)
elif criterion.lower() in ('maximin', 'm'):
H = _lhsmaximin(n, samples, iterations, 'maximin')
elif criterion.lower() in ('centermaximin', 'cm'):
H = _lhsmaximin(n, samples, iterations, 'centermaximin')
elif criterion.lower() in ('correlate', 'corr'):
H = _lhscorrelate(n, samples, iterations)
return H
def _lhsclassic(n, samples):
# Generate the intervals
cut = np.linspace(0, 1, samples + 1)
# Fill points uniformly in each interval
u = np.random.rand(samples, n)
a = cut[:samples]
b = cut[1:samples + 1]
rdpoints = np.zeros_like(u)
for j in range(n):
rdpoints[:, j] = u[:, j]*(b-a) + a
# Make the random pairings
H = np.zeros_like(rdpoints)
for j in range(n):
order = np.random.permutation(range(samples))
H[:, j] = rdpoints[order, j]
return H
def _lhscentered(n, samples):
# Generate the intervals
cut = np.linspace(0, 1, samples + 1)
# Fill points uniformly in each interval
u = np.random.rand(samples, n)
a = cut[:samples]
b = cut[1:samples + 1]
_center = (a + b)/2
# Make the random pairings
H = np.zeros_like(u)
for j in range(n):
H[:, j] = np.random.permutation(_center)
return H
def _lhsmaximin(n, samples, iterations, lhstype):
maxdist = 0
# Maximize the minimum distance between points
for i in range(iterations):
if lhstype=='maximin':
Hcandidate = _lhsclassic(n, samples)
else:
Hcandidate = _lhscentered(n, samples)
d = _pdist(Hcandidate)
if maxdist<np.min(d):
maxdist = np.min(d)
H = Hcandidate.copy()
return H
def _lhscorrelate(n, samples, iterations):
mincorr = np.inf
# Minimize the components correlation coefficients
for i in range(iterations):
# Generate a random LHS
Hcandidate = _lhsclassic(n, samples)
R = np.corrcoef(Hcandidate)
if np.max(np.abs(R[R!=1]))<mincorr:
mincorr = np.max(np.abs(R-np.eye(R.shape[0])))
print('new candidate solution found with max,abs corrcoef = {}'.format(mincorr))
H = Hcandidate.copy()
return H
def _pdist(x):
"""
Calculate the pair-wise point distances of a matrix
:param x : 2d-array
An m-by-n array of scalars, where there are m points in n dimensions.
:return d : array
A 1-by-b array of scalars, where b = m*(m - 1)/2. This array contains
all the pair-wise point distances, arranged in the order (1, 0),
(2, 0), ..., (m-1, 0), (2, 1), ..., (m-1, 1), ..., (m-1, m-2).
:Example:
>>> x = np.array([[0.1629447, 0.8616334],
... [0.5811584, 0.3826752],
... [0.2270954, 0.4442068],
... [0.7670017, 0.7264718],
... [0.8253975, 0.1937736]])
>>> _pdist(x)
array([ 0.6358488, 0.4223272, 0.6189940, 0.9406808, 0.3593699,
0.3908118, 0.3087661, 0.6092392, 0.6486001, 0.5358894])
"""
x = np.atleast_2d(x)
assert len(x.shape)==2, 'Input array must be 2d-dimensional'
m, n = x.shape
if m<2:
return []
d = []
for i in range(m - 1):
for j in range(i + 1, m):
d.append((sum((x[j, :] - x[i, :])**2))**0.5)
return np.array(d)
################################################################################