def Importance_Sampling(n=100):
erro = 10
while erro >= 0.01:
contador_IS = []
for i in range(1,n+1):
#Gerador de quasi-aleatório com distribuição beta
quasi = chaospy.Beta(0.9,1,0,1).sample(100,rule='Halton')
p = np.array(random.choices(quasi,k=100)) #Escolhe um valor da lista quasi
x = beta.pdf(p,0.9,1)
contador_IS.append(np.mean(f(p)/x))
#Calcula o erro padrão
erro = np.std(contador_IS)/np.sqrt(i)
#Verifica se o erro é menor que 1% a partir da 2 iteração
if i > 10:
if erro < 0.01:
break
resultado = np.mean(contador_IS)
return resultado
def jacobi(order, alpha, beta, lower=-1, upper=1, physicist=False):
"""
Gauss-Jacobi quadrature rule.
Compute the sample points and weights for Gauss-Jacobi quadrature. The
sample points are the roots of the nth degree Jacobi polynomial. These
sample points and weights correctly integrate polynomials of degree
:math:`2N-1` or less.
Gaussian quadrature come in two variants: physicist and probabilist. For
Gauss-Jacobi physicist means a weight function
:math:`(1-x)^\alpha (1+x)^\beta` and
weights that sum to :math`2^{\alpha+\beta}`, and probabilist means a weight
function is :math:`B(\alpha, \beta) x^{\alpha-1}(1-x)^{\beta-1}` (where
:math:`B` is the beta normalizing constant) which sum to 1.
Args:
order (int):
The quadrature order.
alpha (float):
First Jakobi shape parameter.
beta (float):
Second Jakobi shape parameter.
lower (float):
Lower bound for the integration interval.
upper (float):
Upper bound for the integration interval.
physicist (bool):
Use physicist weights instead of probabilist.
Returns:
abscissas (numpy.ndarray):
The ``order+1`` quadrature points for where to evaluate the model
function with.
weights (numpy.ndarray):
The quadrature weights associated with each abscissas.
Examples:
>>> abscissas, weights = chaospy.quadrature.jacobi(3, alpha=2, beta=2)
>>> abscissas
array([[-0.69474659, -0.25056281, 0.25056281, 0.69474659]])
>>> weights
array([0.09535261, 0.40464739, 0.40464739, 0.09535261])
See also:
:func:`chaospy.quadrature.gaussian`
"""
order = int(order)
coefficients = chaospy.construct_recurrence_coefficients(
order=order, dist=chaospy.Beta(alpha + 1, beta + 1, lower, upper))
[abscissas], [weights] = chaospy.coefficients_to_quadrature(coefficients)
weights *= 2**(alpha + beta) if physicist else 1
return abscissas[numpy.newaxis], weights
def gegenbauer(
order,
alpha,
lower=-1,
upper=1,
physicist=False,
normed=False,
retall=False,
):
"""
Gegenbauer polynomials.
Args:
order (int):
The polynomial order.
alpha (float):
Gegenbauer shape parameter.
lower (float):
Lower bound for the integration interval.
upper (float):
Upper bound for the integration interval.
physicist (bool):
Use physicist weights instead of probabilist.
Examples:
>>> polynomials, norms = chaospy.expansion.gegenbauer(4, 1, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-0.25, q0**3-0.5*q0, q0**4-0.75*q0**2+0.0625])
>>> norms
array([1. , 0.25 , 0.0625 , 0.015625 , 0.00390625])
>>> chaospy.expansion.gegenbauer(3, 1, physicist=True)
polynomial([1.0, 2.0*q0, 4.0*q0**2-0.5, 8.0*q0**3-2.0*q0])
>>> chaospy.expansion.gegenbauer(3, 1, lower=0.5, upper=1.5, normed=True).round(3)
polynomial([1.0, 4.0*q0-4.0, 16.0*q0**2-32.0*q0+15.0,
64.0*q0**3-192.0*q0**2+184.0*q0-56.0])
"""
multiplier = 1
if physicist:
multiplier = numpy.arange(1, order + 1)
multiplier = 2 * (multiplier + alpha - 1) / multiplier
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order,
chaospy.Beta(alpha + 0.5, alpha + 0.5, lower, upper),
multiplier=multiplier,
)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def chebyshev_2(
order,
lower=-1,
upper=1,
physicist=False,
normed=False,
retall=False,
):
"""
Chebyshev polynomials of the second kind.
Args:
order (int):
The quadrature order.
lower (float):
Lower bound for the integration interval.
upper (float):
Upper bound for the integration interval.
physicist (bool):
Use physicist weights instead of probabilist.
Returns:
(numpoly.ndpoly, numpy.ndarray):
Chebyshev polynomial expansion. Norms of the orthogonal
expansion on the form ``E(orth**2, dist)``.
Examples:
>>> polynomials, norms = chaospy.expansion.chebyshev_2(4, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-0.25, q0**3-0.5*q0, q0**4-0.75*q0**2+0.0625])
>>> norms
array([1. , 0.25 , 0.0625 , 0.015625 , 0.00390625])
>>> chaospy.expansion.chebyshev_2(3, physicist=True)
polynomial([1.0, 2.0*q0, 4.0*q0**2-0.5, 8.0*q0**3-2.0*q0])
>>> chaospy.expansion.chebyshev_2(3, lower=0.5, upper=1.5, normed=True).round(3)
polynomial([1.0, 4.0*q0-4.0, 16.0*q0**2-32.0*q0+15.0,
64.0*q0**3-192.0*q0**2+184.0*q0-56.0])
"""
multiplier = 2 if physicist else 1
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order, chaospy.Beta(1.5, 1.5, lower, upper), multiplier=multiplier
)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def chebyshev_1(
order,
lower=-1,
upper=1,
physicist=False,
normed=False,
retall=False,
):
"""
Chebyshev polynomials of the first kind.
Args:
order (int):
The polynomial order.
lower (float):
Lower bound for the integration interval.
upper (float):
Upper bound for the integration interval.
physicist (bool):
Use physicist weights instead of probabilist.
Returns:
(numpoly.ndpoly, numpy.ndarray):
Chebyshev polynomial expansion. Norms of the orthogonal
expansion on the form ``E(orth**2, dist)``.
Examples:
>>> polynomials, norms = chaospy.expansion.chebyshev_1(4, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-0.5, q0**3-0.75*q0, q0**4-q0**2+0.125])
>>> norms
array([1. , 0.5 , 0.125 , 0.03125 , 0.0078125])
>>> chaospy.expansion.chebyshev_1(3, physicist=True)
polynomial([1.0, q0, 2.0*q0**2-1.0, 4.0*q0**3-2.5*q0])
>>> chaospy.expansion.chebyshev_1(3, lower=0.5, upper=1.5, normed=True).round(3)
polynomial([1.0, 2.828*q0-2.828, 11.314*q0**2-22.627*q0+9.899,
45.255*q0**3-135.765*q0**2+127.279*q0-36.77])
"""
multiplier = 1 + numpy.arange(order).astype(bool) if physicist else 1
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order, chaospy.Beta(0.5, 0.5, lower, upper), multiplier=multiplier
)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def jacobi(
order,
alpha,
beta,
lower=-1,
upper=1,
physicist=False,
normed=False,
retall=False,
):
"""
Jacobi polynomial expansion.
Examples:
>>> polynomials, norms = chaospy.expansion.jacobi(4, 0.5, 0.5, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-0.5, q0**3-0.75*q0, q0**4-q0**2+0.125])
>>> norms
array([1. , 0.5 , 0.125 , 0.03125 , 0.0078125])
>>> chaospy.expansion.jacobi(3, 0.5, 0.5, physicist=True).round(4)
polynomial([1.0, 1.5*q0, 2.5*q0**2-0.8333, 4.375*q0**3-2.1146*q0])
>>> chaospy.expansion.jacobi(3, 1.5, 0.5, normed=True)
polynomial([1.0, 2.0*q0, 4.0*q0**2-1.0, 8.0*q0**3-4.0*q0])
"""
multiplier = 1
if physicist:
multiplier = numpy.arange(1, order + 1)
multiplier = ((2 * multiplier + alpha + beta - 1) *
(2 * multiplier + alpha + beta) /
(2 * multiplier * (multiplier + alpha + beta)))
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order,
chaospy.Beta(alpha, beta, lower=lower, upper=upper),
multiplier=multiplier,
)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
output_filename = params["out_file"]["default"]
output_columns = ["IC_prev_avg_max", "IC_ex_max"]
encoder = uq.encoders.GenericEncoder(template_fname=HOME + '/corona.template',
delimiter='$',
target_filename='corona_in.json')
decoder = uq.decoders.SimpleCSV(target_filename=output_filename,
output_columns=output_columns)
# Add the SC app (automatically set as current app)
campaign.add_app(name="sc", params=params, encoder=encoder, decoder=decoder)
# Create the sampler
vary = {
"seed": cp.DiscreteUniform(2**14, 2**16),
"lockdown_effect": cp.Beta(alpha=14, beta=42),
"phase_interval": cp.Gamma(shape=25, scale=2),
"uptake": cp.Beta(alpha=16, beta=2),
# "Rzero": cp.Gamma(shape=100,scale=.025),
# "duration_infectiousness": cp.Gamma(shape=25,scale=.2),
# "shape_exposed_time": cp.Gamma(shape=17.5,scale=1),
# "intervention_effect_var_inv": cp.Gamma(shape=2,scale=.05)
}
#sampler = uq.sampling.SCSampler(vary=vary, polynomial_order=3,
# quadrature_rule='G', sparse=False)
sampler = uq.sampling.MCSampler(vary=vary, n_mc_samples=2000)
# Associate the sampler with the campaign
campaign.set_sampler(sampler)
output_filename = params["out_file"]["default"]
output_columns = ["IC_prev_avg_max", "IC_ex_max"]
encoder = uq.encoders.GenericEncoder(template_fname=HOME + '/corona.template',
delimiter='$',
target_filename='corona_in.json')
decoder = uq.decoders.SimpleCSV(target_filename=output_filename,
output_columns=output_columns)
campaign.add_app(name="mc", params=params, encoder=encoder, decoder=decoder)
# Create the sampler
vary = {
"seed": cp.DiscreteUniform(2**14, 2**16),
"intervention_effect": cp.Beta(alpha=38, beta=70),
"uptake": cp.Beta(alpha=16, beta=2),
"Rzero": cp.Gamma(shape=100, scale=.025),
"duration_infectiousness": cp.Gamma(shape=25, scale=.2),
"shape_exposed_time": cp.Gamma(shape=17.5, scale=1),
"intervention_effect_var_inv": cp.Gamma(shape=2, scale=.05)
}
# For estimation of the cdf and heatmap
sampler = uq.sampling.RandomSampler(vary=vary, max_num=1e2)
# For the computation of the Sobol indices
# sampler = uq.sampling.MCSampler(vary=vary, n_mc_samples=100)
# Associate the sampler with the campaign
campaign.set_sampler(sampler)
import pytest
import chaospy
from chaospy.recurrence import RECURRENCE_ALGORITHMS
ANALYTICAL_DISTRIBUTIONS = {
"beta": chaospy.Beta(4, 2),
"expon": chaospy.Exponential(1),
"gamma": chaospy.Gamma(2, 2),
"lognorm": chaospy.LogNormal(-10, 0.1),
"normal": chaospy.Normal(2, 3),
"student": chaospy.StudentT(df=25, mu=0.5),
"uniform": chaospy.Uniform(-1, 2),
}
@pytest.fixture(params=RECURRENCE_ALGORITHMS)
def recurrence_algorithm(request):
"""Parameterization of name of recurrence algorithms."""
yield request.param
@pytest.fixture(params=ANALYTICAL_DISTRIBUTIONS.keys())
def analytical_distribution(request):
"""Parameterization of distribution with analytical TTR methods."""
return ANALYTICAL_DISTRIBUTIONS[request.param]
"""Testing polynomial related to distributions."""
import chaospy
import numpy
import pytest
DISTRIBUTIONS = {
"discrete": chaospy.DiscreteUniform(-10, 10),
"normal": chaospy.Normal(0, 1),
"uniform": chaospy.Uniform(-1, 1),
"exponential": chaospy.Exponential(1),
"gamma": chaospy.Gamma(1),
"beta": chaospy.Beta(3, 3, lower=-1, upper=1),
"mvnormal": chaospy.MvNormal([0], [1]),
"custom": chaospy.UserDistribution(
cdf=lambda x: (x+1)/2,
pdf=lambda x: 1/2.,
lower=lambda: -1,
upper=lambda: 1,
ppf=lambda q: 2*q-1,
mom=lambda k: ((k+1.)%2)/(k+1),
ttr=lambda k: (0., k*k/(4.*k*k-1)),
),
}
BUILDERS = {
"stieltjes": chaospy.expansion.stieltjes,
"cholesky": chaospy.expansion.cholesky,
# "gram_schmidt": chaospy.expansion.gram_schmidt,
}
@pytest.fixture(params=DISTRIBUTIONS)
import sobol_seq import chaospy import matplotlib.pyplot as plt sobolSeq = sobol_seq.i4_sobol_generate(2, 25000) x = sobolSeq[:, 0] y = sobolSeq[:, 1] plt.scatter(x, y, s=0.25) plt.show() sobolSeq = sobol_seq.i4_sobol_generate(2, 4675) x = sobolSeq[:, 0] y = sobolSeq[:, 1] plt.scatter(x, y, s=1) plt.show() a = 1 b = 1.5 distribution = chaospy.Beta(a, b) betaSeq = distribution.sample(25000, 'S') plt.hist(betaSeq, bins=100) plt.show()
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Thu Jul 2 09:41:32 2020
@author: federica
"""
import numpy as np
import chaospy as cp
import matplotlib.pyplot as plt
plt.rcParams.update({'font.size': 20})
plt.rcParams['figure.figsize'] = 16, 20
# Contact tracing
beta_tpE = cp.Beta(alpha=2, beta=6)
beta_tcr = cp.Beta(alpha=10, beta=2)
gamma_trI = cp.Gamma(shape=2, scale=.2)
x_CT = np.linspace(0, 1.5, 151)
# Flattening the curve
beta_int1 = cp.Beta(alpha=38, beta=70)
beta_up = cp.Beta(alpha=16, beta=2)
x_FT = np.linspace(0, 1, 101)
# Intermittent lockdown & Phased Opening
beta_lockeffect = cp.Beta(alpha=14, beta=42)
gamma_locklength = cp.Gamma(shape=20, scale=2)
gamma_liftlength = cp.Gamma(shape=15, scale=1)
output_filename = params["out_file"]["default"]
output_columns = ["IC_prev_avg_max", "IC_ex_max"]
encoder = uq.encoders.GenericEncoder(template_fname=HOME + '/corona.template',
delimiter='$',
target_filename='corona_in.json')
decoder = uq.decoders.SimpleCSV(target_filename=output_filename,
output_columns=output_columns)
campaign.add_app(name='mc', params=params, encoder=encoder, decoder=decoder)
# Create the sampler
vary = {
"seed": cp.DiscreteUniform(2**14, 2**16),
"trace_prob_E": cp.Beta(alpha=2, beta=6),
"trace_rate_I": cp.Gamma(shape=2, scale=.2),
"trace_contact_reduction": cp.Beta(alpha=10, beta=2),
"Rzero": cp.Gamma(shape=100, scale=.025),
"duration_infectiousness": cp.Gamma(shape=25, scale=.2),
"shape_exposed_time": cp.Gamma(shape=17.5, scale=1),
"intervention_effect_var_inv": cp.Gamma(shape=2, scale=.05)
}
# For estimation of the cdf and heatmap
sampler = uq.sampling.RandomSampler(vary=vary, max_num=1e3)
# For the computation of the Sobol indices
# sampler = uq.sampling.MCSampler(vary=vary, n_mc_samples=100)
# Associate the sampler with the campaign
