def laguerre(
order,
alpha=0.0,
physicist=False,
normed=False,
retall=False,
):
"""
Examples:
>>> polynomials, norms = chaospy.expansion.laguerre(3, retall=True)
>>> polynomials
polynomial([1.0, q0-1.0, q0**2-4.0*q0+2.0, q0**3-9.0*q0**2+18.0*q0-6.0])
>>> norms
array([ 1., 1., 4., 36.])
>>> chaospy.expansion.laguerre(3, physicist=True).round(5)
polynomial([1.0, -q0+1.0, 0.5*q0**2-2.0*q0+2.0,
-0.16667*q0**3+1.5*q0**2-5.33333*q0+4.66667])
>>> chaospy.expansion.laguerre(3, alpha=2, normed=True).round(3)
polynomial([1.0, 0.577*q0-1.732, 0.204*q0**2-1.633*q0+2.449,
0.053*q0**3-0.791*q0**2+3.162*q0-3.162])
"""
multiplier = -1.0 / numpy.arange(1, order + 1) if physicist else 1.0
_, [polynomials], [norms] = chaospy.recurrence.analytical_stieltjes(
order, chaospy.Gamma(alpha + 1), multiplier=multiplier)
if normed:
polynomials = chaospy.true_divide(polynomials, numpy.sqrt(norms))
norms[:] = 1.0
return (polynomials, norms) if retall else polynomials
def plot_figures():
"""Plot figures for multivariate distribution section."""
rc("figure", figsize=[8., 4.])
rc("figure.subplot", left=.08, top=.95, right=.98)
seed(1000)
Q1 = cp.Gamma(2)
Q2 = cp.Normal(0, Q1)
Q = cp.J(Q1, Q2)
#end
subplot(121)
s, t = meshgrid(linspace(0, 5, 200), linspace(-6, 6, 200))
contourf(s, t, Q.pdf([s, t]), 50)
xlabel("$q_1$")
ylabel("$q_2$")
subplot(122)
Qr = Q.sample(500)
scatter(*Qr)
xlabel("$Q_1$")
ylabel("$Q_2$")
axis([0, 5, -6, 6])
savefig("mv1.png")
clf()
Q2 = cp.Gamma(1)
Q1 = cp.Normal(Q2**2, Q2 + 1)
Q = cp.J(Q1, Q2)
#end
subplot(121)
s, t = meshgrid(linspace(-4, 7, 200), linspace(0, 3, 200))
contourf(s, t, Q.pdf([s, t]), 30)
xlabel("$q_1$")
ylabel("$q_2$")
subplot(122)
Qr = Q.sample(500)
scatter(*Qr)
xlabel("$Q_1$")
ylabel("$Q_2$")
axis([-4, 7, 0, 3])
savefig("mv2.png")
clf()
def test_compare_scipy_Gamma():
shape = 3.0
dist_cp = cp.Gamma(shape=shape, scale=1.0, shift=0)
r0, r1 = dist_cp.range()
x = np.linspace(r0, r1, 300, endpoint=True)
pdf_sp = stats.gamma.pdf(x, shape, scale=1.0, loc=0)
cdf_sp = stats.gamma.cdf(x, shape, scale=1.0, loc=0)
np.testing.assert_allclose(pdf_sp, dist_cp.pdf(x))
np.testing.assert_allclose(cdf_sp, dist_cp.cdf(x))
def laguerre(order, alpha=0.0, physicist=False):
r"""
Generalized Gauss-Laguerre quadrature rule.
Compute the sample points and weights for Gauss-Laguerre quadrature. The
sample points are the roots of the nth degree Laguerre 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-Laguerre physicist means a weight function :math:`x^\alpha e^{-x}`
and weights that sum to :math`\Gamma(\alpha+1)`, and probabilist means a
weight function is :math:`x^\alpha e^{-x}` and sum to 1.
Args:
order (int):
The quadrature order.
alpha (float):
Shape parameter. Defaults to non-generalized Laguerre if 0.
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.laguerre(2)
>>> abscissas
array([[0.41577456, 2.29428036, 6.28994508]])
>>> weights
array([0.71109301, 0.27851773, 0.01038926])
See also:
:func:`chaospy.quadrature.gaussian`
"""
order = int(order)
coefficients = chaospy.construct_recurrence_coefficients(
order=order, dist=chaospy.Gamma(alpha + 1))
[abscissas], [weights] = chaospy.coefficients_to_quadrature(coefficients)
weights *= gamma(alpha + 1) if physicist else 1
return abscissas[numpy.newaxis], weights
output_columns = ["IC_prev_avg_max", "IC_ex_max"]
encoder = uq.encoders.GenericEncoder(template_fname=HOME + '/corona.template',
delimiter='$',
target_filename='corona_in.json')
decoder = uq.decoders.SimpleCSV(target_filename=output_filename,
output_columns=output_columns)
# Add the SC app (automatically set as current app)
campaign.add_app(name="sc", params=params, encoder=encoder, decoder=decoder)
# Create the sampler
vary = {
"seed": cp.DiscreteUniform(2**14, 2**16),
"lockdown_effect": cp.Beta(alpha=14, beta=42),
"phase_interval": cp.Gamma(shape=25, scale=2),
"uptake": cp.Beta(alpha=16, beta=2),
# "Rzero": cp.Gamma(shape=100,scale=.025),
# "duration_infectiousness": cp.Gamma(shape=25,scale=.2),
# "shape_exposed_time": cp.Gamma(shape=17.5,scale=1),
# "intervention_effect_var_inv": cp.Gamma(shape=2,scale=.05)
}
#sampler = uq.sampling.SCSampler(vary=vary, polynomial_order=3,
# quadrature_rule='G', sparse=False)
sampler = uq.sampling.MCSampler(vary=vary, n_mc_samples=2000)
# Associate the sampler with the campaign
campaign.set_sampler(sampler)
# Will draw all (of the finite set of samples)
template_fname= HOME + '/corona.template',
delimiter='$',
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),
"lockdown_effect": cp.Beta(alpha=14, beta=42),
"lockdown_length": cp.Gamma(shape=20, scale=2),
"lift_length": cp.Gamma(shape=15, scale=1),
"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
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),
"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)
}
# 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)
"""
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)
gamma_phase_interval = cp.Gamma(shape=25, scale=2)
"""Test dependent distributions with 1-D components."""
from pytest import raises
import numpy
import chaospy
DIST1 = chaospy.Uniform(1, 2)
DIST2 = chaospy.Gamma(DIST1)
JOINT1 = chaospy.J(DIST1, DIST2)
JOINT2 = chaospy.J(DIST2, DIST1)
def test_1d_stochastic_dependencies():
"""Ensure ``stochastic_dependencies`` behaves as expected for dependent 1-D distributions."""
assert not DIST1.stochastic_dependent
assert DIST2.stochastic_dependent
assert JOINT1.stochastic_dependent
assert JOINT2.stochastic_dependent
def test_1d_dependent_bounds():
"""Ensure lower and upper bounds works for dependent 1-D distributions."""
assert numpy.isclose(DIST2.lower, 0)
assert numpy.isclose(DIST2.upper, 35.84367486)
assert numpy.allclose(JOINT1.lower, [1, 0])
assert numpy.allclose(JOINT1.upper, [2, 35.84367486])
assert numpy.allclose(JOINT2.lower, [0, 1])
assert numpy.allclose(JOINT2.upper, [35.84367486, 2])
def test_1d_dependent_mapping():
"""Ensure inverse and forward behaves as expected for dependent 1-D distributions."""
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)
}
# 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
campaign.set_sampler(sampler)
# Will draw all (of the finite set of samples)
campaign.draw_samples()
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),
"lockdown_length": cp.Gamma(shape=20, scale=2),
"lift_length": cp.Gamma(shape=15, scale=1),
"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.MCSampler(vary=vary, n_mc_samples=2000)
# Associate the sampler with the campaign
campaign.set_sampler(sampler)
# Will draw all (of the finite set of samples)
campaign.draw_samples()
"""Test dependent distributions with 2-D components."""
from pytest import raises
import numpy
import chaospy
DIST1 = chaospy.J(chaospy.Uniform(1, 2), chaospy.Uniform(2, 4))
DIST2 = chaospy.J(chaospy.Gamma(DIST1[0]), chaospy.Gamma(DIST1[1]))
JOINT1 = chaospy.J(DIST1, DIST2)
JOINT2 = chaospy.J(DIST2, DIST1)
def test_2d_stochastic_dependencies():
"""Ensure ``stochastic_dependencies`` behaves as expected for dependent 2-D distributions."""
assert not DIST1.stochastic_dependent
assert DIST2.stochastic_dependent
assert JOINT1.stochastic_dependent
assert JOINT2.stochastic_dependent
def test_2d_dependencies():
"""Ensure 2-D dependencies behaves as expected."""
grid1 = numpy.array([[0, 0, 1, 1], [0, 1, 0, 1], [1, 1, 1, 1],
[1, 1, 1, 1]])
grid2 = numpy.array([[1, 1, 1, 1], [1, 1, 1, 1], [0, 0, 1, 1],
[0, 1, 0, 1]])
inv_map1 = numpy.array(
[[1, 1, 2, 2], [2, 4, 2, 4],
[32.2369909, 32.2369909, 35.84367486, 35.84367486],
[35.84367486, 41.71021463, 35.84367486, 41.71021463]])
inv_map2 = numpy.array(
[[32.2369909, 32.2369909, 35.84367486, 35.84367486],