def uniform(self, a=0, b=1):
# set distributions of the input parameters
builder = ParameterBuilder()
up = builder.defineUncertainParameters()
for idim in range(self.numDims):
up.new().isCalled("x%i" % idim).withUniformDistribution(a, b)
return up.andGetResult()
def normal(self, mu=0, sigma=1, alpha=0.001):
# set distributions of the input parameters
builder = ParameterBuilder()
up = builder.defineUncertainParameters()
for idim in range(self.numDims):
up.new().isCalled("x%i" % idim).withNormalDistribution(
mu, sigma, 0.001)
return builder.andGetResult()
def buildSin2Params(dist_type):
dist = generateDistribution(dist_type, alpha=0.01)
parameterBuilder = ParameterBuilder()
up = parameterBuilder.defineUncertainParameters()
up.new().isCalled("x1").withDistribution(dist)
return parameterBuilder.andGetResult()
def buildBoreholeParams(dist_type):
boreholeLimits = [[0.05, 0.15], [100, 50000], [63070, 115600], [990, 1110],
[63.1, 116], [700, 820], [1120, 1680], [9855, 12045]]
boreholeParamNames = ["r_w", "r", "T_u", "H_u", "T_l", "H_l", "L", "K_w"]
parameterBuilder = ParameterBuilder()
up = parameterBuilder.defineUncertainParameters()
for k in range(8):
xlim = boreholeLimits[k]
dist = generateDistribution(dist_type, xlim)
up.new().isCalled(boreholeParamNames[k]).withDistribution(dist)
return parameterBuilder.andGetResult()
def multivariate_normal(self, mu=0, cov=None, a=0, b=1):
# set distributions of the input parameters
mu = np.array([mu] * numDims)
if cov is None:
# use standard values
diag = np.eye(numDims) * 0.005
offdiag = np.abs(np.eye(numDims) - 1) * 0.001
cov = diag + offdiag
# estimate the density
builder = ParameterBuilder()
up = builder.defineUncertainParameters()
names = ", ".join(["x%i" for i in range(numDims)])
up.new().isCalled(names).withMultivariateNormalDistribution(
mu, cov, 0, 1)
return builder.andGetResult()
def marginalize(self):
"""
NOTE: just returns the marginalized active subset of params
"""
# marginalize the distribution
margDistList = []
margTransformations = []
activeParams = self.activeParams()
distributions = activeParams.getIndependentJointDistribution(
).getDistributions()
transformations = activeParams.getJointTransformation(
).getTransformations()
if len(distributions) == len(activeParams):
return self
else:
for dist, trans in zip(distributions, transformations):
# check if current parameter is independent
if dist.getDim() == 1:
margDistList.append(dist)
margTransformations.append(trans)
else:
# marginalize the densities and update the transformations
innertrans = trans.getTransformations()
for idim in range(dist.getDim()):
margDist = dist.marginalizeToDimX(idim)
margDistList.append(margDist)
# update transformations
if isinstance(innertrans[idim],
RosenblattTransformation):
margTransformations.append(
RosenblattTransformation(margDist))
else:
a, b = margDist.getBounds()
margTransformations.append(
LinearTransformation(a, b))
assert len(margDistList) == len(
margTransformations) == activeParams.getDim()
# update the parameter setting
from pysgpp.extensions.datadriven.uq.parameters.ParameterBuilder import ParameterBuilder
builder = ParameterBuilder()
up = builder.defineUncertainParameters()
for name, dist, trans in zip(activeParams.getNames(), margDistList,
margTransformations):
up.new().isCalled(name).withDistribution(dist)\
.withTransformation(trans)
return builder.andGetResult()
def refineGrid(self, grid, knowledge, params=None, qoi="_", refinets=[0]):
# check if this method is used in the right context
if grid.getType(
) not in polyGridTypes + bsplineGridTypes + linearGridTypes:
raise AttributeError('Grid type %s is not supported' %
grid.getType())
if params is None:
# define standard uniform params
uncertainParams = ParameterBuilder().defineUncertainParameters()
for idim in range(grid.getStorage().getDimension()):
uncertainParams.new().isCalled(
"x_%i" % idim).withUniformDistribution(0, 1)
params = uncertainParams.andGetResult()
# get refinement candidates
if self.verbose:
print("compute ranking")
B = self.candidates(grid, knowledge, params, qoi, refinets)
# now do the refinement
return self.__refine(grid, B, simulate=False)
def setUpClass(cls):
super(MonteCarloStrategyTest, cls).setUpClass()
builder = ParameterBuilder()
up = builder.defineUncertainParameters()
up.new().isCalled('x1').withUniformDistribution(0, 1)
up.new().isCalled('x2').withUniformDistribution(0, 1)
cls.params = builder.andGetResult()
cls.numDims = cls.params.getStochasticDim()
cls.samples = np.random.random((10000, 1))
cls.grid = Grid.createPolyGrid(cls.numDims, 2)
cls.grid.getGenerator().regular(1)
gs = cls.grid.getStorage()
# interpolate parabola
nodalValues = np.zeros(gs.getSize())
x = DataVector(cls.numDims)
for i in range(gs.getSize()):
gs.getCoordinates(gs.getPoint(i), x)
nodalValues[i] = 16 * (1 - x[0]) * (1 - x[1])
cls.alpha = hierarchize(cls.grid, nodalValues)
from pysgpp.extensions.datadriven.uq.dists.MultivariateNormal import MultivariateNormal
from pysgpp import Grid, DataVector, DataMatrix
from pysgpp.extensions.datadriven.uq.operations import (hierarchize,
evalSGFunctionMulti)
from pysgpp.extensions.datadriven.uq.operations.forcePositivity import OperationMakePositive
from pysgpp.extensions.datadriven.uq.quadrature import doQuadrature
from pysgpp.extensions.datadriven.uq.parameters.ParameterBuilder import ParameterBuilder
from pysgpp.extensions.datadriven.uq.dists.SGDEdist import SGDEdist
mu = np.array([0.5, 0.5])
cov = np.array([[0.1, 0.04], [0.04, 0.1]]) / 5.
dist = MultivariateNormal(mu, cov, 0, 1)
# setup 2d case
builder = ParameterBuilder()
up = builder.defineUncertainParameters()
up.new().isCalled("x").withLognormalDistribution(0.5, 0.2, alpha=0.001)
up.new().isCalled("y").withBetaDistribution(3, 3, 0, 1)
disst = builder.andGetResult().getIndependentJointDistribution()
class myDist(object):
def pdf(self, x):
if x[0] < 0.5 and x[1] < 0.5:
return 1.
elif x[0] > 0.5 and x[1] > 0.5:
return -1.
elif x[0] > 0.5 and x[1] < 0.5:
return 2.
else:
def test_variance_opt(self):
# parameters
level = 4
gridConfig = RegularGridConfiguration()
gridConfig.type_ = GridType_Linear
gridConfig.maxDegree_ = 2 # max(2, level + 1)
gridConfig.boundaryLevel_ = 0
gridConfig.dim_ = 2
# mu = np.ones(gridConfig.dim_) * 0.5
# cov = np.diag(np.ones(gridConfig.dim_) * 0.1 / 10.)
# dist = MultivariateNormal(mu, cov, 0, 1) # problems in 3d/l2
# f = lambda x: dist.pdf(x)
def f(x):
return np.prod(4 * x * (1 - x))
def f(x):
return np.arctan(
50 *
(x[0] - .35)) + np.pi / 2 + 4 * x[1]**3 + np.exp(x[0] * x[1] -
1)
# --------------------------------------------------------------------------
# define parameters
paramsBuilder = ParameterBuilder()
up = paramsBuilder.defineUncertainParameters()
for idim in range(gridConfig.dim_):
up.new().isCalled("x_%i" % idim).withBetaDistribution(3, 3, 0, 1)
params = paramsBuilder.andGetResult()
U = params.getIndependentJointDistribution()
T = params.getJointTransformation()
# --------------------------------------------------------------------------
grid = pysgpp.Grid.createGrid(gridConfig)
gs = grid.getStorage()
grid.getGenerator().regular(level)
nodalValues = np.ndarray(gs.getSize())
p = DataVector(gs.getDimension())
for i in range(gs.getSize()):
gp = gs.getCoordinates(gs.getPoint(i), p)
nodalValues[i] = f(p.array())
# --------------------------------------------------------------------------
alpha_vec = pysgpp.DataVector(nodalValues)
pysgpp.createOperationHierarchisation(grid).doHierarchisation(
alpha_vec)
alpha = alpha_vec.array()
checkInterpolation(grid, alpha, nodalValues, epsilon=1e-13)
# --------------------------------------------------------------------------
quad = AnalyticEstimationStrategy()
mean = quad.mean(grid, alpha, U, T)["value"]
var = quad.var(grid, alpha, U, T, mean)["value"]
if self.verbose:
print("mean: %g" % mean)
print("var : %g" % var)
print("-" * 80)
# drop arbitrary grid points and compute the mean and the variance
# -> just use leaf nodes for simplicity
bilinearForm = BilinearGaussQuadratureStrategy(grid.getType())
bilinearForm.setDistributionAndTransformation(U.getDistributions(),
T.getTransformations())
linearForm = LinearGaussQuadratureStrategy(grid.getType())
linearForm.setDistributionAndTransformation(U.getDistributions(),
T.getTransformations())
i = np.random.randint(0, gs.getSize())
gpi = gs.getPoint(i)
# --------------------------------------------------------------------------
# check refinement criterion
ranking = ExpectationValueOptRanking()
mean_rank = ranking.rank(grid, gpi, alpha, params)
if self.verbose:
print("rank mean: %g" % (mean_rank, ))
# --------------------------------------------------------------------------
# check refinement criterion
ranking = VarianceOptRanking()
var_rank = ranking.rank(grid, gpi, alpha, params)
if self.verbose:
print("rank var: %g" % (var_rank, ))
# --------------------------------------------------------------------------
# remove one grid point and update coefficients
toBeRemoved = IndexList()
toBeRemoved.push_back(i)
ixs = gs.deletePoints(toBeRemoved)
gpsj = []
new_alpha = np.ndarray(gs.getSize())
for j in range(gs.getSize()):
new_alpha[j] = alpha[ixs[j]]
gpsj.append(gs.getPoint(j))
# --------------------------------------------------------------------------
# compute the mean and the variance of the new grid
mean_trunc = quad.mean(grid, new_alpha, U, T)["value"]
var_trunc = quad.var(grid, new_alpha, U, T, mean_trunc)["value"]
basis = getBasis(grid)
# compute the covariance
A, _ = bilinearForm.computeBilinearFormByList(gs, [gpi], basis, gpsj,
basis)
b, _ = linearForm.computeLinearFormByList(gs, gpsj, basis)
mean_uwi_phii = np.dot(new_alpha, A[0, :])
mean_phii, _ = linearForm.getLinearFormEntry(gs, gpi, basis)
mean_uwi = np.dot(new_alpha, b)
cov_uwi_phii = mean_uwi_phii - mean_phii * mean_uwi
# compute the variance of phi_i
firstMoment, _ = linearForm.getLinearFormEntry(gs, gpi, basis)
secondMoment, _ = bilinearForm.getBilinearFormEntry(
gs, gpi, basis, gpi, basis)
var_phii = secondMoment - firstMoment**2
# update the ranking
var_estimated = var_trunc + alpha[i]**2 * var_phii + 2 * alpha[
i] * cov_uwi_phii
mean_diff = np.abs(mean_trunc - mean)
var_diff = np.abs(var_trunc - var)
if self.verbose:
print("-" * 80)
print("diff: |var - var_estimated| = %g" %
(np.abs(var - var_estimated), ))
print("diff: |var - var_trunc| = %g = %g = var opt ranking" %
(var_diff, var_rank))
print("diff: |mean - mean_trunc| = %g = %g = mean opt ranking" %
(mean_diff, mean_rank))
self.assertTrue(np.abs(var - var_estimated) < 1e-14)
self.assertTrue(np.abs(mean_diff - mean_rank) < 1e-14)
self.assertTrue(np.abs(var_diff - var_rank) < 1e-14)
def tesst_squared(self):
# parameters
level = 3
gridConfig = RegularGridConfiguration()
gridConfig.type_ = GridType_Linear
gridConfig.maxDegree_ = 2 # max(2, level + 1)
gridConfig.boundaryLevel_ = 0
gridConfig.dim_ = 2
def f(x):
return np.prod(8 * x * (1 - x))
# --------------------------------------------------------------------------
# define parameters
paramsBuilder = ParameterBuilder()
up = paramsBuilder.defineUncertainParameters()
for idim in range(gridConfig.dim_):
up.new().isCalled("x_%i" % idim).withUniformDistribution(0, 1)
params = paramsBuilder.andGetResult()
U = params.getIndependentJointDistribution()
T = params.getJointTransformation()
# --------------------------------------------------------------------------
grid = pysgpp.Grid.createGrid(gridConfig)
gs = grid.getStorage()
grid.getGenerator().regular(level)
nodalValues = np.ndarray(gs.getSize())
weightedNodalValues = np.ndarray(gs.getSize())
p = DataVector(gs.getDimension())
for i in range(gs.getSize()):
gp = gs.getCoordinates(gs.getPoint(i), p)
nodalValues[i] = f(p.array())**2
weightedNodalValues[i] = f(p.array())**2 * U.pdf(
T.unitToProbabilistic(p))
# --------------------------------------------------------------------------
alpha_vec = pysgpp.DataVector(nodalValues)
pysgpp.createOperationHierarchisation(grid).doHierarchisation(
alpha_vec)
alpha = alpha_vec.array()
checkInterpolation(grid, alpha, nodalValues, epsilon=1e-13)
# --------------------------------------------------------------------------
alpha_vec = pysgpp.DataVector(weightedNodalValues)
pysgpp.createOperationHierarchisation(grid).doHierarchisation(
alpha_vec)
weightedAlpha = alpha_vec.array()
checkInterpolation(grid,
weightedAlpha,
weightedNodalValues,
epsilon=1e-13)
# --------------------------------------------------------------------------
# np.random.seed(1234567)
i = np.random.randint(0, gs.getSize())
gpi = gs.getPoint(i)
gs.getCoordinates(gpi, p)
print(evalSGFunction(grid, alpha, p.array()))
print(evalSGFunctionBasedOnParents(grid, alpha, gpi))
# --------------------------------------------------------------------------
# check refinement criterion
ranking = SquaredSurplusRanking()
squared_surplus_rank = ranking.rank(grid, gpi, weightedAlpha, params)
if self.verbose:
print("rank squared surplus: %g" % (squared_surplus_rank, ))
# --------------------------------------------------------------------------
# check refinement criterion
ranking = AnchoredMeanSquaredOptRanking()
anchored_mean_squared_rank = ranking.rank(grid, gpi, alpha, params)
if self.verbose:
print("rank mean squared : %g" % (anchored_mean_squared_rank, ))
def test_anchored_variance_opt(self):
# parameters
level = 4
gridConfig = RegularGridConfiguration()
gridConfig.type_ = GridType_Linear
gridConfig.maxDegree_ = 2 # max(2, level + 1)
gridConfig.boundaryLevel_ = 0
gridConfig.dim_ = 2
# mu = np.ones(gridConfig.dim_) * 0.5
# cov = np.diag(np.ones(gridConfig.dim_) * 0.1 / 10.)
# dist = MultivariateNormal(mu, cov, 0, 1) # problems in 3d/l2
# f = lambda x: dist.pdf(x)
def f(x):
return np.prod(4 * x * (1 - x))
def f(x):
return np.arctan(
50 *
(x[0] - .35)) + np.pi / 2 + 4 * x[1]**3 + np.exp(x[0] * x[1] -
1)
# --------------------------------------------------------------------------
# define parameters
paramsBuilder = ParameterBuilder()
up = paramsBuilder.defineUncertainParameters()
for idim in range(gridConfig.dim_):
up.new().isCalled("x_%i" % idim).withBetaDistribution(3, 3, 0, 1)
params = paramsBuilder.andGetResult()
U = params.getIndependentJointDistribution()
T = params.getJointTransformation()
# --------------------------------------------------------------------------
grid = pysgpp.Grid.createGrid(gridConfig)
gs = grid.getStorage()
grid.getGenerator().regular(level)
nodalValues = np.ndarray(gs.getSize())
p = DataVector(gs.getDimension())
for i in range(gs.getSize()):
gp = gs.getCoordinates(gs.getPoint(i), p)
nodalValues[i] = f(p.array())
# --------------------------------------------------------------------------
alpha_vec = pysgpp.DataVector(nodalValues)
pysgpp.createOperationHierarchisation(grid).doHierarchisation(
alpha_vec)
alpha = alpha_vec.array()
checkInterpolation(grid, alpha, nodalValues, epsilon=1e-13)
# --------------------------------------------------------------------------
i = np.random.randint(0, gs.getSize())
gpi = gs.getPoint(i)
# --------------------------------------------------------------------------
# check refinement criterion
ranking = AnchoredVarianceOptRanking()
var_rank = ranking.rank(grid, gpi, alpha, params)
if self.verbose:
print("rank anchored var: %g" % (var_rank, ))
# --------------------------------------------------------------------------
# compute the mean and the variance of the new grid
x = DataVector(gs.getDimension())
gs.getCoordinates(gpi, x)
x = x.array()
uwxi = evalSGFunction(grid, alpha, x) - alpha[i]
fx = U.pdf(T.unitToProbabilistic(x))
var_rank_estimated = np.abs(
(fx - fx**2) * (-alpha[i]**2 - 2 * alpha[i] * uwxi))
if self.verbose:
print("rank anchored var: %g" % (var_rank_estimated, ))
if self.verbose:
print("-" * 80)
print("diff: |var - var_estimated| = %g" %
(np.abs(var_rank - var_rank_estimated), ))
# Copyright (C) 2008-today The SG++ project
# This file is part of the SG++ project. For conditions of distribution and
# use, please see the copyright notice provided with SG++ or at
# sgpp.sparsegrids.org
import matplotlib.pyplot as plt
import numpy as np
from scipy import fftpack
from pysgpp.extensions.datadriven.uq.parameters.ParameterBuilder import ParameterBuilder
from pysgpp.extensions.datadriven.uq.plot.plot1d import plotFunction1d
builder = ParameterBuilder()
up = builder.defineUncertainParameters()
up.new().isCalled('x').withUniformDistribution(-np.pi, np.pi)
params = builder.andGetResult()
def f(x):
return np.arctan(x) # np.arctan(50 * (x - .35)) # + np.exp(x - 1)
class FourierSeries(object):
def __init__(self, x, y, x_steps=256):
self.n = y.size // 2
ut = np.fft.rfft(y) * np.array([(1. / (2. * self.n))] +
[(1. / self.n)
for i in range(1, self.n)] +
[(1. / (2. * self.n))])
self.f, self.g = (ut.real, -ut.imag[1:-1])
