def mean(self, grid, alpha, U, T):
r"""
Extraction of the expectation the given sparse grid function
interpolating the product of function value and pdf.
\int\limits_{[0, 1]^d} f_N(x) * pdf(x) dx
"""
# extract correct pdf for moment estimation
vol, W = self._extractPDFforMomentEstimation(U, T)
D = T.getTransformations()
# compute the integral of the product
gs = grid.getStorage()
acc = DataVector(gs.size())
acc.setAll(1.)
tmp = DataVector(gs.size())
err = 0
# run over all dimensions
for i, dims in enumerate(W.getTupleIndices()):
dist = W[i]
trans = D[i]
# get the objects needed for integration the current dimensions
gpsi, basisi = project(grid, dims)
if isinstance(dist, SGDEdist):
# if the distribution is given as a sparse grid function we
# need to compute the bilinear form of the grids
# accumulate objects needed for computing the bilinear form
gpsj, basisj = project(dist.grid, range(len(dims)))
# compute the bilinear form
bf = BilinearGaussQuadratureStrategy()
A, erri = bf.computeBilinearFormByList(gpsi, basisi,
gpsj, basisj)
# weight it with the coefficient of the density function
self.mult(A, dist.alpha, tmp)
else:
# the distribution is given analytically, handle them
# analytically in the integration of the basis functions
if isinstance(dist, Dist) and len(dims) > 1:
raise AttributeError('analytic quadrature not supported for multivariate distributions')
if isinstance(dist, Dist):
dist = [dist]
trans = [trans]
lf = LinearGaussQuadratureStrategy(dist, trans)
tmp, erri = lf.computeLinearFormByList(gpsi, basisi)
# print error stats
# print "%s: %g -> %g" % (str(dims), err, err + D[i].vol() * erri)
# import ipdb; ipdb.set_trace()
# accumulate the error
err += D[i].vol() * erri
# accumulate the result
acc.componentwise_mult(tmp)
moment = alpha.dotProduct(acc)
return vol * moment, err
def initQuadratureStrategy(self, grid):
if self.linearForm is None:
self.linearForm = LinearGaussQuadratureStrategy(grid.getType())
if self.bilinearForm is None:
self.bilinearForm = BilinearGaussQuadratureStrategy(grid.getType())
if self.trilinearForm is None:
self.trilinearForm = TrilinearGaussQuadratureStrategy(grid.getType())
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)
class AnalyticEstimationStrategy(SparseGridEstimationStrategy):
def __init__(self):
super(self.__class__, self).__init__()
# system matrices for mean and mean^2
self.A_mean = {}
self.A_var = {}
self.linearForm = None
self.bilinearForm = None
self.trilinearForm = None
def initQuadratureStrategy(self, grid):
if self.linearForm is None:
self.linearForm = LinearGaussQuadratureStrategy(grid.getType())
if self.bilinearForm is None:
self.bilinearForm = BilinearGaussQuadratureStrategy(grid.getType())
if self.trilinearForm is None:
self.trilinearForm = TrilinearGaussQuadratureStrategy(
grid.getType())
def getSystemMatrixForMean(self, grid, W, D):
self.initQuadratureStrategy(grid)
hash_value = (grid.hash_hexdigest(), hash(tuple(W)), hash(tuple(D)))
if hash_value not in self.A_var:
self.A_mean[hash_value] = self.computeSystemMatrixForMean(
grid, W, D)
return self.A_mean[hash_value]
def computeSystemMatrixForMeanProjected(self, gs, gpsi, basisi, dist,
trans, dims):
if isinstance(dist, SGDEdist):
# if the distribution is given as a sparse grid function we
# need to compute the bilinear form of the grids
# accumulate objects needed for computing the bilinear form
assert len(dims) == dist.grid.getStorage().getDimension()
gpsj, basisj = project(dist.grid, list(range(len(dims))))
# compute the bilinear form
# -> the measure needs to be uniform, since it is already
# encoded in the sparse grid density
self.bilinearForm.setDistributionAndTransformation(
[Uniform(0, 1)] * gs.getDimension(), None)
A, erri = self.bilinearForm.computeBilinearFormByList(
gs, gpsi, basisi, gpsj, basisj)
# weight it with the coefficient of the density function
tmp = A.dot(dist.alpha)
else:
# the distribution is given analytically, handle them
# analytically in the integration of the basis functions
if isinstance(dist, Dist) and len(dims) > 1:
# print "WARNINING: results are just approximated -> not independent random variables"
# marginalize the densities and continue
marg_dists = [None] * len(dims)
for i, idim in enumerate(dims):
marg_dists[i] = dist.marginalizeToDimX(idim)
dist = marg_dists
trans = trans.getTransformations()
if isinstance(dist, Dist):
dist = [dist]
trans = [trans]
self.linearForm.setDistributionAndTransformation(dist, trans)
tmp, erri = self.linearForm.computeLinearFormByList(
gs, gpsi, basisi)
return tmp, erri
def computeSystemMatrixForMeanList(self, gs, gps, basis, W, D):
# compute the integral of the product
A_mean = np.ones(len(gps))
err = 0
# run over all dimensions
for i, dims in enumerate(W.getTupleIndices()):
dist = W[i]
trans = D[i]
# get the objects needed for integration the current dimensions
gps_projected = projectList(gps, dims)
A_idim, erri = self.computeSystemMatrixForMeanProjected(
gs, gps_projected, basis, dist, trans, dims)
# print error stats
# print "%s: %g -> %g" % (str(dims), err, err + D[i].vol() * erri)
# import ipdb; ipdb.set_trace()
# accumulate the error
err += erri
# accumulate the result
A_mean *= A_idim
return A_mean, err
def computeSystemMatrixForMean(self, grid, W, D):
# compute the integral of the product
gs = grid.getStorage()
gps = [None] * gs.getSize()
for i in range(gs.getSize()):
gps[i] = gs.getPoint(i)
basis = getBasis(grid)
return self.computeSystemMatrixForMeanList(gs, gps, basis, W, D)
def getSystemMatrixForVariance(self, grid, W, D):
self.initQuadratureStrategy(grid)
hash_value = (grid.hash_hexdigest(), hash(tuple(W)), hash(tuple(D)))
if hash_value not in self.A_var:
self.A_var[hash_value] = self.computeSystemMatrixForVariance(
grid, W, D)
return self.A_var[hash_value]
def computeSystemMatrixForVarianceProjected(self, gs, gpsi, basisi, gpsj,
basisj, dist, trans, dims):
if isinstance(dist, SGDEdist):
# project distribution on desired dimensions
# get the objects needed for integrating
# the current dimensions
gpsk, basisk = project(dist.grid, list(range(len(dims))))
# compute the bilinear form
# -> the measure needs to be uniform, since it is already
# encoded in the sparse grid density
self.trilinearForm.setDistributionAndTransformation(
[Uniform(0, 1)] * gs.getDimension(), None)
A_idim, erri = self.trilinearForm.computeTrilinearFormByList(
gs, gpsk, basisk, dist.alpha, gpsi, basisi, gpsj, basisj)
else:
# the distribution is given analytically, handle them
# analytically in the integration of the basis functions
if isinstance(dist, Dist) and len(dims) > 1:
# print "WARNINING: results are just approximated -> not independent random variables"
# marginalize the densities and continue
marg_dists = [None] * len(dims)
for i, idim in enumerate(dims):
marg_dists[i] = dist.marginalizeToDimX(idim)
dist = marg_dists
trans = trans.getTransformations()
if isinstance(dist, Dist):
dist = [dist]
trans = [trans]
self.bilinearForm.setDistributionAndTransformation(dist, trans)
A_idim, erri = self.bilinearForm.computeBilinearFormByList(
gs, gpsi, basisi, gpsj, basisj)
return A_idim, erri
def computeSystemMatrixForVarianceList(self, gs, gpsi, basisi, gpsj,
basisj, W, D):
# compute the integral of the product times the pdf
A_var = np.ones((len(gpsi), len(gpsj)))
err = 0
for i, dims in enumerate(W.getTupleIndices()):
dist = W[i]
trans = D[i]
# get the objects needed for integrating
# the current dimensions
gpsi_projected = projectList(gpsi, dims)
gpsj_projected = projectList(gpsj, dims)
A_idim, erri = self.computeSystemMatrixForVarianceProjected(
gs, gpsi_projected, basisi, gpsj_projected, basisj, dist,
trans, dims)
# accumulate the results
A_var *= A_idim
# accumulate the error
err += np.mean(A_var) * erri
return A_var, err
def computeSystemMatrixForVariance(self, grid, W, D):
# compute the integral of the product
gs = grid.getStorage()
gps = [None] * gs.getSize()
for i in range(gs.getSize()):
gps[i] = gs.getPoint(i)
basis = getBasis(grid)
return self.computeSystemMatrixForVarianceList(gs, gps, basis, gps,
basis, W, D)
def mean(self, grid, alpha, U, T):
r"""
Extraction of the expectation the given sparse grid function
interpolating the product of function value and pdf.
\int\limits_{[0, 1]^d} f_N(x) * pdf(x) dx
"""
# extract correct pdf for moment estimation
vol, W, D = self._extractPDFforMomentEstimation(U, T)
A_mean, err = self.getSystemMatrixForMean(grid, W, D)
moment = vol * np.dot(alpha, A_mean)
return {
"value": moment,
"err": err,
"confidence_interval": (moment, moment)
}
def secondMoment(self, grid, alpha, U, T):
r"""
Extraction of the second moment of the given sparse grid function
interpolating the product of function value and pdf.
\int\limits_{[0, 1]^d} f(x)^2 * pdf(x) dx
"""
# extract correct pdf for moment estimation
vol, W, D = self._extractPDFforMomentEstimation(U, T)
A_var, err = self.getSystemMatrixForVariance(grid, W, D)
moment = vol * np.dot(alpha, np.dot(A_var, alpha))
return {
"value": moment,
"err": err,
"confidence_interval": (moment, moment)
}
def var(self, grid, alpha, U, T, mean):
r"""
Extraction of variance of the given sparse grid function
interpolating the product of function value and pdf.
\int\limits_{[0, 1]^d} (f(x) - E(f))^2 * pdf(x) dx
"""
moment = self.secondMoment(grid, alpha, U, T)
var = moment["value"] - mean**2
return {
"value": var,
"err": moment["err"],
"confidence_interval": (var, var)
}
def mean(self, grid, alpha, U, T):
r"""
Extraction of the expectation the given sparse grid function
interpolating the product of function value and pdf.
\int\limits_{[0, 1]^d} f_N(x) * pdf(x) dx
"""
# extract correct pdf for moment estimation
vol, W = self._extractPDFforMomentEstimation(U, T)
D = T.getTransformations()
# compute the integral of the product
gs = grid.getStorage()
acc = DataVector(gs.size())
acc.setAll(1.)
tmp = DataVector(gs.size())
err = 0
# run over all dimensions
for i, dims in enumerate(W.getTupleIndices()):
dist = W[i]
trans = D[i]
# get the objects needed for integration the current dimensions
gpsi, basisi = project(grid, dims)
if isinstance(dist, SGDEdist):
# if the distribution is given as a sparse grid function we
# need to compute the bilinear form of the grids
# accumulate objects needed for computing the bilinear form
gpsj, basisj = project(dist.grid, range(len(dims)))
# compute the bilinear form
bf = BilinearGaussQuadratureStrategy()
A, erri = bf.computeBilinearFormByList(gpsi, basisi, gpsj,
basisj)
# weight it with the coefficient of the density function
self.mult(A, dist.alpha, tmp)
else:
# the distribution is given analytically, handle them
# analytically in the integration of the basis functions
if isinstance(dist, Dist) and len(dims) > 1:
raise AttributeError(
'analytic quadrature not supported for multivariate distributions'
)
if isinstance(dist, Dist):
dist = [dist]
trans = [trans]
lf = LinearGaussQuadratureStrategy(dist, trans)
tmp, erri = lf.computeLinearFormByList(gpsi, basisi)
# print error stats
# print "%s: %g -> %g" % (str(dims), err, err + D[i].vol() * erri)
# import ipdb; ipdb.set_trace()
# accumulate the error
err += D[i].vol() * erri
# accumulate the result
acc.componentwise_mult(tmp)
moment = alpha.dotProduct(acc)
return vol * moment, err