class WeightedL2BFRanking(Ranking):
def __init__(self):
super(self.__class__, self).__init__()
self._estimationStrategy = AnalyticEstimationStrategy()
self.initialized = False
def update(self, grid, v, gpi, params, *args, **kws):
"""
Compute ranking for variance estimation
\argmax_{i \in \A} |v_i| \sqrt{E[\varphi_i^2]}
@param grid: Grid grid
@param v: numpy array coefficients
"""
# update the quadrature operations
if not self.initialized:
self._estimationStrategy.initQuadratureStrategy(grid)
U = params.getIndependentJointDistribution()
T = params.getJointTransformation()
self.vol, self.W, self.D = self._estimationStrategy._extractPDFforMomentEstimation(U, T)
self.initialized = True
# get maximum coefficient of ancestors
vi = 0.0
gs = grid.getStorage()
for _, gpp in parents(grid, gpi):
if gs.isContaining(gpp):
ix = gs.getSequenceNumber(gpp)
vi = max(vi, np.abs(v[ix]))
# prepare data
basisi = getBasis(grid)
# compute the second moment for the current grid point
secondMoment, _ = self._estimationStrategy.computeSystemMatrixForVarianceList(gs,
[gpi], basisi,
[gpi], basisi,
self.W, self.D)
secondMoment = max(0.0, self.vol * secondMoment[0])
# update the ranking
return np.abs(vi * np.sqrt(secondMoment))
class ExpectationValueOptRanking(Ranking):
def __init__(self):
super(self.__class__, self).__init__()
self._estimationStrategy = AnalyticEstimationStrategy()
self.initialized = False
self._linearForm = LinearGaussQuadratureStrategy()
def compute_mean_phii(self, gs, gpi, basisi):
mean_phii, _ = self._estimationStrategy.computeSystemMatrixForMeanList(
gs, [gpi], basisi, self.W, self.D)
return mean_phii[0]
def update(self, grid, v, gpi, params, *args, **kws):
"""
Compute ranking for variance estimation
\argmax_{i \in \A} | |v_i| E(\varphi_i)
@param grid: Grid grid
@param v: numpy array coefficients
@param admissibleSet: AdmissibleSet
"""
# update the quadrature operations
if not self.initialized:
self._estimationStrategy.initQuadratureStrategy(grid)
U = params.getIndependentJointDistribution()
T = params.getJointTransformation()
self.vol, self.W, self.D = self._estimationStrategy._extractPDFforMomentEstimation(
U, T)
self.initialized = True
# prepare data
gs = grid.getStorage()
# compute stiffness matrix for next run
basis = getBasis(grid)
ix = gs.getSequenceNumber(gpi)
mean_phii = self.compute_mean_phii(gs, gpi, basis)
return np.abs(v[ix]) * mean_phii
def testMonteCarlo(self):
# initialize pdf and transformation
U = self.params.getIndependentJointDistribution()
T = self.params.getJointTransformation()
strategy = MonteCarloStrategy(samples=self.samples, ixs=[0])
mean_mc = strategy.mean(self.grid, self.alpha, U, T)["value"]
var_mc = strategy.var(self.grid, self.alpha, U, T, mean_mc)["value"]
strategy = AnalyticEstimationStrategy()
mean_ana = strategy.mean(self.grid, self.alpha, U, T)["value"]
var_ana = strategy.var(self.grid, self.alpha, U, T, mean_ana)["value"]
assert np.abs(mean_mc - mean_ana) < 1e-1
assert np.abs(var_mc - var_ana) < 1e-1
def __init__(self):
super(self.__class__, self).__init__()
self._estimationStrategy = AnalyticEstimationStrategy()
self.initialized = False
class VarianceOptRanking(Ranking):
def __init__(self):
super(self.__class__, self).__init__()
self._estimationStrategy = AnalyticEstimationStrategy()
self.initialized = False
def compute_mean_uwi_phii(self, gs, gpswi, w, gpi, basis):
A, _ = self._estimationStrategy.computeSystemMatrixForVarianceList(
gs, gpswi, basis, [gpi], basis, self.W, self.D)
return np.dot(w, A)
def compute_mean_uwi(self, gs, gpswi, w, basis):
b, _ = self._estimationStrategy.computeSystemMatrixForMeanList(
gs, gpswi, basis, self.W, self.D)
return np.dot(w, b)
def compute_mean_phii(self, gs, gpi, basisi):
mean_phii, _ = self._estimationStrategy.computeSystemMatrixForMeanList(
gs, [gpi], basisi, self.W, self.D)
return mean_phii[0]
def compute_var_phii(self, gs, gpi, basisi, mean_phii):
A_var, _ = self._estimationStrategy.computeSystemMatrixForVarianceList(
gs, [gpi], basisi, [gpi], basisi, self.W, self.D)
return A_var[0, 0] - mean_phii**2
def update(self, grid, v, gpi, params, *args, **kws):
"""
Compute ranking for variance estimation
\argmax_{i \in \A} | -v_i^2 V(\varphi_i) - 2 v_i Cov(u_indwli \varphi_i)|
@param grid: Grid grid
@param v: numpy array coefficients
@param admissibleSet: AdmissibleSet
"""
# update the quadrature operations
if not self.initialized:
self._estimationStrategy.initQuadratureStrategy(grid)
U = params.getIndependentJointDistribution()
T = params.getJointTransformation()
self.vol, self.W, self.D = self._estimationStrategy._extractPDFforMomentEstimation(
U, T)
self.initialized = True
# prepare data
gs = grid.getStorage()
basis = getBasis(grid)
# load coefficients and vectors and matrices for variance and mean
# estimation
w = np.ndarray(gs.getSize() - 1)
ix = gs.getSequenceNumber(gpi)
# prepare list of grid points
gpswi = []
jx = 0
for j in range(gs.getSize()):
gpj = gs.getPoint(j)
if gpi.getHash() != gpj.getHash():
gpswi.append(gpj)
w[jx] = v[j]
jx += 1
# compute the covariance
mean_uwi_phii = self.compute_mean_uwi_phii(gs, gpswi, w, gpi, basis)
mean_phii = self.compute_mean_phii(gs, gpi, basis)
mean_uwi = self.compute_mean_uwi(gs, gpswi, w, basis)
cov_uwi_phii = mean_uwi_phii - mean_phii * mean_uwi
# compute the variance of phi_i
var_phii = self.compute_var_phii(gs, gpi, basis, mean_phii)
# update the ranking
return np.abs(-v[ix]**2 * var_phii - 2 * v[ix] * cov_uwi_phii)
def __init__(self):
super(self.__class__, self).__init__()
self._estimationStrategy = AnalyticEstimationStrategy()
self.initialized = False
self._linearForm = LinearGaussQuadratureStrategy()
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)
