def errorEstimationTerr(cdf_value, *args):
"""
Return total error given moments/variances
"""
assembledEstimationsList = packedList(args)
return assembledEstimationsList[0] + normalInverseCDF(
cdf_value[0]) * np.sqrt(assembledEstimationsList[1])
def minimalCostStatisticalError(inputDict, newLevels):
"""Compute sample numbers to satisfy tolerance on statistical error with minimal cost."""
# Variances
variances = inputDict["blendedVariances"]
if variances is None:
# Variance was not blended
hierarchy = inputDict["oldHierarchy"]
if inputDict["parametersForModel"] is not None:
# We have a valid model
variance_params = inputDict["parametersForModel"][1]
variance_model = inputDict["models"][1]
variances = [variance_model(variance_params, l) for l in newLevels]
# Use sample estimation for first level
variances[0] = inputDict["estimations"][-1][0] * hierarchy[0][1]
else:
# No model: fall back to sample estimations
variances = [v * n[1] for v, n in zip(inputDict["estimations"][-1], hierarchy)]
# Costs
cost_parameters = inputDict["costParameters"]
if cost_parameters is not None:
# We have a valid model
cost_model = inputDict["costModel"]
costs = [cost_model(cost_parameters, l) for l in newLevels]
# Use sample estimation for first level
costs[0] = inputDict["costEstimations"][0]
else:
# No model: fall back to sample estimations
costs = inputDict["costEstimations"]
# Compute this term - sum_{k=0}^L sqrt(C_k V_k) for later use
cost_variance_sum = 0.0
for i, cost in enumerate(costs):
cost_variance_sum += np.sqrt(cost * variances[i])
# Factor for sample numbers, independent of level
cdf_value = normalInverseCDF(inputDict["errorParameters"][0][0])
tolerance = inputDict["tolerances"][0] * inputDict["splittingParameter"]
constantFactor = (cdf_value / tolerance) ** 2 * cost_variance_sum
# Compute number of samples
new_samples = []
for i in range(len(newLevels)):
if i > len(costs) - 1:
# No data for this level. Set sample number to zero.
# The hierarchy optimiser enforces the minimal number of samples per level
new_samples.append(0)
continue
variance_level = variances[i]
new_sample_nb = constantFactor * np.sqrt(variance_level / costs[i])
new_samples.append(int(np.ceil(new_sample_nb)))
return new_samples
def errorEstimationStatError(cdfValue, globalEstimations):
"""
Accept the summation over the variances divided by number of samples over all
levels and return its square root as the statistical error
"""
if cdfValue is None:
cdfValue = 1 # default behavior
else:
cdfValue = normalInverseCDF(
cdfValue[0]) # [Pisaroni et al.,CMLMC,pag.25]
error = cdfValue * np.sqrt(globalEstimations[0])
# Ensure no NumPy type
return float(error)
def test_normalInverseCDF(self):
correct_inverse_cdf_values = [
-2.3263478740408408, -1.180559456612439, -0.7461851862161866,
-0.4215776353171568, -0.13689839042801627, 0.13689839042801613,
0.4215776353171568, 0.7461851862161862, 1.180559456612439,
2.3263478740408408
]
cdf_values = np.linspace(0.01, 0.99, num=10)
inverse_cdf_values = [
tools.normalInverseCDF(value) for value in cdf_values
]
for i in range(0, len(inverse_cdf_values)):
self.assertEqual(inverse_cdf_values[i],
correct_inverse_cdf_values[i])
def errorEstimationStatError(ignore, *args):
"""
Accept the summation over the variances divided by number of samples over all
levels and return its square root as the statistical error
"""
assembledEstimationsList = packedList(args)
# TODO - Think of better place for this assertion
assert (
len(assembledEstimationsList) == 1
), "length of assembledEstimationsList passed to errorEstimationStatError is not 1"
if ignore is None:
cdf_value = 1 # default behavior
else:
cdf_value = normalInverseCDF(
ignore[0]) # [Pisaroni et al.,CMLMC,pag.25]
return cdf_value * np.sqrt(assembledEstimationsList[0])
def setUp(self):
self.variableDimension = int(self._randomGenerator.uniform(1, 11, 1))
self.mean = self._randomGenerator.uniform(-10, 10,
self.variableDimension)
self.variance = self._randomGenerator.uniform(0.1, 0.5,
self.variableDimension)
# Set number of samples
# Accepted probability that estimated expectation does not satisfy tolerance
failureProbability = 10**-4
requiredStD = normalInverseCDF(1 -
failureProbability / 2) / self.tolerance
# Choose number of samples to get this failure probability
self.numberOfSamples = int(np.max(self.variance * requiredStD**2))
# Safety factor for higher-order moments
self.numberOfSamples = self.numberOfSamples**2
# Safety bounds
self.numberOfSamples = max(10**3, min(10**6, self.numberOfSamples))