def __init__(self, chaosPOD, dim, defectSizes, detection, simulationSize):
super(PODaggrChaos, self).__init__(dim, defectSizes.shape[0])
self.chaosPOD = chaosPOD
self.dim = dim
self.defectSizes = defectSizes
self.defectNumber = len(defectSizes)
self.simulationSize = simulationSize
self.detection = detection
# get the sample of coefficient using the coef distribution
# used to compute the POD for a given point
sampleCoefs = chaosPOD.getCoefficientDistribution().getSample(
simulationSize)
# get some result from the polynomial chaos to build a vectoriel
# chaos function that return the signal values for all chaos with
# different coefficients for one specific point
chaosResult = chaosPOD.getPolynomialChaosResult()
reducedBasis = chaosResult.getReducedBasis()
transformation = chaosResult.getTransformation()
chaosFunctionCol = []
for i, coefs in enumerate(sampleCoefs):
standardChaosFunction = ot.LinearCombinationFunction(
reducedBasis, coefs)
chaosFunctionCol.append(
ot.ComposedFunction(standardChaosFunction, transformation))
self.chaosFunction = ot.AggregatedFunction(chaosFunctionCol)
#
# .. math::
# f = (f_1, \dots, f_n)
#
# %%
from __future__ import print_function
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
import math as m
ot.Log.Show(ot.Log.NONE)
# %%
# assume a list of functions to aggregate
functions = list()
functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'],
['x1^2 + x2', 'x1 + x2 + x3']))
functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'],
['x1 + 2 * x2 + x3', 'x1 + x2 - x3']))
# %%
# create the aggregated function
function = ot.AggregatedFunction(functions)
# %%
# evaluate the function
x = [1.0, 2.0, 3.0]
y = function(x)
print('x=', x, 'y=', y)
lin31 = func3.isLinearlyDependent(1)
assert (not lin3)
assert (not lin30)
assert (lin31)
# 4. 2D non linear function
func4 = ot.SymbolicFunction(['x1', 'x2'], ['3*x1*x2'])
lin4 = func4.isLinear()
lin40 = func4.isLinearlyDependent(0)
lin41 = func4.isLinearlyDependent(1)
assert (not lin4)
assert (lin40)
assert (lin41)
# 5. Aggregated function
func5 = ot.AggregatedFunction([func2, func4])
lin5 = func5.isLinear()
lin50 = func5.isLinearlyDependent(0)
lin51 = func5.isLinearlyDependent(1)
assert (not lin5)
assert (lin50)
assert (lin51)
# 6. Python function
def pyFunc6(x):
x1, x2 = x
result = [3 * x1 - 2 * x2 + 5, -x1 + x2, 3 * x1 * x2]
return result
import openturns as ot
ot.TESTPREAMBLE()
dim = 2
x = [2.0 + i for i in range(dim)]
print("x=", x)
factory = ot.ConstantBasisFactory(dim)
print("factory=", factory)
basis = factory.build()
print("basis=", basis)
f = ot.AggregatedFunction(basis)
y = f(x)
print("y=", y)
factory = ot.LinearBasisFactory(dim)
print("factory=", factory)
basis = factory.build()
print("basis=", basis)
f = ot.AggregatedFunction(basis)
y = f(x)
print("y=", y)
factory = ot.QuadraticBasisFactory(dim)
print("factory=", factory)
basis = factory.build()
def run(self):
"""
Build the POD models.
Notes
-----
This method build the polynomial chaos model. First the censored data
are filtered if needed. The Box Cox transformation is performed if it is
enabled. Then it builds the POD models, the Monte Carlo simulation is
performed for each given defect sizes. The confidence interval is
computed by simulating new coefficients of the polynomial chaos, then
Monte Carlo simulations are performed.
"""
# run the chaos algorithm and get result if not given
if not self._userChaos:
if self._verbose:
print('Start build polynomial chaos model...')
self._algoChaos = self._buildChaosAlgo(self._input, self._signals)
self._algoChaos.run()
if self._verbose:
print('Polynomial chaos model completed')
self._chaosResult = self._algoChaos.getResult()
# get the metamodel
self._chaosPred = self._chaosResult.getMetaModel()
# get the basis, coef and transformation, needed for the confidence interval
self._chaosCoefs = self._chaosResult.getCoefficients()
self._reducedBasis = self._chaosResult.getReducedBasis()
self._transformation = self._chaosResult.getTransformation()
self._basisFunction = ot.ComposedFunction(ot.AggregatedFunction(
self._reducedBasis), self._transformation)
# compute the residuals and stderr
inputSize = self._input.getSize()
basisSize = self._reducedBasis.getSize()
self._residuals = self._signals - self._chaosPred(self._input) # residuals
self._stderr = np.sqrt(np.sum(np.array(self._residuals)**2) / (inputSize - basisSize - 1))
# Check the quality of the chaos model
R2 = self.getR2()
Q2 = self.getQ2()
if self._verbose:
print('Polynomial chaos validation R2 (>0.8) : {:0.4f}'.format(R2))
print('Polynomial chaos validation Q2 (>0.8) : {:0.4f}'.format(Q2))
# Compute the POD values for each defect sizes
self.POD = self._computePOD(self._defectSizes, self._chaosCoefs)
# create the interpolate function
interpModel = interp1d(self._defectSizes, self.POD, kind='linear')
self._PODmodel = ot.PythonFunction(1, 1, interpModel)
####################### confidence interval ############################
dof = inputSize - basisSize - 1
varEpsilon = (ot.ChiSquare(dof).inverse() * dof * self._stderr**2).getRealization()[0]
gramBasis = ot.Matrix(self._basisFunction(self._input)).computeGram()
covMatrix = gramBasis.solveLinearSystem(ot.IdentityMatrix(basisSize)) * varEpsilon
self._coefsDist = ot.Normal(np.hstack(self._chaosCoefs), ot.CovarianceMatrix(covMatrix.getImplementation()))
coefsRandom = self._coefsDist.getSample(self._simulationSize)
self._PODPerDefect = ot.Sample(self._simulationSize, self._defectNumber)
for i, coefs in enumerate(coefsRandom):
self._PODPerDefect[i, :] = self._computePOD(self._defectSizes, coefs)
if self._verbose:
updateProgress(i, self._simulationSize, 'Computing POD per defect')