def trainData():
"""
Generate training data
"""
qBound = [[-1, 1], [-1, 1]]
x1_ = sampling.trainSample(sampleType='GQ',
GQdistType='Unif',
qInfo=qBound[0],
nSamp=4)
x2_ = sampling.trainSample(sampleType='GQ',
GQdistType='Unif',
qInfo=qBound[1],
nSamp=4)
xTrain = reshaper.vecs2grid([x1_.q, x2_.q])
yTrain_mean = np.asarray([
-0.0169906, -0.0191095, -0.0167435, -0.0172338, -0.0203195,
-0.020089, -0.0184691, -0.0188843, -0.0164581, -0.0200013,
-0.0186512, -0.0159343, -0.0185975, -0.0155899, -0.0178921,
-0.018329
])
yTrain_sdev = np.asarray([
0.00131249, 0.00104324, 0.00085491, 0.00099751, 0.00094231,
0.00102579, 0.0010804, 0.00089567, 0.00081245, 0.0011208,
0.00110756, 0.00126673, 0.00108875, 0.00145115, 0.00098541,
0.00130559
])
return qBound, xTrain, yTrain_mean, yTrain_sdev
def fEx(p, sampleType, n, qInfo, fExName):
"""
Generate synthetic training data
"""
# (a) xTrain
nSamp = n[0] * n[1]
xi = []
q = []
qBound = []
if sampleType[0] == 'LHS' and sampleType[1] == 'LHS':
if distType == ['Unif'] * p:
qBound = qInfo
xi = sampling.LHS_sampling(nSamp, [[-1, 1]] * p)
xTrain = np.zeros((nSamp, p))
for i in range(p):
xTrain[:, i] = pce.mapFromUnit(xi[:, i], qBound[i])
fEx_ = analyticTestFuncs.fEx2D(xTrain[:, 0], xTrain[:, 1],
fExName, 'comp')
else:
raise ValueError(
"LHS works only when all q have 'Unif' distribution.")
else:
for i in range(p):
samps = sampling.trainSample(sampleType=sampleType[i],
GQdistType=distType[i],
qInfo=qInfo[i],
nSamp=n[i])
q.append(samps.q)
xTrain = reshaper.vecs2grid(q)
fEx_ = analyticTestFuncs.fEx2D(q[0], q[1], fExName, 'tensorProd')
return xTrain, fEx_
def lagInt_Quads2Line_test():
"""
Test lagInt_Quads2Line().
The test samples of (q1,q2) are generated along a defined line q2=a*q1+b
in the admissible space of q1-a2.
The training samples are drawn in the usual way, covering the admissible space of q1-q2.
"""
#----- SETTINGS --------------------------------------------------------------
nNodes = [9, 9] #number of training samples for q1, q2
sampType = [
'GLL', #type of training samples for q1, q2
'unifSpaced'
]
qBound = [
[-0.75, 1.5], #admissible range of q1,q2
[-0.8, 2.5]
] #Note that the line should be confined in this space
lineDef = {
'start':
[1.4,
2.3], #coordinates of the line's starting point in the q1-q2 plane
'end': [-0.7,
-0.2], #coordinates of the line's end point in the q1-q2 plane
'noPtsLine': 100 #number of the test samples
}
#-----------------------------------------------------------------------------
p = len(nNodes)
# Generate the training samples
qNodes = []
for i in range(p):
qNodes_ = sampling.trainSample(sampleType=sampType[i],
qInfo=qBound[i],
nSamp=nNodes[i])
qNodes.append(qNodes_.q)
# Evaluate the simulator at the training samples
fNodes = analyticTestFuncs.fEx2D(qNodes[0], qNodes[1], 'type1',
'tensorProd').val
# Construct the lagrange interpolation and evalautes it at the test points over the line
qLine, fLine = lagInt_Quads2Line(fNodes, qNodes, lineDef)
# Plots
plt.figure(figsize=(8, 5))
plt.plot(qLine[0],
fLine,
'-ob',
mfc='none',
label='Lagrange Interpolation')
fLine_ex = analyticTestFuncs.fEx2D(qLine[0], qLine[1], 'type1',
'comp').val #exact response
plt.plot(qLine[0], fLine_ex, '-xr', label='Exact Value')
plt.xlabel(r'$q_1$', fontsize=16)
plt.ylabel('Response', fontsize=14)
plt.legend(loc='best')
plt.grid(alpha=0.4)
plt.show()
def trainData(xInfo, n, noiseType, trainSamplyType, distType, fType):
"""
Create training data D={X,Y}
"""
X_ = sampling.trainSample(sampleType=trainSampleType,
GQdistType=distType,
qInfo=xInfo,
nSamp=n)
x = X_.q
sdV = noiseGen(n, noiseType)
y = fEx(x, fType, xInfo) + sdV * np.random.randn(n)
return x, y, sdV
def moments(self, distType, qInfo):
"""
Mean and variance of f(q) estimated by the Monte-Carlo approach
(These can be used as reference values instead of the analytical values)
Args:
`distType`: List of length 2
The i-th value (string) specifies the distribution type of the i-th parameter
`qInfo`: List of length 2
Information about the parameter range or distribution.
* If `q` is Gaussian ('Norm' or 'normRand') => qInfo=[mean,sdev]
* Otherwise, qInfo=[min(q),max(q)]=admissible range of q
Returns:
`mean`: float
Expected value of f(q) estimated by the Monte-Carlo method
`var`: float
Variance of f(q) estimated by the Monte-Carlo method
"""
nMC = 100000 #number of MC samples
print(
'... Reference moments are calculated by the Monte-Carlo method with %d samples'
% nMC)
qMC = []
p = len(distType)
if p != 2:
raise ValueError("distType should have length 2")
for i in range(p):
if distType[i] == 'Unif':
sampleType_ = 'unifRand'
elif distType[i] == 'Norm':
sampleType_ = 'normRand'
else:
raise ValueError("Invalid distType for parameter %d" % i)
samps = sampling.trainSample(sampleType=sampleType_,
GQdistType=distType[i],
qInfo=qInfo[i],
nSamp=nMC)
qMC.append(samps.q)
self.fVal_mc = fEx2D(qMC[0], qMC[1], self.typ, 'comp').val
self.mean = np.mean(self.fVal_mc)
self.var = np.mean(self.fVal_mc**2.) - self.mean**2.
def lagInt_1d_test():
"""
Test Lagrange inerpolation over a 1D parameter space.
"""
#----- SETTINGS -------------------
nNodes = 15 #number of training samples (nodes)
qBound = [-1, 3] #range over which the samples are taken
nTest = 100 #number of test points
sampType = 'GLL' #Type of samples, see trainSample class in samping.py
fType = 'type1' #Type of model function used as simulator
#----------------------------------
# Create the training samples and evaluate the simulator at each sample
samps_ = sampling.trainSample(sampleType=sampType,
qInfo=qBound,
nSamp=nNodes)
qNodes = samps_.q
fNodes = analyticTestFuncs.fEx1D(qNodes, fType, qBound).val
# Generate the test samples
qTestFull = np.linspace(qBound[0], qBound[1], nTest)
qTest = np.linspace(min(qNodes), max(qNodes), nTest)
# Construct the Lagrange interpolation and evaluate it at the test points
fInterpTest = lagInt(fNodes=fNodes, qNodes=[qNodes], qTest=[qTest]).val
# Plot
fTestFull = analyticTestFuncs.fEx1D(qTestFull, fType, qBound).val
plt.figure(figsize=(12, 7))
plt.plot(qTestFull, fTestFull, '--r', lw=2, label='Exact f(q)')
plt.plot(qTest,
fInterpTest,
'-b',
lw=2,
label='f(q) by Lagrange Interpolation')
plt.plot(qNodes, fNodes, 'oc', markersize='8', label='Nodes')
plt.legend(loc='best', fontsize=17)
plt.xticks(fontsize=15)
plt.yticks(fontsize=15)
plt.grid()
plt.xlabel(r'$q$', fontsize=26)
plt.ylabel(r'$f(q)$', fontsize=26)
plt.show()
def lagInt_2d_test():
"""
Test Lagrange inerpolation over a 2D parameter space.
"""
#----- SETTINGS --------------------------------------------------------------
nNodes = [
5, 4
] #number of training samples nodes in space of parameters q1, q2
sampType = [
'GLL', #Method of drawing samples for q1, q2
'unifSpaced'
]
qBound = [
[-0.75, 1.5], # admissible range of parameters
[-0.5, 2.5]
]
# Settings of the exact response surface
domRange = [
[-2, 2], #domain range for q1, q2
[-3, 3]
]
nTest = [100, 101] #number of test samples
#-----------------------------------------------------------------------------
p = len(nNodes)
# Create the training samples over each parameter space
qNodes = []
for i in range(p):
qNodes_ = sampling.trainSample(sampleType=sampType[i],
qInfo=qBound[i],
nSamp=nNodes[i])
qNodes.append(qNodes_.q)
# Evaluate the simulator at each joint sample
fNodes = analyticTestFuncs.fEx2D(qNodes[0], qNodes[1], 'type1',
'tensorProd').val
# Generate the test samples
qTestList = []
for i in range(p):
qTest_ = sampling.testSample(sampleType='unifSpaced',
qBound=qBound[i],
nSamp=nTest[i])
qTestList.append(qTest_.q)
# Construct the Lagrange interpolation and evaluate it at the test samples
fTest = lagInt(fNodes=fNodes,
qNodes=qNodes,
qTest=qTestList,
liDict={
'testRule': 'tensorProd'
}).val
# Evaluate the exact model response over domRange
qTestFull = []
for i in range(p):
qTestFull_ = np.linspace(domRange[i][0], domRange[i][1], nTest[i])
qTestFull.append(qTestFull_)
fTestFull = analyticTestFuncs.fEx2D(qTestFull[0], qTestFull[1], 'type1',
'tensorProd').val
fTestFullGrid = fTestFull.reshape((nTest[0], nTest[1]), order='F').T
fTestGrid = fTest.reshape((nTest[0], nTest[1]), order='F').T
# Plots
plt.figure(figsize=(16, 8))
plt.subplot(1, 2, 1)
ax = plt.gca()
CS1 = plt.contour(qTestFull[0], qTestFull[1], fTestFullGrid, 35)
plt.clabel(CS1,
inline=True,
fontsize=15,
colors='k',
fmt='%0.2f',
rightside_up=True,
manual=False)
qNodesGrid = reshaper.vecs2grid(qNodes)
plt.plot(qNodesGrid[:, 0], qNodesGrid[:, 1], 'o', color='r', markersize=6)
plt.xlabel(r'$q_1$', fontsize=25)
plt.ylabel(r'$q_2$', fontsize=25)
plt.xticks(fontsize=17)
plt.yticks(fontsize=17)
plt.title('Exact Response Surface')
plt.subplot(1, 2, 2)
ax = plt.gca()
CS2 = plt.contour(qTestList[0], qTestList[1], fTestGrid, 20)
plt.clabel(CS2,
inline=True,
fontsize=15,
colors='k',
fmt='%0.2f',
rightside_up=True,
manual=False)
plt.plot(qNodesGrid[:, 0], qNodesGrid[:, 1], 'o', color='r', markersize=6)
plt.xlabel(r'$q_1$', fontsize=25)
plt.ylabel(r'$q_2$', fontsize=25)
plt.xticks(fontsize=17)
plt.yticks(fontsize=17)
plt.title('Response Surface by Lagrange Interpolation')
plt.xlim(domRange[0])
plt.ylim(domRange[1])
plt.show()
def lagInt_3d_test():
"""
Test Lagrange inerpolation over a 3D parameter space.
"""
#----- SETTINGS -------------------
nNodes = [8, 7, 6] #number of training samples for q1, q2, q3
sampType = [
'GLL', #Type of samples for q1, q2, q3
'unifSpaced',
'Clenshaw'
]
qBound = [
[-0.75, 1.5], #range of parameters q1, q2, q3
[-0.5, 2.5],
[1, 3]
]
nTest = [10, 11, 12] #number of test samples for q1, q2, q3
fOpts = {'a': 7, 'b': 0.1} #parameters in Ishigami function
#----------------------------------
p = len(nNodes)
# Generate the training samples
qNodes = []
for i in range(p):
qNodes_ = sampling.trainSample(sampleType=sampType[i],
qInfo=qBound[i],
nSamp=nNodes[i])
qNodes.append(qNodes_.q)
# Run the simulator at the training samples
fNodes = analyticTestFuncs.fEx3D(qNodes[0], qNodes[1], qNodes[2],
'Ishigami', 'tensorProd', fOpts).val
# Create the test samples and run the simultor at them
qTest = []
for i in range(p):
qTest_ = sampling.testSample(sampleType='unifSpaced',
qBound=qBound[i],
nSamp=nTest[i])
qTest.append(qTest_.q)
fTestEx = analyticTestFuncs.fEx3D(qTest[0], qTest[1], qTest[2], 'Ishigami',
'tensorProd', fOpts).val
# Construct the Lagrange interpolation an evaluate it at the test samples
fInterp = lagInt(fNodes=fNodes,
qNodes=qNodes,
qTest=qTest,
liDict={
'testRule': 'tensorProd'
}).val
# Plot
plt.figure(figsize=(14, 8))
plt.subplot(2, 1, 1)
fInterp_ = fInterp.reshape(np.asarray(np.prod(np.asarray(nTest))),
order='F')
plt.plot(fInterp_, '-ob', mfc='none', label='Lagrange Interpolation')
plt.plot(fTestEx, '--xr', ms=5, label='Exact Value')
plt.ylabel(r'$f(q_1,q_2,q_3)$', fontsize=18)
plt.xlabel(r'Test Sample Number', fontsize=14)
plt.legend(loc='best', fontsize=14)
plt.grid(alpha=0.4)
plt.subplot(2, 1, 2)
plt.plot(abs(fInterp_ - fTestEx), '-sk')
plt.ylabel(r'$|f_{Interp}(q)-f_{Exact}(q)|$', fontsize=15)
plt.xlabel(r'Test Sample Number', fontsize=14)
plt.grid(alpha=0.4)
plt.show()
def lagIntAtGQs_2d_test():
"""
Test pce2pce_GQ(...) for 2D uncertain parameter space
"""
#------ SETTINGS ----------------------------------------------------
#Space 1
nSamp1 = [6, 10] #number of samples in PCE1, parameter 1,2
space1 = [
[-2, 1.5], #admissible space of PCE1 (both parameters)
[-3, 2.5]
]
sampleType1 = ['GLL', 'unifRand'] #see trainSample class in sampling.py
#Space 2
nSamp2 = [4, 5] #number of samples in PCE2, parameter 1,2
space2 = [
[-0.5, 1], #admissible space of PCEw (both parameters)
[-2., 1.5]
]
#Test samples
nTest = [100, 101] #number of test samples of parameter 1,2
#model function
fType = 'type1' #Type of simulator
#---------------------------------------------------------------------
p = 2
distType2 = ['Unif', 'Unif']
#(1) Generate samples from space 1
q1 = []
for i in range(p):
q1_ = sampling.trainSample(sampleType=sampleType1[i],
qInfo=space1[i],
nSamp=nSamp1[i])
space1[i] = [
min(q1_.q), max(q1_.q)
] #correction for uniform samples (otherwise contours are not plotted properly)
q1.append(q1_.q)
#Response values at the GL points
fVal1 = analyticTestFuncs.fEx2D(q1[0], q1[1], fType, 'tensorProd').val
#(2) Lagrange interpolation from samples 1 to GQ nodes on space 2
q2, xi2, fVal2 = lagIntAtGQs(fVal1, q1, space1, nSamp2, space2, distType2)
#(3) Construct a PCE on space 2
pceDict = {
'p': p,
'sampleType': 'GQ',
'pceSolveMethod': 'Projection',
'truncMethod': 'TP',
'distType': distType2
}
pce2 = pce(fVal=fVal2, xi=xi2, pceDict=pceDict, nQList=nSamp2)
#(4) Evaluate the surrogates: Lagrange interpolation over space 1
# PCE over space 2
#test samples
qTest1 = []
xiTest2 = []
qTest2 = []
for i in range(p):
testSamps1 = sampling.testSample('unifSpaced',
qBound=space1[i],
nSamp=nTest[i])
qTest1.append(testSamps1.q)
testSamps2 = sampling.testSample('unifSpaced',
GQdistType=distType2[i],
qBound=space2[i],
nSamp=nTest[i])
xiTest2.append(testSamps2.xi)
qTest2.append(testSamps2.q)
#evaluation
#space 1
fTest1_ex = analyticTestFuncs.fEx2D(qTest1[0], qTest1[1], fType,
'tensorProd').val
fTest1 = lagInt(fNodes=fVal1,
qNodes=q1,
qTest=qTest1,
liDict={
'testRule': 'tensorProd'
}).val
#space 2
pceEval2 = pceEval(coefs=pce2.coefs,
xi=xiTest2,
distType=distType2,
kSet=pce2.kSet)
fTest2 = pceEval2.pceVal
#(5) 2d contour plots
plt.figure(figsize=(20, 8))
plt.subplot(1, 3, 1)
ax = plt.gca()
fTest_Grid = fTest1_ex.reshape(nTest, order='F').T
CS1 = plt.contour(qTest1[0], qTest1[1], fTest_Grid,
35) #,cmap=plt.get_cmap('viridis'))
plt.clabel(CS1,
inline=True,
fontsize=13,
colors='k',
fmt='%0.2f',
rightside_up=True,
manual=False)
plt.xlabel('q1')
plt.ylabel('q2')
plt.title('Exact response surface over space 1')
#
plt.subplot(1, 3, 2)
ax = plt.gca()
fTest1_Grid = fTest1.reshape(nTest, order='F').T
CS2 = plt.contour(qTest1[0], qTest1[1], fTest1_Grid,
35) #,cmap=plt.get_cmap('viridis'))
plt.clabel(CS2,
inline=True,
fontsize=13,
colors='k',
fmt='%0.2f',
rightside_up=True,
manual=False)
q1Grid = reshaper.vecs2grid(q1)
plt.plot(q1Grid[:, 0], q1Grid[:, 1], 'ob', markersize=6)
q2_ = reshaper.vecs2grid(q2)
plt.plot(q2_[:, 0], q2_[:, 1], 'sr', markersize=6)
plt.xlabel('q1')
plt.ylabel('q2')
plt.title(
'Response surface by Lagrange Int.\n over space-1 using blue circles')
#
plt.subplot(1, 3, 3)
ax = plt.gca()
fTest2_Grid = fTest2.reshape(nTest, order='F').T
CS3 = plt.contour(qTest2[0], qTest2[1], fTest2_Grid,
20) #,cmap=plt.get_cmap('viridis'))
plt.clabel(CS3,
inline=True,
fontsize=13,
colors='k',
fmt='%0.2f',
rightside_up=True,
manual=False)
plt.plot(q2_[:, 0], q2_[:, 1], 'sr', markersize=6)
plt.xlabel('q1')
plt.ylabel('q2')
plt.title('Response surface by PCE over space-2 \n using red squares')
plt.xlim(space1[0][:])
plt.ylim(space1[1][:])
plt.show()
def lagIntAtGQs_1d_test():
"""
lagIntAtGQs test for 1D parameter
"""
#------ SETTINGS --------------------
#space 1
nSampMod1 = [7] #number of samples in PCE1
space1 = [[-0.5, 2.]] #admissible space of param in PCE1
sampleType1 = 'GLL' #see trainSample class in sampling.py
#space 2
distType2 = ['Unif'] #distribution type of the RV
nSampMod2 = [5] #number of samples in PCE2
space2 = [[0.0, 1.5]] #admissible space of param in PCE2
#model function
fType = 'type1' #Type of simulator
#test samples
nTest = 100 #number of test samples
#------------------------------------
#(1) Generates samples from SpaceMod1
qInfo_ = space1[0]
samps1 = sampling.trainSample(sampleType=sampleType1,
qInfo=qInfo_,
nSamp=nSampMod1[0])
q1 = samps1.q
xi1 = samps1.xi
qBound1 = samps1.qBound
#Evaluate the simulator at samples1
fEx = analyticTestFuncs.fEx1D(q1, fType, qInfo_)
fVal1 = fEx.val
#(2) Lagrange interpolation from samples 1 to GQ nodes on space 2
q2, xi2, fVal2 = lagIntAtGQs(fVal1, [q1], space1, nSampMod2, space2,
distType2)
#(3) Construct a PCE over space 2 using the GQ nodes
pceDict = {
'p': 1,
'sampleType': 'GQ',
'pceSolveMethod': 'Projection',
'distType': distType2
}
pce2 = pce(fVal=fVal2, xi=xi2[:, None], pceDict=pceDict)
#(4) Evaluate the surrogates: Lagrange interpolation over space 1
# PCE over space 2
testSamps1 = sampling.testSample(sampleType='unifSpaced',
qBound=space1[0],
nSamp=nTest)
qTest1 = testSamps1.q
fTest1_ex = analyticTestFuncs.fEx1D(qTest1, fType, space1).val
fTest1 = lagInt(fNodes=fVal1, qNodes=[q1], qTest=[qTest1]).val
#
testSamps2 = sampling.testSample(sampleType='unifSpaced',
qBound=space2[0],
nSamp=nTest)
qTest2 = testSamps2.q
xiTest2 = testSamps2.xi
pcePred_ = pceEval(coefs=pce2.coefs, xi=[xiTest2], distType=distType2)
fTest2 = pcePred_.pceVal
#(5) Plot
plt.figure(figsize=(15, 8))
plt.plot(qTest1, fTest1_ex, '--k', lw=2, label=r'Exact $f(q)$')
plt.plot(q1,
fVal1,
'ob',
markersize=8,
label='Original samples over space1')
plt.plot(qTest1, fTest1, '-b', lw=2, label='Lagrange Int. over space 1')
plt.plot(q2, fVal2, 'sr', markersize=8, label='GQ samples over space2')
plt.plot(qTest2, fTest2, '-r', lw=2, label='PCE over space 2')
plt.xlabel(r'$q$', fontsize=26)
plt.ylabel(r'$f(q)$', fontsize=26)
plt.xticks(fontsize=20)
plt.yticks(fontsize=20)
plt.grid(alpha=0.4)
plt.legend(loc='best', fontsize=20)
plt.show()
def ppce_cnstrct_1d(self):
"""
Constructing a probabilistic PCE over a 1D parameter space
"""
print('... Probabilistic PCE for 1D input parameter.')
p = self.p
ppceDict = self.ppceDict
qTrain = self.qTrain
yTrain = self.yTrain
noiseSdev = self.noiseV
# (0) Assignments
nGQ = ppceDict['nGQtest']
qInfo = ppceDict['qInfo']
nMC = ppceDict['nMC']
nw_ = int(nMC / 10)
distType = ppceDict['distType']
# Make a dict for GPR
gprOpts = {
'nIter': ppceDict['nIter_gpr'],
'lr': ppceDict['lr_gpr'],
'convPlot': ppceDict['convPlot_gpr']
}
standardizeYTrain_ = False
if 'standardizeYTrain_gpr' in ppceDict.keys():
gprOpts.update(
{'standardizeYTrain': ppceDict['standardizeYTrain_gpr']})
standardizeYTrain_ = True
# (1) Generate test points that are Gauss quadratures chosen based on the
# distribution of q (gPCE rule)
sampsGQ = sampling.trainSample(sampleType='GQ',
GQdistType=distType,
qInfo=qInfo,
nSamp=nGQ)
qTest = sampsGQ.q
# (2) Construct GPR surrogate based on training data
gpr_ = gpr_torch.gpr(qTrain[:, None], yTrain[:, None], noiseSdev,
qTest[:, None], gprOpts)
post_f = gpr_.post_f
post_obs = gpr_.post_y
shift_ = 0.0
scale_ = 1.0
if standardizeYTrain_:
shift_ = gpr_.shift[0] # 0: single-response
scale_ = gpr_.scale[0]
# (3) Use samples of GPR tested at GQ nodes to construct a PCE
fMean_list = []
fVar_list = []
pceDict = {
'p': p,
'sampleType': 'GQ',
'pceSolveMethod': 'Projection',
'distType': [distType]
}
for j in range(nMC):
# Draw a sample for f(q) from GPR surrogate
f_ = post_obs.sample().numpy() * scale_ + shift_
# Construct PCE for the drawn sample
pce_ = pce(fVal=f_, xi=[], pceDict=pceDict, verbose=False) # GP+TP
fMean_list.append(pce_.fMean)
fVar_list.append(pce_.fVar)
if ((j + 1) % nw_ == 0):
print(
"...... ppce repetition for finding samples of the PCE coefficients, iter = %d/%d"
% (j + 1, nMC))
# (4) Outputs
fMean_list = np.asarray(fMean_list)
fVar_list = np.asarray(fVar_list)
# Optional outputs: only used for gprPlot
optOut = {'post_f': post_f, 'post_obs': post_obs, 'qTest': [qTest]}
self.fMean_samps = fMean_list
self.fVar_samps = fVar_list
self.optOut = optOut
def ppce_cnstrct_pd(self):
"""
Constructing a probabilistic PCE over a p-D parameter space, p>1
"""
p = self.p
print('... Probabilistic PCE for %d-D input parameter.' % p)
ppceDict = self.ppceDict
qTrain = self.qTrain
yTrain = self.yTrain
noiseSdev = self.noiseV
# (0) Assignments
nGQ = ppceDict['nGQtest']
qInfo = ppceDict['qInfo']
nMC = ppceDict['nMC']
nw_ = int(nMC / 10)
distType = ppceDict['distType']
# Make a dict for gpr (do NOT change)
gprOpts = {
'nIter': ppceDict['nIter_gpr'],
'lr': ppceDict['lr_gpr'],
'convPlot': ppceDict['convPlot_gpr']
}
standardizeYTrain_ = False
if 'standardizeYTrain_gpr' in ppceDict.keys():
gprOpts.update(
{'standardizeYTrain': ppceDict['standardizeYTrain_gpr']})
standardizeYTrain_ = True
# Make a dict for PCE (do NOT change)
# Always use TP truncation with GQ sampling (hence Projection method)
pceDict = {
'p': p,
'truncMethod': 'TP',
'sampleType': 'GQ',
'pceSolveMethod': 'Projection',
'distType': distType
}
# (1) Generate test points that are Gauss quadratures chosen based on
# the distribution of q (gPCE rule)
qTestList = []
for i in range(p):
sampsGQ = sampling.trainSample(sampleType='GQ',
GQdistType=distType[i],
qInfo=qInfo[i],
nSamp=nGQ[i])
qTestList.append(sampsGQ.q)
qTestGrid = reshaper.vecs2grid(qTestList)
# (2) Construct GPR surrogate based on training data
gpr_ = gpr_torch.gpr(qTrain, yTrain[:, None], noiseSdev, qTestGrid,
gprOpts)
post_f = gpr_.post_f
post_obs = gpr_.post_y
shift_ = 0.0
scale_ = 1.0
if standardizeYTrain_:
shift_ = gpr_.shift[0] # 0: single-response
scale_ = gpr_.scale[0]
# optional: plot constructed response surface
# gpr_torch.gprPlot().torch2d_3dSurf(qTrain,yTrain,qTestList,post_obs,shift=shift_,scale=scale_)
# (3) Use samples of GPR tested at GQ nodes to construct a PCE
fMean_list = []
fVar_list = []
for j in range(nMC):
# Draw a sample for f(q) from GPR surrogate
f_ = post_obs.sample().numpy() * scale_ + shift_
# Construct PCE for the drawn sample
pce_ = pce(fVal=f_,
nQList=nGQ,
xi=[],
pceDict=pceDict,
verbose=False)
fMean_list.append(pce_.fMean)
fVar_list.append(pce_.fVar)
if ((j + 1) % nw_ == 0):
print(
"...... ppce repetition for finding samples of the PCE coefficients, iter = %d/%d"
% (j + 1, nMC))
# (4) Outputs
fMean_list = np.asarray(fMean_list)
fVar_list = np.asarray(fVar_list)
# Optional outputs: only used for gprPlot
optOut = {'post_f': post_f, 'post_obs': post_obs, 'qTest': qTestList}
self.optOut = optOut
self.fMean_samps = fMean_list
self.fVar_samps = fVar_list
def sobol_2par_unif_test():
"""
Test for sobol when we have 2 uncertain parameters q1, q2.
Sobol indices are computed for f(q1,q2)=q1**2.+q1*q2 that is analyticTestFuncs.fEx2D('type3').
Indices are computed from the following methods:
* Method1: Direct computation by UQit
* Method2: First a PCE is constructed and then its values are used to compute Sobol indices
* Method3: Analytical expressions (reference values)
"""
#--------------------------
#------- SETTINGS
n=[101, 100] #number of samples for q1 and q2, Method1
qBound=[[-3,1], #admissible range of parameters
[-1,2]]
nQpce=[5,6] #number of GQ points for Method2
#--------------------------
fType='type3' #type of analytical function
p=len(n)
distType=['Unif']*p
#(1) Samples from parameters space
q=[]
pdf=[]
for i in range(p):
q.append(np.linspace(qBound[i][0],qBound[i][1],n[i]))
pdf.append(np.ones(n[i])/(qBound[i][1]-qBound[i][0]))
#(2) Compute function value at the parameter samples
fEx_=analyticTestFuncs.fEx2D(q[0],q[1],fType,'tensorProd')
fEx=np.reshape(fEx_.val,n,'F')
#(3) Compute Sobol indices direct numerical integration
sobol_=sobol(q,fEx,pdf)
Si=sobol_.Si
STi=sobol_.STi
Sij=sobol_.Sij
#(4) Construct a PCE and then use the predictions of that in numerical integration
#for computing Sobol indices.
#Generate observations at Gauss-Legendre points
xi=[]
qpce=[]
for i in range(p):
samps=sampling.trainSample(sampleType='GQ',GQdistType=distType[i],qInfo=qBound[i],nSamp=nQpce[i])
xi.append(samps.xi)
qpce.append(samps.q)
fVal_pceCnstrct=analyticTestFuncs.fEx2D(qpce[0],qpce[1],fType,'tensorProd').val
#Construct the PCE
xiGrid=reshaper.vecs2grid(xi)
pceDict={'p':2,'sampleType':'GQ','truncMethod':'TP','pceSolveMethod':'Projection',
'distType':distType}
pce_=pce(fVal=fVal_pceCnstrct,nQList=nQpce,xi=xiGrid,pceDict=pceDict)
#Use the PCE to predict at test samples from parameter space
qpceTest=[]
xiTest=[]
for i in range(p):
testSamps=sampling.testSample('unifSpaced',GQdistType=distType[i],qBound=qBound[i],nSamp=n[i])
xiTest.append(testSamps.xi)
qpceTest.append(testSamps.q)
fPCETest_=pceEval(coefs=pce_.coefs,kSet=pce_.kSet,xi=xiTest,distType=distType)
fPCETest=fPCETest_.pceVal
#compute Sobol indices
sobolPCE_=sobol(qpceTest,fPCETest,pdf)
Si_pce=sobolPCE_.Si
Sij_pce=sobolPCE_.Sij
#(5) Exact Sobol indices (analytical expressions)
if fType=='type3':
fEx_.sobol(qBound)
Si_ex=fEx_.Si
STi_ex=fEx_.STi
Sij_ex=fEx_.Sij
#(6) results
print(' > Main Indices by UQit:\n\t S1=%g, S2=%g, S12=%g' %(Si[0],Si[1],Sij[0]))
print(' > Main indice by gPCE+Numerical Integration:\n\t S1=%g, S2=%g, S12=%g' %(Si_pce[0],Si_pce[1],Sij_pce[0]))
print(' > Main Analytical Reference:\n\t S1=%g, S2=%g, S12=%g' %(Si_ex[0],Si_ex[1],Sij_ex[0]))
print(' > Total Indices by UQit:\n\t ST1=%g, ST2=%g' %(STi[0],STi[1]))
print(' > Total Analytical Reference:\n\t ST1=%g, ST2=%g' %(STi_ex[0],STi_ex[1]))
def pce_3d_test():
"""
Test PCE for 3D uncertain parameter
"""
#----- SETTINGS------------
distType = ['Unif', 'Unif', 'Unif'] #distribution type of the parameters
qInfo = [
[-0.75, 1.5], #range of parameters
[-0.5, 2.5],
[1.0, 3.0]
]
nQ = [6, 5, 4] #number of parameter samples in the 3 dimensions
funOpt = {'a': 7, 'b': 0.1} #parameters in the Ishigami function
#PCE options
truncMethod = 'TO' #'TP'=Tensor Product
#'TO'=Total Order
sampleType = 'GQ' #'GQ'=Gauss Quadrature nodes
#other types: see trainSample in sampling.py
pceSolveMethod = 'Regression' #'Regression': for any combination of sample points and truncation methods
#'Projection': only for 'GQ'+'TP'
nTest = [5, 4, 3] #number of test samples for the parameters
if truncMethod == 'TO':
LMax = 10 #max polynomial order in each parameter direction
#--------------------
p = len(distType)
#Assemble the pceDict
pceDict = {
'p': p,
'truncMethod': truncMethod,
'sampleType': sampleType,
'pceSolveMethod': pceSolveMethod,
'distType': distType
}
if truncMethod == 'TO':
pceDict.update({'LMax': LMax})
#Generate training data
xi = []
q = []
qBound = []
for i in range(p):
samps = sampling.trainSample(sampleType=sampleType,
GQdistType=distType[i],
qInfo=qInfo[i],
nSamp=nQ[i])
xi.append(samps.xi)
q.append(samps.q)
qBound.append(samps.qBound)
fEx = analyticTestFuncs.fEx3D(q[0], q[1], q[2], 'Ishigami', 'tensorProd',
funOpt)
fVal = fEx.val
#Construct the PCE
xiGrid = reshaper.vecs2grid(xi)
pce_ = pce(fVal=fVal, xi=xiGrid, pceDict=pceDict, nQList=nQ)
fMean = pce_.fMean
fVar = pce_.fVar
pceCoefs = pce_.coefs
kSet = pce_.kSet
#Convergence of the PCE terms
convPlot(coefs=pceCoefs, distType=distType, kSet=kSet)
#Exact moments of the Ishigami function
fEx.moments(qInfo=qBound)
m = fEx.mean
v = fEx.var
#Compare the moments estimated by PCE with the exact analytical values
print(writeUQ.printRepeated('-', 50))
print('\t\t Exact \t\t PCE')
print('E[f]: ', m, fMean)
print('V[f]: ', v, fVar)
print(writeUQ.printRepeated('-', 50))
#Compare the PCE predictions at test points with the exact values of the model response
qTest = []
xiTest = []
for i in range(p):
testSamps = sampling.testSample('unifSpaced',
GQdistType=distType[i],
qInfo=qInfo[i],
qBound=qBound[i],
nSamp=nTest[i])
qTest.append(testSamps.q)
xiTest.append(testSamps.xi)
fVal_test_ex = analyticTestFuncs.fEx3D(qTest[0], qTest[1], qTest[2],
'Ishigami', 'tensorProd',
funOpt).val
#PCE prediction at test points
pcePred_ = pceEval(coefs=pceCoefs, xi=xiTest, distType=distType, kSet=kSet)
fVal_test_pce = pcePred_.pceVal
#Plot the exact and PCE response values
nTest_ = np.prod(np.asarray(nTest))
fVal_test_pce_ = fVal_test_pce.reshape(nTest_, order='F')
err = np.linalg.norm(fVal_test_pce_ - fVal_test_ex)
plt.figure(figsize=(10, 4))
plt.plot(fVal_test_pce_, '-ob', mfc='none', ms=5, label='Exact')
plt.plot(fVal_test_ex, '-xr', ms=5, label='PCE')
plt.xlabel('Index of test samples, k')
plt.ylabel('Model response')
plt.legend(loc='best')
plt.grid(alpha=0.4)
plt.show()
print('||fEx(q)-fPCE(q)|| % = ', err * 100)
def pce_1d_test():
"""
Test PCE for 1D uncertain parameter
"""
#--- settings -------------------------
#Parameter settings
distType = 'Norm' #distribution type of the parameter
if distType == 'Unif':
qInfo = [-2, 4.0] #parameter range only if 'Unif'
fType = 'type1' #Type of test exact model function
elif distType == 'Norm':
qInfo = [.5, 0.9] #[m,v] for 'Norm' q~N(m,v^2)
fType = 'type2' #Type of test exact model function
n = 20 #number of training samples
nTest = 200 #number of test sample sin the parameter space
#PCE Options
sampleType = 'GQ' #'GQ'=Gauss Quadrature nodes
#''= any other sample => only 'Regression' can be selected
# see `trainSample` class in sampling.py
pceSolveMethod = 'Projection' #'Regression': for any combination of sample points
#'Projection': only for GQ
LMax_ = 10 #number of terms (=K) in PCE (Only used with Regresson method)
#(LMax will be over written by nSamples if it is provided for 'GQ'+'Projection')
#-------------------------------------
#(0) Make the pceDict
pceDict = {
'p': 1,
'sampleType': sampleType,
'pceSolveMethod': pceSolveMethod,
'LMax': LMax_,
'distType': [distType]
}
#(1) Generate training data
samps = sampling.trainSample(sampleType=sampleType,
GQdistType=distType,
qInfo=qInfo,
nSamp=n)
q = samps.q
xi = samps.xi
qBound = samps.qBound
fEx = analyticTestFuncs.fEx1D(q, fType, qInfo)
f = fEx.val
#(2) Compute the exact moments (as the reference data)
fEx.moments(qInfo)
fMean_ex = fEx.mean
fVar_ex = fEx.var
#(3) Construct the PCE
pce_ = pce(fVal=f, xi=xi[:, None], pceDict=pceDict)
fMean = pce_.fMean #E[f(q)] estimated by PCE
fVar = pce_.fVar #V[f(q)] estimated by PCE
pceCoefs = pce_.coefs #Coefficients in the PCE
#(4) Compare moments: exact vs. PCE estimations
print(writeUQ.printRepeated('-', 70))
print('-------------- Exact -------- PCE --------- Error % ')
print('Mean of f(q) = %g\t%g\t%g' % (fMean_ex, fMean,
(fMean - fMean_ex) / fMean_ex * 100.))
print('Var of f(q) = %g\t%g\t%g' % (fVar_ex, fVar,
(fVar - fVar_ex) / fVar_ex * 100.))
print(writeUQ.printRepeated('-', 70))
#(5) Plots
# Plot convergence of the PCE coefficients
convPlot(coefs=pceCoefs, distType=distType)
#
#(6) Evaluate the PCE at test samples
# Test samples
testSamps = sampling.testSample('unifSpaced',
GQdistType=distType,
qInfo=qInfo,
qBound=qBound,
nSamp=nTest)
qTest = testSamps.q
xiTest = testSamps.xi
fTest = analyticTestFuncs.fEx1D(qTest, fType,
qInfo).val #exact response at test samples
#Prediction by PCE at test samples
pcePred_ = pceEval(coefs=pceCoefs, xi=[xiTest], distType=[distType])
fPCE = pcePred_.pceVal
#Exact and PCE response surface
plt.figure(figsize=(12, 5))
ax = plt.gca()
plt.plot(qTest, fTest, '-k', lw=2, label=r'Exact $f(q)$')
plt.plot(q, f, 'ob', label=sampleType + ' Training Samples')
plt.plot(qTest, fPCE, '-r', lw=2, label='PCE')
plt.plot(qTest,
fMean * np.ones(len(qTest)),
'-b',
label=r'$\mathbb{E}[f(q)]$')
ax.fill_between(qTest,
fMean + 1.96 * mt.sqrt(fVar) * np.ones(len(qTest)),
fMean - 1.96 * mt.sqrt(fVar) * np.ones(len(qTest)),
color='powderblue',
alpha=0.4)
plt.plot(qTest,
fMean + 1.96 * mt.sqrt(fVar) * np.ones(len(qTest)),
'--b',
label=r'$\mathbb{E}[f(q)]\pm 95\%CI$')
plt.plot(qTest, fMean - 1.96 * mt.sqrt(fVar) * np.ones(len(qTest)), '--b')
plt.title('Example of 1D PCE for random variable of type %s' % distType)
plt.xlabel(r'$q$', fontsize=19)
plt.ylabel(r'$f(q)$', fontsize=19)
plt.xticks(fontsize=18)
plt.yticks(fontsize=18)
plt.grid(alpha=0.3)
plt.legend(loc='best', fontsize=17)
plt.show()
def pce_2d_test():
"""
Test PCE for 2D uncertain parameter
"""
#---- SETTINGS------------
#Parameters specifications
distType = ['Norm', 'Norm'] #distribution type of the parameters q1, q2
qInfo = [
[-2, 1], #info on parameters
[-2, 0.4]
]
nQ = [7, 6] #number of training samples of parameters
nTest = [121,
120] #number of test points in parameter spaces to evaluate PCE
#PCE Options
truncMethod = 'TO' #'TP'=Tensor Product
#'TO'=Total Order
sampleType = [
'GQ', 'GQ'
] #'GQ'=Gauss Quadrature nodes ('Projection' or 'Regression')
#For other type of samples, see sampling.py, trainSample => only 'Regression' can be used
#'LHS': Latin Hypercube Sampling (only when all distType='Unif')
fType = 'type1' #Type of the exact model response, 'type1', 'type2', 'type3', 'Rosenbrock'
pceSolveMethod = 'Regression' #'Regression': for any combination of sampling and truncation methods
#'Projection': only for 'GQ'+'TP'
if truncMethod == 'TO':
LMax = 8 #max polynomial order in each parameter dimention
#------------------------
p = len(distType)
#Assemble the pceDict
pceDict = {
'p': p,
'truncMethod': truncMethod,
'sampleType': sampleType,
'pceSolveMethod': pceSolveMethod,
'distType': distType
}
if truncMethod == 'TO':
pceDict.update({'LMax': LMax, 'pceSolveMethod': 'Regression'})
#Generate the training data
xi = []
q = []
qBound = []
if sampleType[0] == 'LHS' and sampleType[1] == 'LHS':
if distType == ['Unif'] * p:
qBound = qInfo
xi = sampling.LHS_sampling(nQ[0] * nQ[1], [[-1, 1]] * p)
for i in range(p):
q.append(pce.mapFromUnit(xi[:, i], qBound[i]))
fEx_ = analyticTestFuncs.fEx2D(q[0], q[1], fType, 'comp')
xiGrid = xi
else:
raise ValueError(
"LHS works only when all q have 'Unif' distribution.")
else:
for i in range(p):
samps = sampling.trainSample(sampleType=sampleType[i],
GQdistType=distType[i],
qInfo=qInfo[i],
nSamp=nQ[i])
q.append(samps.q)
xi.append(samps.xi)
qBound.append(samps.qBound)
fEx_ = analyticTestFuncs.fEx2D(q[0], q[1], fType, 'tensorProd')
xiGrid = reshaper.vecs2grid(xi)
fVal = fEx_.val
#Construct the PCE
pce_ = pce(fVal=fVal, xi=xiGrid, pceDict=pceDict, nQList=nQ)
fMean = pce_.fMean
fVar = pce_.fVar
pceCoefs = pce_.coefs
kSet = pce_.kSet
#Plot the convergence indicator of the PCE
convPlot(coefs=pceCoefs, distType=distType, kSet=kSet)
#Generate test samples for the parameters and evaluate the exact response surface at them
qTest = []
xiTest = []
for i in range(p):
testSamps = sampling.testSample('unifSpaced',
GQdistType=distType[i],
qInfo=qInfo[i],
qBound=qBound[i],
nSamp=nTest[i])
qTest_ = testSamps.q
xiTest_ = testSamps.xi
qTest.append(qTest_)
xiTest.append(xiTest_)
fTest = analyticTestFuncs.fEx2D(qTest[0], qTest[1], fType,
'tensorProd').val
#Evaluate PCE at the test samples
pcePred_ = pceEval(coefs=pceCoefs, xi=xiTest, distType=distType, kSet=kSet)
fPCE = pcePred_.pceVal
#Use MC method to directly estimate reference values for the mean and varaiance of f(q)
fEx_.moments(distType, qInfo)
fMean_mc = fEx_.mean
fVar_mc = fEx_.var
#Compare the PCE estimates for moments of f(q) with the reference values from MC
print(writeUQ.printRepeated('-', 70))
print('------------ MC -------- PCE --------- Error % ')
print('Mean of f(q) = %g\t%g\t%g' % (fMean_mc, fMean,
(fMean - fMean_mc) / fMean_mc * 100.))
print('Var of f(q) = %g\t%g\t%g' % (fVar_mc, fVar,
(fVar - fVar_mc) / fVar_mc * 100.))
print(writeUQ.printRepeated('-', 70))
#Plot the exact and PCE response surfaces as contours in the parameters space
# Create 2D grid from the test samples and plot the contours of response surface over it
fTestGrid = fTest.reshape(nTest, order='F')
fErrorGrid = (abs(fTestGrid - fPCE))
# 2D grid from the sampled parameters
if sampleType[0] == 'LHS' and sampleType[1] == 'LHS':
qGrid = reshaper.vecsGlue(q[0], q[1])
else:
qGrid = reshaper.vecs2grid(q)
plt.figure(figsize=(21, 8))
plt.subplot(1, 3, 1)
ax = plt.gca()
CS1 = plt.contour(qTest[0], qTest[1], fTestGrid.T, 40)
plt.clabel(CS1,
inline=True,
fontsize=13,
colors='k',
fmt='%0.2f',
rightside_up=True,
manual=False)
plt.plot(qGrid[:, 0], qGrid[:, 1], 'o', color='r', markersize=7)
plt.xlabel(r'$q_1$')
plt.ylabel(r'$q_2$')
plt.title('Exact Response')
plt.subplot(1, 3, 2)
ax = plt.gca()
CS2 = plt.contour(qTest[0], qTest[1], fPCE.T, 40)
plt.clabel(CS2,
inline=True,
fontsize=13,
colors='k',
fmt='%0.2f',
rightside_up=True,
manual=False)
plt.plot(qGrid[:, 0], qGrid[:, 1], 'o', color='r', markersize=7)
plt.xlabel(r'$q_1$')
plt.ylabel(r'$q_2$')
plt.title('PCE Response')
plt.subplot(1, 3, 3)
ax = plt.gca()
CS3 = plt.contour(qTest[0], qTest[1], fErrorGrid.T, 40)
plt.clabel(CS3,
inline=True,
fontsize=13,
colors='k',
fmt='%0.2f',
rightside_up=True,
manual=False)
plt.xlabel(r'$q_1$')
plt.ylabel(r'$q_2$')
plt.plot(qGrid[:, 0], qGrid[:, 1], 'o', color='r', markersize=7)
plt.title('|Exact-Surrogate|')
plt.show()