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 trainDataGen(p,sampleType,n,qBound,fExName,noiseType):
"""
Generate Training Data
"""
# (a) xTrain
if sampleType=='grid':
nSamp=n[0]*n[1]
gridList=[]
for i in range(p):
#grid_=torch.linspace(qBound[i][0],qBound[i][1],n[i]) #torch
grid_=np.linspace(qBound[i][0],qBound[i][1],n[i])
gridList.append(grid_)
xTrain=reshaper.vecs2grid(gridList)
# xTrain = gpytorch.utils.grid.create_data_from_grid(gridList) #torch
elif sampleType=='random':
nSamp=n #number of random samples
xTrain=sampling.LHS_sampling(n,qBound)
# (b) Observation noise
#noiseSdev=torch.ones(nTot).mul(0.1) #torch
noiseSdev=noiseGen(nSamp,noiseType,xTrain,fExName)
#yTrain = torch.sin(mt.pi*xTrain[:,0])*torch.cos(.25*mt.pi*xTrain[:,1])+
# torch.randn_like(xTrain[:,0]).mul(0.1) #torch
# (c) Training response
yTrain=analyticTestFuncs.fEx2D(xTrain[:,0],xTrain[:,1],fExName,'comp').val
yTrain_noiseFree=yTrain
yTrain=yTrain_noiseFree+noiseSdev*np.random.randn(nSamp)
return xTrain,yTrain,noiseSdev,yTrain_noiseFree
def noiseGen(n, noiseType, xTrain, fExName):
"""
Generate a 1D numpy array of standard deviation of the observation noise
"""
if noiseType == 'h**o':
sd = 0.2 #(non-zero, to avoid instabilities)
sdV = sd * np.ones(n)
elif noiseType == 'hetero':
sdV = 0.1 * (analyticTestFuncs.fEx2D(xTrain[:, 0], xTrain[:, 1],
fExName, 'comp').val + 0.001)
return sdV
def noiseGen(n,noiseType,xTrain,fExName):
"""
Generate a 1D numpy array of standard deviations of the observation noise
"""
if noiseType=='h**o':
sd=0.2 # noise standard deviation (Note: non-zero, to avoid instabilities)
sdV=sd*np.ones(n)
elif noiseType=='hetero':
#sdMin=0.01
#sdMax=0.5
#sdV=sdMin+(sdMax-sdMin)*np.linspace(0.0,1.0,n)
#sdV=0.15*np.ones(n)
sdV=0.1*(analyticTestFuncs.fEx2D(xTrain[:,0],xTrain[:,1],fExName,'comp').val+0.001)
return sdV
def trainDataGen(p, sampleType, n, qBound, fExName, noiseType):
"""
Generate Training Data
"""
# (a) xTrain
if sampleType == 'grid':
nSamp = n[0] * n[1]
gridList = []
for i in range(p):
grid_ = np.linspace(qBound[i][0], qBound[i][1], n[i])
gridList.append(grid_)
xTrain = reshaper.vecs2grid(gridList)
elif sampleType == 'random':
nSamp = n # number of random samples
xTrain = sampling.LHS_sampling(n, qBound)
# (b) Observation noise
noiseSdev = noiseGen(nSamp, noiseType, xTrain, fExName)
# (c) Training response
yTrain = analyticTestFuncs.fEx2D(xTrain[:, 0], xTrain[:, 1], fExName,
'comp').val
yTrain_noiseFree = yTrain
yTrain = yTrain_noiseFree + noiseSdev * np.random.randn(nSamp)
return xTrain, yTrain, noiseSdev, yTrain_noiseFree
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 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 gprTorch_2d_singleTask_test():
"""
Test for GPR for 2d input
"""
##
def plot_trainData(n,fSamples,noiseSdev,yTrain):
"""
Plot the noisy training data which are used in GPR.
"""
plt.figure(figsize=(10,5))
x_=np.zeros(n)
for i in range(n):
x_[i]=i+1
for i in range(500): #only for plottig possible realizations
noise_=noiseSdev*np.random.randn(n)
plt.plot(x_,fSamples+noise_,'.',color='steelblue',alpha=0.4,markersize=1)
plt.errorbar(x_,fSamples,yerr=1.96*abs(noiseSdev),ls='none',capsize=5,ecolor='k',
elinewidth=4,label=r'$95\%$ CI in Obs.')
plt.plot(x_,fSamples,'o' ,markersize=6,markerfacecolor='lime',
markeredgecolor='salmon',label='Mean Observation')
plt.plot(x_,yTrain ,'xr' ,markersize=6,label='Sample Observation')
plt.legend(loc='best',fontsize=15)
plt.ylabel('QoI',fontsize=17)
plt.xlabel('Simulation Index',fontsize=17)
plt.xticks(fontsize=15)
plt.yticks(fontsize=15)
plt.title('Training data with associated confidence')
plt.show()
##
def trainDataGen(p,sampleType,n,qBound,fExName,noiseType):
"""
Generate Training Data
"""
# (a) xTrain
if sampleType=='grid':
nSamp=n[0]*n[1]
gridList=[]
for i in range(p):
#grid_=torch.linspace(qBound[i][0],qBound[i][1],n[i]) #torch
grid_=np.linspace(qBound[i][0],qBound[i][1],n[i])
gridList.append(grid_)
xTrain=reshaper.vecs2grid(gridList)
# xTrain = gpytorch.utils.grid.create_data_from_grid(gridList) #torch
elif sampleType=='random':
nSamp=n #number of random samples
xTrain=sampling.LHS_sampling(n,qBound)
# (b) Observation noise
#noiseSdev=torch.ones(nTot).mul(0.1) #torch
noiseSdev=noiseGen(nSamp,noiseType,xTrain,fExName)
#yTrain = torch.sin(mt.pi*xTrain[:,0])*torch.cos(.25*mt.pi*xTrain[:,1])+
# torch.randn_like(xTrain[:,0]).mul(0.1) #torch
# (c) Training response
yTrain=analyticTestFuncs.fEx2D(xTrain[:,0],xTrain[:,1],fExName,'comp').val
yTrain_noiseFree=yTrain
yTrain=yTrain_noiseFree+noiseSdev*np.random.randn(nSamp)
return xTrain,yTrain,noiseSdev,yTrain_noiseFree
##
def noiseGen(n,noiseType,xTrain,fExName):
"""
Generate a 1D numpy array of standard deviations of the observation noise
"""
if noiseType=='h**o':
sd=0.2 # noise standard deviation (Note: non-zero, to avoid instabilities)
sdV=sd*np.ones(n)
elif noiseType=='hetero':
#sdMin=0.01
#sdMax=0.5
#sdV=sdMin+(sdMax-sdMin)*np.linspace(0.0,1.0,n)
#sdV=0.15*np.ones(n)
sdV=0.1*(analyticTestFuncs.fEx2D(xTrain[:,0],xTrain[:,1],fExName,'comp').val+0.001)
return sdV
#
#----- SETTINGS
qBound=[[-2,2],[-2,2]] #Admissible range of parameters
fExName='type1' #Type of simulator in analyticTestFuncs.fEx2D
#'type1', 'type2', 'type3', 'Rosenbrock'
sampleType='random' #'random' or 'grid': type of training samples
if sampleType=='grid':
n=[9,9] #number of training samples in each input dimension
elif sampleType=='random':
n=100 #total number of training samples drawn randomly
noiseType='hetero' #noise type: 'h**o'=homoscedastic, 'hetero'=heterscedastic
#options for GPR
nIter_=1000 #number of iterations in optimization of GPR hyperparameters
lr_ =0.05 #learning rate in the optimization of the hyperparameters
convPlot_=True #plot convergence of optimization of GPR hyperparameters
nTest=[21,20] #number of test points in each parameter dimension
#------------------------------------------------
#(0) Assemble the gprOpts dict
gprOpts={'nIter':nIter_,'lr':lr_,'convPlot':convPlot_}
#(1) Generate training data
p=len(qBound) #dimension of the input
xTrain,yTrain,noiseSdev,yTrain_noiseFree=trainDataGen(p,sampleType,n,qBound,fExName,noiseType)
nSamp=yTrain.shape[0]
plot_trainData(nSamp,yTrain_noiseFree,noiseSdev,yTrain)
#(2) Create the test samples
xTestList=[]
for i in range(p):
#grid_=torch.linspace(qBound[i][0],qBound[i][1],20) #torch
grid_=np.linspace(qBound[i][0],qBound[i][1],nTest[i])
xTestList.append(grid_)
xTest=reshaper.vecs2grid(xTestList)
#(3) Construct the GPR based on the training data and make predictions at the test samples
gpr_=gpr(xTrain,yTrain[:,None],noiseSdev,xTest,gprOpts)
post_f=gpr_.post_f
post_obs=gpr_.post_y
# Predicted mean and variance of the posteriors at the test grid
fP_=gprPost(post_f,nTest)
fP_.torchPost()
post_f_mean=fP_.mean
post_f_sdev=fP_.sdev
lower_f=fP_.ciL
upper_f=fP_.ciU
obsP_=gprPost(post_obs,nTest)
obsP_.torchPost()
post_obs_mean=obsP_.mean
post_obs_sdev=obsP_.sdev
lower_obs=obsP_.ciL
upper_obs=obsP_.ciU
# Plots
with torch.no_grad():
fig = plt.figure(figsize=(16,4))
ax = fig.add_subplot(141)
fEx_test=analyticTestFuncs.fEx2D(xTest[:,0],xTest[:,1],fExName,'comp').val
CS0=ax.contour(xTestList[0],xTestList[1],fEx_test.reshape((nTest[0],nTest[1]),order='F').T,levels=40)
ax.clabel(CS0, inline=True, fontsize=15,colors='k',fmt='%0.2f',rightside_up=True,manual=False)
ax.plot(xTrain[:,0],xTrain[:,1],'or')
ax.set_title(r'Exact $f(q)$')
ax = fig.add_subplot(142)
CS1=ax.contour(xTestList[0],xTestList[1],(post_f_mean).T,levels=40)
ax.clabel(CS1, inline=True, fontsize=15,colors='k',fmt='%0.2f',rightside_up=True,manual=False)
ax.plot(xTrain[:,0],xTrain[:,1],'or')
ax.set_title(r'Mean Posterior of $f(q)$')
ax = fig.add_subplot(143)
CS2=ax.contour(xTestList[0],xTestList[1],upper_obs.T,levels=40)
ax.clabel(CS2, inline=True, fontsize=15,colors='k',fmt='%0.2f',rightside_up=True,manual=False)
ax.plot(xTrain[:,0],xTrain[:,1],'or')
ax.set_title(r'Upper Confidence for Observations')
ax = fig.add_subplot(144)
CS2=ax.contour(xTestList[0],xTestList[1],lower_obs.T,levels=40)
ax.clabel(CS2, inline=True, fontsize=15,colors='k',fmt='%0.2f',rightside_up=True,manual=False)
ax.plot(xTrain[:,0],xTrain[:,1],'or')
ax.set_title(r'Lower Confidence for Observations')
plt.show()
#2dplot
pltOpts={'title':'Mean posterior of f(q)',
'xlab':r'$q_1$',
'ylab':r'$q_2$'}
gprPlot(pltOpts).torch2d_2dcont(xTrain,xTestList,post_f_mean)
#3d plot
gprPlot().torch2d_3dSurf(xTrain,yTrain,xTestList,post_obs)
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_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()
