def test_two_outputs():
f = ot.SymbolicFunction(['x'], ['x * sin(x)', 'x * cos(x)'])
sampleX = [[1.0], [2.0], [3.0], [4.0], [5.0], [6.0], [7.0], [8.0]]
sampleY = f(sampleX)
basis = ot.Basis([
ot.SymbolicFunction(['x'], ['x']),
ot.SymbolicFunction(['x'], ['x^2'])
])
covarianceModel = ot.SquaredExponential([1.0])
covarianceModel.setActiveParameter([])
covarianceModel = ot.TensorizedCovarianceModel([covarianceModel] * 2)
algo = ot.KrigingAlgorithm(sampleX, sampleY, covarianceModel, basis)
algo.run()
result = algo.getResult()
mm = result.getMetaModel()
assert mm.getOutputDimension() == 2, "wrong output dim"
ott.assert_almost_equal(mm(sampleX), sampleY)
# Check the conditional covariance
reference_covariance = ot.Matrix([[4.4527, 0.0, 8.34404, 0.0],
[0.0, 2.8883, 0.0, 5.41246],
[8.34404, 0.0, 15.7824, 0.0],
[0.0, 5.41246, 0.0, 10.2375]])
ott.assert_almost_equal(
result([[9.5], [10.0]]).getCovariance() - reference_covariance,
ot.Matrix(4, 4), 0.0, 2e-2)
def runPCA(self):
# Perform PCA
data = np.array(self.sample).T
# Make the column-mean zero
columnmean = data.mean(axis=0)
for i in range(self.verticesNumber):
data[:, i] = data[:, i] - columnmean[i]
# Compute SVD of the matrix
mymatrix = ot.Matrix(data)
singular_values, U, VT = mymatrix.computeSVD(True)
V = VT.transpose()
# Truncate
VL = V[:, 0:self.numberOfComponents]
# Project
self.principalComponents = np.array(mymatrix * VL)
# Compute explained variance
explained_variance = ot.Point(self.verticesNumber)
for i in range(self.verticesNumber):
explained_variance[i] = singular_values[i]**2
n_samples = self.processSample.getSize()
explained_variance /= n_samples - 1
# Compute total variance
total_var = explained_variance.norm1()
# Compute explained variance ratio
explained_variance_ratio = explained_variance / total_var
# Truncate
self.explained_variance_ratio = explained_variance_ratio[0:self.numberOfComponents]
def _PODgaussModelCl(self, defects, intercept, slope, stderr, detection):
class buildPODModel():
def __init__(self, intercept, slope, sigmaEpsilon, detection):
self.intercept = intercept
self.slope = slope
self.sigmaEpsilon = sigmaEpsilon
self.detection = detection
def PODmodel(self, x):
t = (self.detection -
(self.intercept + self.slope * x)) / self.sigmaEpsilon
return ot.DistFunc.pNormal(t, True)
N = defects.getSize()
X = ot.Sample(N, [1, 0])
X[:, 1] = defects
X = ot.Matrix(X)
covMatrix = X.computeGram(True).solveLinearSystem(ot.IdentityMatrix(2))
sampleNormal = ot.Normal([0, 0],
ot.CovarianceMatrix(
covMatrix.getImplementation())).getSample(
self._simulationSize)
sampleSigmaEpsilon = (ot.Chi(N - 2).inverse() * np.sqrt(N - 2) *
stderr).getSample(self._simulationSize)
PODcoll = []
for i in range(self._simulationSize):
sigmaEpsilon = sampleSigmaEpsilon[i][0]
interceptSimu = sampleNormal[i][0] * sigmaEpsilon + intercept
slopeSimu = sampleNormal[i][1] * sigmaEpsilon + slope
PODcoll.append(
buildPODModel(interceptSimu, slopeSimu, sigmaEpsilon,
detection).PODmodel)
return PODcoll
def convert_hmat_to_matrix(hmat):
res = ot.Matrix(hmat.getNbRows(), hmat.getNbColumns())
for i in range(hmat.getNbRows()):
x = ot.Point(hmat.getNbColumns())
x[i] = 1.0
y = ot.Point(hmat.getNbRows())
hmat.gemv('N', 1.0, x, 0.0, y)
for j in range(hmat.getNbColumns()):
res[i, j] = y[j]
return res
def gradient(self, point):
shape = np.array(point).shape
# Check passed argument (should be 1D array)
if len(shape) == 1 and shape[0] == self.__inputs_dimension:
result = self.__p7_model.grad(point)
self.__calls_number += 1
return ot.Matrix(result).transpose()
else:
raise ValueError('Wrong shape %s of input array, required: (%s,)' %
(shape, self.__inputs_dimension))
def computeDurbinWatsonTest(x, residuals, hypothesis="Equal"):
# Parameters:
# hypothesis : string
# "Equal" : hypothesis is autocorrelation is 0
# "Less" : hypothesis is autocorrelation is less than 0
# "Greater" : hypothesis is autocorrelation is greater than 0
nx = x.getSize()
dim = x.getDimension()
residuals = np.array(residuals)
# statistic Durbin Watson
dw = np.sum(np.diff(np.hstack(residuals))**2) / np.sum(residuals**2)
# Normal approxiimation of DW to compute the pvalue
X = ot.Matrix(nx, dim + 1)
X[:, 0] = np.ones((nx, 1))
X[:, 1] = x
B = ot.Matrix(nx, dim + 1)
B[0, 1] = x[0][0] - x[1][0]
B[nx - 1, 1] = x[nx - 1][0] - x[nx - 2][0]
for i in range(nx - 2):
B[i + 1, 1] = -x[i][0] + 2 * x[i + 1][0] - x[i + 2][0]
XtX = X.computeGram()
XBQt = ot.SquareMatrix(XtX.solveLinearSystem(B.transpose() * X))
P = 2 * (nx - 1) - XBQt.computeTrace()
XBTrace = ot.SquareMatrix(XtX.solveLinearSystem(B.computeGram(),
False)).computeTrace()
Q = 2 * (3 * nx - 4) - 2 * XBTrace + ot.SquareMatrix(
XBQt * XBQt).computeTrace()
dmean = P / (nx - (dim + 1))
dvar = 2.0 / ((nx - (dim + 1)) * (nx - (dim + 1) + 2)) * (Q - P * dmean)
# compute the pvalue with respect to hypothesis
# Default pvalue is for hypothesis == "Equal"
# complementary CDF of standard normal distribution
pValue = 2 * ot.DistFunc.pNormal(np.abs(dw - dmean) / np.sqrt(dvar), True)
if hypothesis == "Less":
pValue = 1 - pValue / 2
elif hypothesis == "Greater":
pValue = pValue / 2
return pValue
def test_model(myModel, test_grad=True, x1=None, x2=None):
print('myModel = ', myModel)
spatialDimension = myModel.getInputDimension()
dimension = myModel.getOutputDimension()
active = myModel.getActiveParameter()
print('active=', active)
print('parameter=', myModel.getParameter())
print('parameterDescription=', myModel.getParameterDescription())
if x1 is None and x2 is None:
x1 = ot.Point(spatialDimension)
x2 = ot.Point(spatialDimension)
for j in range(spatialDimension):
x1[j] = -1.0 - j
x2[j] = 3.0 + 2.0 * j
eps = 1e-5
print('myModel(', x1, ', ', x2, ')=', repr(myModel(x1, x2)))
grad = myModel.partialGradient(x1, x2)
print('dCov =', repr(grad))
if (dimension == 1):
gradfd = ot.Point(spatialDimension)
for j in range(spatialDimension):
x1_d = ot.Point(x1)
x1_d[j] = x1_d[j] + eps
gradfd[j] = (myModel(x1_d, x2)[0, 0] - myModel(x1, x2)[0, 0]) / eps
else:
gradfd = ot.Matrix(spatialDimension, dimension * dimension)
covarianceX1X2 = myModel(x1, x2)
# Symmetrize matrix
covarianceX1X2.getImplementation().symmetrize()
centralValue = ot.Point(covarianceX1X2.getImplementation())
# Loop over the shifted points
for i in range(spatialDimension):
currentPoint = ot.Point(x1)
currentPoint[i] += eps
localCovariance = myModel(currentPoint, x2)
localCovariance.getImplementation().symmetrize()
currentValue = ot.Point(
localCovariance.getImplementation())
for j in range(currentValue.getSize()):
gradfd[i, j] = (currentValue[j] - centralValue[j]) / eps
print('dCov (FD)=', repr(gradfd))
if test_grad:
pGrad = myModel.parameterGradient(x1, x2)
precision = ot.PlatformInfo.GetNumericalPrecision()
ot.PlatformInfo.SetNumericalPrecision(4)
print('dCov/dP=', pGrad)
ot.PlatformInfo.SetNumericalPrecision(precision)
def hyperplane(coefs):
"""
Generate a linear NMF from its coefficients
"""
dim = len(coefs)
constant = [0.]
center = [0.] * len(coefs)
linear = ot.Matrix(1, dim)
for i in range(dim):
linear[0, i] = coefs[i]
function = ot.LinearFunction(center, constant, linear)
return function
def test_model(myModel):
print('myModel = ', myModel)
spatialDimension = myModel.getSpatialDimension()
dimension = myModel.getDimension()
active = myModel.getActiveParameter()
print('active=', active)
print('parameter=', myModel.getParameter())
print('parameterDescription=', myModel.getParameterDescription())
x1 = ot.NumericalPoint(spatialDimension)
x2 = ot.NumericalPoint(spatialDimension)
for j in range(spatialDimension):
x1[j] = -1.0 - j
x2[j] = 3.0 + 2.0 * j
eps = 1e-5
if (dimension == 1):
print('myModel(', x1, ', ', x2, ')=', repr(myModel(x1, x2)))
grad = myModel.partialGradient(x1, x2)
print('dCov =', repr(grad))
gradfd = ot.NumericalPoint(spatialDimension)
for j in range(spatialDimension):
x1_d = ot.NumericalPoint(x1)
x1_d[j] = x1_d[j] + eps
gradfd[j] = (myModel(x1_d, x2)[0, 0] - myModel(x1, x2)[0, 0]) / eps
print('dCov (FD)=', repr(gradfd))
else:
print('myModel(', x1, ', ', x2, ')=', repr(myModel(x1, x2)))
grad = myModel.partialGradient(x1, x2)
print('dCov =', repr(grad))
gradfd = ot.Matrix(spatialDimension, dimension * dimension)
covarianceX1X2 = myModel(x1, x2)
# Symmetrize matrix
covarianceX1X2.getImplementation().symmetrize()
centralValue = ot.NumericalPoint(covarianceX1X2.getImplementation())
# Loop over the shifted points
for i in range(spatialDimension):
currentPoint = ot.NumericalPoint(x1)
currentPoint[i] += eps
localCovariance = myModel(currentPoint, x2)
localCovariance.getImplementation().symmetrize()
currentValue = ot.NumericalPoint(
localCovariance.getImplementation())
for j in range(currentValue.getSize()):
gradfd[i, j] = (currentValue[j] - centralValue[j]) / eps
print('dCov (FD)=', repr(gradfd))
def _PODgaussModel(self, defects, stderr, linearModel):
X = ot.NumericalSample(defects.getSize(), [1, 0])
X[:, 1] = defects
X = ot.Matrix(X)
# compute the prediction variance of the linear regression model
def predictionVariance(x):
Y = ot.NumericalPoint([1.0, x])
gramX = X.computeGram()
return stderr**2 * (1. + ot.dot(Y, gramX.solveLinearSystem(Y)))
# function to compute the POD(defect)
def PODmodel(x):
t = (self._detectionBoxCox - linearModel(x[0])) / np.sqrt(predictionVariance(x[0]))
# DistFunc.pNormal(t,True) = complementary CDF of the Normal(0,1)
return [ot.DistFunc.pNormal(t,True)]
return PODmodel
def test_one_input_one_output():
sampleSize = 6
dimension = 1
f = ot.SymbolicFunction(['x0'], ['x0 * sin(x0)'])
X = ot.Sample(sampleSize, dimension)
X2 = ot.Sample(sampleSize, dimension)
for i in range(sampleSize):
X[i, 0] = 3.0 + i
X2[i, 0] = 2.5 + i
X[0, 0] = 1.0
X[1, 0] = 3.0
X2[0, 0] = 2.0
X2[1, 0] = 4.0
Y = f(X)
Y2 = f(X2)
# create algorithm
basis = ot.ConstantBasisFactory(dimension).build()
covarianceModel = ot.SquaredExponential([1e-02], [4.50736])
algo = ot.KrigingAlgorithm(X, Y, covarianceModel, basis)
algo.run()
# perform an evaluation
result = algo.getResult()
ott.assert_almost_equal(result.getMetaModel()(X), Y)
ott.assert_almost_equal(result.getResiduals(), [1.32804e-07], 1e-3, 1e-3)
ott.assert_almost_equal(result.getRelativeErrors(), [5.20873e-21])
# Kriging variance is 0 on learning points
covariance = result.getConditionalCovariance(X)
covariancePoint = ot.Point(covariance.getImplementation())
theoricalVariance = ot.Point(sampleSize * sampleSize)
ott.assert_almost_equal(covariance,
ot.Matrix(sampleSize, sampleSize),
8.95e-7, 8.95e-7)
# Covariance per marginal & extract variance component
coll = result.getConditionalMarginalCovariance(X)
var = [mat[0, 0] for mat in coll]
ott.assert_almost_equal(var, [0]*sampleSize, 1e-14, 1e-14)
# Variance per marginal
var = result.getConditionalMarginalVariance(X)
ott.assert_almost_equal(var, ot.Point(sampleSize), 1e-14, 1e-14)
def getQ2(self):
"""
Accessor to the Q2 value.
Returns
-------
Q2 : float
The Q2 value computed analytically.
"""
basisMatrix = ot.Matrix(self._basisFunction(self._input))
gramBasis = basisMatrix.computeGram()
H = basisMatrix * gramBasis.solveLinearSystem(basisMatrix.transpose())
Hdiag = np.vstack(np.array(H).diagonal())
fittedSignals = np.array(self._chaosPred(self._input))
delta = np.array(self._signals - fittedSignals) / (1. - Hdiag)
return 1 - np.mean(delta**2) / self._signals.computeVariance()[0]
def test_model(myModel):
print("myModel = ", myModel)
spatialDimension = myModel.getSpatialDimension()
dimension = myModel.getDimension()
x1 = ot.NumericalPoint(spatialDimension)
x2 = ot.NumericalPoint(spatialDimension)
for j in range(spatialDimension):
x1[j] = -1.0 - j
x2[j] = 3.0 + 2.0 * j
eps = 1e-5
if (dimension == 1):
print("myModel(", x1, ", ", x2, ")=", myModel(x1, x2))
grad = myModel.partialGradient(x1, x2)
print("dCov =", grad)
gradfd = ot.NumericalPoint(spatialDimension)
for j in range(spatialDimension):
x1_d = ot.NumericalPoint(x1)
x1_d[j] = x1_d[j] + eps
gradfd[j] = (myModel(x1_d, x2)[0, 0] - myModel(x1, x2)[0, 0]) / eps
print("dCov (FD)=", gradfd)
else:
print("myModel(", x1, ", ", x2, ")=", repr(myModel(x1, x2)))
grad = myModel.partialGradient(x1, x2)
print("dCov =", repr(grad))
gradfd = ot.Matrix(spatialDimension, dimension * dimension)
covarianceX1X2 = myModel(x1, x2);
# Symmetrize matrix
covarianceX1X2.getImplementation().symmetrize()
centralValue = ot.NumericalPoint(covarianceX1X2.getImplementation())
# Loop over the shifted points
for i in range(spatialDimension):
currentPoint = ot.NumericalPoint(x1)
currentPoint[i] += eps
localCovariance = myModel(currentPoint, x2);
localCovariance.getImplementation().symmetrize()
currentValue = ot.NumericalPoint(localCovariance.getImplementation())
for j in range(currentValue.getSize()):
gradfd[i, j] = (currentValue[j] - centralValue[j]) / eps;
print("dCov (FD)=", repr(gradfd))
sampler.setBurnIn(2000)
# get a realization
realization = sampler.getRealization()
print('y1=', realization)
# try to generate a sample
sampleSize = 1000
sample = sampler.getSample(sampleSize)
x_mu = sample.computeMean()
x_sigma = sample.computeStandardDeviation()
# compute covariance
x_cov = sample.computeCovariance()
P = ot.Matrix(obsSize, chainDim)
for i in range(obsSize):
for j in range(chainDim):
P[i, j] = p[i, j]
Qn = P.transpose() * P + Q0
Qn_inv = ot.SquareMatrix(chainDim)
for j in range(chainDim):
I_j = [0] * chainDim
I_j[j] = 1.0
Qn_inv_j = Qn.solveLinearSystem(I_j)
for i in range(chainDim):
Qn_inv[i, j] = Qn_inv_j[i]
sigma_exp = [0] * chainDim
for i in range(chainDim):
sigma_exp[i] = m.sqrt(Qn_inv[i, i])
def test_one_input_one_output():
sampleSize = 6
dimension = 1
f = ot.SymbolicFunction(['x0'], ['x0 * sin(x0)'])
X = ot.Sample(sampleSize, dimension)
X2 = ot.Sample(sampleSize, dimension)
for i in range(sampleSize):
X[i, 0] = 3.0 + i
X2[i, 0] = 2.5 + i
X[0, 0] = 1.0
X[1, 0] = 3.0
X2[0, 0] = 2.0
X2[1, 0] = 4.0
Y = f(X)
Y2 = f(X2)
# create covariance model
basis = ot.ConstantBasisFactory(dimension).build()
covarianceModel = ot.SquaredExponential()
# create algorithm
algo = ot.KrigingAlgorithm(X, Y, covarianceModel, basis)
# set sensible optimization bounds and estimate hyperparameters
algo.setOptimizationBounds(ot.Interval(X.getMin(), X.getMax()))
algo.run()
# perform an evaluation
result = algo.getResult()
ott.assert_almost_equal(result.getMetaModel()(X), Y)
ott.assert_almost_equal(result.getResiduals(), [1.32804e-07], 1e-3, 1e-3)
ott.assert_almost_equal(result.getRelativeErrors(), [5.20873e-21])
# Kriging variance is 0 on learning points
covariance = result.getConditionalCovariance(X)
nullMatrix = ot.Matrix(sampleSize, sampleSize)
ott.assert_almost_equal(covariance, nullMatrix, 0.0, 1e-13)
# Kriging variance is non-null on validation points
validCovariance = result.getConditionalCovariance(X2)
values = ot.Matrix([[
0.81942182, -0.35599947, -0.17488593, 0.04622401, -0.03143555,
0.04054783
],
[
-0.35599947, 0.20874735, 0.10943841, -0.03236419,
0.02397483, -0.03269184
],
[
-0.17488593, 0.10943841, 0.05832917, -0.01779918,
0.01355719, -0.01891618
],
[
0.04622401, -0.03236419, -0.01779918, 0.00578327,
-0.00467674, 0.00688697
],
[
-0.03143555, 0.02397483, 0.01355719, -0.00467674,
0.0040267, -0.00631173
],
[
0.04054783, -0.03269184, -0.01891618, 0.00688697,
-0.00631173, 0.01059488
]])
ott.assert_almost_equal(validCovariance - values, nullMatrix, 0.0, 1e-7)
# Covariance per marginal & extract variance component
coll = result.getConditionalMarginalCovariance(X)
var = [mat[0, 0] for mat in coll]
ott.assert_almost_equal(var, [0] * sampleSize, 1e-14, 1e-13)
# Variance per marginal
var = result.getConditionalMarginalVariance(X)
ott.assert_almost_equal(var, ot.Sample(sampleSize, 1), 1e-14, 1e-13)
# Prediction accuracy
ott.assert_almost_equal(Y2, result.getMetaModel()(X2), 0.3, 0.0)
sampler.setBurnIn(2000)
# get a realization
realization = sampler.getRealization()
print('y1=', realization)
# try to generate a sample
sampleSize = 1000
sample = sampler.getSample(sampleSize)
x_mu = sample.computeMean()
x_sigma = sample.computeStandardDeviation()
# compute covariance
x_cov = sample.computeCovariance()
P = ot.Matrix(obsSize, chainDim)
for i in range(obsSize):
for j in range(chainDim):
P[i, j] = p[i, j]
Qn = P.transpose() * P + Q0
Qn_inv = ot.SquareMatrix(chainDim)
for j in range(chainDim):
I_j = [0] * chainDim
I_j[j] = 1.0
Qn_inv_j = Qn.solveLinearSystem(I_j)
for i in range(chainDim):
Qn_inv[i, j] = Qn_inv_j[i]
sigma_exp = [0] * chainDim
for i in range(chainDim):
sigma_exp[i] = m.sqrt(Qn_inv[i, i])
p4[0] = 102.
p4[1] = 202.
s1[2] = p4
myStudy.add('mySample', s1)
# Add a point with a description
pDesc = ot.PointWithDescription(p1)
desc = pDesc.getDescription()
desc[0] = 'x'
desc[1] = 'y'
desc[2] = 'z'
pDesc.setDescription(desc)
myStudy.add(pDesc)
# Add a matrix
matrix = ot.Matrix(2, 3)
matrix[0, 0] = 0
matrix[0, 1] = 1
matrix[0, 2] = 2
matrix[1, 0] = 3
matrix[1, 1] = 4
matrix[1, 2] = 5
myStudy.add('m', matrix)
# Create a Point that we will try to reinstaciate after reloading
point = ot.Point(2, 1000.)
point.setName('point')
myStudy.add('point', point)
# Create a Simulation::Result
simulationResult = ot.ProbabilitySimulationResult(ot.ThresholdEvent(), 0.5,
from __future__ import print_function
import openturns as ot
t_names = [
'Matrix', 'SquareMatrix', 'TriangularMatrix', 'SymmetricMatrix',
'CovarianceMatrix', 'CorrelationMatrix'
]
t_names.extend([
'ComplexMatrix', 'HermitianMatrix', 'TriangularComplexMatrix',
'SquareComplexMatrix'
])
for i, iname in enumerate(t_names):
# try conversion
ref = ot.Matrix([[1.0, 0.0], [0.0, 0.5]])
a = getattr(ot, iname)(ref)
print('a=', a)
# try scalar mul
try:
s = 5.
ats = a * s
print('a*s=', ats)
sta = s * a
print('s*a=', sta)
except:
print('no scalar mul for', iname)
# try scalar div
try:
sampleSize = 3
X = ot.Sample(sampleSize, 1)
for i in range(sampleSize):
X[i, 0] = i + 1.0
Y = ot.Sample(sampleSize, 1)
phis = []
for j in range(basisSize):
phis.append(ot.SymbolicFunction(['x'], ['x^' + str(j + 1)]))
basis = ot.Basis(phis)
for i in range(basisSize):
print(ot.FunctionCollection(basis)[i](X))
proxy = ot.DesignProxy(X, basis)
full = range(basisSize)
design = proxy.computeDesign(full)
print(design)
proxy.setWeight([0.5] * sampleSize)
design = proxy.computeDesign(full)
print(design)
proxy = ot.DesignProxy(ot.Matrix(design))
full = range(basisSize)
design = proxy.computeDesign(full)
print(design)
def test_model(myModel, test_partial_grad=True, x1=None, x2=None):
inputDimension = myModel.getInputDimension()
dimension = myModel.getOutputDimension()
if x1 is None and x2 is None:
x1 = ot.Point(inputDimension)
x2 = ot.Point(inputDimension)
for j in range(inputDimension):
x1[j] = -1.0 - j
x2[j] = 3.0 + 2.0 * j
else:
x1 = ot.Point(x1)
x2 = ot.Point(x2)
if myModel.isStationary():
ott.assert_almost_equal(myModel(x1 - x2), myModel(x1, x2), 1e-14,
1e-14)
ott.assert_almost_equal(myModel(x2 - x1), myModel(x1, x2), 1e-14,
1e-14)
eps = 1e-3
mesh = ot.IntervalMesher([9] * inputDimension).build(
ot.Interval([-10] * inputDimension, [10] * inputDimension))
C = myModel.discretize(mesh)
if dimension == 1:
# Check that discretize & computeAsScalar provide the
# same values
vertices = mesh.getVertices()
for j in range(len(vertices)):
for i in range(j, len(vertices)):
ott.assert_almost_equal(
C[i, j], myModel.computeAsScalar(vertices[i], vertices[j]),
1e-14, 1e-14)
else:
# Check that discretize & operator() provide the
# same values
vertices = mesh.getVertices()
localMatrix = ot.SquareMatrix(dimension)
for j in range(len(vertices)):
for i in range(j, len(vertices)):
for localJ in range(dimension):
for localI in range(dimension):
localMatrix[localI, localJ] = C[i * dimension + localI,
j * dimension + localJ]
ott.assert_almost_equal(localMatrix,
myModel(vertices[i], vertices[j]),
1e-14, 1e-14)
if test_partial_grad:
grad = myModel.partialGradient(x1, x2)
if (dimension == 1):
gradfd = ot.Matrix(inputDimension, 1)
for j in range(inputDimension):
x1_g = ot.Point(x1)
x1_d = ot.Point(x1)
x1_g[j] = x1_d[j] + eps
x1_d[j] = x1_d[j] - eps
gradfd[j, 0] = (myModel.computeAsScalar(x1_g, x2) -
myModel.computeAsScalar(x1_d, x2)) / (2 * eps)
else:
gradfd = ot.Matrix(inputDimension, dimension * dimension)
covarianceX1X2 = myModel(x1, x2)
centralValue = ot.Point(covarianceX1X2.getImplementation())
# Loop over the shifted points
for i in range(inputDimension):
currentPoint = ot.Point(x1)
currentPoint[i] += eps
localCovariance = myModel(currentPoint, x2)
currentValue = ot.Point(localCovariance.getImplementation())
for j in range(currentValue.getSize()):
gradfd[i, j] = (currentValue[j] - centralValue[j]) / eps
ott.assert_almost_equal(grad, gradfd, 1e-5, 1e-5,
"in " + myModel.getClassName() + " grad")
from matplotlib import pyplot as plt
from openturns.viewer import View
from math import sqrt
domain = ot.Interval(-1.0, 1.0)
basis = ot.OrthogonalProductFunctionFactory([ot.FourierSeriesFactory()])
basisSize = 10
experiment = ot.GaussProductExperiment(basis.getMeasure(), [20])
mustScale = False
threshold = 0.001
factory = ot.KarhunenLoeveQuadratureFactory(domain, experiment, basis,
basisSize, mustScale, threshold)
model = ot.AbsoluteExponential([1.0])
ev = ot.NumericalPoint()
functions = factory.build(model, ev)
g = ot.Graph()
g.setXTitle("$t$")
g.setYTitle("$\sqrt{\lambda_n}\phi_n$")
for i in range(functions.getSize()):
g.add((functions.build(i) * ot.LinearNumericalMathFunction(
ot.NumericalPoint(domain.getDimension()),
ot.NumericalPoint(1, sqrt(ev[i])), ot.Matrix(
1, domain.getDimension()))).draw(-1.0, 1.0, 256))
g.setColors(ot.Drawable.BuildDefaultPalette(functions.getSize()))
fig = plt.figure(figsize=(6, 4))
plt.suptitle("P1 approx. of KL expansion for $C(s,t)=e^{-|s-t|}$")
axis = fig.add_subplot(111)
axis.set_xlim(auto=True)
View(g, figure=fig, axes=[axis], add_legend=False)
#! /usr/bin/env python
import openturns as ot
import pickle
from io import BytesIO
obj_list = []
obj_list.append(ot.Point([1.6, -8.7]))
obj_list.append(ot.Sample([[4.6, -3.7], [8.4, 6.3]]))
obj_list.append(ot.Description(['x', 'y', 'z']))
obj_list.append(ot.Indices([1, 2, 4]))
obj_list.append(ot.Matrix([[1, 2], [3, 4]]))
obj_list.append(ot.SymbolicFunction(['x1', 'x2'], ['y1=x1+x2']))
src = BytesIO()
for obj in obj_list:
pickle.dump(obj, src)
src.seek(0)
for obj in obj_list:
obj2 = pickle.load(src)
print(('object: ' + str(obj)))
print(('same: ' + str(obj2 == obj) + '\n'))
def test_model(myModel, test_partial_grad=True, x1=None, x2=None):
inputDimension = myModel.getInputDimension()
dimension = myModel.getOutputDimension()
if x1 is None and x2 is None:
x1 = ot.Point(inputDimension)
x2 = ot.Point(inputDimension)
for j in range(inputDimension):
x1[j] = -1.0 - j
x2[j] = 3.0 + 2.0 * j
else:
x1 = ot.Point(x1)
x2 = ot.Point(x2)
if myModel.isStationary():
ott.assert_almost_equal(myModel(x1 - x2), myModel(x1, x2), 1e-14,
1e-14)
ott.assert_almost_equal(myModel(x2 - x1), myModel(x1, x2), 1e-14,
1e-14)
eps = 1e-3
mesh = ot.IntervalMesher([7] * inputDimension).build(
ot.Interval([-10] * inputDimension, [10] * inputDimension))
C = myModel.discretize(mesh)
if dimension == 1:
# Check that discretize & computeAsScalar provide the
# same values
vertices = mesh.getVertices()
for j in range(len(vertices)):
for i in range(j, len(vertices)):
ott.assert_almost_equal(
C[i, j], myModel.computeAsScalar(vertices[i], vertices[j]),
1e-14, 1e-14)
else:
# Check that discretize & operator() provide the same values
vertices = mesh.getVertices()
localMatrix = ot.SquareMatrix(dimension)
for j in range(len(vertices)):
for i in range(j, len(vertices)):
for localJ in range(dimension):
for localI in range(dimension):
localMatrix[localI, localJ] = C[i * dimension + localI,
j * dimension + localJ]
ott.assert_almost_equal(localMatrix,
myModel(vertices[i], vertices[j]),
1e-14, 1e-14)
# Now we suppose that discretize is ok
# we look at crossCovariance of (vertices, vertices) which should return the same values
C.getImplementation().symmetrize()
crossCov = myModel.computeCrossCovariance(vertices, vertices)
ott.assert_almost_equal(
crossCov, C, 1e-14, 1e-14,
"in " + myModel.getClassName() + "::computeCrossCovariance")
# Now crossCovariance(sample, sample) is ok
# Let us validate crossCovariance(Sample, point) with 1st column(s) of previous calculations
crossCovSamplePoint = myModel.computeCrossCovariance(vertices, vertices[0])
crossCovCol = crossCov.reshape(crossCov.getNbRows(), dimension)
ott.assert_almost_equal(
crossCovSamplePoint, crossCovCol, 1e-14, 1e-14,
"in " + myModel.getClassName() + "::computeCrossCovarianceSamplePoint")
if test_partial_grad:
grad = myModel.partialGradient(x1, x2)
if (dimension == 1):
gradfd = ot.Matrix(inputDimension, 1)
for j in range(inputDimension):
x1_g = ot.Point(x1)
x1_d = ot.Point(x1)
x1_g[j] = x1_d[j] + eps
x1_d[j] = x1_d[j] - eps
gradfd[j, 0] = (myModel.computeAsScalar(x1_g, x2) -
myModel.computeAsScalar(x1_d, x2)) / (2 * eps)
else:
gradfd = ot.Matrix(inputDimension, dimension * dimension)
covarianceX1X2 = myModel(x1, x2)
centralValue = ot.Point(covarianceX1X2.getImplementation())
# Loop over the shifted points
for i in range(inputDimension):
currentPoint = ot.Point(x1)
currentPoint[i] += eps
localCovariance = myModel(currentPoint, x2)
currentValue = ot.Point(localCovariance.getImplementation())
for j in range(currentValue.getSize()):
gradfd[i, j] = (currentValue[j] - centralValue[j]) / eps
ott.assert_almost_equal(grad, gradfd, 1e-5, 1e-5,
"in " + myModel.getClassName() + " grad")
collFactories = [
ot.UniformFactory(),
ot.NormalFactory(),
ot.TriangularFactory(),
ot.ExponentialFactory(),
ot.GammaFactory()
]
#, TrapezoidalFactory()
result, norms = distribution.project(collFactories)
print("projections=", result)
print("norms=", norms)
# ------------------------------ Multivariate tests ------------------------------#
# 2D RandomMixture
collection = [ot.Normal(0.0, 1.0)] * 3
weightMatrix = ot.Matrix(2, 3)
weightMatrix[0, 0] = 1.0
weightMatrix[0, 1] = -2.0
weightMatrix[0, 2] = 1.0
weightMatrix[1, 0] = 1.0
weightMatrix[1, 1] = 1.0
weightMatrix[1, 2] = -3.0
# Build the RandomMixture
distribution2D = ot.RandomMixture(collection, weightMatrix)
print("distribution = ", distribution2D)
print("range = ", distribution2D.getRange())
print("mean = ", distribution2D.getMean())
print("cov = ", distribution2D.getCovariance())
print("sigma = ", distribution2D.getStandardDeviation())
distribution2D.setBlockMin(3)
#! /usr/bin/env python
import openturns as ot
ref_values = [[1.0, 0.0], [0.0, 0.5]]
mats = [ot.Matrix(ref_values),
ot.SquareMatrix(ref_values),
ot.TriangularMatrix(ref_values),
ot.SymmetricMatrix(ref_values),
ot.CovarianceMatrix(ref_values),
ot.CorrelationMatrix(ref_values)]
mats.extend([
ot.ComplexMatrix(ref_values),
ot.HermitianMatrix(ref_values),
ot.TriangularComplexMatrix(ref_values),
ot.SquareComplexMatrix(ref_values)])
for a in mats:
# try conversion
ref = ot.Matrix([[1.0, 0.0], [0.0, 0.5]])
iname = a.__class__.__name__
print('a=', a)
# try scalar mul
try:
s = 5.
ats = a * s
print('a*s=', ats)
sta = s * a
# .. math::
# f : \underline{X} \mapsto \underline{\underline{A}} ( \underline{X} - \underline{b} ) + \underline{c} + \frac{1}{2} \underline{X}^T \times \underline{\underline{\underline{M}}} \times \underline{X}
#
# %%
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)
# %%
# create a quadratic function
inputDimension = 3
outputDimension = 2
center = [1.0] * inputDimension
constant = [-1.0, 2.0] # c
linear = ot.Matrix(inputDimension, outputDimension) # A
quadratic = ot.SymmetricTensor(inputDimension, outputDimension) # M
quadratic[0, 0, 1] = 3.0
function = ot.QuadraticFunction(center, constant, linear, quadratic)
x = [7.0, 8.0, 9.0]
print(function(x))
# %%
# draw y1 with x1=2.0, x2=1.0, x0 in [0, 2]
graph = ot.ParametricFunction(
function, [1, 2], [2.0, 1.0]).getMarginal(1).draw(0.0, 2.0)
view = viewer.View(graph)
plt.show()
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.NumericalMathFunction(
ot.NumericalMathFunction(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.NumericalSample(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')
import openturns as ot
from openturns.viewer import View
center = [0.0]
constant = [3.0]
linear = ot.Matrix([[2.0]])
f = ot.LinearFunction(center, constant, linear)
graph = f.draw(0.0, 10.0)
graph.setTitle('$y=2x+3$')
View(graph, figure_kwargs={'figsize': (8, 4)}, add_legend=True)
# %%
# We see that there is a large covariance matrix diagonal.
#
# Let us compute a 95% confidence interval for the solution :math:`\theta^\star`.
# %%
print(distributionPosterior.computeBilateralConfidenceIntervalWithMarginalProbability(0.95)[0])
# %%
# The confidence interval is *very* large. In order to clarify the situation, we compute the Jacobian matrix of the model at the candidate point.
# %%
mycf.setParameter(thetaPrior)
thetaDim = len(thetaPrior)
jacobianMatrix = ot.Matrix(nbobs,thetaDim)
for i in range(nbobs):
jacobianMatrix[i,:] = mycf.parameterGradient(Qobs[i]).transpose()
print(jacobianMatrix[0:5,:])
# %%
# The rank of the problem can be seen from the singular values of the Jacobian matrix.
# %%
print(jacobianMatrix.computeSingularValues())
# %%
# We can see that there are two singular values which are relatively close to zero.
#
# This explains why the Jacobian matrix is close to being rank-degenerate.
#
print("determinant=%.6f" % determinant)
ev = ot.ScalarCollection(2)
ev = matrix1.computeEigenValues()
print("ev=" + repr(ev))
if matrix1.isPositiveDefinite():
isSPD = "true"
else:
isSPD = "false"
print("isSPD=", isSPD)
matrix2 = matrix1.computeCholesky()
print("matrix2=" + repr(matrix2))
b = ot.Matrix(2, 3)
b[0, 0] = 5.0
b[1, 0] = 0.0
b[0, 1] = 10.0
b[1, 1] = 1.0
b[0, 2] = 15.0
b[1, 2] = 2.0
result2 = matrix1.solveLinearSystem(b, True)
print("result2=" + repr(result2))
matrix3 = ot.CovarianceMatrix(3)
matrix3[1, 0] = float('nan')
try:
print("ev=", matrix3.computeSingularValues())
except:
print("ok")