def test_SquaredExponential(self):
# With 2 principal components
setup_HDRenv()
# Test with no outlier in the band
xmin = 0.0
step = 0.1
n = 100
timeGrid = ot.RegularGrid(xmin, step, n + 1)
amplitude = [7.0]
scale = [1.5]
covarianceModel = ot.SquaredExponential(scale, amplitude)
process = ot.GaussianProcess(covarianceModel, timeGrid)
nbTrajectories = 50
processSample = process.getSample(nbTrajectories)
# KL decomposition
reduction = othdr.KarhunenLoeveDimensionReductionAlgorithm(
processSample, 2)
reduction.run()
reducedComponents = reduction.getReducedComponents()
# Distribution fit in reduced space
ks = ot.KernelSmoothing()
reducedDistribution = ks.build(reducedComponents)
hdr = othdr.ProcessHighDensityRegionAlgorithm(processSample,
reducedComponents,
reducedDistribution,
[0.95, 0.5])
hdr.run()
graph = hdr.draw()
otv.View(graph)
def test_NonZeroMean(self):
# Create the KL result
numberOfVertices = 10
interval = ot.Interval(-1.0, 1.0)
mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval)
covariance = ot.SquaredExponential()
zeroProcess = ot.GaussianProcess(covariance, mesh)
# Define a trend function
f = ot.SymbolicFunction(["t"], ["30 * t"])
fTrend = ot.TrendTransform(f, mesh)
# Add it to the process
process = ot.CompositeProcess(fTrend, zeroProcess)
# Sample
sampleSize = 100
processSample = process.getSample(sampleSize)
threshold = 0.0
algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold)
algo.run()
klresult = algo.getResult()
# Create the KL reduction
meanField = processSample.computeMean()
klreduce = ot.KarhunenLoeveReduction(klresult)
# Generate a trajectory and reduce it
field = process.getRealization()
values = field.getValues()
reducedValues = klreduce(values)
ott.assert_almost_equal(values, reducedValues)
def test_KarhunenLoeveValidationMultidimensional(self):
# Create the KL result
numberOfVertices = 20
interval = ot.Interval(-1.0, 1.0)
mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval)
outputDimension = 2
univariateCovariance = ot.SquaredExponential()
covarianceCollection = [univariateCovariance] * outputDimension
multivariateCovariance = ot.TensorizedCovarianceModel(
covarianceCollection)
process = ot.GaussianProcess(multivariateCovariance, mesh)
sampleSize = 100
sampleSize = 10
processSample = process.getSample(sampleSize)
threshold = 1.0e-7
algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold)
algo.run()
klresult = algo.getResult()
# Create the validation
validation = ot.KarhunenLoeveValidation(processSample, klresult)
# Check residuals
residualProcessSample = validation.computeResidual()
assert (type(residualProcessSample) is ot.ProcessSample)
# Check standard deviation
residualSigmaField = validation.computeResidualStandardDeviation()
zeroSample = ot.Sample(numberOfVertices, outputDimension)
ott.assert_almost_equal(residualSigmaField, zeroSample)
# Check graph
graph = validation.drawValidation()
if False:
from openturns.viewer import View
View(graph).save('validation2.png')
def plotCovarianceModel(myCovarianceModel, myTimeGrid, nbTrajectories):
'''Plots the given number of trajectories with given covariance model.'''
process = ot.GaussianProcess(myCovarianceModel, myTimeGrid)
sample = process.getSample(nbTrajectories)
graph = sample.drawMarginal(0)
graph.setTitle("")
return graph
def test_KarhunenLoeveValidation(self):
# Create the KL result
numberOfVertices = 20
interval = ot.Interval(-1.0, 1.0)
mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval)
covariance = ot.SquaredExponential()
process = ot.GaussianProcess(covariance, mesh)
sampleSize = 100
processSample = process.getSample(sampleSize)
threshold = 1.0e-7
algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold)
algo.run()
klresult = algo.getResult()
# Create validation
validation = ot.KarhunenLoeveValidation(processSample, klresult)
# Check residuals
residualProcessSample = validation.computeResidual()
assert (type(residualProcessSample) is ot.ProcessSample)
# Check standard deviation
residualSigmaField = validation.computeResidualStandardDeviation()
exact = ot.Sample(numberOfVertices, 1)
#ott.assert_almost_equal(residualSigmaField, exact)
# Check mean
residualMean = validation.computeResidualMean()
exact = ot.Sample(numberOfVertices, 1)
#ott.assert_almost_equal(residualMean, exact)
# Check graph
graph0 = validation.drawValidation()
graph1 = residualProcessSample.drawMarginal(0)
graph2 = residualMean.drawMarginal(0)
graph3 = residualSigmaField.drawMarginal(0)
graph4 = validation.drawObservationWeight(0)
graph5 = validation.drawObservationQuality()
if 0:
from openturns.viewer import View
View(graph0).save('validation1.png')
View(graph1).save('validation1-residual.png')
View(graph2).save('validation1-residual-mean.png')
View(graph3).save('validation1-residual-stddev.png')
View(graph4).save('validation1-indiv-weight.png')
View(graph5).save('validation1-indiv-quality.png')
def test_ZeroMean(self):
# Create the KL result
numberOfVertices = 10
interval = ot.Interval(-1.0, 1.0)
mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval)
covariance = ot.SquaredExponential()
process = ot.GaussianProcess(covariance, mesh)
sampleSize = 10
processSample = process.getSample(sampleSize)
threshold = 0.0
algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold)
algo.run()
klresult = algo.getResult()
# Create the KL reduction
meanField = processSample.computeMean()
klreduce = ot.KarhunenLoeveReduction(klresult)
# Generate a trajectory and reduce it
field = process.getRealization()
values = field.getValues()
reducedValues = klreduce(values)
ott.assert_almost_equal(values, reducedValues)
def test_trend(self):
N = 100
M = 1000
P = 10
mean = ot.SymbolicFunction("x", "sign(x)")
cov = ot.SquaredExponential([1.0], [0.1])
mesh = ot.IntervalMesher([N]).build(ot.Interval(-2.0, 2.0))
process = ot.GaussianProcess(ot.TrendTransform(mean, mesh), cov, mesh)
sample = process.getSample(M)
algo = ot.KarhunenLoeveSVDAlgorithm(sample, 1e-6)
algo.run()
result = algo.getResult()
trend = ot.TrendTransform(
ot.P1LagrangeEvaluation(sample.computeMean()), mesh)
sample2 = process.getSample(P)
sample2.setName('reduction of sign(x) w/o trend')
reduced1 = ot.KarhunenLoeveReduction(result)(sample2)
reduced2 = ot.KarhunenLoeveReduction(result, trend)(sample2)
g = sample2.drawMarginal(0)
g.setColors(["red"])
g1 = reduced1.drawMarginal(0)
g1.setColors(["blue"])
drs = g1.getDrawables()
for i, d in enumerate(drs):
d.setLineStyle("dashed")
drs[i] = d
g1.setDrawables(drs)
g.add(g1)
g2 = reduced2.drawMarginal(0)
g2.setColors(["green"])
drs = g2.getDrawables()
for i, d in enumerate(drs):
d.setLineStyle("dotted")
drs[i] = d
g2.setDrawables(drs)
g.add(g2)
if 0:
from openturns.viewer import View
View(g).save('reduction.png')
# %%
# Input model
print("Create the input process")
# Domain bound
a = 1
# Reference correlation length
b = 0.5
# Number of vertices in the mesh
N = 100
# Bandwidth of the smoothers
h = 0.05
mesh = ot.IntervalMesher([N - 1]).build(ot.Interval(-a, a))
covariance_X = ot.AbsoluteExponential([b])
process_X = ot.GaussianProcess(covariance_X, mesh)
# %%
# for some pretty graphs
def drawKL(scaledKL, KLev, mesh, title="Scaled KL modes"):
graph_modes = scaledKL.drawMarginal()
graph_modes.setTitle(title + " scaled KL modes")
graph_modes.setXTitle('$x$')
graph_modes.setYTitle(r'$\sqrt{\lambda_i}\phi_i$')
data_ev = [[i, KLev[i]] for i in range(scaledKL.getSize())]
graph_ev = ot.Graph()
graph_ev.add(ot.Curve(data_ev))
graph_ev.add(ot.Cloud(data_ev))
graph_ev.setTitle(title + " KL eigenvalues")
graph_ev.setXTitle('$k$')
# which mesh is of box of dimension 2
myIndices = ot.Indices([80, 40])
myMesher = ot.IntervalMesher(myIndices)
lowerBound = [0., 0.]
upperBound = [2., 1.]
myInterval = ot.Interval(lowerBound, upperBound)
myMesh = myMesher.build(myInterval)
# Define a bidimensional temporal Gaussian process on the mesh
# with independent components
amplitude = [1.0, 1.0]
scale = [0.2, 0.3]
myCovModel = ot.ExponentialModel(scale, amplitude)
myXtProcess_temp = ot.GaussianProcess(myCovModel, myMesh)
# Non linear transformation of myXtProcess
# to get a positive process
g2 = ot.SymbolicFunction(['x1', 'x2'], ['x1^2', 'abs(x2)'])
myDynTransform = ot.ValueFunction(g2, 2)
myXtProcess = ot.CompositeProcess(myDynTransform, myXtProcess_temp)
# Create the image Y
myYtProcess = ot.CompositeProcess(myDynFunc, myXtProcess)
# Get the antecedent : myXtProcess
print('My antecedent process = ', myYtProcess.getAntecedent())
# Get the dynamical function
# which performs the transformation
# Set Numerical precision to 4
ot.PlatformInfo.SetNumericalPrecision(4)
sampleSize = 40
inputDimension = 1
# Create the function to estimate
model = ot.SymbolicFunction(["x0"], ["x0"])
X = ot.Sample(sampleSize, inputDimension)
for i in range(sampleSize):
X[i, 0] = 3.0 + (8.0 * i) / sampleSize
Y = model(X)
# Add a small noise to data
Y += ot.GaussianProcess(ot.AbsoluteExponential([0.1], [0.2]),
ot.Mesh(X)).getRealization().getValues()
basis = ot.LinearBasisFactory(inputDimension).build()
# Case of a misspecified covariance model
covarianceModel = ot.DiracCovarianceModel(inputDimension)
print("===================================================\n")
algo = ot.GeneralLinearModelAlgorithm(X, Y, covarianceModel, basis)
algo.run()
result = algo.getResult()
print("\ncovariance (dirac, optimized)=", result.getCovarianceModel())
print("trend (dirac, optimized)=", result.getTrendCoefficients())
print("===================================================\n")
# Now without estimating covariance parameters
basis = ot.LinearBasisFactory(inputDimension).build()
from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) # %% # Create a time grid tMin = 0.0 timeStep = 0.1 n = 100 tgrid = ot.RegularGrid(tMin, timeStep, n) # %% # Create a normal process amplitude = [5.0] scale = [3.0] model = ot.ExponentialModel(scale, amplitude) process = ot.GaussianProcess(model, tgrid) # %% # Create the 1-d domain A: [2.,5.] lowerBound = [2.0] upperBound = [5.0] domain = ot.Interval(lowerBound, upperBound) # %% # Create an event from a Process and a Domain event = ot.ProcessEvent(process, domain) # %% # Create the Monte-Carlo algorithm montecarlo = ot.ProbabilitySimulationAlgorithm(event)
# # %% from __future__ import print_function import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) # %% # Create a Gaussian process numberOfVertices = 20 interval = ot.Interval(-1.0, 1.0) mesh = ot.IntervalMesher([numberOfVertices - 1]).build(interval) covariance = ot.SquaredExponential() process = ot.GaussianProcess(covariance, mesh) # %% # decompose it using KL-SVD sampleSize = 100 processSample = process.getSample(sampleSize) threshold = 1.0e-7 algo = ot.KarhunenLoeveSVDAlgorithm(processSample, threshold) algo.run() klresult = algo.getResult() # %% # Instanciate the validation service validation = ot.KarhunenLoeveValidation(processSample, klresult) # %%
def _exec(self, X):
f = ot.Function(
ot.PiecewiseLinearEvaluation(
[x[0] for x in self.getInputMesh().getVertices()], X))
outputValues = ot.Sample(0, 1)
for t in self.getOutputMesh().getVertices():
kernel = ot.Normal(t[0], 0.05)
def pdf(X):
return [kernel.computePDF(X)]
weight = ot.Function(ot.PythonFunction(1, 1, pdf))
outputValues.add(
self.algo_.integrate(weight * f, kernel.getRange()))
return outputValues
N = 5
X = ot.GaussianProcess(ot.GeneralizedExponential([0.1], 1.0), mesh)
f = ot.FieldFunction(GaussianConvolution())
Y = ot.CompositeProcess(f, X)
x_graph = X.getSample(N).drawMarginal(0)
y_graph = Y.getSample(N).drawMarginal(0)
fig = plt.figure(figsize=(10, 4))
plt.suptitle("Composite process")
x_axis = fig.add_subplot(121)
y_axis = fig.add_subplot(122)
View(x_graph, figure=fig, axes=[x_axis], add_legend=False)
View(y_graph, figure=fig, axes=[y_axis], add_legend=False)
amplitude = [1.0, 2.0, 3.0]
# %%
# We define a spatial correlation :
spatialCorrelation = ot.CorrelationMatrix(3)
spatialCorrelation[0, 1] = 0.8
spatialCorrelation[0, 2] = 0.6
spatialCorrelation[1, 2] = 0.1
# %%
# The covariance model is now created with :
myCovarianceModel = ot.ExponentialModel(scale, amplitude, spatialCorrelation)
# %%
# Eventually, the process is built with :
process = ot.GaussianProcess(myCovarianceModel, time_grid)
# %%
# The dimension d of the process may be retrieved by
dim = process.getOutputDimension()
print("Dimension : %d" % dim)
# %%
# The underlying mesh of the process is obtained with the `getMesh` method :
mesh = process.getMesh()
# %%
# We have access to peculiar data of the mesh such as the corners :
minMesh = mesh.getVertices().getMin()[0]
maxMesh = mesh.getVertices().getMax()[0]
#
# %%
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 process
grid = ot.RegularGrid(0.0, 0.1, 10)
amplitude = [5.0]
scale = [0.2]
covModel = ot.ExponentialModel(scale, amplitude)
X = ot.GaussianProcess(covModel, grid)
# %%
# Draw a sample
sample = X.getSample(6)
sample.setName('X')
graph = sample.drawMarginal(0)
view = viewer.View(graph)
# %%
# Define a trend function
f = ot.SymbolicFunction(['t'], ['30*t'])
fTrend = ot.TrendTransform(f, grid)
# %%
# Add it to the process
tmax = myMesh.getEnd()
# Create the collection of HermitianMatrix
covariance = ot.CovarianceMatrix(N)
for k in range(N):
s = myMesh.getValue(k)
for l in range(k + 1):
t = myMesh.getValue(l)
covariance[k, l] = C(s, t)
# Create the covariance model
myCovarianceModel = ot.UserDefinedCovarianceModel(myMesh, covariance)
# Create the non stationary Gaussian process with
# that covariance model
myProcess = ot.GaussianProcess(myCovarianceModel, myMesh)
# Create a sample of fields
size = 10**4
myFieldSample = myProcess.getSample(size)
# Build a covariance model factory
myFactory = ot.NonStationaryCovarianceModelFactory()
# Estimation on a the ProcessSample
myEstimatedModel = myFactory.build(myFieldSample)
# Define the python function associated to myCovarianceModel
def covMod(X):
# Sample [int64] indexing
s0 = ot.Sample(5, 3)
idx = np.array([1, 3, 4])
print('sample[[int64]]:', s0[idx])
s0[idx] = ot.Normal(3).getSample(3)
print('sample[[int64]]=Sample:', s0)
# generic int64 indexing
s0 = ot.Description(5, 'aa')
idx = np.int64(2)
print('Description[int64]:', s0[idx])
s0[idx] = 'zou'
print('Description[int64]=str', s0[idx])
# generic [int64] indexing
s0 = ot.Description(5, 'aa')
idx = np.array([1, 3, 4])
print('Description[[int64]]:', s0[idx])
s0[idx] = ['zou'] * 3
print('Description[[int64]]=str', s0[idx])
# Field int64 indexing
mesher = ot.IntervalMesher([10, 5])
mesh = mesher.build(ot.Interval([0.0, 0.0], [2.0, 1.0]))
process = ot.GaussianProcess(ot.ExponentialModel([0.2] * 2, [1.0]), mesh)
field = process.getRealization()
idx = np.int64(2)
print('Field[int64]', field[idx])
field[idx] = [6.0]
print('Field[int64]=Point', field[idx])
amplitude = [1.0] * outputDimension
# Scale values
scale = [1.0] * inputDimension
tmin = 0.0
step = 0.1
n = 11
myTimeGrid = ot.RegularGrid(tmin, step, n)
size = 25
# Second order model with parameters
myCovModel = ot.ExponentialModel(scale, amplitude)
print("myCovModel = ", myCovModel)
myProcess1 = ot.GaussianProcess(myCovModel, myTimeGrid)
print("myProcess1 = ", myProcess1)
print("is stationary? ", myProcess1.isStationary())
myProcess1.setSamplingMethod(ot.GaussianProcess.CHOLESKY)
print("mean over ", size, " realizations = ",
myProcess1.getSample(size).computeMean())
myProcess1.setSamplingMethod(ot.GaussianProcess.GIBBS)
print("mean over ", size, " realizations = ",
myProcess1.getSample(size).computeMean())
# With constant trend
trend = ot.TrendTransform(ot.SymbolicFunction("t", "4.0"), myTimeGrid)
myProcess2 = ot.GaussianProcess(trend, myCovModel, myTimeGrid)
myProcess2.setSamplingMethod(ot.GaussianProcess.GIBBS)
print("myProcess2 = ", myProcess2)
print("is stationary? ", myProcess2.isStationary())
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
# Create a bivariate Gaussian process
myMesh = ot.IntervalMesher([39, 39]).build(ot.Interval([0.0] * 2, [1.0] * 2))
myCov = ot.GeneralizedExponential(2 * [0.1], 1.3)
myProcess = ot.GaussianProcess(myCov, myMesh)
myField = myProcess.getRealization()
graph = myField.drawMarginal(0, False)
fig = plt.figure(figsize=(8, 4))
plt.suptitle("A field")
axis = fig.add_subplot(111)
axis.set_xlim(auto=True)
View(graph, figure=fig, axes=[axis], add_legend=True)
ot.Log.Show(ot.Log.NONE) # %% # define a covariance model defaultDimension = 1 # Amplitude values amplitude = [1.0] * defaultDimension # Scale values scale = [1.0] * defaultDimension # Covariance model myModel = ot.AbsoluteExponential(scale, amplitude) # %% # define a mesh tmin = 0.0 step = 0.1 n = 11 myTimeGrid = ot.RegularGrid(tmin, step, n) # %% # create the process process = ot.GaussianProcess(myModel, myTimeGrid) print(process) # %% # draw a sample sample = process.getSample(6) graph = sample.drawMarginal(0) view = viewer.View(graph) plt.show()
import openturns as ot from matplotlib import pyplot as plt from openturns.viewer import View # Create a process on a regular grid myGrid = ot.RegularGrid(0.0, 0.1, 100) amplitude = [5.0] scale = [0.2] myCovModel = ot.ExponentialModel(scale, amplitude) myXProcess = ot.GaussianProcess(myCovModel, myGrid) # Create a trend fTrend = ot.SymbolicFunction(["t"], ["1+2*t+t^2"]) fTemp = ot.TrendTransform(fTrend, myGrid) # Add the trend to the process and get a field myYProcess = ot.CompositeProcess(fTemp, myXProcess) myYField = myYProcess.getRealization() # Create a TrendFactory myBasisSequenceFactory = ot.LARS() myFittingAlgorithm = ot.KFold() func1 = ot.SymbolicFunction(["t"], ["1"]) func2 = ot.SymbolicFunction(["t"], ["t"]) func3 = ot.SymbolicFunction(["t"], ["t^2"]) myBasis = ot.Basis([func1, func2, func3]) myTrendFactory = ot.TrendFactory(myBasisSequenceFactory, myFittingAlgorithm) # Estimate the trend
# %% # Define a 2-d mesh indices = ot.Indices([40, 20]) mesher = ot.IntervalMesher(indices) lowerBound = [0., 0.] upperBound = [2., 1.] interval = ot.Interval(lowerBound, upperBound) mesh = mesher.build(interval) # %% # Create the covariance model amplitude = [1.0, 2.0, 3.0] scale = [4.0, 5.0] spatialCorrelation = ot.CorrelationMatrix(3) spatialCorrelation[0, 1] = 0.8 spatialCorrelation[0, 2] = 0.6 spatialCorrelation[1, 2] = 0.1 covmodel = ot.ExponentialModel(scale, amplitude, spatialCorrelation) # Create a normal process process = ot.GaussianProcess(covmodel, mesh) # %% # Create a domain A in R^3: [0.8; 1.2]*[1.5; 1.6]*[0.5; 0.7] lowerBound = [0.8, 1.5, 0.5] upperBound = [1.2, 1.6, 0.7] domain = ot.Interval(lowerBound, upperBound) # Create the event event = ot.ProcessEvent(process, domain)
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
from math import sqrt
mesh = ot.IntervalMesher([128]).build(ot.Interval(-1.0, 1.0))
threshold = 0.001
model = ot.AbsoluteExponential([1.0])
sample = ot.GaussianProcess(model, mesh).getSample(100)
algo = ot.KarhunenLoeveSVDAlgorithm(sample, threshold)
algo.run()
ev = algo.getResult().getEigenValues()
modes = algo.getResult().getScaledModesAsProcessSample()
g = modes.drawMarginal(0)
g.setXTitle("$t$")
g.setYTitle("$\sqrt{\lambda_n}\phi_n$")
fig = plt.figure(figsize=(6, 4))
plt.suptitle("SVD 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
from __future__ import print_function
import openturns as ot
mesh = ot.IntervalMesher([9]).build(ot.Interval(-1.0, 1.0))
cov1D = ot.AbsoluteExponential([1.0])
algo = ot.KarhunenLoeveP1Algorithm(mesh, cov1D, 0.0)
algo.run()
result = algo.getResult()
lambd = result.getEigenvalues()
KLModes = result.getModesAsProcessSample()
print("KL modes=", KLModes)
print("KL eigenvalues=", lambd)
process = ot.GaussianProcess(cov1D, KLModes.getMesh())
coefficients = result.project(process.getSample(10))
print("KL coefficients=", coefficients)
KLFunctions = result.getModes()
print("KL functions=", KLFunctions)
print("KL lift=", result.lift(coefficients[0]))
print("KL lift as field=", result.liftAsField(coefficients[0]))
R = ot.CorrelationMatrix(2)
R[0, 1] = 0.5
scale = [1.0]
amplitude = [1.0, 2.0]
cov2D = ot.ExponentialModel(scale, amplitude, R)
algo = ot.KarhunenLoeveP1Algorithm(mesh, cov2D, 0.0)
algo.run()
result = algo.getResult()
lambd = result.getEigenvalues()
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if ot.WhiteNoise().__class__.__name__ == 'Process':
# default to Gaussian for the interface class
process = ot.GaussianProcess()
elif ot.WhiteNoise().__class__.__name__ == 'DiscreteMarkovChain':
process = ot.WhiteNoise()
process.setTransitionMatrix(ot.SquareMatrix([[0.0,0.5,0.5],[0.7,0.0,0.3],[0.8,0.0,0.2]]))
origin = 0
process.setOrigin(origin)
else:
process = ot.WhiteNoise()
process.setTimeGrid(ot.RegularGrid(0.0, 0.02, 50))
process.setDescription(['$x$'])
sample = process.getSample(6)
sample_graph = sample.drawMarginal(0)
sample_graph.setTitle(str(process))
fig = plt.figure(figsize=(10, 4))
sample_axis = fig.add_subplot(111)
View(sample_graph, figure=fig, axes=[sample_axis], add_legend=False)
amplitude = [1.0] * dimension scale = [1] * dimension covarianceModel = ot.GeneralizedExponential(scale, amplitude, 2) # %% # We define the time grid on which we want to sample the Gaussian process. # %% tmin = 0.0 step = 0.01 n = 10001 timeGrid = ot.RegularGrid(tmin, step, n) # %% # Finally we define the Gaussian process. process = ot.GaussianProcess(covarianceModel, timeGrid) print(process) # %% # Basics on the HMatrix algebra # ----------------------------- # # The `HMatrix` framework uses efficient linear algebra techniques to speed-up the # (Cholesky) factorization of the covariance matrix. # This method can be tuned with several parameters. We should concentrate on the easiest ones. # %% # We set the sampling method to `HMAT` (default is the classical/dense case). process.setSamplingMethod(ot.GaussianProcess.HMAT) # %%
domain = ot.Interval(-1.0, 1.0)
basis = ot.OrthogonalProductPolynomialFactory([ot.LegendreFactory()])
basisSize = 5
experiment = ot.LHSExperiment(basis.getMeasure(), 100)
mustScale = False
threshold = 0.0001
model = ot.AbsoluteExponential([1.0])
algo = ot.KarhunenLoeveQuadratureAlgorithm(
domain, model, experiment, basis, basisSize, mustScale, threshold)
algo.run()
result = algo.getResult()
lambd = result.getEigenValues()
KLModes = result.getModesAsProcessSample()
print("KL modes=", KLModes)
print("KL eigenvalues=", lambd)
process = ot.GaussianProcess(model, KLModes.getMesh())
sample = process.getSample(10)
coefficients = result.project(sample)
print("KL coefficients=", coefficients)
KLFunctions = result.getModes()
print("KL functions=", KLFunctions)
print("KL lift=", result.lift(coefficients[0]))
print("KL lift as field=", result.liftAsField(coefficients[0]))
# Now using Legendre/Gauss quadrature
marginalDegree = 5
algo = ot.KarhunenLoeveQuadratureAlgorithm(
domain, model, marginalDegree, threshold)
algo.run()
result = algo.getResult()
lambd = result.getEigenValues()
KLModes = result.getScaledModesAsProcessSample()
#! /usr/bin/env python
from __future__ import print_function
import openturns as ot
ot.TESTPREAMBLE()
mesh = ot.IntervalMesher([9]).build(ot.Interval(-1.0, 1.0))
# 1D mesh, 1D covariance, uniform weight, automatic centering, more samples
# than vertices
cov1D = ot.AbsoluteExponential([1.0])
sample = ot.GaussianProcess(cov1D, mesh).getSample(16)
algo = ot.KarhunenLoeveSVDAlgorithm(sample, 0.0)
algo.run()
result = algo.getResult()
lambd = result.getEigenvalues()
KLModes = result.getModesAsProcessSample()
print("KL modes=", KLModes)
print("KL eigenvalues=", lambd)
coefficients = result.project(sample)
print("KL coefficients=", coefficients)
KLFunctions = result.getModes()
print("KL functions=", KLFunctions)
print("KL lift=", result.lift(coefficients[0]))
print("KL lift as field=", result.liftAsField(coefficients[0]))
# 1D mesh, 1D covariance, uniform weight, automatic centering
sample = ot.GaussianProcess(cov1D, mesh).getSample(6)
algo = ot.KarhunenLoeveSVDAlgorithm(sample, 0.0)
algo.run()
result = algo.getResult()
lambd = result.getEigenvalues()
# Define a bi dimensional mesh as a box
myIndices = ot.Indices([40, 20])
myMesher = ot.IntervalMesher(myIndices)
lowerBound = [0.0, 0.0]
upperBound = [2.0, 1.0]
myInterval = ot.Interval(lowerBound, upperBound)
myMesh = myMesher.build(myInterval)
# Define a scalar temporal Gaussian process on the mesh
# this process is stationary
# myXproc R^2 --> R
amplitude = [1.0]
scale = [0.2, 0.2]
myCovModel = ot.ExponentialModel(scale, amplitude)
myXproc = ot.GaussianProcess(myCovModel, myMesh)
# Transform myXproc to make its variance depend on the vertex (s,t)
# and to get a positive process
# thanks to the spatial function g
# myXtProcess R --> R
g = ot.SymbolicFunction(['x1'], ['exp(x1)'])
myDynTransform = ot.ValueFunction(g, 2)
myXtProcess = ot.CompositeProcess(myDynTransform, myXproc)
myField = myXtProcess.getRealization()
graphMarginal1 = ot.KernelSmoothing().build(myField.getValues()).drawPDF()
graphMarginal1.setTitle("")
graphMarginal1.setXTitle("X")
graphMarginal1.setLegendPosition("")
# %%
import openturns as ot
from openturns.viewer import View
# %%
# First build a process to generate the input data.
# We assemble a 4-d process from functional and Gaussian processes.
T = 3.0
NT = 32
tg = ot.RegularGrid(0.0, T / NT, NT)
f1 = ot.SymbolicFunction(['t'], ['sin(t)'])
f2 = ot.SymbolicFunction(['t'], ['cos(t)^2'])
coeff1_dist = ot.Normal([1.0] * 2, [0.6] * 2, ot.CorrelationMatrix(2))
p1 = ot.FunctionalBasisProcess(coeff1_dist, ot.Basis([f1, f2]), tg)
p2 = ot.GaussianProcess(ot.SquaredExponential([1.0], [T / 4.0]), tg)
coeff3_dist = ot.ComposedDistribution([ot.Uniform(), ot.Normal()])
f1 = ot.SymbolicFunction(["t"], ["1", "0"])
f2 = ot.SymbolicFunction(["t"], ["0", "1"])
p3 = ot.FunctionalBasisProcess(coeff3_dist, ot.Basis([f1, f2]))
X = ot.AggregatedProcess([p1, p2, p3])
X.setMesh(tg)
# %%
# Draw some input trajectories from our process
ot.RandomGenerator.SetSeed(0)
x = X.getSample(10)
graph = x.drawMarginal(0)
graph.setTitle(f'{x.getSize()} input trajectories')
_ = View(graph)