def drawSample(self, inputSample, outputSample):
"""
Draw the sample of an event.
The points inside and outside the event are colored.
Parameters
----------
inputSample: an ot.Sample
The input 2D sample.
outputSample: an ot.Sample
The output 1D sample.
Returns
-------
None.
"""
if inputSample.getDimension() != 2:
raise ValueError("The input dimension of the input sample "
"is equal to %d but should be 2." %
(inputSample.getDimension()))
#
threshold = self.event.getThreshold()
g = self.event.getFunction()
operator = self.event.getOperator()
#
sampleSize = outputSample.getSize()
insideIndices = []
outsideIndices = []
for i in range(sampleSize):
y = outputSample[i, 0]
isInside = operator(y, threshold)
if isInside:
insideIndices.append(i)
else:
outsideIndices.append(i)
#
insideSample = ot.Sample([inputSample[i] for i in insideIndices])
outsideSample = ot.Sample([inputSample[i] for i in outsideIndices])
#
description = g.getInputDescription()
title = "Points X s.t. g(X) %s %s" % (operator, threshold)
graph = ot.Graph(title, description[0], description[1], True, "")
if len(insideIndices) > 0:
cloud = ot.Cloud(insideSample, self.insideEventPointColor,
"fsquare", "In")
graph.add(cloud)
if len(outsideIndices) > 0:
cloud = ot.Cloud(outsideSample, self.outsideEventPointColor,
"fsquare", "Out")
graph.add(cloud)
graph.setLegendPosition("topright")
return graph
def draw(dist, Y):
g = ot.Graph()
g.setAxes(True)
g.setGrid(True)
c = ot.Cloud(dist.getSample(10000))
c.setColor("red")
c.setPointStyle("bullet")
g.add(c)
c = ot.Cloud(Y)
c.setColor("black")
c.setPointStyle("bullet")
g.add(c)
g.setBoundingBox(ot.Interval(
Y.getMin()-0.5*Y.computeRange(), Y.getMax()+0.5*Y.computeRange()))
return g
def myPolynomialCurveFittingGraph(total_degree, x_train, y_train):
"""Returns the graphics for a polynomial curve fitting
with given total degree"""
responseSurface, basis = myPolynomialDataFitting(total_degree, x_train,
y_train)
# Graphics
n_test = 100
x_test = linearSample(0, 1, n_test)
ypredicted_test = responseSurface(basis(x_test))
# Graphics
graph = ot.Graph("Polynomial curve fitting", "x", "y", True, "topright")
# The "unknown" function
curve = g.draw(0, 1)
curve.setColors(["green"])
graph.add(curve)
# Training set
cloud = ot.Cloud(x_train, y_train)
cloud.setPointStyle("circle")
cloud.setLegend("N=%d" % (x_train.getSize()))
graph.add(cloud)
# Predictions
curve = ot.Curve(x_test, ypredicted_test)
curve.setLegend("Polynomial Degree = %d" % (total_degree))
curve.setColor("red")
graph.add(curve)
return graph
def DrawUnivariateSampleDistribution(sample, distribution):
"""
Draw a unidimensional sample and its distribution.
Parameters
----------
sample : ot.Sample
A dimension 1 sample.
distribution : ot.Distribution
A dimension 1 distribution.
Returns
-------
graph : ot.Graph
The PDF plot and the sample on the X axis.
"""
if sample.getDimension()!=1:
raise ValueError("Expect a 1 dimension sample, but dimension is %d" % (sample.getDimension()))
if distribution.getDimension()!=1:
raise ValueError("Expect a 1 dimension distribution, but dimension is %d" % (distribution.getDimension()))
graph = distribution.drawPDF()
# Add points on X axis
sample_size = sample.getSize()
data = ot.Sample(sample_size, 2)
data[:, 0] = sample
cloud = ot.Cloud(data)
graph.add(cloud)
return graph
def drawLevelSetContour2D(distribution,
numberOfPointsInXAxis,
alpha,
threshold,
sampleSize=500):
'''
Compute the minimum volume LevelSet of measure equal to alpha and get the
corresponding density value (named threshold).
Generate a sample of the distribution and draw it.
Draw a contour plot for the distribution, where the PDF is equal to threshold.
'''
sample = distribution.getSample(sampleSize)
X1min = sample[:, 0].getMin()[0]
X1max = sample[:, 0].getMax()[0]
X2min = sample[:, 1].getMin()[0]
X2max = sample[:, 1].getMax()[0]
xx = ot.Box([numberOfPointsInXAxis], ot.Interval([X1min],
[X1max])).generate()
yy = ot.Box([numberOfPointsInXAxis], ot.Interval([X2min],
[X2max])).generate()
xy = ot.Box([numberOfPointsInXAxis, numberOfPointsInXAxis],
ot.Interval([X1min, X2min], [X1max, X2max])).generate()
data = distribution.computePDF(xy)
graph = ot.Graph('', 'X1', 'X2', True, 'topright')
labels = ["%.2f%%" % (100 * alpha)]
contour = ot.Contour(xx, yy, data, [threshold], labels)
contour.setColor('black')
graph.setTitle("%.2f%% of the distribution, sample size = %d" %
(100 * alpha, sampleSize))
graph.add(contour)
cloud = ot.Cloud(sample)
graph.add(cloud)
return graph
def drawBidimensionalSample(sample, title):
n = sample.getSize()
graph = ot.Graph("%s, size=%d" % (title, n), r"$X_1$", r"$X_2$", True, '')
#cloud = ot.Cloud(sample)
cloud = ot.Cloud(sample, "blue", "fsquare", "")
graph.add(cloud)
return graph
def _inliers_outliers(self, sample=None, inliers=True):
"""Inliers or outliers cloud drawing.
:param sample: Sample of size (n_samples, n_dims).
:type sample: :class:`openturns.Sample`
:param bool inliers: Whether to draw inliers or outliers.
:return: OpenTURNS Cloud or Pair object if :attr:`dim` > 2.
:rtype: :class:`openturns.Cloud` or :class:`openturns.Pairs`
"""
if inliers:
idx = self.computeOutlierIndices(False)
legend = "Inliers at alpha=%.4f" % (self.outlierAlpha)
marker_color = 'blue'
else:
idx = self.computeOutlierIndices()
legend = "Outliers at alpha=%.4f" % (self.outlierAlpha)
marker_color = 'red'
if sample is None:
sample = np.array(self.sample)
else:
sample = np.asarray(sample)
sample = sample[idx, :]
if sample.size == 0:
return
if sample.shape[1] == 2:
cloud = ot.Cloud(sample, marker_color, self.data_marker, legend)
else:
cloud = ot.Pairs(sample, '', self.sample.getDescription(),
marker_color, self.data_marker)
return cloud
def plotMyBasicKriging(krigResult, xMin, xMax, X, Y, level = 0.95):
'''
Given a kriging result, plot the data, the kriging metamodel
and a confidence interval.
'''
samplesize = X.getSize()
meta = krigResult.getMetaModel()
graphKriging = meta.draw(xMin, xMax)
graphKriging.setLegends(["Kriging"])
# Create a grid of points and evaluate the function and the kriging
nbpoints = 50
xGrid = linearSample(xMin,xMax,nbpoints)
yFunction = g(xGrid)
yKrig = meta(xGrid)
# Compute the conditional covariance
epsilon = ot.Point(nbpoints,1.e-8)
conditionalVariance = krigResult.getConditionalMarginalVariance(xGrid)+epsilon
conditionalVarianceSample = ot.Sample([[cv] for cv in conditionalVariance])
conditionalSigma = sqrt(conditionalVarianceSample)
# Compute the quantile of the Normal distribution
alpha = 1-(1-level)/2
quantileAlpha = ot.DistFunc.qNormal(alpha)
# Graphics of the bounds
epsilon = 1.e-8
dataLower = [yKrig[i,0] - quantileAlpha * conditionalSigma[i,0] for i in range(nbpoints)]
dataUpper = [yKrig[i,0] + quantileAlpha * conditionalSigma[i,0] for i in range(nbpoints)]
# Coordinates of the vertices of the Polygons
vLow = [[xGrid[i,0],dataLower[i]] for i in range(nbpoints)]
vUp = [[xGrid[i,0],dataUpper[i]] for i in range(nbpoints)]
# Compute the Polygon graphics
boundsPoly = plot_kriging_bounds(vLow,vUp,nbpoints)
boundsPoly.setLegend("95% bounds")
# Validate the kriging metamodel
mmv = ot.MetaModelValidation(xGrid, yFunction, meta)
Q2 = mmv.computePredictivityFactor()[0]
# Plot the function
graphFonction = ot.Curve(xGrid,yFunction)
graphFonction.setLineStyle("dashed")
graphFonction.setColor("magenta")
graphFonction.setLineWidth(2)
graphFonction.setLegend("Function")
# Draw the X and Y observed
cloudDOE = ot.Cloud(X, Y)
cloudDOE.setPointStyle("circle")
cloudDOE.setColor("red")
cloudDOE.setLegend("Data")
# Assemble the graphics
graph = ot.Graph()
graph.add(boundsPoly)
graph.add(graphFonction)
graph.add(cloudDOE)
graph.add(graphKriging)
graph.setLegendPosition("bottomright")
graph.setAxes(True)
graph.setGrid(True)
graph.setTitle("Size = %d, Q2=%.2f%%" % (samplesize,100*Q2))
graph.setXTitle("X")
graph.setYTitle("Y")
return graph
def _drawObservationsVsInputs1Dimension(self, inputObservations,
outputObservations, outputAtPrior,
outputAtPosterior):
"""
Plots the observed output of the model depending
on the observed input before and after calibration.
Can manage only 1D samples.
"""
xDim = inputObservations.getDimension()
if (xDim != 1):
raise TypeError('Input observations are not 1D.')
yDim = outputObservations.getDimension()
xdescription = inputObservations.getDescription()
ydescription = outputObservations.getDescription()
graph = ot.Graph("", xdescription[0], ydescription[0], True,
"topright")
# Observations
if (yDim == 1):
cloud = ot.Cloud(inputObservations, outputObservations)
cloud.setColor(self.observationColor)
cloud.setLegend("Observations")
graph.add(cloud)
else:
raise TypeError('Output observations are not 1D.')
# Model outputs before calibration
yPriorDim = outputAtPrior.getDimension()
if (yPriorDim == 1):
cloud = ot.Cloud(inputObservations, outputAtPrior)
cloud.setColor(self.priorColor)
cloud.setLegend("Prior")
graph.add(cloud)
else:
raise TypeError('Output prior predictions are not 1D.')
# Model outputs after calibration
yPosteriorDim = outputAtPosterior.getDimension()
if (yPosteriorDim == 1):
cloud = ot.Cloud(inputObservations, outputAtPosterior)
cloud.setColor(self.posteriorColor)
cloud.setLegend("Posterior")
graph.add(cloud)
else:
raise TypeError('Output posterior predictions are not 1D.')
return graph
def plotXvsY(sampleX, sampleY, figsize=(15, 3)):
dimX = sampleX.getDimension()
inputdescr = sampleX.getDescription()
fig = pl.figure(figsize=figsize)
for i in range(dimX):
ax = fig.add_subplot(1, dimX, i + 1)
graph = ot.Graph('', inputdescr[i], 'Y', True, '')
cloud = ot.Cloud(sampleX[:, i], sampleY)
graph.add(cloud)
_ = ot.viewer.View(graph, figure=fig, axes=[ax])
return fig
def plot_exact_model():
graph = ot.Graph('', 'x', '', True, '')
y_test = model(x_test)
curveModel = ot.Curve(x_test, y_test)
curveModel.setLineStyle("solid")
curveModel.setColor("black")
graph.add(curveModel)
cloud = ot.Cloud(Xtrain, Ytrain)
cloud.setColor("black")
cloud.setPointStyle("fsquare")
graph.add(cloud)
return graph
def drawLevelSet1D(distribution, levelSet, alpha, threshold, sampleSize = 100):
'''
Draw a 1D sample included in a given levelSet.
The sample is generated from the distribution.
'''
inLevelSample = computeSampleInLevelSet(distribution,levelSet,sampleSize)
cloudSample = from1Dto2Dsample(inLevelSample)
graph = distribution.drawPDF()
mycloud = ot.Cloud(cloudSample)
graph.add(mycloud)
graph.setTitle("%.2f%% of the distribution, sample size = %d, " % (100*alpha, sampleSize))
return graph
def _drawObservationsVsPredictions1Dimension(self, outputObservations,
outputAtPrior,
outputAtPosterior):
"""
Plots the output of the model depending
on the output observations before and after calibration.
Can manage only 1D samples.
"""
yDim = outputObservations.getDimension()
ydescription = outputObservations.getDescription()
xlabel = "%s Observations" % (ydescription[0])
ylabel = "%s Predictions" % (ydescription[0])
graph = ot.Graph("", xlabel, ylabel, True, "topleft")
# Plot the diagonal
if (yDim == 1):
curve = ot.Curve(outputObservations, outputObservations)
curve.setColor(self.observationColor)
graph.add(curve)
else:
raise TypeError('Output observations are not 1D.')
# Plot the predictions before
yPriorDim = outputAtPrior.getDimension()
if (yPriorDim == 1):
cloud = ot.Cloud(outputObservations, outputAtPrior)
cloud.setColor(self.priorColor)
cloud.setLegend("Prior")
graph.add(cloud)
else:
raise TypeError('Output prior predictions are not 1D.')
# Plot the predictions after
yPosteriorDim = outputAtPosterior.getDimension()
if (yPosteriorDim == 1):
cloud = ot.Cloud(outputObservations, outputAtPosterior)
cloud.setColor(self.posteriorColor)
cloud.setLegend("Posterior")
graph.add(cloud)
else:
raise TypeError('Output posterior predictions are not 1D.')
return graph
def drawIFS(f_i,
skip=100,
iterations=1000,
batch_size=1,
name="IFS",
color="blue"):
# Any set of initial points should work in theory
initialPoints = ot.Normal(2).getSample(batch_size)
# Compute the contraction factor of each function
all_r = [m.sqrt(abs(f[1].computeDeterminant())) for f in f_i]
# Find the box counting dimension, ie the value s such that r_1^s+...+r_n^s-1=0
equation = "-1.0"
for r in all_r:
equation += "+" + str(r) + "^s"
dim = len(f_i)
s = ot.Brent().solve(ot.SymbolicFunction("s", equation), 0.0, 0.0,
-m.log(dim) / m.log(max(all_r)))
# Add a small perturbation to sample even the degenerated transforms
probabilities = [r**s + 1e-2 for r in all_r]
# Build the sampling distribution
support = [[i] for i in range(dim)]
choice = ot.UserDefined(support, probabilities)
currentPoints = initialPoints
points = ot.Sample(0, 2)
# Convert the f_i into LinearEvaluation to benefit from the evaluation over
# a Sample
phi_i = [ot.LinearEvaluation([0.0] * 2, f[0], f[1]) for f in f_i]
# Burning phase
for i in range(skip):
index = int(round(choice.getRealization()[0]))
currentPoints = phi_i[index](currentPoints)
# Iteration phase
for i in range(iterations):
index = int(round(choice.getRealization()[0]))
currentPoints = phi_i[index](currentPoints)
points.add(currentPoints)
# Draw the IFS
graph = ot.Graph()
graph.setTitle(name)
graph.setXTitle("x")
graph.setYTitle("y")
graph.setGrid(True)
cloud = ot.Cloud(points)
cloud.setColor(color)
cloud.setPointStyle("dot")
graph.add(cloud)
return graph, s
def build(self, dataX, dataY):
logLikelihood = ot.NumericalMathFunction(ReducedLogLikelihood(dataX, dataY))
xlb = np.linspace(self.lambdaMin_,self.lambdaMax_,num=500)
lambdax = [logLikelihood([x])[0] for x in xlb]
algo = ot.TNC(logLikelihood)
algo.setStartingPoint([xlb[np.array(lambdax).argmax()]])
algo.setBoundConstraints(ot.Interval(self.lambdaMin_, self.lambdaMax_))
algo.setOptimizationProblem(ot.BoundConstrainedAlgorithmImplementationResult.MAXIMIZATION)
algo.run()
optimalLambda = algo.getResult().getOptimizer()[0]
# graph
optimalLogLikelihood = algo.getResult().getOptimalValue()
graph = logLikelihood.draw(0.01 * optimalLambda, 10.0 * optimalLambda)
c = ot.Cloud([[optimalLambda, optimalLogLikelihood]])
c.setColor("red")
c.setPointStyle("circle")
graph.add(c)
return ot.BoxCoxTransform([optimalLambda]), graph
def _inliers_outliers(self, sample, inliers=True):
"""Inliers or outliers cloud drawing."""
# Perform selection
if inliers:
idx = self.inlier_indices
legend = "Inliers at alpha=%.4f" % (self.outlierAlpha)
marker_color = self.inlier_color
else:
idx = self.outlier_indices
legend = "Outliers at alpha=%.4f" % (self.outlierAlpha)
marker_color = self.outlier_color
sample_selection = sample[idx, :]
if sample_selection.getSize() == 0:
return
cloud = ot.Cloud(sample_selection, marker_color, self.data_marker,
legend)
return cloud
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('$k$')
graph_ev.setYTitle(r'$\lambda_i$')
graph_ev.setAxes(True)
graph_ev.setGrid(True)
graph_ev.setLogScale(2)
bb = graph_ev.getBoundingBox()
lower = bb.getLowerBound()
lower[1] = 1.0e-7
bb = ot.Interval(lower, bb.getUpperBound())
graph_ev.setBoundingBox(bb)
return graph_modes, graph_ev
# %%
sigmaObservationNoiseH = 0.1 # (m)
noiseH = ot.Normal(0.,sigmaObservationNoiseH)
sampleNoiseH = noiseH.getSample(nbobs)
Hobs = outputH + sampleNoiseH
# %%
# Plot the Y observations versus the X observations.
# %%
Qobs = inputSample[:,0]
# %%
graph = ot.Graph("Observations","Q (m3/s)","H (m)",True)
cloud = ot.Cloud(Qobs,Hobs)
graph.add(cloud)
view = viewer.View(graph)
# %%
# Setting the calibration parameters
# ----------------------------------
# %%
# Define the value of the reference values of the :math:`\theta` parameter. In the bayesian framework, this is called the mean of the *prior* normal distribution. In the data assimilation framework, this is called the *background*.
# %%
KsInitial = 20.
ZvInitial = 49.
ZmInitial = 51.
thetaPrior = [KsInitial, ZvInitial, ZmInitial]
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
ot.RandomGenerator.SetSeed(0)
factory = ot.GeometricFactory()
ref = factory.build()
dimension = ref.getDimension()
if dimension <= 2:
sample = ref.getSample(50)
distribution = factory.build(sample)
if dimension == 1:
distribution.setDescription(['$t$'])
pdf_graph = distribution.drawPDF(256)
cloud = ot.Cloud(sample, ot.Sample(sample.getSize(), 1))
cloud.setColor('blue')
cloud.setPointStyle('fcircle')
pdf_graph.add(cloud)
fig = plt.figure(figsize=(10, 4))
plt.suptitle(str(distribution))
pdf_axis = fig.add_subplot(111)
View(pdf_graph, figure=fig, axes=[pdf_axis], add_legend=False)
else:
sample = ref.getSample(500)
distribution.setDescription(['$t_0$', '$t_1$'])
pdf_graph = distribution.drawPDF([256] * 2)
cloud = ot.Cloud(sample)
cloud.setColor('red')
cloud.setPointStyle('fcircle')
pdf_graph.add(cloud)
fig = plt.figure(figsize=(10, 4))
plt.suptitle(str(distribution))
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
size = 6
ot.RandomGenerator.SetSeed(3)
# Data
x1 = ot.Uniform(1.0, 9.0).getSample(size)
y1 = ot.Uniform(0.0, 120.0).getSample(size)
graph = ot.Graph("Non significant Spearman coefficient", "u", "v", True, "")
cloud1 = ot.Cloud(x1, y1)
cloud1.setPointStyle("diamond")
cloud1.setColor("orange")
cloud1.setLineWidth(2)
graph.add(cloud1)
size = 150 - size
# Data
x2 = ot.Uniform(1.0, 9.0).getSample(size)
y2 = ot.Uniform(0.0, 120.0).getSample(size)
# Merge with previous data
x = ot.Sample(x1)
y = ot.Sample(y1)
x.add(x2)
y.add(y2)
# Quadratic model
algo = ot.QuadraticLeastSquares(x, y)
algo.run()
quadratic = algo.getMetaModel()
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
# %%
# Create 2-d samples to visualize
N = 500
R = ot.CorrelationMatrix(2)
R[0, 1] = -0.7
# 2d N(1,1) with correlation
sample1 = ot.Normal([1.0] * 2, [1.0] * 2, R).getSample(N)
sample2 = ot.Normal(2).getSample(N) # 2d N(0,1) independent
# %%
# Create cloud drawables
cloud1 = ot.Cloud(sample1, 'blue', 'fsquare', 'First Cloud')
cloud2 = ot.Cloud(sample2, 'red', 'fsquare', 'Second Cloud')
# Then, assemble it into a graph
myGraph2d = ot.Graph('2d clouds', 'x1', 'x2', True, 'topright')
myGraph2d.add(cloud1)
myGraph2d.add(cloud2)
view = viewer.View(myGraph2d)
# %%
# Create a 3-d sample
mean = [0.0] * 3
sigma = [2.0, 1.5, 1.0]
R = ot.CorrelationMatrix(3)
R[0, 1] = 0.8
R[1, 2] = -0.5
from openturns.viewer import View
f = ot.SymbolicFunction(['x'], ['17-exp(0.1*(x-1.0))'])
graph = f.draw(0.0, 12.0)
dist = ot.Normal([5.0, 15.0], [1.0, 0.25], ot.IdentityMatrix(2))
N = 1000
sample = dist.getSample(N)
sample1 = ot.Sample(0, 2)
sample2 = ot.Sample(0, 2)
for X in sample:
x, y = X
if f([x])[0] > y:
sample1.add(X)
else:
sample2.add(X)
cloud = ot.Cloud(sample1)
cloud.setColor('green')
cloud.setPointStyle('square')
graph.add(cloud)
cloud = ot.Cloud(sample2)
cloud.setColor('red')
cloud.setPointStyle('square')
graph.add(cloud)
graph.setTitle('Monte Carlo simulation (Pf=0.048, N=1000)')
graph.setLegends(['domain Df', 'simulations'])
graph.setLegendPosition('topright')
View(graph)
# %%
# Scale the design proportionnally to the standard deviation of each component.
# %%
covariance = rv.getCovariance()
scaling = [m.sqrt(covariance[i, i]) for i in range(dim)]
print('scaling=', scaling)
sample *= scaling
# %%
# Center the design around the mean point of the distribution.
# %%
center = rv.getMean()
print('center=', center)
sample += center
# %%
# Draw the design as well as the distribution iso-values.
# %%
graph = distribution.drawPDF()
doe = ot.Cloud(sample)
doe.setColor('red')
doe.setLegend('design')
graph.add(doe)
view = viewer.View(graph)
plt.show()
# %%
size = 64
distribution = ot.ComposedDistribution(
[ot.Uniform(lowerbound[0], upperbound[0])] * dim)
experiment = ot.LowDiscrepancyExperiment(
ot.SobolSequence(), distribution, size)
solver = ot.MultiStart(ot.Cobyla(problem), experiment.generate())
# %%
# Visualize the starting points of the optimization algorithm
# -----------------------------------------------------------
# %%
startingPoints = solver.getStartingSample()
graph = rastrigin.draw(lowerbound, upperbound, [100]*dim)
graph.setTitle("Rastrigin function")
cloud = ot.Cloud(startingPoints)
cloud.setPointStyle("bullet")
cloud.setColor("black")
graph.add(cloud)
graph.setLegends([""])
# sphinx_gallery_thumbnail_number = 2
view = viewer.View(graph)
# %%
# We see that the starting points are well spread accross the input domain of the function.
# %%
# Solve the optimization problem
# ------------------------------
# %%
degrees = range(5, 12)
q2 = ot.Sample(len(degrees), 2)
for maximumDegree in degrees:
ot.ResourceMap.SetAsUnsignedInteger("FunctionalChaosAlgorithm-MaximumTotalDegree",maximumDegree)
print("Maximum total degree =", maximumDegree)
algo = ot.FunctionalChaosAlgorithm(inputSample, outputSample)
algo.run()
result = algo.getResult()
metamodel = result.getMetaModel()
for outputIndex in range(2):
val = ot.MetaModelValidation(inputTest, outputTest[:,outputIndex], metamodel.getMarginal(outputIndex))
q2[maximumDegree - degrees[0], outputIndex] = val.computePredictivityFactor()[0]
# %%
graph = ot.Graph("Predictivity","Total degree","Q2",True)
cloud = ot.Cloud([[d] for d in degrees], q2[:,0])
cloud.setLegend("Output #0")
cloud.setPointStyle("bullet")
graph.add(cloud)
cloud = ot.Cloud([[d] for d in degrees], q2[:,1])
cloud.setLegend("Output #1")
cloud.setColor("red")
cloud.setPointStyle("bullet")
graph.add(cloud)
graph.setLegendPosition("topright")
view = viewer.View(graph)
plt.show()
# %%
# We see that a total degree lower than 9 is not sufficient to describe the first output with good predictivity. However, the coefficient of predictivity drops when the total degree gets greater than 12.
# The predictivity of the second output seems to be much less satisfactory: a little more work would be required to improve the metamodel.
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
ot.RandomGenerator.SetSeed(0)
factory = ot.TriangularFactory()
ref = factory.build()
dimension = ref.getDimension()
if dimension <= 2:
sample = ref.getSample(50)
distribution = factory.build(sample)
if dimension == 1:
distribution.setDescription(['$t$'])
pdf_graph = distribution.drawPDF(256)
cloud = ot.Cloud(sample, ot.NumericalSample(sample.getSize(), 1))
cloud.setColor('blue')
cloud.setPointStyle('fcircle')
pdf_graph.add(cloud)
fig = plt.figure(figsize=(10, 4))
plt.suptitle(str(distribution))
pdf_axis = fig.add_subplot(111)
View(pdf_graph, figure=fig, axes=[pdf_axis], add_legend=False)
else:
sample = ref.getSample(500)
distribution.setDescription(['$t_0$', '$t_1$'])
pdf_graph = distribution.drawPDF([256]*2)
cloud = ot.Cloud(sample)
cloud.setColor('red')
cloud.setPointStyle('fcircle')
pdf_graph.add(cloud)
fig = plt.figure(figsize=(10, 4))
plt.suptitle(str(distribution))
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
f = ot.SymbolicFunction(['x'], ['sin(x)'])
a = -2.5
b = 4.5
algo = ot.Fejer1([20])
value, nodes = algo.integrateWithNodes(f, ot.Interval(a, b))
g = f.draw(a, b, 512)
lower = ot.Cloud(nodes, ot.Sample(nodes.getSize(), 1))
lower.setColor("magenta")
lower.setPointStyle('circle')
g.add(lower)
g.setTitle(r"GaussLegendre example: $\int_{-5/2}^{9/2}\sin(t)\,dt=$" +
str(value[0]))
fig = plt.figure(figsize=(8, 4))
axis = fig.add_subplot(111)
axis.set_xlim(auto=True)
View(g, figure=fig, axes=[axis], add_legend=False)
import openturns as ot
from openturns.viewer import View
N = 20
ot.RandomGenerator.SetSeed(10)
x = ot.Uniform(0.0, 10.0).getSample(N)
f = ot.SymbolicFunction(['x'], ['5*x+10'])
y = f(x) + ot.Normal(0.0, 5.0).getSample(N)
graph = f.draw(0.0, 10.0)
graph.setTitle(
'... because the rank transformation turns any monotonic trend\ninto a linear relation for which Pearson\'s correlation is relevant'
)
graph.setXTitle('u')
graph.setYTitle('v')
cloud = ot.Cloud(x, y)
cloud.setPointStyle('circle')
cloud.setColor('orange')
graph.add(cloud)
View(graph)
delta = xexact - xoptim
absoluteError = delta.norm()
absoluteError
# %%
# We see that the algorithm found an accurate approximation of the solution.
# %%
result.getOptimalValue() # f(x*)
# %%
result.getEvaluationNumber()
# %%
graph = rosenbrock.draw(lowerbound, upperbound, [100] * 2)
cloud = ot.Cloud(ot.Sample([x0, xoptim]))
cloud.setColor("black")
cloud.setPointStyle("bullet")
graph.add(cloud)
graph.setTitle("Rosenbrock function")
view = viewer.View(graph)
# %%
# We see that the algorithm had to start from the top left of the banana and go to the top right.
# %%
graph = result.drawOptimalValueHistory()
view = viewer.View(graph)
# %%
# The function value history make the path of the algorithm clear. In the first step, the algorithm went in the valley, which made the function value decrease rapidly. Once there, the algorithm had to follow the bottom of the valley so that the function decreased but slowly. In the final steps, the algorithm found the neighbourhood of the minimum so that the local convergence could take place.
ref_func_with_error = ot.SymbolicFunction(['x', 'eps'], ['x * sin(x) + eps'])
ref_func = ot.ParametricFunction(ref_func_with_error, [1], [0.0])
x = np.vstack(np.linspace(xmin, xmax, n_pt))
ot.RandomGenerator.SetSeed(1235)
eps = ot.Normal(0, 1.5).getSample(n_pt)
X = ot.Sample(n_pt, 2)
X[:, 0] = x
X[:, 1] = eps
if with_error:
y = np.array(ref_func_with_error(X))
else:
y = np.array(ref_func(x))
# %%
graph = ref_func.draw(xmin, xmax, 200)
cloud = ot.Cloud(x, y)
cloud.setColor('red')
cloud.setPointStyle('bullet')
graph.add(cloud)
graph.setLegends(["Function", "Data"])
graph.setLegendPosition("topleft")
graph.setTitle("Sample size = %d" % (n_pt))
view = viewer.View(graph)
# %%
# Create the kriging algorithm
# ----------------------------
# 1. basis
ot.ResourceMap.SetAsBool(
'GeneralLinearModelAlgorithm-UseAnalyticalAmplitudeEstimate', True)
