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 test_two_inputs_one_output():
# Kriging use case
inputDimension = 2
# Learning data
levels = [8, 5]
box = ot.Box(levels)
inputSample = box.generate()
# Scale each direction
inputSample *= 10.0
model = ot.SymbolicFunction(['x', 'y'], ['cos(0.5*x) + sin(y)'])
outputSample = model(inputSample)
# Validation
sampleSize = 10
inputValidSample = ot.ComposedDistribution(
2 * [ot.Uniform(0, 10.0)]).getSample(sampleSize)
outputValidSample = model(inputValidSample)
# 2) Definition of exponential model
# The parameters have been calibrated using TNC optimization
# and AbsoluteExponential models
scales = [5.33532, 2.61534]
amplitude = [1.61536]
covarianceModel = ot.SquaredExponential(scales, amplitude)
# 3) Basis definition
basis = ot.ConstantBasisFactory(inputDimension).build()
# 4) Kriging algorithm
algo = ot.KrigingAlgorithm(inputSample, outputSample, covarianceModel,
basis)
algo.run()
result = algo.getResult()
# Get meta model
metaModel = result.getMetaModel()
outData = metaModel(inputValidSample)
# 5) Errors
# Interpolation
ott.assert_almost_equal(outputSample, metaModel(inputSample), 3.0e-5,
3.0e-5)
# 6) Kriging variance is 0 on learning points
covariance = result.getConditionalCovariance(inputSample)
ott.assert_almost_equal(covariance, ot.SquareMatrix(len(inputSample)),
7e-7, 7e-7)
# Covariance per marginal & extract variance component
coll = result.getConditionalMarginalCovariance(inputSample)
var = [mat[0, 0] for mat in coll]
ott.assert_almost_equal(var, [0] * len(var), 0.0, 1e-13)
# Variance per marginal
var = result.getConditionalMarginalVariance(inputSample)
ott.assert_almost_equal(var, ot.Point(len(inputSample)), 0.0, 1e-13)
# Estimation
ott.assert_almost_equal(outputValidSample, outData, 1.e-1, 1e-1)
def drawLimitState(self, bounds, nX=50, nY=50):
"""
Draw the limit state of an event.
Parameters
----------
event : an ot.Event
The event we want to draw.
bounds: an ot.Interval
The lower and upper bounds of the interval.
nX : an int
The number of points in the X axis.
nY : an int
The number of points in the Y axis.
Returns
-------
None.
"""
if bounds.getDimension() != 2:
raise ValueError("The input dimension of the bounds "
"is equal to %d but should be 2." %
(bounds.getDimension()))
#
threshold = self.event.getThreshold()
g = self.event.getFunction()
#
boxExperiment = ot.Box([nX, nY], bounds)
inputSample = boxExperiment.generate()
outputSample = g(inputSample)
#
description = g.getInputDescription()
graph = ot.Graph("Limit state surface", description[0], description[1],
True, "")
# Create the contour
levels = ot.Point([threshold])
labels = [str(threshold)]
drawLabels = True
lowerBound = bounds.getLowerBound()
upperBound = bounds.getUpperBound()
x = LinearSample(lowerBound[0], upperBound[0], nX + 2)
y = LinearSample(lowerBound[1], upperBound[1], nY + 2)
contour = ot.Contour(x, y, outputSample, levels, labels, drawLabels)
graph.add(contour)
return graph
def drawLimitStateCrossCut(self, bounds, i=0, j=1, nX=50, nY=50):
"""
Draw the cross-cut of the limit state of an event on a cross-cut.
Parameters
----------
bounds: an ot.Interval
The lower and upper bounds of the cross-cut interval.
i : int
The index of the first marginal of the cross-cut.
j : int
The index of the second marginal of the cross-cut.
nX : an int
The number of points in the X axis.
nY : an int
The number of points in the Y axis.
Returns
-------
graph : ot.Graph
The plot of the (i, j) cross cut.
"""
if bounds.getDimension() != 2:
raise ValueError("The input dimension of the bounds "
"is equal to %d but should be 2." %
(bounds.getDimension()))
#
threshold = self.event.getThreshold()
description = self.g.getInputDescription()
boxExperiment = ot.Box([nX, nY], bounds)
reducedInputSample = boxExperiment.generate()
crosscutFunction = self.buildCrossCutFunction(i, j)
outputSample = crosscutFunction(reducedInputSample)
#
graph = ot.Graph("Limit state surface", description[i], description[j],
True, "")
# Create the contour
levels = ot.Point([threshold])
labels = [str(threshold)]
drawLabels = True
lowerBound = bounds.getLowerBound()
upperBound = bounds.getUpperBound()
x = LinearSample(lowerBound[0], upperBound[0], nX + 2)
y = LinearSample(lowerBound[1], upperBound[1], nY + 2)
contour = ot.Contour(x, y, outputSample, levels, labels, drawLabels)
graph.add(contour)
return graph
import openturns as ot
from openturns.viewer import View
# Box
d = ot.Box([3, 4, 5])
s = d.generate()
s.setDescription(["X1", "X2", "X3"])
g = ot.Graph()
g.setTitle("Box experiment")
g.setGridColor("black")
p = ot.Pairs(s)
g.add(p)
View(g)
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
# Generate sample with the given plane
levels = [4, 4]
myPlane = ot.Box(levels)
sample = myPlane.generate()
# Create an empty graph
graph = ot.Graph("Box plane", "x1", "x2", True, "")
# Create the cloud
cloud = ot.Cloud(sample, "blue", "fsquare", "")
# Then, draw it
graph.add(cloud)
fig = plt.figure(figsize=(4, 4))
axis = fig.add_subplot(111)
axis.set_xlim(auto=True)
View(graph, figure=fig, axes=[axis], add_legend=False)
#! /usr/bin/env python
import openturns as ot
levels = [4] * 2
experiment = ot.Box(levels)
print('experiment = ', experiment)
sample = experiment.generate()
print('sample = ', repr(sample))
experiment = ot.Box(levels, ot.Interval([7.0, 8.0], [9.0, 16.0]))
print('experiment = ', experiment)
sample = experiment.generate()
print('sample = ', repr(sample))
s = 10.0
event = ot.ThresholdEvent(vectorY, ot.Greater(), s)
# %%
# This event can easily be represented with a 1D curve as it is a branch of an hyperbole.
# If :math:`y = x_1 x_2 = 10.0`, then the boundary of the domain of failure is the curve :
#
# .. math::
#
# h : x_1 \mapsto \frac{10.0}{x_1}
#
# %%
# We shall represent this curve using a :class:`~openturns.Contour` object.
nx, ny = 15, 15
xx = ot.Box([nx], ot.Interval([0.0], [10.0])).generate()
yy = ot.Box([ny], ot.Interval([-10.0], [10.0])).generate()
inputData = ot.Box([nx, ny], ot.Interval([0.0, -10.0],
[10.0, 10.0])).generate()
outputData = f(inputData)
mycontour = ot.Contour(xx, yy, outputData, [10.0], ["10.0"])
myGraph = ot.Graph("Representation of the failure domain", r"$X_1$", r"$X_2$",
True, "")
myGraph.add(mycontour)
# %%
texts = [r" Event : $\mathcal{D} = \{Y \geq 10.0\}$"]
myText = ot.Text([[4.0, 4.0]], texts)
myText.setTextSize(1)
myGraph.add(myText)
view = otv.View(myGraph)
def draw(self, drawInliers=False, drawOutliers=True):
"""
Draw the high density regions.
Parameters
----------
drawInliers : bool
If True, draw inliers points.
drawOutliers : bool
If True, draw outliers points.
"""
plabels = self.sample.getDescription()
# Bivariate space
grid = ot.GridLayout(self.dim, self.dim)
# Axis are created and stored top to bottom, left to right
for i in range(self.dim):
for j in range(self.dim):
if i >= j: # lower triangle
graph = ot.Graph("", "", "", True, "topright")
if i == j: # diag
marginal_distribution = self.distribution.getMarginal(i)
curve = marginal_distribution.drawPDF()
graph.add(curve)
if drawInliers:
marginal_sample = self.sample[self.inlier_indices, i]
data = ot.Sample(marginal_sample.getSize(), 2)
data[:, 0] = marginal_sample
cloud = ot.Cloud(data)
cloud.setColor(self.inlier_color)
graph.add(cloud)
if drawOutliers:
marginal_sample = self.sample[self.outlier_indices, i]
data = ot.Sample(marginal_sample.getSize(), 2)
data[:, 0] = marginal_sample
cloud = ot.Cloud(data)
cloud.setColor(self.outlier_color)
graph.add(cloud)
elif i > j: # lower corners
# Use a regular grid to compute probability response surface
X1min = self.sample[:, j].getMin()[0]
X1max = self.sample[:, j].getMax()[0]
X2min = self.sample[:, i].getMin()[0]
X2max = self.sample[:, i].getMax()[0]
xx = ot.Box([self.numberOfPointsInXAxis],
ot.Interval([X1min], [X1max])).generate()
yy = ot.Box([self.numberOfPointsInXAxis],
ot.Interval([X2min], [X2max])).generate()
xy = ot.Box(
[
self.numberOfPointsInXAxis,
self.numberOfPointsInXAxis
],
ot.Interval([X1min, X2min], [X1max, X2max]),
).generate()
data = self.distribution.getMarginal([j, i]).computePDF(xy)
# Label using percentage instead of probability
n_contours = len(self.alphaLevels)
labels = [
"%.0f %%" % (self.alphaLevels[i] * 100)
for i in range(n_contours)
]
contour = ot.Contour(xx, yy, data, self.pvalues,
ot.Description(labels))
contour.setColor(self.contour_color)
graph.add(contour)
sample_ij = self.sample[:, [j, i]]
if drawInliers:
cloud = self._drawInliers(sample_ij)
if cloud is not None:
graph.add(cloud)
if drawOutliers:
cloud = self._drawOutliers(sample_ij)
if cloud is not None:
graph.add(cloud)
if j == 0 and i > 0:
graph.setYTitle(plabels[i])
if i == self.dim - 1:
graph.setXTitle(plabels[j])
if i >= j: # lower triangle
graph.setLegends([""])
grid.setGraph(i, j, graph)
return grid
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
# Create a function
f = ot.Function(["x", "y"], ["z"], ["exp(-sin(cos(y)^2*x^2+sin(x)^2*y^2))"])
# Generate the data for the curves to be drawn
nX = 75
nY = 75
inputData = ot.Box([nX, nY]).generate()
inputData *= [10.0] * 2
inputData += [-5.0] * 2
data = f(inputData)
levels = [(0.5 + i) / 5 for i in range(5)]
# Create an empty graph
graph = ot.Graph("Complex iso lines", "u1", "u2", True, "")
# Create the contour
contour = ot.Contour(nX + 2, nY + 2, data)
contour.setLevels(levels)
# Then, draw it
graph.add(contour)
fig = plt.figure(figsize=(4, 4))
plt.suptitle("Complex iso lines example")
axis = fig.add_subplot(111)
axis.set_xlim(auto=True)
View(graph, figure=fig, axes=[axis], add_legend=False)
twoSample = ot.Sample(size, 1)
for i in range(size):
oneSample[i, 0] = 7.0 * sin(-3.5 + (6.5 * i) / (size - 1.0)) + 2.0
twoSample[i,
0] = -2.0 * oneSample[i, 0] + 3.0 + 0.05 * sin(oneSample[i, 0])
test = ot.LinearModelAlgorithm(oneSample, twoSample)
result = ot.LinearModelResult(test.getResult())
print("trend coefficients = ", result.getCoefficients())
print("Fit y ~ 1 + 0.1 x + 10 x^2 model using 100 points")
ot.RandomGenerator.SetSeed(0)
size = 100
# Define a linespace from 0 to 10 with size points
# We use a Box expermient ==> remove 0 & 1 points
experiment = ot.Box([size - 2])
X = experiment.generate()
# X is defined in [0,1]
X *= [10]
# Stack X2
X2 = ot.Sample(X)
for i in range(size):
X2[i, 0] = X[i, 0] * X2[i, 0]
X.stack(X2)
# Define y
Y = ot.Sample(size, 1)
for i in range(size):
Y[i, 0] = 1.0 + 0.1 * X[i, 0] + 10.0 * \
X[i, 0] * X[i, 0] + 0.1 * ot.DistFunc.rNormal()
test = ot.LinearModelAlgorithm(X, Y)
result = test.getResult()
# model
branin = ot.SymbolicFunction(['x1', 'x2'], [
'((x2-(5.1/(4*_pi^2))*x1^2+5*x1/_pi-6)^2+10*(1-1/8*_pi)*cos(x1)+10-54.8104)/51.9496',
'0.96'
])
transfo = ot.SymbolicFunction(['u1', 'u2'], ['15*u1-5', '15*u2'])
model = ot.ComposedFunction(branin, transfo)
# problem
problem = ot.OptimizationProblem()
problem.setObjective(model)
bounds = ot.Interval([0.0] * dim, [1.0] * dim)
problem.setBounds(bounds)
# design
experiment = ot.Box([1, 1])
inputSample = experiment.generate()
modelEval = model(inputSample)
outputSample = modelEval.getMarginal(0)
# first kriging model
covarianceModel = ot.SquaredExponential([0.3007, 0.2483], [0.981959])
basis = ot.ConstantBasisFactory(dim).build()
kriging = ot.KrigingAlgorithm(inputSample, outputSample, covarianceModel,
basis)
noise = list(map(lambda x: x[1], modelEval))
kriging.setNoise(noise)
kriging.run()
# algo
algo = ot.EfficientGlobalOptimization(problem, kriging.getResult())
def drawContour(self, drawData=False, drawOutliers=True):
"""Draw contour.
If :attr:`drawData`, the whole sample is drawn. Otherwise, depending on
:attr:`drawOutliers` it will either show the outliers or the inliers
only.
:param bool drawData: Plot inliers and outliers.
:param bool drawOutliers: Whether to draw inliers or outliers.
:returns: figure, axes and OpenTURNS Graph object.
:rtypes: Matplotlib figure instances, Matplotlib AxesSubplot instances,
:class:`openturns.Graph`
"""
plabels = self.sample.getDescription()
# Bivariate space
fig = plt.figure(figsize=(10, 10))
sub_ax = [] # Axis stored as a list
sub_graph = []
# Axis are created and stored top to bottom, left to right
for i in range(self.dim):
for j in range(self.dim):
k = i + j * self.dim + 1
if i <= j: # lower triangle
ax = fig.add_subplot(self.dim, self.dim, k)
graph = ot.Graph('', '', '', True, 'topright')
if i == j: # diag
pdf_graph = self.distribution.getMarginal(i).drawPDF()
graph.add(pdf_graph)
elif i < j: # lower corners
# Use a regular grid to compute probability response surface
X1min = self.sample[:, i].getMin()[0]
X1max = self.sample[:, i].getMax()[0]
X2min = self.sample[:, j].getMin()[0]
X2max = self.sample[:, j].getMax()[0]
xx = ot.Box([self.numberOfPointsInXAxis],
ot.Interval([X1min], [X1max])).generate()
yy = ot.Box([self.numberOfPointsInXAxis],
ot.Interval([X2min], [X2max])).generate()
xy = ot.Box([self.numberOfPointsInXAxis, self.numberOfPointsInXAxis],
ot.Interval([X1min, X2min], [X1max, X2max])).generate()
data = self.distribution.getMarginal([i, j]).computePDF(xy)
# Label using percentage instead of probability
n_contours = len(self.contoursAlpha)
labels = ["%.0f %%" % (self.contoursAlpha[i] * 100)
for i in range(n_contours)]
contour = ot.Contour(xx, yy, data, self.pvalues,
ot.Description(labels))
contour.setColor('black')
graph.add(contour)
sample_ = np.array(self.sample)[:, [i, j]]
if drawData:
inliers_ = self.drawInliers(sample=sample_)
outliers_ = self.drawOutliers(sample=sample_)
if inliers_ is not None:
graph.add(inliers_)
if outliers_ is not None:
graph.add(outliers_)
elif drawOutliers:
outliers_ = self.drawOutliers(sample=sample_)
if outliers_ is not None:
graph.add(outliers_)
else:
inliers_ = self.drawInliers(sample=sample_)
if inliers_ is not None:
graph.add(inliers_)
if i == 0:
graph.setYTitle(plabels[j])
if j == (self.dim - 1):
graph.setXTitle(plabels[i])
graph.setLegends([''])
sub_graph.append(ot.viewer.View(graph, figure=fig, axes=[ax]))
sub_ax.append(ax)
return fig, sub_ax, sub_graph
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
nrows = 3
ncols = 4
# coordinates of grid
grid = ot.Box([5, 5], ot.Interval([0.0] * 2, [6.0] * 2))
sample = grid.generate()
grid_x = sample.getMarginal(0)
grid_y = sample.getMarginal(1)
#plt.rc('text', usetex=True)
q_values = [1.0, 0.75, 0.5]
fig = plt.figure()
index = 1
for i in range(nrows):
q = q_values[i]
enumerate = ot.HyperbolicAnisotropicEnumerateFunction(2, q)
for j in range(ncols):
ax = fig.add_subplot(nrows, ncols, index, aspect=1.0)
ax.plot(grid_x, grid_y, 'xr')
strataIndex = j + 3
strata_x, strata_y = [], []
strataCardinal = enumerate.getStrataCumulatedCardinal(strataIndex)
for ii in range(strataCardinal):
x = enumerate(ii)
strata_x.append(x[0])
strata_y.append(x[1])
print("mean = ", distribution2D.getMean())
print("cov = ", distribution2D.getCovariance())
print("sigma = ", distribution2D.getStandardDeviation())
distribution2D.setBlockMin(3)
distribution2D.setBlockMax(10)
# Build a grid for validation
xMin = distribution2D.getRange().getLowerBound()[0]
xMax = distribution2D.getRange().getUpperBound()[0]
yMin = distribution2D.getRange().getLowerBound()[1]
yMax = distribution2D.getRange().getUpperBound()[1]
# Number of points of discretization
nx = 4
ny = 4
boxParameters = [nx, ny]
boxGrid = ot.Box(boxParameters)
grid = boxGrid.generate()
# scaling box grid
scaleFactor = [0.25 * (xMax - xMin), 0.25 * (yMax - yMin)]
grid *= scaleFactor
# translating
translateFactor = distribution2D.getMean()[0:2]
grid += translateFactor
# Compute PDF
# parameters for theoritical PDF, obtained thanks to Maple
factor = sqrt(2) / (20 * pi)
for index in range(grid.getSize()):
point = grid[index]
PDF = distribution2D.computePDF(point)
# Very small values are not very accurate on x86, skip them
if (PDF < 1.e-12):
sample += [5.0, 8.0] graph = drawBidimensionalSample(sample, "Factorial") view = viewer.View(graph) # %% # Composite design # ---------------- # %% experiment = ot.Composite(2, levels) sample = experiment.generate() sample *= 2.0 sample += [5.0, 8.0] graph = drawBidimensionalSample(sample, "Composite") view = viewer.View(graph) # %% # Grid design # ----------- # # %% levels = [3, 4] experiment = ot.Box(levels) sample = experiment.generate() sample *= 2.0 sample += [5.0, 8.0] graph = drawBidimensionalSample(sample, "Box") view = viewer.View(graph) plt.show()
graphModel0.setXTitle(r'$x_1$')
graphModel0.setYTitle(r'$x_2$')
graphModel0.setTitle(r'Isolines of the model : $Y = f(X)$, first marginal')
# %%
# We represent the second marginal of `vecY`.
graphModel1 = f.draw(0, 1, 1, [0.0, 0.0], [-5.0, -5.0], [5.0, 5.0])
graphModel1.setXTitle(r'$x_1$')
graphModel1.setYTitle(r'$x_2$')
graphModel1.setTitle(r'Isolines of the model : $Y = f(X)$, second marginal')
# %%
# We shall now represent the curves delimiting the domain of interest :
#
nx, ny = 15, 15
xx = ot.Box([nx], ot.Interval([-5.0], [5.0])).generate()
yy = ot.Box([ny], ot.Interval([-5.0], [5.0])).generate()
inputData = ot.Box([nx, ny], ot.Interval([-5.0, -5.0], [5.0, 5.0])).generate()
outputData = f(inputData)
# %%
# The contour line associated with the 0.0 value for the first marginal.
mycontour0 = ot.Contour(xx, yy, outputData.getMarginal(0), [0.0], ["0.0"])
mycontour0.setColor("black")
mycontour0.setLineStyle("dashed")
graphModel0.add(mycontour0)
# %%
# The contour line associated with the 1.0 value for the first marginal.
mycontour1 = ot.Contour(xx, yy, outputData.getMarginal(0), [1.0], ["1.0"])
mycontour1.setColor("black")
# %%
# Graphically, both the metamodel and the exact function look the same. The metamodel also has three
# basins around the minima and the value of the metamodel at each minimum location is comparable to
# the exact value of -0.979476. We have roughly two correct digits for each isoline.
# %%
# Standard deviation
# ------------------
#
# We finally take a look at the standard deviation in the :math:`[0,1] \times [0,1]` domain. It may be
# seen as a measure of the error of the metamodel.
# %%
# We discretize the domain with 22 points (N inside points and 2 endpoints) :
N = 20
inputData = ot.Box([N, N]).generate()
# %%
# We compute the conditional variance of the model and take the square root to get the deviation :
condCov = result.getConditionalMarginalVariance(inputData, 0)
condCovSd = sqrt(condCov)
# %%
# As we have previously done we build contours with the following levels ans labels :
levels = [0.01, 0.025, 0.050, 0.075, 0.1, 0.125, 0.150, 0.175]
labels = ['0.01', '0.025', '0.050', '0.075', '0.1', '0.125', '0.150', '0.175']
contour = ot.Contour(N + 2, N + 2, condCovSd)
graph = ot.Graph('', 'x', 'y', True, '')
graph.add(contour)
# %%
import openturns as ot
from math import cos, sin
ot.TESTPREAMBLE()
try:
# Fix numerical precision
ot.PlatformInfo.SetNumericalPrecision(2)
# Kriging use case
spatialDimension = 2
# Learning data
levels = [8, 5]
box = ot.Box(levels)
inputSample = box.generate()
# Scale each direction
inputSample *= 10
# Define model
model = ot.Function(['x', 'y'], ['z'], ['cos(0.5*x) + sin(y)'])
outputSample = model(inputSample)
# 2) Definition of exponential model
covarianceModel = ot.SquaredExponential([1.988, 0.924], [3.153])
# 3) Basis definition
basisCollection = ot.BasisCollection(
1,
ot.ConstantBasisFactory(spatialDimension).build())
#! /usr/bin/env python
from __future__ import print_function
import openturns as ot
levels = [4] * 2
myPlane = ot.Box(levels)
print('myPlane = ', myPlane)
sample = myPlane.generate()
print('sample = ', repr(sample))
myPlane = ot.Box(levels, ot.Interval([7.0, 8.0], [9.0, 16.0]))
print('myPlane = ', myPlane)
sample = myPlane.generate()
print('sample = ', repr(sample))