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"""
Gaussian Process Regression : generate trajectories from the metamodel
======================================================================
"""
# %%
# The main goal of this example is to show how to simulate new trajectories from a Gaussian Process Regression metamodel.
# %%
# Introduction
# ------------
#
# We consider the sine function:
#
# .. math::
# \model(x) = \sin(x)
#
#
# for any :math:`x\in[0,12]`.
#
# We want to create a metamodel of this function. This is why we create a sample of :math:`n` observations of the function:
#
# .. math::
# y_i = \model(x_i)
#
#
# for :math:`i=1,...,7`, where :math:`x_i` is the i-th input and :math:`y_i` is the corresponding output.
#
# We consider the seven following inputs :
#
# ============ === === === === ===== ==== ======
# :math:`i` 1 2 3 4 5 6 7
# ============ === === === === ===== ==== ======
# :math:`x_i` 1 3 4 6 7.9 11 11.5
# ============ === === === === ===== ==== ======
#
# We are going to consider a Gaussian Process Regression metamodel with:
#
# * a constant trend,
# * a Matern covariance model.
#
# In the :doc:`/auto_surrogate_modeling/gaussian_process_regression/plot_gpr_1d` example, we detail the estimation of this metamodel.
# Refer to it for further details: we only focus here on the simulation of new trajectories.
#
import openturns as ot
import openturns.viewer as otv
import openturns.experimental as otexp
# %%
# Creation of the metamodel
# -------------------------
#
# We define the function `g`, the training sample `(x_train, y_train)` and the test sample `(x_test, y_tst)`.
# %%
g = ot.SymbolicFunction(["x"], ["sin(x)"])
# %%
x_train = ot.Sample([[x] for x in [1.0, 3.0, 4.0, 6.0, 7.9, 11.0, 11.5]])
y_train = g(x_train)
n_train = x_train.getSize()
n_train
# %%
xmin = 0.0
xmax = 12.0
n_test = 101
step = (xmax - xmin) / (n_test - 1)
myRegularGrid = ot.RegularGrid(xmin, step, n_test)
x_test = myRegularGrid.getVertices()
y_test = g(x_test)
# %%
# In order to observe the function and the location of the points in the input design of experiments, we define the following function which plots the data.
# %%
def plot_1d_data(x_data, y_data, type="Curve", legend=None, color=None, linestyle=None):
"""Plot the data (x_data,y_data) as a Cloud/Curve"""
if type == "Curve":
graphF = ot.Curve(x_data, y_data)
else:
graphF = ot.Cloud(x_data, y_data)
if legend is not None:
graphF.setLegend(legend)
if color is not None:
graphF.setColor(color)
if linestyle is not None:
graphF.setLineStyle(linestyle)
return graphF
# %%
# Here, we draw the model and the train sample.
graph = ot.Graph("Model and Train sample", "X", "Y", True, "")
graph.add(
plot_1d_data(
x_test, y_test, type="Curve", legend="model", color="black", linestyle="dashed"
)
)
graph.add(
plot_1d_data(x_train, y_train, type="Cloud", legend="train sample", color="red")
)
graph.setLegendPosition("upper right")
view = otv.View(graph)
# %%
# We use the :class:`~openturns.ConstantBasisFactory` class to define the trend and the :class:`~openturns.MaternModel` class to define the covariance model.
# This Matérn model is based on the regularity parameter :math:`\nu=3/2`.
# %%
dimension = 1
basis = ot.ConstantBasisFactory(dimension).build()
covarianceModel = ot.MaternModel([1.0] * dimension, 1.5)
# %%
# We estimate the Gaussian process :math:`Y` with the class :class:`~openturns.experimental.GaussianProcessFitter`.
fitter_algo = otexp.GaussianProcessFitter(x_train, y_train, covarianceModel, basis)
fitter_algo.run()
fitter_result = fitter_algo.getResult()
print(fitter_result)
# %%
# We observe that the `scale` and `amplitude` hyper-parameters have been optimized by the
# :meth:`~openturns.experimental.GaussianProcessFitter.run` method, while the :math:`\nu`
# parameter has remained unchanged, as expected.
#
# Then, we condition the gaussian process to make it interpolate the data set using the class
# :class:`~openturns.experimental.GaussianProcessRegression`.
gpr_algo = otexp.GaussianProcessRegression(fitter_result)
gpr_algo.run()
gpr_result = gpr_algo.getResult()
print(gpr_result)
# %%
# We get the metamodel and the predictions on the test sample.
gpr_metamodel = gpr_result.getMetaModel()
y_test_MM = gpr_metamodel(x_test)
# %%
# The following function plots the Gaussian Process Regression predictions on the test sample.
graph = ot.Graph("Gaussian process regression metamodel", "X", "Y", True, "")
graph.add(
plot_1d_data(
x_test, y_test, type="Curve", legend="model", color="black", linestyle="dashed"
)
)
graph.add(
plot_1d_data(x_train, y_train, type="Cloud", legend="train sample", color="red")
)
graph.add(
plot_1d_data(
x_test, y_test_MM, type="Curve", legend="GPR", color="blue", linestyle="solid"
)
)
graph.setLegendPosition("upper right")
view = otv.View(graph)
# %%
# Simulate new trajectories
# -------------------------
#
# In order to generate new trajectories of the conditioned Gaussian process, we use the class
# :class:`~openturns.experimental.ConditionedGaussianProcess`, which provides a :class:`~openturns.Process`.
# It is created from the result of the Gaussian Process Regression algorithm.
process = otexp.ConditionedGaussianProcess(gpr_result, myRegularGrid)
# %%
# The method :meth:`~openturns.Process.getSample` method returns a :class:`~openturns.ProcessSample`.
# %%
# sphinx_gallery_thumbnail_number = 3
trajectories = process.getSample(10)
type(trajectories)
graph = trajectories.drawMarginal()
graph.add(
plot_1d_data(
x_test,
y_test,
type="Curve",
legend="model",
color="black",
linestyle="dashed",
)
)
graph.add(
plot_1d_data(x_train, y_train, type="Cloud", legend="train sample", color="red")
)
graph.setXTitle("X")
graph.setYTitle("Y")
graph.setLegendPosition("upper right")
graph.setTitle("10 simulated trajectories")
view = otv.View(graph)
# %%
# Display all figures.
otv.View.ShowAll()
|
