Simulate new trajectories from a kriging metamodel¶
The main goal of this example is to show how to simulate new trajectories from a kriging metamodel.
We consider the sine function:
for any .
We want to create a metamodel of this function. This is why we create a sample of observations of the function:
for , where is the i-th input and is the corresponding output.
We consider the seven following inputs :
We are going to consider a kriging metamodel with a
a Matern covariance model.
Creation of the metamodel¶
We begin by defining the function
g as a symbolic function. Then we define the
x_train variable which contains the inputs of the design of experiments of the training step. Then we compute the
y_train corresponding outputs. The variable
n_train is the size of the training design of experiments.
import numpy as np import openturns as ot
g = ot.SymbolicFunction(['x'], ['sin(x)'])
x_train = ot.Sample([1.,3.,4.,6.,7.9,11., 11.5],1) y_train = g(x_train) n_train = x_train.getSize() n_train
In order to compare the function and its metamodel, we use a test (i.e. validation) design of experiments made of a regular grid of 100 points from 0 to 12. Then we convert this grid into a
Sample and we compute the outputs of the function on this sample.
xmin = 0. xmax = 12. n_test = 100 step = (xmax-xmin)/(n_test-1) myRegularGrid = ot.RegularGrid(xmin, step, n_test) x_test_coord = myRegularGrid.getValues() x_test = ot.Sample(x_test_coord,1) 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 functions which plots the data.
def plot_data_train(x_train,y_train): '''Plot the data (x_train,y_train) as a Cloud, in red''' graph_train = ot.Cloud(x_train,y_train) graph_train.setColor("red") graph_train.setLegend("Data") return graph_train
def plot_data_test(x_test,y_test): '''Plot the data (x_test,y_test) as a Curve, in dashed black''' graphF = ot.Curve(x_test,y_test) graphF.setLegend("Exact") graphF.setColor("black") graphF.setLineStyle("dashed") return graphF
graph = ot.Graph() graph.add(plot_data_test(x_test,y_test)) graph.add(plot_data_train(x_train,y_train)) graph.setAxes(True) graph.setXTitle("X") graph.setYTitle("Y") graph.setLegendPosition("topright") graph
We use the
ConstantBasisFactory class to define the trend and the
MaternModel class to define the covariance model. This Matérn model is based on the regularity parameter .
dimension = 1 basis = ot.ConstantBasisFactory(dimension).build() covarianceModel = ot.MaternModel([1.]*dimension, 1.5) algo = ot.KrigingAlgorithm(x_train, y_train, covarianceModel, basis) algo.run() krigingResult = algo.getResult() krigingResult
KrigingResult(covariance models=MaternModel(scale=[0.318568], amplitude=[0.822262], nu=1.5), covariance coefficients=0 : [ 1.13905 ]
1 : [ 1.01761 ]
2 : [ -1.76279 ]
3 : [ -0.559147 ]
4 : [ 1.78757 ]
5 : [ -1.61945 ]
6 : [ -0.00284256 ], basis=[Basis( [class=LinearEvaluation name=Unnamed center= constant= linear=[[ 0 ]]] )], trend coefficients=[[0.00736738]])
We observe that the
amplitude hyper-parameters have been optimized by the
run method. Then we get the metamodel with
getMetaModel and evaluate the outputs of the metamodel on the test design of experiments.
krigeageMM = krigingResult.getMetaModel() y_test_MM = krigeageMM(x_test)
The following function plots the kriging data.
def plot_data_kriging(x_test,y_test_MM): '''Plots (x_test,y_test_MM) from the metamodel as a Curve, in blue''' graphK = ot.Curve(x_test,y_test_MM) graphK.setColor("blue") graphK.setLegend("Kriging") return graphK
graph = ot.Graph() graph.add(plot_data_test(x_test,y_test)) graph.add(plot_data_train(x_train,y_train)) graph.add(plot_data_kriging(x_test,y_test_MM)) graph.setAxes(True) graph.setXTitle("X") graph.setYTitle("Y") graph.setLegendPosition("topright") graph
Simulate new trajectories¶
In order to generate new trajectories of the conditioned gaussian process, we couild technically use the
KrigingRandomVector class, because it provides the
getSample method that we need. However, the
KrigingRandomVector class was more specifically designed to create a
RandomVector so that it can feed, for example, a function which has a field as input argument.
This is why we use the
ConditionedGaussianProcess, which provides a
n_test = 100 step = (xmax-xmin)/(n_test-1) myRegularGrid = ot.RegularGrid(xmin, step, n_test) vertices = myRegularGrid.getVertices()
If we directly use the
vertices values, we get:
RuntimeError: InternalException : Error: the matrix is not definite positive.
Indeed, some points in
vertices are also in
x_train. This is why the conditioned covariance matrix is singular at these points.
This is why we define the following function which deletes points in
vertices which are also found in
def deleteCommonValues(x_train,x_test): ''' Delete from x_test the values which are in x_train so that values in x_test have no interect with x_train. ''' x_test_filtered = x_test # Initialize for x_train_value in x_train: print("Checking %s" % (x_train_value)) indices = np.argwhere(x_test==x_train_value) if len(indices) == 1: print(" Delete %s" % (x_train_value)) x_test_filtered = np.delete(x_test_filtered, indices[0, 0]) else: print(" OK") return x_test_filtered
vertices_filtered = deleteCommonValues(np.array(x_train.asPoint()),np.array(vertices.asPoint()))
Checking 1.0 OK Checking 3.0 OK Checking 4.0 Delete 4.0 Checking 6.0 OK Checking 7.9 OK Checking 11.0 OK Checking 11.5 OK
evaluationMesh = ot.Mesh(ot.Sample(vertices_filtered,1))
process = ot.ConditionedGaussianProcess(krigingResult, evaluationMesh)
trajectories = process.getSample(10) type(trajectories)
getSample method returns a
ProcessSample. By comparison, the
getSample method of a
KrigingRandomVector would return a
graph = trajectories.drawMarginal() graph.add(plot_data_test(x_test,y_test)) graph.add(plot_data_train(x_train,y_train)) graph.setAxes(True) graph.setXTitle("X") graph.setYTitle("Y") graph.setLegendPosition("topright") graph.setTitle("10 simulated trajectories") graph
Metamodeling with Gaussian processes, Bertrand Iooss, EDF R&D, 2014, www.gdr-mascotnum.fr/media/sssamo14_iooss.pdf