.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_meta_modeling/kriging_metamodel/plot_gpr_cantilever_beam.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_meta_modeling_kriging_metamodel_plot_gpr_cantilever_beam.py: Gaussian Process Regression : cantilever beam model =================================================== .. GENERATED FROM PYTHON SOURCE LINES 7-9 In this example, we create a Gaussian Process Regression (GPR) metamodel of the :ref:`cantilever beam `. We use a squared exponential covariance kernel for the Gaussian process. In order to estimate the hyper-parameters, we use a design of experiments of size 20. .. GENERATED FROM PYTHON SOURCE LINES 12-14 Definition of the model ----------------------- .. GENERATED FROM PYTHON SOURCE LINES 16-28 .. code-block:: Python from openturns.usecases import cantilever_beam import openturns as ot import openturns.experimental as otexp import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) # Reset default settings ot.ResourceMap.Reload() ot.RandomGenerator.SetSeed(0) .. GENERATED FROM PYTHON SOURCE LINES 29-30 We load the cantilever beam use case : .. GENERATED FROM PYTHON SOURCE LINES 30-32 .. code-block:: Python cb = cantilever_beam.CantileverBeam() .. GENERATED FROM PYTHON SOURCE LINES 33-34 We define the function which evaluates the output depending on the inputs. .. GENERATED FROM PYTHON SOURCE LINES 34-36 .. code-block:: Python model = cb.model .. GENERATED FROM PYTHON SOURCE LINES 37-38 Then we define the distribution of the input random vector. .. GENERATED FROM PYTHON SOURCE LINES 38-40 .. code-block:: Python myDistribution = cb.distribution .. GENERATED FROM PYTHON SOURCE LINES 41-43 Create the design of experiments -------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 45-47 We consider a simple Monte-Carlo sample as a design of experiments. This is why we generate an input sample using the `getSample` method of the distribution. Then we evaluate the output using the `model` function. .. GENERATED FROM PYTHON SOURCE LINES 49-53 .. code-block:: Python sampleSize_train = 20 X_train = myDistribution.getSample(sampleSize_train) Y_train = model(X_train) .. GENERATED FROM PYTHON SOURCE LINES 54-55 The following figure presents the distribution of the vertical deviations Y on the training sample. We observe that the large deviations occur less often. .. GENERATED FROM PYTHON SOURCE LINES 57-63 .. code-block:: Python histo = ot.HistogramFactory().build(Y_train).drawPDF() histo.setXTitle("Vertical deviation (cm)") histo.setTitle("Distribution of the vertical deviation") histo.setLegends([""]) view = viewer.View(histo) .. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_gpr_cantilever_beam_001.png :alt: Distribution of the vertical deviation :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_gpr_cantilever_beam_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 64-66 Create the metamodel -------------------- .. GENERATED FROM PYTHON SOURCE LINES 68-71 In order to create the GPR metamodel, we first select a constant trend with the `ConstantBasisFactory` class. Then we use a squared exponential covariance kernel. The `SquaredExponential` kernel has one amplitude coefficient and 4 scale coefficients. This is because this covariance kernel is anisotropic : each of the 4 input variables is associated with its own scale coefficient. .. GENERATED FROM PYTHON SOURCE LINES 73-76 .. code-block:: Python basis = ot.ConstantBasisFactory(cb.dim).build() covarianceModel = ot.SquaredExponential(cb.dim) .. GENERATED FROM PYTHON SOURCE LINES 77-79 Typically, the optimization algorithm is quite good at setting sensible optimization bounds. In this case, however, the range of the input domain is extreme. .. GENERATED FROM PYTHON SOURCE LINES 81-84 .. code-block:: Python print("Lower and upper bounds of X_train:") print(X_train.getMin(), X_train.getMax()) .. rst-class:: sphx-glr-script-out .. code-block:: none Lower and upper bounds of X_train: [6.50185e+10,256.24,2.50948,1.3515e-07] [7.20029e+10,363.287,2.59143,1.59996e-07] .. GENERATED FROM PYTHON SOURCE LINES 85-87 We need to manually define sensible optimization bounds. Note that since the amplitude parameter is computed analytically (this is possible when the output dimension is 1), we only need to set bounds on the scale parameter. .. GENERATED FROM PYTHON SOURCE LINES 89-93 .. code-block:: Python scaleOptimizationBounds = ot.Interval( [1.0, 1.0, 1.0, 1.0e-10], [1.0e11, 1.0e3, 1.0e1, 1.0e-5] ) .. GENERATED FROM PYTHON SOURCE LINES 94-98 Finally, we use the `GaussianProcessFitter` and `GaussianProcessRegression` classes to create the GPR metamodel. It requires a training sample, a covariance kernel and a trend basis as input arguments. We need to set the initial scale parameter for the optimization. The upper bound of the input domain is a sensible choice here. We must not forget to actually set the optimization bounds defined above. .. GENERATED FROM PYTHON SOURCE LINES 100-105 .. code-block:: Python covarianceModel.setScale(X_train.getMax()) fitter_algo = otexp.GaussianProcessFitter(X_train, Y_train, covarianceModel, basis) fitter_algo.setOptimizationBounds(scaleOptimizationBounds) .. GENERATED FROM PYTHON SOURCE LINES 106-109 The `run` method optimizes the metamodel hyperparameters. We can then print the constant trend of the metamodel, estimated using the least squares method. .. GENERATED FROM PYTHON SOURCE LINES 111-118 .. code-block:: Python fitter_algo.run() fitter_result = fitter_algo.getResult() gpr_algo = otexp.GaussianProcessRegression(fitter_result) gpr_algo.run() gpr_result = gpr_algo.getResult() gprMetamodel = gpr_result.getMetaModel() .. GENERATED FROM PYTHON SOURCE LINES 119-120 The `getTrendCoefficients` method returns the coefficients of the trend. .. GENERATED FROM PYTHON SOURCE LINES 122-124 .. code-block:: Python print(gpr_result.getTrendCoefficients()) .. rst-class:: sphx-glr-script-out .. code-block:: none [0.425125] .. GENERATED FROM PYTHON SOURCE LINES 125-126 We can also print the hyperparameters of the covariance model, which have been estimated by maximizing the likelihood. .. GENERATED FROM PYTHON SOURCE LINES 128-130 .. code-block:: Python gpr_result.getCovarianceModel() .. raw:: html

SquaredExponential(scale=[7.20029e+10,363.4,2.59143,1.59996e-07], amplitude=[0.376081])



.. GENERATED FROM PYTHON SOURCE LINES 131-133 Validate the metamodel ---------------------- .. GENERATED FROM PYTHON SOURCE LINES 135-136 We finally want to validate the GPR metamodel. This is why we generate a validation sample with size 100 and we evaluate the output of the model on this sample. .. GENERATED FROM PYTHON SOURCE LINES 138-142 .. code-block:: Python sampleSize_test = 100 X_test = myDistribution.getSample(sampleSize_test) Y_test = model(X_test) .. GENERATED FROM PYTHON SOURCE LINES 143-144 The `MetaModelValidation` classe makes the validation easy. To create it, we use the validation samples and the metamodel. .. GENERATED FROM PYTHON SOURCE LINES 146-148 .. code-block:: Python val = ot.MetaModelValidation(Y_test, gprMetamodel(X_test)) .. GENERATED FROM PYTHON SOURCE LINES 149-150 The `computeR2Score` computes the R2 score. .. GENERATED FROM PYTHON SOURCE LINES 152-155 .. code-block:: Python R2 = val.computeR2Score()[0] print(R2) .. rst-class:: sphx-glr-script-out .. code-block:: none 0.9999676759393747 .. GENERATED FROM PYTHON SOURCE LINES 156-157 The residuals are the difference between the model and the metamodel. .. GENERATED FROM PYTHON SOURCE LINES 159-166 .. code-block:: Python r = val.getResidualSample() graph = ot.HistogramFactory().build(r).drawPDF() graph.setXTitle("Residuals (cm)") graph.setTitle("Distribution of the residuals") graph.setLegends([""]) view = viewer.View(graph) .. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_gpr_cantilever_beam_002.png :alt: Distribution of the residuals :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_gpr_cantilever_beam_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 167-168 We observe that the negative residuals occur with nearly the same frequency of the positive residuals: this is a first sign of good quality. .. GENERATED FROM PYTHON SOURCE LINES 170-171 The `drawValidation` method allows one to compare the observed outputs and the metamodel outputs. .. GENERATED FROM PYTHON SOURCE LINES 173-174 sphinx_gallery_thumbnail_number = 3 .. GENERATED FROM PYTHON SOURCE LINES 174-179 .. code-block:: Python graph = val.drawValidation() graph.setTitle("R2 = %.2f%%" % (100 * R2)) view = viewer.View(graph) plt.show() .. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_gpr_cantilever_beam_003.png :alt: R2 = 100.00% :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_gpr_cantilever_beam_003.png :class: sphx-glr-single-img .. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_gpr_cantilever_beam.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_gpr_cantilever_beam.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_gpr_cantilever_beam.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_gpr_cantilever_beam.zip `