.. 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_kriging_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_kriging_cantilever_beam.py: Kriging : cantilever beam model =============================== .. GENERATED FROM PYTHON SOURCE LINES 7-9 In this example, we create a Kriging 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 is 20. .. GENERATED FROM PYTHON SOURCE LINES 12-14 Definition of the model ----------------------- .. GENERATED FROM PYTHON SOURCE LINES 16-23 .. code-block:: Python from openturns.usecases import cantilever_beam import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 24-25 We load the cantilever beam use case : .. GENERATED FROM PYTHON SOURCE LINES 25-27 .. code-block:: Python cb = cantilever_beam.CantileverBeam() .. GENERATED FROM PYTHON SOURCE LINES 28-29 We define the function which evaluates the output depending on the inputs. .. GENERATED FROM PYTHON SOURCE LINES 29-31 .. code-block:: Python model = cb.model .. GENERATED FROM PYTHON SOURCE LINES 32-33 Then we define the distribution of the input random vector. .. GENERATED FROM PYTHON SOURCE LINES 33-36 .. code-block:: Python dim = cb.dim # number of inputs myDistribution = cb.distribution .. GENERATED FROM PYTHON SOURCE LINES 37-39 Create the design of experiments -------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 41-43 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 45-49 .. code-block:: Python sampleSize_train = 20 X_train = myDistribution.getSample(sampleSize_train) Y_train = model(X_train) .. GENERATED FROM PYTHON SOURCE LINES 50-51 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 53-59 .. 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_kriging_cantilever_beam_001.png :alt: Distribution of the vertical deviation :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_cantilever_beam_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 60-62 Create the metamodel -------------------- .. GENERATED FROM PYTHON SOURCE LINES 64-68 In order to create the Kriging metamodel, we first select a constant trend with the :class:`~openturns.ConstantBasisFactory` class. Then we use a squared exponential covariance kernel. The :class:`~openturns.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 70-73 .. code-block:: Python basis = ot.ConstantBasisFactory(dim).build() covarianceModel = ot.SquaredExponential(dim) .. GENERATED FROM PYTHON SOURCE LINES 74-76 Typically, the optimization algorithm is quite good at setting optimization bounds. In this case, however, the range of the input domain is extreme. .. GENERATED FROM PYTHON SOURCE LINES 78-81 .. 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: [6.52677e+10,244.612,2.50392,1.34039e-07] [7.35161e+10,354.916,2.59295,1.56347e-07] .. GENERATED FROM PYTHON SOURCE LINES 82-84 We need to manually define 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 86-90 .. 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 91-95 Finally, we use the :class:`~openturns.KrigingAlgorithm` class to create the Kriging 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 sensitive choice here. We must not forget to actually set the optimization bounds defined above. .. GENERATED FROM PYTHON SOURCE LINES 97-102 .. code-block:: Python covarianceModel.setScale(X_train.getMax()) algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizationBounds(scaleOptimizationBounds) .. GENERATED FROM PYTHON SOURCE LINES 103-106 The `run` method has optimized the hyperparameters of the metamodel. We can then print the constant trend of the metamodel, which have been estimated using the least squares method. .. GENERATED FROM PYTHON SOURCE LINES 108-112 .. code-block:: Python algo.run() result = algo.getResult() krigingMetamodel = result.getMetaModel() .. GENERATED FROM PYTHON SOURCE LINES 113-114 The `getTrendCoefficients` method returns the coefficients of the trend. .. GENERATED FROM PYTHON SOURCE LINES 116-118 .. code-block:: Python print(result.getTrendCoefficients()) .. rst-class:: sphx-glr-script-out .. code-block:: none [0.45244] .. GENERATED FROM PYTHON SOURCE LINES 119-120 We can also print the hyperparameters of the covariance model, which have been estimated by maximizing the likelihood. .. GENERATED FROM PYTHON SOURCE LINES 122-124 .. code-block:: Python result.getCovarianceModel() .. raw:: html

SquaredExponential(scale=[7.35161e+10,354.916,2.59295,1.56347e-07], amplitude=[0.350094])



.. GENERATED FROM PYTHON SOURCE LINES 125-127 Validate the metamodel ---------------------- .. GENERATED FROM PYTHON SOURCE LINES 129-130 We finally want to validate the Kriging 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 132-136 .. code-block:: Python sampleSize_test = 100 X_test = myDistribution.getSample(sampleSize_test) Y_test = model(X_test) .. GENERATED FROM PYTHON SOURCE LINES 137-139 The :class:`~openturns.MetaModelValidation` class makes surrogate model validation easy. To create it, we use the validation samples and the metamodel. .. GENERATED FROM PYTHON SOURCE LINES 141-144 .. code-block:: Python metamodelPredictions = krigingMetamodel(X_test) val = ot.MetaModelValidation(Y_test, metamodelPredictions) .. GENERATED FROM PYTHON SOURCE LINES 145-147 The :meth:`~openturns.MetaModelValidation.computeR2Score` method computes the R2 coefficient of determination. .. GENERATED FROM PYTHON SOURCE LINES 149-152 .. code-block:: Python r2Score = val.computeR2Score()[0] print(r2Score) .. rst-class:: sphx-glr-script-out .. code-block:: none 0.9999090577204818 .. GENERATED FROM PYTHON SOURCE LINES 153-154 The residuals are the difference between the model and the metamodel. .. GENERATED FROM PYTHON SOURCE LINES 156-163 .. 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_kriging_cantilever_beam_002.png :alt: Distribution of the residuals :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_cantilever_beam_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 164-165 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 167-169 The :meth:`~openturns.MetaModelValidation.drawValidation` method allows one to compare the observed outputs and the metamodel outputs. .. GENERATED FROM PYTHON SOURCE LINES 171-176 .. code-block:: Python graph = val.drawValidation() graph.setTitle("R2 = %.2f%%" % (100 * r2Score)) view = viewer.View(graph) plt.show() # sphinx_gallery_thumbnail_number = 3 .. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_cantilever_beam_003.png :alt: R2 = 99.99% :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_cantilever_beam_003.png :class: sphx-glr-single-img .. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_kriging_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_kriging_cantilever_beam.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_kriging_cantilever_beam.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_kriging_cantilever_beam.zip `