.. 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 6-8 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 11-13 Definition of the model ----------------------- .. GENERATED FROM PYTHON SOURCE LINES 15-22 .. 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 23-24 We load the cantilever beam use case : .. GENERATED FROM PYTHON SOURCE LINES 24-26 .. code-block:: Python cb = cantilever_beam.CantileverBeam() .. GENERATED FROM PYTHON SOURCE LINES 27-28 We define the function which evaluates the output depending on the inputs. .. GENERATED FROM PYTHON SOURCE LINES 28-30 .. code-block:: Python model = cb.model .. GENERATED FROM PYTHON SOURCE LINES 31-32 Then we define the distribution of the input random vector. .. GENERATED FROM PYTHON SOURCE LINES 32-35 .. code-block:: Python dim = cb.dim # number of inputs myDistribution = cb.distribution .. GENERATED FROM PYTHON SOURCE LINES 36-38 Create the design of experiments -------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 40-42 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 44-48 .. code-block:: Python sampleSize_train = 20 X_train = myDistribution.getSample(sampleSize_train) Y_train = model(X_train) .. GENERATED FROM PYTHON SOURCE LINES 49-50 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 52-58 .. 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 59-61 Create the metamodel -------------------- .. GENERATED FROM PYTHON SOURCE LINES 63-66 In order to create the Kriging 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 68-71 .. code-block:: Python basis = ot.ConstantBasisFactory(dim).build() covarianceModel = ot.SquaredExponential(dim) .. GENERATED FROM PYTHON SOURCE LINES 72-74 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 76-79 .. 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: [6.51899e+10,240.933,2.50312,1.31237e-07] [7.40235e+10,387.288,2.59677,1.61993e-07] .. GENERATED FROM PYTHON SOURCE LINES 80-82 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 84-88 .. 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 89-93 Finally, we use the `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 sensible choice here. We must not forget to actually set the optimization bounds defined above. .. GENERATED FROM PYTHON SOURCE LINES 95-100 .. 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 101-104 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 106-110 .. code-block:: Python algo.run() result = algo.getResult() krigingMetamodel = result.getMetaModel() .. GENERATED FROM PYTHON SOURCE LINES 111-112 The `getTrendCoefficients` method returns the coefficients of the trend. .. GENERATED FROM PYTHON SOURCE LINES 114-116 .. code-block:: Python print(result.getTrendCoefficients()) .. rst-class:: sphx-glr-script-out .. code-block:: none [0.359039] .. GENERATED FROM PYTHON SOURCE LINES 117-118 We can also print the hyperparameters of the covariance model, which have been estimated by maximizing the likelihood. .. GENERATED FROM PYTHON SOURCE LINES 120-122 .. code-block:: Python result.getCovarianceModel() .. raw:: html

SquaredExponential(scale=[7.40235e+10,387.288,2.59677,1.61993e-07], amplitude=[0.402853])



.. GENERATED FROM PYTHON SOURCE LINES 123-125 Validate the metamodel ---------------------- .. GENERATED FROM PYTHON SOURCE LINES 127-128 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 130-134 .. code-block:: Python sampleSize_test = 100 X_test = myDistribution.getSample(sampleSize_test) Y_test = model(X_test) .. GENERATED FROM PYTHON SOURCE LINES 135-136 The `MetaModelValidation` classe makes the validation easy. To create it, we use the validation samples and the metamodel. .. GENERATED FROM PYTHON SOURCE LINES 138-140 .. code-block:: Python val = ot.MetaModelValidation(X_test, Y_test, krigingMetamodel) .. GENERATED FROM PYTHON SOURCE LINES 141-142 The `computePredictivityFactor` computes the Q2 factor. .. GENERATED FROM PYTHON SOURCE LINES 144-147 .. code-block:: Python Q2 = val.computePredictivityFactor()[0] print(Q2) .. rst-class:: sphx-glr-script-out .. code-block:: none 0.9999093255530566 .. GENERATED FROM PYTHON SOURCE LINES 148-149 The residuals are the difference between the model and the metamodel. .. GENERATED FROM PYTHON SOURCE LINES 151-158 .. 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 159-160 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 162-163 The `drawValidation` method allows one to compare the observed outputs and the metamodel outputs. .. GENERATED FROM PYTHON SOURCE LINES 165-166 sphinx_gallery_thumbnail_number = 3 .. GENERATED FROM PYTHON SOURCE LINES 166-171 .. code-block:: Python graph = val.drawValidation() graph.setTitle("Q2 = %.2f%%" % (100 * Q2)) view = viewer.View(graph) plt.show() .. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_cantilever_beam_003.png :alt: Q2 = 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 `