.. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` 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 the cantilever beam model ================================= In this example, we create a kriging metamodel of the :ref:`cantilever beam `. We use a squared exponential covariance model for the kriging. In order to estimate the hyper-parameters, we use a design of experiments which size is 20. Definition of the model ----------------------- .. code-block:: default import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) We load the cantilever beam use case : .. code-block:: default from openturns.usecases import cantilever_beam as cantilever_beam cb = cantilever_beam.CantileverBeam() We define the function which evaluates the output depending on the inputs. .. code-block:: default model = cb.model Then we define the distribution of the input random vector. .. code-block:: default dim = cb.dim # number of inputs myDistribution = cb.distribution We use a transformation because data contain very large values. .. code-block:: default transformation = myDistribution.getIsoProbabilisticTransformation() Create the design of experiments -------------------------------- We consider a simple Monte-Carlo sampling 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. .. code-block:: default sampleSize_train = 20 X_train = myDistribution.getSample(sampleSize_train) Y_train = model(X_train) The following figure presents the distribution of the vertical deviations Y on the training sample. We observe that the large deviations occur less often. .. code-block:: default 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:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_cantilever_beam_001.png :alt: Distribution of the vertical deviation :class: sphx-glr-single-img Create the metamodel -------------------- In order to create the kriging metamodel, we first select a constant trend with the `ConstantBasisFactory` class. Then we use a squared exponential covariance model. Finally, we use the `KrigingAlgorithm` class to create the kriging metamodel, taking the training sample, the covariance model and the trend basis as input arguments. .. code-block:: default dimension = myDistribution.getDimension() basis = ot.ConstantBasisFactory(dimension).build() covarianceModel = ot.SquaredExponential([1.]*dimension, [1.0]) algo = ot.KrigingAlgorithm(transformation(X_train), Y_train, covarianceModel, basis) algo.run() result = algo.getResult() krigingMetamodel = result.getMetaModel() 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. .. code-block:: default result.getTrendCoefficients() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none [class=Point name=Unnamed dimension=1 values=[0.166863]] We can also print the hyperparameters of the covariance model, which have been estimated by maximizing the likelihood. .. code-block:: default result.getCovarianceModel() .. raw:: html

SquaredExponential(scale=[5.25598,4.89996,4.56237,7.84169], amplitude=[0.0394524])

Validate the metamodel ---------------------- We finally want to validate the kriging metamodel. This is why we generate a validation sample which size is equal to 100 and we evaluate the output of the model on this sample. .. code-block:: default sampleSize_test = 100 X_test = myDistribution.getSample(sampleSize_test) Y_test = model(X_test) The `MetaModelValidation` classe makes the validation easy. To create it, we use the validation samples and the metamodel. .. code-block:: default val = ot.MetaModelValidation(transformation(X_test), Y_test, krigingMetamodel) The `computePredictivityFactor` computes the Q2 factor. .. code-block:: default Q2 = val.computePredictivityFactor() Q2 .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0.9912968960793251 Since the Q2 is larger than 95%, we can say that the quality is acceptable. The residuals are the difference between the model and the metamodel. .. code-block:: default r = val.getResidualSample() graph = ot.HistogramFactory().build(r).drawPDF() view = viewer.View(graph) .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_cantilever_beam_002.png :alt: y0 PDF :class: sphx-glr-single-img We observe that the negative residuals occur with nearly the same frequency of the positive residuals: this is a first sign of good quality. Furthermore, the residuals are most of the times contained in the [-1,1] interval, which is a sign of quality given the amplitude of the output (approximately from 5 to 25 cm). The `drawValidation` method allows to compare the observed outputs and the metamodel outputs. .. code-block:: default graph = val.drawValidation() graph.setTitle("Q2 = %.2f%%" % (100*Q2)) view = viewer.View(graph) plt.show() .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_cantilever_beam_003.png :alt: Q2 = 99.13% :class: sphx-glr-single-img We observe that the metamodel predictions are close to the model outputs, since most red points are close to the diagonal. However, when we consider extreme deviations (i.e. less than 10 or larger than 20), then the quality is less obvious. Given that the kriging metamodel quality is sensitive to the design of experiments, it might be interesting to consider a Latin Hypercube Sampling (LHS) design to further improve the predictions quality. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.213 seconds) .. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_kriging_cantilever_beam.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. 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-jupyter :download:`Download Jupyter notebook: plot_kriging_cantilever_beam.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_