.. 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_beam_trend.py" .. LINE NUMBERS ARE GIVEN BELOW. .. 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_beam_trend.py: Choose the trend basis of a kriging metamodel ============================================= .. GENERATED FROM PYTHON SOURCE LINES 6-13 The goal of this example is to show how to configure the trend in a Kriging metamodel. This is why this example focuses on the three available trends: * `ConstantBasisFactory`, * `LinearBasisFactory`, * `QuadraticBasisFactory`. For this purpose, we use the :ref:`cantilever beam ` example. .. GENERATED FROM PYTHON SOURCE LINES 15-17 Definition of the model ----------------------- .. GENERATED FROM PYTHON SOURCE LINES 19-25 .. code-block:: default from openturns.usecases import cantilever_beam as cantilever_beam import openturns as ot from openturns.viewer import View ot.RandomGenerator.SetSeed(0) ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 26-27 We load the use case : .. GENERATED FROM PYTHON SOURCE LINES 27-29 .. code-block:: default cb = cantilever_beam.CantileverBeam() .. GENERATED FROM PYTHON SOURCE LINES 30-31 We define the function which evaluates the output depending on the inputs. .. GENERATED FROM PYTHON SOURCE LINES 31-33 .. code-block:: default model = cb.model .. GENERATED FROM PYTHON SOURCE LINES 34-35 Then we define the distribution of the input random vector. .. GENERATED FROM PYTHON SOURCE LINES 35-38 .. code-block:: default dimension = cb.dim # number of inputs myDistribution = cb.distribution .. GENERATED FROM PYTHON SOURCE LINES 39-41 Create the design of experiments -------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 43-44 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. .. GENERATED FROM PYTHON SOURCE LINES 46-50 .. code-block:: default sampleSize_train = 10 X_train = myDistribution.getSample(sampleSize_train) Y_train = model(X_train) .. GENERATED FROM PYTHON SOURCE LINES 51-53 Create the metamodel -------------------- .. GENERATED FROM PYTHON SOURCE LINES 55-57 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 59-62 .. code-block:: default basis = ot.ConstantBasisFactory(dimension).build() covarianceModel = ot.SquaredExponential(dimension) .. GENERATED FROM PYTHON SOURCE LINES 63-65 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 67-70 .. code-block:: default print("Lower and upper bounds of X_train:") print(X_train.getMin(), X_train.getMax()) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none Lower and upper bounds of X_train: [6.50185e+10,262.654,2.50948,1.40294e-07] [6.88439e+10,323.088,2.59143,1.5807e-07] .. GENERATED FROM PYTHON SOURCE LINES 71-73 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 75-78 .. code-block:: default 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 79-83 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 85-89 .. code-block:: default covarianceModel.setScale(X_train.getMax()) algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizationBounds(scaleOptimizationBounds) .. GENERATED FROM PYTHON SOURCE LINES 90-91 Run the algorithm and get the result. .. GENERATED FROM PYTHON SOURCE LINES 93-97 .. code-block:: default algo.run() result = algo.getResult() krigingWithConstantTrend = result.getMetaModel() .. GENERATED FROM PYTHON SOURCE LINES 98-99 The `getTrendCoefficients` method returns the coefficients of the trend. .. GENERATED FROM PYTHON SOURCE LINES 101-103 .. code-block:: default print(result.getTrendCoefficients()) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none [[0.139566]] .. GENERATED FROM PYTHON SOURCE LINES 104-105 The constant trend always has only one coefficient (if there is one single output). .. GENERATED FROM PYTHON SOURCE LINES 107-109 .. code-block:: default print(result.getCovarianceModel()) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none SquaredExponential(scale=[6.88439e+10,323.088,2.59143,1.5807e-07], amplitude=[0.182278]) .. GENERATED FROM PYTHON SOURCE LINES 110-112 Setting the trend ----------------- .. GENERATED FROM PYTHON SOURCE LINES 114-123 .. code-block:: default covarianceModel.setScale(X_train.getMax()) basis = ot.LinearBasisFactory(dimension).build() algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizationBounds(scaleOptimizationBounds) algo.run() result = algo.getResult() krigingWithLinearTrend = result.getMetaModel() result.getTrendCoefficients() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none [class=Point name=Unnamed dimension=5 values=[-1.03742e-24,-1.3417e-13,-3.04399e-22,-2.45078e-24,-1.55769e-31]] .. GENERATED FROM PYTHON SOURCE LINES 124-137 The number of coefficients in the linear and quadratic trends depends on the number of inputs, which is equal to .. math:: dim = 4 in the cantilever beam case. We observe that the number of coefficients in the trend is 5, which corresponds to: * 1 coefficient for the constant part, * dim=4 coefficients for the linear part. .. GENERATED FROM PYTHON SOURCE LINES 139-150 .. code-block:: default covarianceModel.setScale(X_train.getMax()) basis = ot.QuadraticBasisFactory(dimension).build() algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizationBounds(scaleOptimizationBounds) algo.run() result = algo.getResult() krigingWithQuadraticTrend = result.getMetaModel() result.getTrendCoefficients() print(algo.getOptimizationBounds()) print(result.getCovarianceModel()) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none [1, 1e+11] [1, 1000] [1, 10] [1e-10, 1e-05] SquaredExponential(scale=[6.88439e+10,323.088,2.59143,1.5807e-07], amplitude=[1.3025e-10]) .. GENERATED FROM PYTHON SOURCE LINES 151-164 This time, the trend has 15 coefficients: * 1 coefficient for the constant part, * 4 coefficients for the linear part, * 10 coefficients for the quadratic part. This is because the number of coefficients in the quadratic part has .. math:: \frac{dim (dim+1)}{2}=\frac{4\times 5}{2}=10 coefficients, associated with the symmetric matrix of the quadratic function. .. GENERATED FROM PYTHON SOURCE LINES 166-168 Validate the metamodel ---------------------- .. GENERATED FROM PYTHON SOURCE LINES 170-171 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 173-178 .. code-block:: default sampleSize_test = 100 X_test = myDistribution.getSample(sampleSize_test) Y_test = model(X_test) .. GENERATED FROM PYTHON SOURCE LINES 179-189 .. code-block:: default def drawMetaModelValidation(X_test, Y_test, krigingMetamodel, title): val = ot.MetaModelValidation(X_test, Y_test, krigingMetamodel) Q2 = val.computePredictivityFactor()[0] graph = val.drawValidation().getGraph(0, 0) graph.setLegends([""]) graph.setLegends(["%s, Q2 = %.2f%%" % (title, 100*Q2), ""]) graph.setLegendPosition("topleft") return graph .. GENERATED FROM PYTHON SOURCE LINES 190-203 .. code-block:: default grid = ot.GridLayout(1, 3) grid.setTitle("Different trends") graphConstant = drawMetaModelValidation( X_test, Y_test, krigingWithConstantTrend, "Constant") graphLinear = drawMetaModelValidation( X_test, Y_test, krigingWithLinearTrend, "Linear") graphQuadratic = drawMetaModelValidation( X_test, Y_test, krigingWithQuadraticTrend, "Quadratic") grid.setGraph(0, 0, graphConstant) grid.setGraph(0, 1, graphLinear) grid.setGraph(0, 2, graphQuadratic) _ = View(grid, figure_kw={'figsize': (13, 4)}) .. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_beam_trend_001.png :alt: Different trends :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_beam_trend_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 204-214 We observe that the three trends perform very well in this case. With more coefficients, the Kriging metamodel is more flexibile and can adjust better to the training sample. This does not mean, however, that the trend coefficients will provide a good fit for the validation sample. The number of parameters in each Kriging metamodel is the following: * the constant trend Kriging has 6 coefficients to estimate: 5 coefficients for the covariance matrix and 1 coefficient for the trend, * the linear trend Kriging has 10 coefficients to estimate: 5 coefficients for the covariance matrix and 5 coefficients for the trend, * the quadratic trend Kriging has 20 coefficients to estimate: 5 coefficients for the covariance matrix and 15 coefficients for the trend. In the cantilever beam example, fitting the metamodel to a training sample with only 10 points is made much easier because the function to approximate is almost linear. In this case, a quadratic trend is not advisable because it can interpolate all points in the training sample. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.266 seconds) .. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_kriging_beam_trend.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_beam_trend.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_kriging_beam_trend.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_