.. 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_isotropic.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_isotropic.py: Kriging with an isotropic covariance function ============================================= In typical machine learning applications, Gaussian process regression/Kriging surrogate models take several inputs, and those inputs are usually heterogeneous (e.g. in the :doc:`cantilever beam ` use case, inputs are various physical quantities). In geostatistical applications however, inputs are typically spatial coordinates, which means one can assume the output varies at the same rate in all directions. This calls for a specific kind of covariance kernel, represented in the library by the :class:`~openturns.IsotropicCovarianceModel` class. .. GENERATED FROM PYTHON SOURCE LINES 20-23 Modeling temperature across a surface ------------------------------------- In this example, we collect temperature data over a floorplan using sensors. .. GENERATED FROM PYTHON SOURCE LINES 25-48 .. code-block:: Python import numpy as np import openturns as ot import matplotlib.pyplot as plt ot.Log.Show(ot.Log.NONE) coordinates = ot.Sample( [ [100.0, 100.0], [500.0, 100.0], [900.0, 100.0], [100.0, 350.0], [500.0, 350.0], [900.0, 350.0], [100.0, 600.0], [500.0, 600.0], [900.0, 600.0], ] ) observations = ot.Sample( [[25.0], [25.0], [10.0], [20.0], [25.0], [20.0], [15.0], [25.0], [25.0]] ) .. GENERATED FROM PYTHON SOURCE LINES 49-50 Let us plot the data. .. GENERATED FROM PYTHON SOURCE LINES 50-61 .. code-block:: Python # Extract coordinates. x = np.array(coordinates[:, 0]) y = np.array(coordinates[:, 1]) # Plot the data with a scatter plot and a color map. fig = plt.figure() plt.scatter(x, y, c=observations, cmap="viridis") plt.colorbar() plt.show() .. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_isotropic_001.png :alt: plot kriging isotropic :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_isotropic_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 62-68 Because we are going to view several Kriging models in this example, we use a function to automate the process of optimizing the scale parameter and producing the metamodel. Since version 1.15 of the library, input data are no longer rescaled by the :class:`~openturns.KrigingAlgorithm` class, so we need to manually set sensible bounds for the scale parameter. .. GENERATED FROM PYTHON SOURCE LINES 68-94 .. code-block:: Python lower = 50.0 upper = 1000.0 def fitKriging(coordinates, observations, covarianceModel, basis): """ Fit the parameters of a Kriging metamodel. """ # Define the Kriging algorithm. algo = ot.KrigingAlgorithm(coordinates, observations, covarianceModel, basis) # Set the optimization bounds for the scale parameter to sensible values # given the data set. scale_dimension = covarianceModel.getScale().getDimension() algo.setOptimizationBounds( ot.Interval([lower] * scale_dimension, [upper] * scale_dimension) ) # Run the Kriging algorithm and extract the fitted surrogate model. algo.run() krigingResult = algo.getResult() krigingMetamodel = krigingResult.getMetaModel() return krigingResult, krigingMetamodel .. GENERATED FROM PYTHON SOURCE LINES 95-96 Let us define a helper function to plot Kriging predictions. .. GENERATED FROM PYTHON SOURCE LINES 96-129 .. code-block:: Python def plotKrigingPredictions(krigingMetamodel): """ Plot the predictions of a Kriging metamodel. """ # Create the mesh of the box [0., 1000.] * [0., 700.] myInterval = ot.Interval([0.0, 0.0], [1000.0, 700.0]) # Define the number of intervals in each direction of the box nx = 20 ny = 20 myIndices = [nx - 1, ny - 1] myMesher = ot.IntervalMesher(myIndices) myMeshBox = myMesher.build(myInterval) # Predict vertices = myMeshBox.getVertices() predictions = krigingMetamodel(vertices) # Format for plot X = np.array(vertices[:, 0]).reshape((ny, nx)) Y = np.array(vertices[:, 1]).reshape((ny, nx)) predictions_array = np.array(predictions).reshape((ny, nx)) # Plot plt.figure() plt.pcolormesh(X, Y, predictions_array, shading="auto") plt.colorbar() plt.show() return .. GENERATED FROM PYTHON SOURCE LINES 130-134 Predict with an anisotropic geometric covariance kernel ------------------------------------------------------- In order to illustrate the usefulness of isotropic covariance kernels, we first perform prediction with an anisotropic geometric kernel. .. GENERATED FROM PYTHON SOURCE LINES 136-143 Keep in mind that, when there are more than one input dimension, the :class:`~openturns.CovarianceModel` classes use a multidimensional scale parameter :math:`\vect{\theta}`. They are anisotropic geometric by default. Our example has two input dimensions, so :math:`\vect{\theta} = (\theta_1, \theta_2)`. .. GENERATED FROM PYTHON SOURCE LINES 143-153 .. code-block:: Python inputDimension = 2 basis = ot.ConstantBasisFactory(inputDimension).build() covarianceModel = ot.SquaredExponential(inputDimension) krigingResult, krigingMetamodel = fitKriging( coordinates, observations, covarianceModel, basis ) plotKrigingPredictions(krigingMetamodel) .. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_isotropic_002.png :alt: plot kriging isotropic :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_isotropic_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 154-157 We see weird vertical columns on the plot. How did this happen? Let us have a look at the optimized scale parameter :math:`\hat{\vect{\theta}} = (\hat{\theta}_1, \hat{\theta}_2)`. .. GENERATED FROM PYTHON SOURCE LINES 157-159 .. code-block:: Python print(krigingResult.getCovarianceModel().getScale()) .. rst-class:: sphx-glr-script-out .. code-block:: none [50,344.874] .. GENERATED FROM PYTHON SOURCE LINES 160-161 The value of :math:`\hat{\theta}_1` is actually equal to the lower bound: .. GENERATED FROM PYTHON SOURCE LINES 161-164 .. code-block:: Python print(lower) .. rst-class:: sphx-glr-script-out .. code-block:: none 50.0 .. GENERATED FROM PYTHON SOURCE LINES 165-167 This means that temperatures are likely to vary a lot along the X axis and much slower across the Y axis based on the observation data. .. GENERATED FROM PYTHON SOURCE LINES 169-174 Predict with an isotropic covariance kernel --------------------------------------------------- If we know that variations of the temperature are isotropic (i.e. with no priviledged direction), we can embed this information within the covariance kernel. .. GENERATED FROM PYTHON SOURCE LINES 174-177 .. code-block:: Python isotropic = ot.IsotropicCovarianceModel(ot.SquaredExponential(), inputDimension) .. GENERATED FROM PYTHON SOURCE LINES 178-183 The :class:`~openturns.IsotropicCovarianceModel` class creates an isotropic version with a given input dimension of a :class:`~openturns.CovarianceModel`. Because is is isotropic, it only needs one scale parameter :math:`\theta_{iso}` and it will make sure :math:`\theta_1 = \theta_2 = \theta_{iso}` at all times during the optimization. .. GENERATED FROM PYTHON SOURCE LINES 183-189 .. code-block:: Python krigingResult, krigingMetamodel = fitKriging( coordinates, observations, isotropic, basis ) print(krigingResult.getCovarianceModel().getScale()) .. rst-class:: sphx-glr-script-out .. code-block:: none [286.426] .. GENERATED FROM PYTHON SOURCE LINES 190-191 Prediction with the isotropic covariance kernel is much more satisfactory. .. GENERATED FROM PYTHON SOURCE LINES 191-194 .. code-block:: Python # sphinx_gallery_thumbnail_number = 3 plotKrigingPredictions(krigingMetamodel) .. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_isotropic_003.png :alt: plot kriging isotropic :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_isotropic_003.png :class: sphx-glr-single-img .. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_kriging_isotropic.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_isotropic.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_kriging_isotropic.py `