.. 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_hyperparameters_optimization.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_hyperparameters_optimization.py: Kriging: configure the optimization solver ========================================== .. GENERATED FROM PYTHON SOURCE LINES 7-44 The goal of this example is to show how to fine-tune the optimization solver used to estimate the hyperparameters of the covariance model of the Kriging metamodel. Introduction ------------ In a Kriging metamodel, there are various types of parameters which are estimated from the data. * The parameters :math:`{\bf \beta}` associated with the deterministic trend. These parameters are computed based on linear least squares. * The parameters of the covariance model. The covariance model has two types of parameters. * The amplitude parameter :math:`\sigma^2` is estimated from the data. If the output dimension is equal to one, this parameter is estimated using the analytic variance estimator which maximizes the likelihood. Otherwise, if output dimension is greater than one or analytical sigma disabled, this parameter is estimated from numerical optimization. * The other parameters :math:`{\bf \theta}\in\mathbb{R}^d` where :math:`d` is the spatial dimension of the covariance model. Often, the parameter :math:`{\bf \theta}` is a scale parameter. This step involves an optimization algorithm. All these parameters are estimated with the :class:`~openturns.GeneralLinearModelAlgorithm` class. The estimation of the :math:`{\bf \theta}` parameters is the step which has the highest CPU cost. Moreover, the maximization of likelihood may be associated with difficulties e.g. many local maximums or even the non convergence of the optimization algorithm. In this case, it might be useful to fine tune the optimization algorithm so that the convergence of the optimization algorithm is, hopefully, improved. Furthermore, there are several situations in which the optimization can be initialized or completely bypassed. Suppose for example that we have already created an initial Kriging metamodel with :math:`N` points and we want to add a single new point. * It might be interesting to initialize the optimization algorithm with the optimum found for the previous Kriging metamodel: this may reduce the number of iterations required to maximize the likelihood. * We may as well completely bypass the optimization step: if the previous covariance model was correctly estimated, the update of the parameters may or may not significantly improve the estimates. This is why the goal of this example is to see how to configure the optimization of the hyperparameters of a Kriging metamodel. .. GENERATED FROM PYTHON SOURCE LINES 46-48 Definition of the model ----------------------- .. GENERATED FROM PYTHON SOURCE LINES 50-54 .. code-block:: Python import openturns as ot ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 55-56 We define the symbolic function which evaluates the output `Y` depending on the inputs `E`, `F`, `L` and `I`. .. GENERATED FROM PYTHON SOURCE LINES 58-60 .. code-block:: Python model = ot.SymbolicFunction(["E", "F", "L", "I"], ["F*L^3/(3*E*I)"]) .. GENERATED FROM PYTHON SOURCE LINES 61-62 Then we define the distribution of the input random vector. .. GENERATED FROM PYTHON SOURCE LINES 64-65 Young's modulus `E` .. GENERATED FROM PYTHON SOURCE LINES 65-78 .. code-block:: Python E = ot.Beta(0.9, 2.27, 2.5e7, 5.0e7) # in N/m^2 E.setDescription("E") # Load F F = ot.LogNormal() # in N F.setParameter(ot.LogNormalMuSigma()([30.0e3, 9e3, 15.0e3])) F.setDescription("F") # Length L L = ot.Uniform(250.0, 260.0) # in cm L.setDescription("L") # Moment of inertia I II = ot.Beta(2.5, 1.5, 310, 450) # in cm^4 II.setDescription("I") .. GENERATED FROM PYTHON SOURCE LINES 79-80 Finally, we define the dependency using a :class:`~openturns.NormalCopula`. .. GENERATED FROM PYTHON SOURCE LINES 82-88 .. code-block:: Python dim = 4 # number of inputs R = ot.CorrelationMatrix(dim) R[2, 3] = -0.2 myCopula = ot.NormalCopula(ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(R)) myDistribution = ot.JointDistribution([E, F, L, II], myCopula) .. GENERATED FROM PYTHON SOURCE LINES 89-91 Create the design of experiments -------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 93-96 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 98-102 .. code-block:: Python sampleSize_train = 10 X_train = myDistribution.getSample(sampleSize_train) Y_train = model(X_train) .. GENERATED FROM PYTHON SOURCE LINES 103-105 Create the metamodel -------------------- .. GENERATED FROM PYTHON SOURCE LINES 107-111 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 model. Finally, we use the :class:`~openturns.KrigingAlgorithm` class to create the Kriging metamodel, taking the training sample, the covariance model and the trend basis as input arguments. .. GENERATED FROM PYTHON SOURCE LINES 113-136 .. code-block:: Python dimension = myDistribution.getDimension() basis = ot.ConstantBasisFactory(dimension).build() # Trick B, v2 x_range = X_train.getMax() - X_train.getMin() print("x_range:") print(x_range) scale_max_factor = 4.0 # Must be > 1, tune this to match your problem scale_min_factor = 0.1 # Must be < 1, tune this to match your problem maximum_scale_bounds = scale_max_factor * x_range minimum_scale_bounds = scale_min_factor * x_range scaleOptimizationBounds = ot.Interval(minimum_scale_bounds, maximum_scale_bounds) print("scaleOptimizationBounds") print(scaleOptimizationBounds) covarianceModel = ot.SquaredExponential([1.0] * dimension, [1.0]) covarianceModel.setScale(maximum_scale_bounds) # Trick A algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizationBounds(scaleOptimizationBounds) algo.run() result = algo.getResult() krigingMetamodel = result.getMetaModel() .. rst-class:: sphx-glr-script-out .. code-block:: none x_range: [1.88116e+07,15170,8.84385,90.3543] scaleOptimizationBounds [1.88116e+06, 7.52464e+07] [1517, 60680.1] [0.884385, 35.3754] [9.03543, 361.417] .. GENERATED FROM PYTHON SOURCE LINES 137-141 The :meth:`~openturns.KrigingAlgorithm.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 143-145 .. code-block:: Python result.getTrendCoefficients() .. raw:: html
class=Point name=Unnamed dimension=1 values=[24.5528]


.. GENERATED FROM PYTHON SOURCE LINES 146-147 We can also print the hyperparameters of the covariance model, which have been estimated by maximizing the likelihood. .. GENERATED FROM PYTHON SOURCE LINES 149-152 .. code-block:: Python basic_covariance_model = result.getCovarianceModel() print(basic_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[6.81799e+07,60680.1,35.3754,361.417], amplitude=[15.8802]) .. GENERATED FROM PYTHON SOURCE LINES 153-155 Get the optimizer algorithm --------------------------- .. GENERATED FROM PYTHON SOURCE LINES 157-161 The :meth:`~openturns.KrigingAlgorithm.getOptimizationAlgorithm` method returns the optimization algorithm used to optimize the :math:`{\bf \theta}` parameters of the :class:`~openturns.SquaredExponential` covariance model. .. GENERATED FROM PYTHON SOURCE LINES 163-165 .. code-block:: Python solver = algo.getOptimizationAlgorithm() .. GENERATED FROM PYTHON SOURCE LINES 166-167 Get the default optimizer. .. GENERATED FROM PYTHON SOURCE LINES 169-172 .. code-block:: Python solverImplementation = solver.getImplementation() solverImplementation.getClassName() .. rst-class:: sphx-glr-script-out .. code-block:: none 'TNC' .. GENERATED FROM PYTHON SOURCE LINES 173-178 The :meth:`~openturns.KrigingAlgorithm.getOptimizationBounds` method returns the bounds. The dimension of these bounds correspond to the spatial dimension of the covariance model. In the metamodeling context, this correspond to the input dimension of the model. .. GENERATED FROM PYTHON SOURCE LINES 180-183 .. code-block:: Python bounds = algo.getOptimizationBounds() bounds.getDimension() .. rst-class:: sphx-glr-script-out .. code-block:: none 4 .. GENERATED FROM PYTHON SOURCE LINES 184-188 .. code-block:: Python lbounds = bounds.getLowerBound() print("lbounds") print(lbounds) .. rst-class:: sphx-glr-script-out .. code-block:: none lbounds [1.88116e+06,1517,0.884385,9.03543] .. GENERATED FROM PYTHON SOURCE LINES 189-193 .. code-block:: Python ubounds = bounds.getUpperBound() print("ubounds") print(ubounds) .. rst-class:: sphx-glr-script-out .. code-block:: none ubounds [7.52464e+07,60680.1,35.3754,361.417] .. GENERATED FROM PYTHON SOURCE LINES 194-195 The :meth:`~openturns.KrigingAlgorithm.getOptimizeParameters` method returns `True` if these parameters are to be optimized. .. GENERATED FROM PYTHON SOURCE LINES 197-201 .. code-block:: Python isOptimize = algo.getOptimizeParameters() print(isOptimize) .. rst-class:: sphx-glr-script-out .. code-block:: none True .. GENERATED FROM PYTHON SOURCE LINES 202-204 Configure the starting point of the optimization ------------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 206-209 The starting point of the optimization is based on the parameters of the covariance model. In the following example, we configure the parameters of the covariance model to the arbitrary values `[12.0, 34.0, 56.0, 78.0]`. .. GENERATED FROM PYTHON SOURCE LINES 211-216 .. code-block:: Python covarianceModel = ot.SquaredExponential([12.0, 34.0, 56.0, 78.0], [1.0]) covarianceModel.setScale(maximum_scale_bounds) # Trick A algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizationBounds(scaleOptimizationBounds) # Trick B .. GENERATED FROM PYTHON SOURCE LINES 217-219 .. code-block:: Python algo.run() .. GENERATED FROM PYTHON SOURCE LINES 220-224 .. code-block:: Python result = algo.getResult() new_covariance_model = result.getCovarianceModel() print(new_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[6.81799e+07,60680.1,35.3754,361.417], amplitude=[15.8802]) .. GENERATED FROM PYTHON SOURCE LINES 225-226 In order to see the difference with the default optimization, we print the previous optimized covariance model. .. GENERATED FROM PYTHON SOURCE LINES 228-230 .. code-block:: Python print(basic_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[6.81799e+07,60680.1,35.3754,361.417], amplitude=[15.8802]) .. GENERATED FROM PYTHON SOURCE LINES 231-232 We observe that this does not change much the values of the parameters in this case. .. GENERATED FROM PYTHON SOURCE LINES 234-236 Disabling the optimization -------------------------- .. GENERATED FROM PYTHON SOURCE LINES 238-241 It is sometimes useful to completely disable the optimization of the parameters. In order to see the effect of this, we first initialize the parameters of the covariance model with the arbitrary values `[12.0, 34.0, 56.0, 78.0]`. .. GENERATED FROM PYTHON SOURCE LINES 243-246 .. code-block:: Python covarianceModel = ot.SquaredExponential([12.0, 34.0, 56.0, 78.0], [91.0]) algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) .. GENERATED FROM PYTHON SOURCE LINES 247-249 The :meth:`~openturns.KrigingAlgorithm.setOptimizeParameters` method can be used to disable the optimization of the parameters. .. GENERATED FROM PYTHON SOURCE LINES 251-253 .. code-block:: Python algo.setOptimizeParameters(False) .. GENERATED FROM PYTHON SOURCE LINES 254-255 Then we run the algorithm and get the result. .. GENERATED FROM PYTHON SOURCE LINES 257-260 .. code-block:: Python algo.run() result = algo.getResult() .. GENERATED FROM PYTHON SOURCE LINES 261-263 We observe that the covariance model is unchanged: the parameters have not been optimized, as required. .. GENERATED FROM PYTHON SOURCE LINES 265-268 .. code-block:: Python updated_covariance_model = result.getCovarianceModel() print(updated_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[12,34,56,78], amplitude=[91]) .. GENERATED FROM PYTHON SOURCE LINES 269-270 The trend, however, is still optimized, using linear least squares. .. GENERATED FROM PYTHON SOURCE LINES 272-274 .. code-block:: Python result.getTrendCoefficients() .. raw:: html
class=Point name=Unnamed dimension=1 values=[12.9635]


.. GENERATED FROM PYTHON SOURCE LINES 275-282 Reuse the parameters from a previous optimization ------------------------------------------------- In this example, we show how to reuse the optimized parameters of a previous Kriging and configure a new one. Furthermore, we disable the optimization so that the parameters of the covariance model are not updated. This make the process of adding a new point very fast: it improves the quality by adding a new point in the design of experiments without paying the price of the update of the covariance model. .. GENERATED FROM PYTHON SOURCE LINES 284-285 Step 1: Run a first Kriging algorithm. .. GENERATED FROM PYTHON SOURCE LINES 287-298 .. code-block:: Python dimension = myDistribution.getDimension() basis = ot.ConstantBasisFactory(dimension).build() covarianceModel = ot.SquaredExponential([1.0] * dimension, [1.0]) covarianceModel.setScale(maximum_scale_bounds) # Trick A algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizationBounds(scaleOptimizationBounds) # Trick B algo.run() result = algo.getResult() covarianceModel = result.getCovarianceModel() print(covarianceModel) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[6.81799e+07,60680.1,35.3754,361.417], amplitude=[15.8802]) .. GENERATED FROM PYTHON SOURCE LINES 299-300 Step 2: Create a new point and add it to the previous training design. .. GENERATED FROM PYTHON SOURCE LINES 302-305 .. code-block:: Python X_new = myDistribution.getSample(20) Y_new = model(X_new) .. GENERATED FROM PYTHON SOURCE LINES 306-309 .. code-block:: Python X_train.add(X_new) X_train.getSize() .. rst-class:: sphx-glr-script-out .. code-block:: none 30 .. GENERATED FROM PYTHON SOURCE LINES 310-313 .. code-block:: Python Y_train.add(Y_new) Y_train.getSize() .. rst-class:: sphx-glr-script-out .. code-block:: none 30 .. GENERATED FROM PYTHON SOURCE LINES 314-315 Step 3: Create an updated Kriging, using the new point with the old covariance parameters. .. GENERATED FROM PYTHON SOURCE LINES 317-325 .. code-block:: Python algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizeParameters(False) algo.run() result = algo.getResult() notUpdatedCovarianceModel = result.getCovarianceModel() print(notUpdatedCovarianceModel) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[6.81799e+07,60680.1,35.3754,361.417], amplitude=[15.8802]) .. GENERATED FROM PYTHON SOURCE LINES 326-335 .. code-block:: Python def printCovarianceParameterChange(covarianceModel1, covarianceModel2): parameters1 = covarianceModel1.getFullParameter() parameters2 = covarianceModel2.getFullParameter() for i in range(parameters1.getDimension()): deltai = abs(parameters1[i] - parameters2[i]) print("Change in the parameter #%d = %s" % (i, deltai)) return .. GENERATED FROM PYTHON SOURCE LINES 336-338 .. code-block:: Python printCovarianceParameterChange(covarianceModel, notUpdatedCovarianceModel) .. rst-class:: sphx-glr-script-out .. code-block:: none Change in the parameter #0 = 0.0 Change in the parameter #1 = 0.0 Change in the parameter #2 = 0.0 Change in the parameter #3 = 0.0 Change in the parameter #4 = 0.0 Change in the parameter #5 = 0.0 .. GENERATED FROM PYTHON SOURCE LINES 339-343 We see that the parameters did not change *at all*: disabling the optimization allows one to keep a constant covariance model. In a practical algorithm, we may, for example, add a block of 10 new points before updating the parameters of the covariance model. At this point, we may reuse the previous covariance model so that the optimization starts from a better point, compared to the parameters default values. This will reduce the cost of the optimization. .. GENERATED FROM PYTHON SOURCE LINES 345-347 Configure the local optimization solver --------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 349-351 The following example shows how to set the local optimization solver. We choose the `SLSQP` algorithm from :class:`~openturns.NLopt`. .. GENERATED FROM PYTHON SOURCE LINES 353-362 .. code-block:: Python problem = solver.getProblem() local_solver = ot.NLopt(problem, "LD_SLSQP") covarianceModel = ot.SquaredExponential([1.0] * dimension, [1.0]) covarianceModel.setScale(maximum_scale_bounds) # Trick A algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizationBounds(scaleOptimizationBounds) # Trick B algo.setOptimizationAlgorithm(local_solver) algo.run() .. GENERATED FROM PYTHON SOURCE LINES 363-366 .. code-block:: Python finetune_covariance_model = result.getCovarianceModel() print(finetune_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[6.81799e+07,60680.1,35.3754,361.417], amplitude=[15.8802]) .. GENERATED FROM PYTHON SOURCE LINES 367-370 .. code-block:: Python printCovarianceParameterChange(finetune_covariance_model, basic_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none Change in the parameter #0 = 0.0 Change in the parameter #1 = 0.0 Change in the parameter #2 = 0.0 Change in the parameter #3 = 0.0 Change in the parameter #4 = 0.0 Change in the parameter #5 = 0.0 .. GENERATED FROM PYTHON SOURCE LINES 371-373 Configure the global optimization solver ---------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 375-382 The following example checks the robustness of the optimization of the Kriging algorithm with respect to the optimization of the likelihood function in the covariance model parameters estimation. We use a :class:`~openturns.MultiStart` algorithm in order to avoid to be trapped by a local minimum. Furthermore, we generate the design of experiments using a :class:`~openturns.LHSExperiment`, which guarantees that the points will fill the space. .. GENERATED FROM PYTHON SOURCE LINES 384-388 .. code-block:: Python sampleSize_train = 10 X_train = myDistribution.getSample(sampleSize_train) Y_train = model(X_train) .. GENERATED FROM PYTHON SOURCE LINES 389-392 First, we create a multivariate distribution, based on independent :class:`~openturns.Uniform` marginals which have the bounds required by the covariance model. .. GENERATED FROM PYTHON SOURCE LINES 394-397 .. code-block:: Python distributions = [ot.Uniform(lbounds[i], ubounds[i]) for i in range(dim)] boundedDistribution = ot.JointDistribution(distributions) .. GENERATED FROM PYTHON SOURCE LINES 398-400 We first generate a Latin Hypercube Sampling (LHS) design made of 25 points in the sample space. This LHS is optimized so as to fill the space. .. GENERATED FROM PYTHON SOURCE LINES 402-412 .. code-block:: Python K = 25 # design size LHS = ot.LHSExperiment(boundedDistribution, K) LHS.setAlwaysShuffle(True) SA_profile = ot.GeometricProfile(10.0, 0.95, 20000) LHS_optimization_algo = ot.SimulatedAnnealingLHS(LHS, ot.SpaceFillingC2(), SA_profile) LHS_optimization_algo.generate() LHS_design = LHS_optimization_algo.getResult() starting_points = LHS_design.getOptimalDesign() starting_points.getSize() .. rst-class:: sphx-glr-script-out .. code-block:: none 25 .. GENERATED FROM PYTHON SOURCE LINES 413-415 We can check that the minimum and maximum in the sample correspond to the bounds of the design of experiments. .. GENERATED FROM PYTHON SOURCE LINES 417-419 .. code-block:: Python print(lbounds, ubounds) .. rst-class:: sphx-glr-script-out .. code-block:: none [1.88116e+06,1517,0.884385,9.03543] [7.52464e+07,60680.1,35.3754,361.417] .. GENERATED FROM PYTHON SOURCE LINES 420-422 .. code-block:: Python starting_points.getMin(), starting_points.getMax() .. rst-class:: sphx-glr-script-out .. code-block:: none (class=Point name=Unnamed dimension=4 values=[2.24878e+06,3846.09,1.17854,21.4222], class=Point name=Unnamed dimension=4 values=[7.29669e+07,58613.2,35.2441,347.446]) .. GENERATED FROM PYTHON SOURCE LINES 423-424 Then we create a :class:`~openturns.MultiStart` algorithm based on the LHS starting points. .. GENERATED FROM PYTHON SOURCE LINES 426-429 .. code-block:: Python solver.setMaximumIterationNumber(10000) multiStartSolver = ot.MultiStart(solver, starting_points) .. GENERATED FROM PYTHON SOURCE LINES 430-431 Finally, we configure the optimization algorithm so as to use the :class:`~openturns.MultiStart` algorithm. .. GENERATED FROM PYTHON SOURCE LINES 433-438 .. code-block:: Python algo = ot.KrigingAlgorithm(X_train, Y_train, covarianceModel, basis) algo.setOptimizationBounds(scaleOptimizationBounds) # Trick B algo.setOptimizationAlgorithm(multiStartSolver) algo.run() .. GENERATED FROM PYTHON SOURCE LINES 439-442 .. code-block:: Python finetune_covariance_model = result.getCovarianceModel() print(finetune_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[6.81799e+07,60680.1,35.3754,361.417], amplitude=[15.8802]) .. GENERATED FROM PYTHON SOURCE LINES 443-445 .. code-block:: Python printCovarianceParameterChange(finetune_covariance_model, basic_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none Change in the parameter #0 = 0.0 Change in the parameter #1 = 0.0 Change in the parameter #2 = 0.0 Change in the parameter #3 = 0.0 Change in the parameter #4 = 0.0 Change in the parameter #5 = 0.0 .. GENERATED FROM PYTHON SOURCE LINES 446-447 We see that there are no changes in the estimated parameters. This shows that the first optimization of the parameters worked fine. .. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_kriging_hyperparameters_optimization.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_hyperparameters_optimization.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_kriging_hyperparameters_optimization.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_kriging_hyperparameters_optimization.zip `