.. 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 6-43 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 `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 45-47 Definition of the model ----------------------- .. GENERATED FROM PYTHON SOURCE LINES 49-53 .. code-block:: Python import openturns as ot ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 54-55 We define the symbolic function which evaluates the output Y depending on the inputs E, F, L and I. .. GENERATED FROM PYTHON SOURCE LINES 57-59 .. code-block:: Python model = ot.SymbolicFunction(["E", "F", "L", "I"], ["F*L^3/(3*E*I)"]) .. GENERATED FROM PYTHON SOURCE LINES 60-61 Then we define the distribution of the input random vector. .. GENERATED FROM PYTHON SOURCE LINES 63-64 Young's modulus E .. GENERATED FROM PYTHON SOURCE LINES 64-77 .. 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 78-79 Finally, we define the dependency using a `NormalCopula`. .. GENERATED FROM PYTHON SOURCE LINES 81-87 .. 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.ComposedDistribution([E, F, L, II], myCopula) .. GENERATED FROM PYTHON SOURCE LINES 88-90 Create the design of experiments -------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 92-94 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 96-100 .. code-block:: Python sampleSize_train = 10 X_train = myDistribution.getSample(sampleSize_train) Y_train = model(X_train) .. GENERATED FROM PYTHON SOURCE LINES 101-103 Create the metamodel -------------------- .. GENERATED FROM PYTHON SOURCE LINES 105-109 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. .. GENERATED FROM PYTHON SOURCE LINES 111-134 .. 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: [2.12636e+07,24296.8,7.35174,106.039] scaleOptimizationBounds [2.12636e+06, 8.50545e+07] [2429.68, 97187.2] [0.735174, 29.407] [10.6039, 424.154] .. GENERATED FROM PYTHON SOURCE LINES 135-138 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 140-142 .. code-block:: Python result.getTrendCoefficients() .. raw:: html
class=Point name=Unnamed dimension=1 values=[18.0423]


.. GENERATED FROM PYTHON SOURCE LINES 143-144 We can also print the hyperparameters of the covariance model, which have been estimated by maximizing the likelihood. .. GENERATED FROM PYTHON SOURCE LINES 146-149 .. code-block:: Python basic_covariance_model = result.getCovarianceModel() print(basic_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[8.50532e+07,22986,29.407,366.658], amplitude=[14.488]) .. GENERATED FROM PYTHON SOURCE LINES 150-152 Get the optimizer algorithm --------------------------- .. GENERATED FROM PYTHON SOURCE LINES 154-155 The `getOptimizationAlgorithm` method returns the optimization algorithm used to optimize the :math:`{\bf \theta}` parameters of the `SquaredExponential` covariance model. .. GENERATED FROM PYTHON SOURCE LINES 157-159 .. code-block:: Python solver = algo.getOptimizationAlgorithm() .. GENERATED FROM PYTHON SOURCE LINES 160-161 Get the default optimizer. .. GENERATED FROM PYTHON SOURCE LINES 163-166 .. code-block:: Python solverImplementation = solver.getImplementation() solverImplementation.getClassName() .. rst-class:: sphx-glr-script-out .. code-block:: none 'TNC' .. GENERATED FROM PYTHON SOURCE LINES 167-169 The `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 171-174 .. code-block:: Python bounds = algo.getOptimizationBounds() bounds.getDimension() .. rst-class:: sphx-glr-script-out .. code-block:: none 4 .. GENERATED FROM PYTHON SOURCE LINES 175-179 .. code-block:: Python lbounds = bounds.getLowerBound() print("lbounds") print(lbounds) .. rst-class:: sphx-glr-script-out .. code-block:: none lbounds [2.12636e+06,2429.68,0.735174,10.6039] .. GENERATED FROM PYTHON SOURCE LINES 180-184 .. code-block:: Python ubounds = bounds.getUpperBound() print("ubounds") print(ubounds) .. rst-class:: sphx-glr-script-out .. code-block:: none ubounds [8.50545e+07,97187.2,29.407,424.154] .. GENERATED FROM PYTHON SOURCE LINES 185-186 The `getOptimizeParameters` method returns `True` if these parameters are to be optimized. .. GENERATED FROM PYTHON SOURCE LINES 188-192 .. code-block:: Python isOptimize = algo.getOptimizeParameters() print(isOptimize) .. rst-class:: sphx-glr-script-out .. code-block:: none True .. GENERATED FROM PYTHON SOURCE LINES 193-195 Configure the starting point of the optimization ------------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 197-199 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.,34.,56.,78.]`. .. GENERATED FROM PYTHON SOURCE LINES 201-206 .. 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 207-209 .. code-block:: Python algo.run() .. GENERATED FROM PYTHON SOURCE LINES 210-214 .. 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=[8.50532e+07,22986,29.407,366.658], amplitude=[14.488]) .. GENERATED FROM PYTHON SOURCE LINES 215-216 In order to see the difference with the default optimization, we print the previous optimized covariance model. .. GENERATED FROM PYTHON SOURCE LINES 218-220 .. code-block:: Python print(basic_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[8.50532e+07,22986,29.407,366.658], amplitude=[14.488]) .. GENERATED FROM PYTHON SOURCE LINES 221-222 We observe that this does not change much the values of the parameters in this case. .. GENERATED FROM PYTHON SOURCE LINES 224-226 Disabling the optimization -------------------------- .. GENERATED FROM PYTHON SOURCE LINES 228-230 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.,34.,56.,78.]`. .. GENERATED FROM PYTHON SOURCE LINES 232-236 .. 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) algo.setOptimizationBounds(scaleOptimizationBounds) # Trick B .. GENERATED FROM PYTHON SOURCE LINES 237-238 The `setOptimizeParameters` method can be used to disable the optimization of the parameters. .. GENERATED FROM PYTHON SOURCE LINES 240-242 .. code-block:: Python algo.setOptimizeParameters(False) .. GENERATED FROM PYTHON SOURCE LINES 243-244 Then we run the algorithm and get the result. .. GENERATED FROM PYTHON SOURCE LINES 246-249 .. code-block:: Python algo.run() result = algo.getResult() .. GENERATED FROM PYTHON SOURCE LINES 250-252 We observe that the covariance model is unchanged: the parameters have not been optimized, as required. .. GENERATED FROM PYTHON SOURCE LINES 254-257 .. 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 258-259 The trend, however, is still optimized, using linear least squares. .. GENERATED FROM PYTHON SOURCE LINES 261-263 .. code-block:: Python result.getTrendCoefficients() .. raw:: html
class=Point name=Unnamed dimension=1 values=[12.0499]


.. GENERATED FROM PYTHON SOURCE LINES 264-271 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 273-274 Step 1: Run a first kriging algorithm. .. GENERATED FROM PYTHON SOURCE LINES 276-287 .. 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=[8.50532e+07,22986,29.407,366.658], amplitude=[14.488]) .. GENERATED FROM PYTHON SOURCE LINES 288-289 Step 2: Create a new point and add it to the previous training design. .. GENERATED FROM PYTHON SOURCE LINES 291-294 .. code-block:: Python X_new = myDistribution.getSample(20) Y_new = model(X_new) .. GENERATED FROM PYTHON SOURCE LINES 295-298 .. 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 299-302 .. 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 303-304 Step 3: Create an updated kriging, using the new point with the old covariance parameters. .. GENERATED FROM PYTHON SOURCE LINES 306-314 .. 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=[8.50532e+07,22986,29.407,366.658], amplitude=[14.488]) .. GENERATED FROM PYTHON SOURCE LINES 315-324 .. 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 325-327 .. 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 .. GENERATED FROM PYTHON SOURCE LINES 328-332 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 334-336 Configure the local optimization solver --------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 338-340 The following example shows how to set the local optimization solver. We choose the SLSQP algorithm from NLOPT. .. GENERATED FROM PYTHON SOURCE LINES 342-351 .. 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 352-355 .. code-block:: Python finetune_covariance_model = result.getCovarianceModel() print(finetune_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[8.50532e+07,22986,29.407,366.658], amplitude=[14.488]) .. GENERATED FROM PYTHON SOURCE LINES 356-359 .. 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 .. GENERATED FROM PYTHON SOURCE LINES 360-362 Configure the global optimization solver ---------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 364-368 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 `MultiStart` algorithm in order to avoid to be trapped by a local minimum. Furthermore, we generate the design of experiments using a `LHSExperiments`, which guarantees that the points will fill the space. .. GENERATED FROM PYTHON SOURCE LINES 370-374 .. code-block:: Python sampleSize_train = 10 X_train = myDistribution.getSample(sampleSize_train) Y_train = model(X_train) .. GENERATED FROM PYTHON SOURCE LINES 375-376 First, we create a multivariate distribution, based on independent `Uniform` marginals which have the bounds required by the covariance model. .. GENERATED FROM PYTHON SOURCE LINES 378-381 .. code-block:: Python distributions = [ot.Uniform(lbounds[i], ubounds[i]) for i in range(dim)] boundedDistribution = ot.ComposedDistribution(distributions) .. GENERATED FROM PYTHON SOURCE LINES 382-383 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 385-395 .. 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 396-397 We can check that the minimum and maximum in the sample correspond to the bounds of the design of experiment. .. GENERATED FROM PYTHON SOURCE LINES 399-401 .. code-block:: Python print(lbounds, ubounds) .. rst-class:: sphx-glr-script-out .. code-block:: none [2.12636e+06,2429.68,0.735174,10.6039] [8.50545e+07,97187.2,29.407,424.154] .. GENERATED FROM PYTHON SOURCE LINES 402-404 .. 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=[3.58268e+06,4126.37,0.875832,25.501], class=Point name=Unnamed dimension=4 values=[8.21336e+07,95739.1,28.6265,414.801]) .. GENERATED FROM PYTHON SOURCE LINES 405-406 Then we create a `MultiStart` algorithm based on the LHS starting points. .. GENERATED FROM PYTHON SOURCE LINES 408-411 .. code-block:: Python solver.setMaximumIterationNumber(10000) multiStartSolver = ot.MultiStart(solver, starting_points) .. GENERATED FROM PYTHON SOURCE LINES 412-413 Finally, we configure the optimization algorithm so as to use the `MultiStart` algorithm. .. GENERATED FROM PYTHON SOURCE LINES 415-420 .. 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 421-424 .. code-block:: Python finetune_covariance_model = result.getCovarianceModel() print(finetune_covariance_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SquaredExponential(scale=[8.50532e+07,22986,29.407,366.658], amplitude=[14.488]) .. GENERATED FROM PYTHON SOURCE LINES 425-427 .. 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 .. GENERATED FROM PYTHON SOURCE LINES 428-429 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 `