.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_meta_modeling/polynomial_chaos_metamodel/plot_functional_chaos_advanced_ctors.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_polynomial_chaos_metamodel_plot_functional_chaos_advanced_ctors.py: Advanced polynomial chaos construction ====================================== .. GENERATED FROM PYTHON SOURCE LINES 6-11 In this example we are going to expose advanced elements in the construction of a polynomial chaos algorithm: - construction of the multivariate orthonormal basis, - truncature strategy of the multivariate orthonormal basis, - evaluation strategy of the approximation coefficients. .. GENERATED FROM PYTHON SOURCE LINES 13-28 In this example, we consider the following function :math:`\mathbb{R}^4 \rightarrow \mathbb{R}`: .. math:: g(\mathbf{x}) = 1+x_1 x_2 + 2 x_3^2+x_4^4 for any :math:`x_1,x_2,x_3,x_4\in\mathbb{R}`. We assume that the inputs have normal, uniform, gamma and beta distributions : .. math:: X_1 \sim \mathcal{N}(0,1), \qquad X_2 \sim \mathcal{U}(-1,1), \qquad X_3 \sim \mathcal{G}(2.75,1), \qquad X_4 \sim \mathcal{B}(2.5,1,-1,2), and :math:`X_1`, :math:`X_2`, :math:`X_3` and :math:`X_4` are independent. .. GENERATED FROM PYTHON SOURCE LINES 30-32 Define the model and the input distribution ------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 34-40 .. code-block:: default from __future__ import print_function import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 41-44 .. code-block:: default model = ot.SymbolicFunction(['x1', 'x2', 'x3', 'x4'], [ '1+x1*x2 + 2*x3^2+x4^4']) .. GENERATED FROM PYTHON SOURCE LINES 45-46 Create a distribution of dimension 4. .. GENERATED FROM PYTHON SOURCE LINES 48-51 .. code-block:: default distribution = ot.ComposedDistribution( [ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0), ot.Beta(2.5, 1.0, -1.0, 2.0)]) .. GENERATED FROM PYTHON SOURCE LINES 52-55 .. code-block:: default inputDimension = distribution.getDimension() inputDimension .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 4 .. GENERATED FROM PYTHON SOURCE LINES 56-58 STEP 1: Construction of the multivariate orthonormal basis ---------------------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 60-61 Create the univariate polynomial family collection which regroups the polynomial families for each direction. .. GENERATED FROM PYTHON SOURCE LINES 63-65 .. code-block:: default polyColl = ot.PolynomialFamilyCollection(inputDimension) .. GENERATED FROM PYTHON SOURCE LINES 66-67 We could use the Krawtchouk and Charlier families (for discrete distributions). .. GENERATED FROM PYTHON SOURCE LINES 69-72 .. code-block:: default polyColl[0] = ot.KrawtchoukFactory() polyColl[1] = ot.CharlierFactory() .. GENERATED FROM PYTHON SOURCE LINES 73-74 We could also use the automatic selection of the polynomial which corresponds to the distribution: this is done with the `StandardDistributionPolynomialFactory` class. .. GENERATED FROM PYTHON SOURCE LINES 76-80 .. code-block:: default for i in range(inputDimension): marginal = distribution.getMarginal(i) polyColl[i] = ot.StandardDistributionPolynomialFactory(marginal) .. GENERATED FROM PYTHON SOURCE LINES 81-82 In our specific case, we use specific polynomial factories. .. GENERATED FROM PYTHON SOURCE LINES 84-90 .. code-block:: default polyColl[0] = ot.HermiteFactory() polyColl[1] = ot.LegendreFactory() polyColl[2] = ot.LaguerreFactory(2.75) # Parameter for the Jacobi factory : 'Probabilty' encoded with 1 polyColl[3] = ot.JacobiFactory(2.5, 3.5, 1) .. GENERATED FROM PYTHON SOURCE LINES 91-92 Create the enumeration function. .. GENERATED FROM PYTHON SOURCE LINES 94-95 The first possibility is to use the `LinearEnumerateFunction`. .. GENERATED FROM PYTHON SOURCE LINES 97-99 .. code-block:: default enumerateFunction = ot.LinearEnumerateFunction(inputDimension) .. GENERATED FROM PYTHON SOURCE LINES 100-101 Another possibility is to use the `HyperbolicAnisotropicEnumerateFunction`, which gives less weight to interactions. .. GENERATED FROM PYTHON SOURCE LINES 103-107 .. code-block:: default q = 0.4 enumerateFunction_1 = ot.HyperbolicAnisotropicEnumerateFunction( inputDimension, q) .. GENERATED FROM PYTHON SOURCE LINES 108-109 Create the multivariate orthonormal basis which is the the cartesian product of the univariate basis. .. GENERATED FROM PYTHON SOURCE LINES 111-114 .. code-block:: default multivariateBasis = ot.OrthogonalProductPolynomialFactory( polyColl, enumerateFunction) .. GENERATED FROM PYTHON SOURCE LINES 115-116 Ask how many polynomials have total degrees equal to k=5. .. GENERATED FROM PYTHON SOURCE LINES 118-121 .. code-block:: default k = 5 enumerateFunction.getStrataCardinal(k) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 56 .. GENERATED FROM PYTHON SOURCE LINES 122-123 Ask how many polynomials have degrees lower or equal to k=5. .. GENERATED FROM PYTHON SOURCE LINES 125-127 .. code-block:: default enumerateFunction.getStrataCumulatedCardinal(k) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 126 .. GENERATED FROM PYTHON SOURCE LINES 128-129 Give the k-th term of the multivariate basis. To calculate its degree, add the integers. .. GENERATED FROM PYTHON SOURCE LINES 131-134 .. code-block:: default k = 5 enumerateFunction(k) .. raw:: html

[2,0,0,0]



.. GENERATED FROM PYTHON SOURCE LINES 135-136 Build a term of the basis as a Function. Generally, we do not need to construct manually any term, all terms are constructed automatically by a strategy of construction of the basis. .. GENERATED FROM PYTHON SOURCE LINES 138-142 .. code-block:: default i = 5 Psi_i = multivariateBasis.build(i) Psi_i .. raw:: html

-0.707107 + 0.707107 * x0^2



.. GENERATED FROM PYTHON SOURCE LINES 143-144 Get the measure mu associated to the multivariate basis. .. GENERATED FROM PYTHON SOURCE LINES 146-149 .. code-block:: default distributionMu = multivariateBasis.getMeasure() distributionMu .. raw:: html

ComposedDistribution(Normal(mu = 0, sigma = 1), Uniform(a = -1, b = 1), Gamma(k = 3.75, lambda = 1, gamma = 0), Beta(alpha = 2.5, beta = 1, a = -1, b = 1), IndependentCopula(dimension = 4))



.. GENERATED FROM PYTHON SOURCE LINES 150-152 STEP 2: Truncature strategy of the multivariate orthonormal basis ----------------------------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 154-155 FixedStrategy : all the polynomials af degree lower or equal to 2 which corresponds to the 15 first ones. .. GENERATED FROM PYTHON SOURCE LINES 157-160 .. code-block:: default p = 15 truncatureBasisStrategy = ot.FixedStrategy(multivariateBasis, p) .. GENERATED FROM PYTHON SOURCE LINES 161-162 SequentialStrategy : among the maximumCardinalBasis = 100 first polynomials of the multivariate basis those verfying the convergence criterion. .. GENERATED FROM PYTHON SOURCE LINES 164-168 .. code-block:: default maximumCardinalBasis = 100 truncatureBasisStrategy_1 = ot.SequentialStrategy( multivariateBasis, maximumCardinalBasis) .. GENERATED FROM PYTHON SOURCE LINES 169-171 CleaningStrategy : among the maximumConsideredTerms = 500 first polynomials, those which have the mostSignificant = 50 most significant contributions with significance criterion significanceFactor equal to :math:`10^{-4}` The `True` boolean indicates if we are interested in the online monitoring of the current basis update (removed or added coefficients). .. GENERATED FROM PYTHON SOURCE LINES 173-179 .. code-block:: default maximumConsideredTerms = 500 mostSignificant = 50 significanceFactor = 1.0e-4 truncatureBasisStrategy_2 = ot.CleaningStrategy( multivariateBasis, maximumConsideredTerms, mostSignificant, significanceFactor, True) .. GENERATED FROM PYTHON SOURCE LINES 180-182 STEP 3: Evaluation strategy of the approximation coefficients ------------------------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 184-185 The technique illustrated is the Least Squares technique where the points come from an design of experiments. Here : the Monte Carlo sampling technique. .. GENERATED FROM PYTHON SOURCE LINES 187-191 .. code-block:: default sampleSize = 100 evaluationCoeffStrategy = ot.LeastSquaresStrategy( ot.MonteCarloExperiment(sampleSize)) .. GENERATED FROM PYTHON SOURCE LINES 192-193 You can specify the approximation algorithm. This is the algorithm that generates a sequence of basis using Least Angle Regression. .. GENERATED FROM PYTHON SOURCE LINES 195-197 .. code-block:: default basisSequenceFactory = ot.LARS() .. GENERATED FROM PYTHON SOURCE LINES 198-199 This algorithm estimates the empirical error on each sub-basis using Leave One Out strategy. .. GENERATED FROM PYTHON SOURCE LINES 201-208 .. code-block:: default fittingAlgorithm = ot.CorrectedLeaveOneOut() # Finally the metamodel selection algorithm embbeded in LeastSquaresStrategy approximationAlgorithm = ot.LeastSquaresMetaModelSelectionFactory( basisSequenceFactory, fittingAlgorithm) evaluationCoeffStrategy_2 = ot.LeastSquaresStrategy( ot.MonteCarloExperiment(sampleSize), approximationAlgorithm) .. GENERATED FROM PYTHON SOURCE LINES 209-210 Try integration. .. GENERATED FROM PYTHON SOURCE LINES 212-216 .. code-block:: default marginalDegrees = [2] * inputDimension evaluationCoeffStrategy_3 = ot.IntegrationStrategy( ot.GaussProductExperiment(distributionMu, marginalDegrees)) .. GENERATED FROM PYTHON SOURCE LINES 217-226 STEP 4: Creation of the Functional Chaos Algorithm -------------------------------------------------- The `FunctionalChaosAlgorithm` class combines * the model : `model` * the distribution of the input random vector : `distribution` * the truncature strategy of the multivariate basis * and the evaluation strategy of the coefficients .. GENERATED FROM PYTHON SOURCE LINES 228-230 .. code-block:: default polynomialChaosAlgorithm = ot.FunctionalChaosAlgorithm( model, distribution, truncatureBasisStrategy, evaluationCoeffStrategy) .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.006 seconds) .. _sphx_glr_download_auto_meta_modeling_polynomial_chaos_metamodel_plot_functional_chaos_advanced_ctors.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_functional_chaos_advanced_ctors.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_functional_chaos_advanced_ctors.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_