.. 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_general_purpose_metamodels_plot_expert_mixture.py: Mixture of experts ================== In this example we are going to approximate a piece wise continuous function using an expert mixture of metamodels. The metamodels will be represented by the family of :math:f_k \forall \in [1, N]: .. math:: \begin{align} f(\underline{x}) = f_1(\underline{x}) \quad \forall \underline{z} \in Class\, 1 \dots f(\underline{x}) = f_k(\underline{x}) \quad \forall \underline{z} \in Class\, k \dots f(\underline{x}) = f_N(\underline{x}) \quad \forall \underline{z} \in Class\, N \end{align} where the N classes are defined by the classifier. Using the supervised mode the classifier partitions the input and output space at once: .. math:: z =(\underline{x}, f( \underline{x})) The classifier is MixtureClassifier based on a MixtureDistribution defined as: .. math:: p(\underline{x}) = \sum_{i=1}^N w_ip_i(\underline{x}) The rule to assign a point to a class is defined as follows: :math:\underline{x} is assigned to the class :math:j=argmax_j \log w_kp_k(\underline{z}). The grade of :math:\underline{x} with respect to the class :math:k is :math:\log w_kp_k(\underline{x}). .. code-block:: default from __future__ import print_function import openturns as ot from matplotlib import pyplot as plt import openturns.viewer as viewer from matplotlib import pylab as plt from openturns.viewer import View import numpy as np ot.Log.Show(ot.Log.NONE) .. code-block:: default dimension = 1 # Define the piecewise model we want to rebuild def piecewise(X): # if x < 0.0: # f = (x+0.75)**2-0.75**2 # else: # f = 2.0-x**2 xarray = np.array(X, copy=False) return np.piecewise(xarray, [xarray < 0, xarray >= 0], [lambda x: x*(x+1.5), lambda x: 2.0 - x*x]) f = ot.PythonFunction(1, 1, func_sample=piecewise) Build a metamodel over each segment .. code-block:: default degree = 5 samplingSize = 100 enumerateFunction = ot.LinearEnumerateFunction(dimension) productBasis = ot.OrthogonalProductPolynomialFactory([ot.LegendreFactory()] * dimension, enumerateFunction) adaptiveStrategy = ot.FixedStrategy(productBasis, enumerateFunction.getStrataCumulatedCardinal(degree)) projectionStrategy = ot.LeastSquaresStrategy(ot.MonteCarloExperiment(samplingSize)) Segment 1: (-1.0; 0.0) .. code-block:: default d1 = ot.Uniform(-1.0, 0.0) fc1 = ot.FunctionalChaosAlgorithm(f, d1, adaptiveStrategy, projectionStrategy) fc1.run() mm1 = fc1.getResult().getMetaModel() graph = mm1.draw(-1.0, -1e-6) view = viewer.View(graph) .. image:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_expert_mixture_001.png :alt: v0 as a function of x0 :class: sphx-glr-single-img Segment 2: (0.0, 1.0) .. code-block:: default d2 = ot.Uniform(0.0, 1.0) fc2 = ot.FunctionalChaosAlgorithm(f, d2, adaptiveStrategy, projectionStrategy) fc2.run() mm2 = fc2.getResult().getMetaModel() graph = mm2.draw(1e-6,1.0) view = viewer.View(graph) .. image:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_expert_mixture_002.png :alt: v0 as a function of x0 :class: sphx-glr-single-img Define the mixture .. code-block:: default R = ot.CorrelationMatrix(2) d1 = ot.Normal([-1.0, -1.0], [1.0]*2, R)# segment 1 d2 = ot.Normal([1.0, 1.0], [1.0]*2, R)# segment 2 weights = [1.0]*2 atoms = [d1, d2] mixture = ot.Mixture(atoms, weights) Create the classifier based on the mixture .. code-block:: default classifier = ot.MixtureClassifier(mixture) Create local experts using the metamodels .. code-block:: default experts = ot.Basis([mm1, mm2]) Create a mixture of experts .. code-block:: default evaluation = ot.ExpertMixture(experts, classifier) moe = ot.Function(evaluation) Draw the mixture of experts .. code-block:: default graph = moe.draw(-1.0, 1.0) view = viewer.View(graph) plt.show() .. image:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_expert_mixture_003.png :alt: v0 as a function of x0 :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.223 seconds) .. _sphx_glr_download_auto_meta_modeling_general_purpose_metamodels_plot_expert_mixture.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_expert_mixture.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: plot_expert_mixture.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _