.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_probabilistic_modeling/stochastic_processes/plot_gaussian_processes_comparison.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_probabilistic_modeling_stochastic_processes_plot_gaussian_processes_comparison.py: Compare covariance models ========================= .. GENERATED FROM PYTHON SOURCE LINES 7-12 The main goal of this example is to briefly review the most important covariance models and compare them in terms of regularity of the trajectories. We first show how to define a covariance model, a temporal grid and a Gaussian process. We first consider the squared exponential covariance model and show how the trajectories are sensitive to its parameters. We show how to define a trend. In the final section, we compare the trajectories from exponential and Matérn covariance models. .. GENERATED FROM PYTHON SOURCE LINES 14-18 References ---------- * Carl Edward Rasmussen and Christopher K. I. Williams (2006) Gaussian Processes for Machine Learning. Chapter 4: "Covariance Functions", www.GaussianProcess.org/gpml .. GENERATED FROM PYTHON SOURCE LINES 20-27 The anisotropic squared exponential model ----------------------------------------- The :class:`~openturns.SquaredExponential` class allows one to define covariance models: * :math:`\sigma\in\mathbb{R}` is the amplitude parameter, * :math:`\boldsymbol{\theta}\in\mathbb{R}^d` is the scale. .. GENERATED FROM PYTHON SOURCE LINES 29-34 .. code-block:: Python from matplotlib import pyplot as plt from openturns.viewer import View import openturns as ot import openturns.viewer as viewer .. GENERATED FROM PYTHON SOURCE LINES 35-36 Amplitude values .. GENERATED FROM PYTHON SOURCE LINES 36-42 .. code-block:: Python amplitude = [3.5] # Scale values scale = [1.5] # Covariance model myModel = ot.SquaredExponential(scale, amplitude) .. GENERATED FROM PYTHON SOURCE LINES 43-58 Gaussian processes ------------------ In order to create a :class:`~openturns.GaussianProcess`, we must have: * a covariance model, * a grid. Optionnally, we can define a trend (we will see that later in the example). By default, the trend is zero. We consider the domain :math:`\mathcal{D}=[0,10]`. We discretize this domain with 100 cells (which corresponds to 101 nodes), with steps equal to 0.1 starting from 0: .. math:: (x_0=x_{min}=0,\:x_1=0.1,\:\ldots,\:x_n=x_{max}=10). .. GENERATED FROM PYTHON SOURCE LINES 60-68 .. code-block:: Python xmin = 0.0 step = 0.1 n = 100 myTimeGrid = ot.RegularGrid(xmin, step, n + 1) graph = myTimeGrid.draw() graph.setTitle("Regular grid") view = viewer.View(graph) .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_001.svg :alt: Regular grid :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_001.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 69-70 Then we create the Gaussian process (by default the trend is zero). .. GENERATED FROM PYTHON SOURCE LINES 72-74 .. code-block:: Python process = ot.GaussianProcess(myModel, myTimeGrid) .. GENERATED FROM PYTHON SOURCE LINES 75-76 Then we generate 10 trajectores with the `getSample` method. This trajectories are in a :class:`~openturns.ProcessSample`. .. GENERATED FROM PYTHON SOURCE LINES 78-82 .. code-block:: Python nbTrajectories = 10 sample = process.getSample(nbTrajectories) type(sample) .. GENERATED FROM PYTHON SOURCE LINES 83-84 We can draw the trajectories with `drawMarginal`. .. GENERATED FROM PYTHON SOURCE LINES 86-91 .. code-block:: Python graph = sample.drawMarginal(0) graph.setTitle("amplitude=%.3f, scale=%.3f" % (amplitude[0], scale[0])) view = viewer.View(graph) .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_002.svg :alt: amplitude=3.500, scale=1.500 :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_002.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 92-93 In order to make the next examples easier, we define a function which plots a given number of trajectories from a Gaussian process based on a covariance model. .. GENERATED FROM PYTHON SOURCE LINES 96-105 .. code-block:: Python def plotCovarianceModel(myCovarianceModel, myTimeGrid, nbTrajectories): """Plots the given number of trajectories with given covariance model.""" process = ot.GaussianProcess(myCovarianceModel, myTimeGrid) sample = process.getSample(nbTrajectories) graph = sample.drawMarginal(0) graph.setTitle("") return graph .. GENERATED FROM PYTHON SOURCE LINES 106-107 The amplitude parameter sets the variance of the process. A greater amplitude increases the chances of getting larger absolute values of the process. .. GENERATED FROM PYTHON SOURCE LINES 109-116 .. code-block:: Python amplitude = [7.0] scale = [1.5] myModel = ot.SquaredExponential(scale, amplitude) graph = plotCovarianceModel(myModel, myTimeGrid, 10) graph.setTitle("amplitude=%.3f, scale=%.3f" % (amplitude[0], scale[0])) view = viewer.View(graph) .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_003.svg :alt: amplitude=7.000, scale=1.500 :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_003.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 117-118 Modifying the scale parameter is here equivalent to stretch or contract the "time" :math:`x`. .. GENERATED FROM PYTHON SOURCE LINES 120-127 .. code-block:: Python amplitude = [3.5] scale = [0.5] myModel = ot.SquaredExponential(scale, amplitude) graph = plotCovarianceModel(myModel, myTimeGrid, 10) graph.setTitle("amplitude=%.3f, scale=%.3f" % (amplitude[0], scale[0])) view = viewer.View(graph) .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_004.svg :alt: amplitude=3.500, scale=0.500 :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_004.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 128-132 Define the trend ---------------- The trend is a deterministic function. With the :class:`~openturns.GaussianProcess` class, the associated process is the sum of a trend and a Gaussian process with zero mean. .. GENERATED FROM PYTHON SOURCE LINES 134-137 .. code-block:: Python f = ot.SymbolicFunction(["x"], ["2*x"]) fTrend = ot.TrendTransform(f, myTimeGrid) .. GENERATED FROM PYTHON SOURCE LINES 138-143 .. code-block:: Python amplitude = [3.5] scale = [1.5] myModel = ot.SquaredExponential(scale, amplitude) process = ot.GaussianProcess(fTrend, myModel, myTimeGrid) .. GENERATED FROM PYTHON SOURCE LINES 144-145 sphinx_gallery_thumbnail_number = 5 .. GENERATED FROM PYTHON SOURCE LINES 145-151 .. code-block:: Python nbTrajectories = 10 sample = process.getSample(nbTrajectories) graph = sample.drawMarginal(0) graph.setTitle("amplitude=%.3f, scale=%.3f" % (amplitude[0], scale[0])) view = viewer.View(graph) .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_005.svg :alt: amplitude=3.500, scale=1.500 :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_005.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 152-166 Other covariance models ----------------------- There are other covariance models. The models which are used more often are the following: * :class:`~openturns.SquaredExponential`. The generated processes can be derivated in mean square at all orders. * :class:`~openturns.MaternModel`. When :math:`\nu\rightarrow+\infty`, it converges to the squared exponential model. This model can be derivated :math:`k` times only if :math:`k<\nu`. In other words, when :math:`\nu` increases, then the trajectories are more and more regular. The particular case :math:`\nu=1/2` is the exponential model. The most commonly used values are :math:`\nu=3/2` and :math:`\nu=5/2`, which produce trajectories that are, in terms of regularity, in between the squared exponential and the exponential models. * :class:`~openturns.ExponentialModel`. The associated process is continuous, but not differentiable. .. GENERATED FROM PYTHON SOURCE LINES 168-170 The Matérn and exponential models --------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 172-179 .. code-block:: Python amplitude = [1.0] scale = [1.0] nu1, nu2, nu3 = 2.5, 1.5, 0.5 myModel1 = ot.MaternModel(scale, amplitude, nu1) myModel2 = ot.MaternModel(scale, amplitude, nu2) myModel3 = ot.MaternModel(scale, amplitude, nu3) .. GENERATED FROM PYTHON SOURCE LINES 180-185 .. code-block:: Python nbTrajectories = 10 graph1 = plotCovarianceModel(myModel1, myTimeGrid, nbTrajectories) graph2 = plotCovarianceModel(myModel2, myTimeGrid, nbTrajectories) graph3 = plotCovarianceModel(myModel3, myTimeGrid, nbTrajectories) .. GENERATED FROM PYTHON SOURCE LINES 186-197 .. code-block:: Python fig = plt.figure(figsize=(20, 6)) ax1 = fig.add_subplot(1, 3, 1) _ = View(graph1, figure=fig, axes=[ax1]) _ = ax1.set_title("Matern 5/2") ax2 = fig.add_subplot(1, 3, 2) _ = View(graph2, figure=fig, axes=[ax2]) _ = ax2.set_title("Matern 3/2") ax3 = fig.add_subplot(1, 3, 3) _ = View(graph3, figure=fig, axes=[ax3]) _ = ax3.set_title("Matern 1/2") .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_006.svg :alt: , Matern 5/2, Matern 3/2, Matern 1/2 :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_006.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 198-199 We see than, when :math:`\nu` increases, then the trajectories are smoother and smoother. .. GENERATED FROM PYTHON SOURCE LINES 201-203 .. code-block:: Python myExpModel = ot.ExponentialModel(scale, amplitude) .. GENERATED FROM PYTHON SOURCE LINES 204-208 .. code-block:: Python graph = plotCovarianceModel(myExpModel, myTimeGrid, nbTrajectories) graph.setTitle("Exponential") view = viewer.View(graph) .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_007.svg :alt: Exponential :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_gaussian_processes_comparison_007.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 209-210 We see that the exponential model produces very irregular trajectories. .. GENERATED FROM PYTHON SOURCE LINES 210-211 .. code-block:: Python viewer.View.ShowAll() .. _sphx_glr_download_auto_probabilistic_modeling_stochastic_processes_plot_gaussian_processes_comparison.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_gaussian_processes_comparison.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_gaussian_processes_comparison.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_gaussian_processes_comparison.zip `