.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_calibration/bayesian_calibration/plot_bayesian_calibration_flooding.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_calibration_bayesian_calibration_plot_bayesian_calibration_flooding.py: Bayesian calibration of the flooding model ========================================== .. GENERATED FROM PYTHON SOURCE LINES 6-15 Abstract -------- The goal of this example is to present the Bayesian calibration of the :ref:`flooding model`. We use the :class:`~openturns.RandomWalkMetropolisHastings` and :class:`~openturns.Gibbs` classes and simulate a sample of the posterior distribution using :ref:`metropolis_hastings`. .. GENERATED FROM PYTHON SOURCE LINES 18-79 Parameters to calibrate ----------------------- The vector of parameters to calibrate is: .. math:: \vect{\theta} = (K_s,Z_v,Z_m). The variables to calibrate are :math:`(K_s,Z_v,Z_m)` and are set to the following values: .. math:: K_s = 30, \qquad Z_v = 50, \qquad Z_m = 55. Observations ------------ In this section, we describe the statistical model associated with the :math:`n` observations. The errors of the water heights are associated with a gaussian distribution with a zero mean and a standard variation equal to: .. math:: \sigma=0.1. Therefore, the observed water heights are: .. math:: H_i = G(Q_i,K_s,Z_v,Z_m) + \epsilon_i for :math:`i=1,...,n` where .. math:: \epsilon \sim \mathcal{N}(0,\sigma^2) and we make the hypothesis that the observation errors are independent. We consider a sample size equal to: .. math:: n=20. The observations are the couples :math:`\{(Q_i,H_i)\}_{i=1,...,n}`, i.e. each observation is a couple made of the flowrate and the corresponding river height. Variables --------- - :math:`Q` : Input. Observed. - :math:`K_s`, :math:`Z_v`, :math:`Z_m` : Input. Calibrated. - :math:`H`: Output. Observed. Analysis -------- In the description of the :ref:`flooding model`, we see that only one parameter can be identified. Hence, calibrating this model requires some regularization. In this example, we use Bayesian methods as a way to regularize the model. .. GENERATED FROM PYTHON SOURCE LINES 81-83 Generate the observations ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 85-93 .. code-block:: Python import pylab as pl from openturns.usecases import flood_model import openturns.viewer as viewer import numpy as np import openturns as ot ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 94-95 A basic implementation of the probabilistic model is available in the usecases module : .. GENERATED FROM PYTHON SOURCE LINES 95-97 .. code-block:: Python fm = flood_model.FloodModel() .. GENERATED FROM PYTHON SOURCE LINES 98-108 We define the model :math:`g` which has 4 inputs and one output H. The nonlinear least squares does not take into account for bounds in the parameters. Therefore, we ensure that the output is computed whatever the inputs. The model fails into two situations: * if :math:`K_s<0`, * if :math:`Z_v-Z_m<0`. In these cases, we return an infinite number. .. GENERATED FROM PYTHON SOURCE LINES 110-124 .. code-block:: Python def functionFlooding(X): L = 5.0e3 B = 300.0 Q, K_s, Z_v, Z_m = X alpha = (Z_m - Z_v) / L if alpha < 0.0 or K_s <= 0.0: H = np.inf else: H = (Q / (K_s * B * np.sqrt(alpha))) ** (3.0 / 5.0) return [H] .. GENERATED FROM PYTHON SOURCE LINES 125-129 .. code-block:: Python g = ot.PythonFunction(4, 1, functionFlooding) g = ot.MemoizeFunction(g) g.setOutputDescription(["H (m)"]) .. GENERATED FROM PYTHON SOURCE LINES 130-131 We load the input distribution for :math:`Q`. .. GENERATED FROM PYTHON SOURCE LINES 131-133 .. code-block:: Python Q = fm.Q .. GENERATED FROM PYTHON SOURCE LINES 134-135 Set the parameters to be calibrated. .. GENERATED FROM PYTHON SOURCE LINES 137-144 .. code-block:: Python K_s = ot.Dirac(30.0) Z_v = ot.Dirac(50.0) Z_m = ot.Dirac(55.0) K_s.setDescription(["Ks (m^(1/3)/s)"]) Z_v.setDescription(["Zv (m)"]) Z_m.setDescription(["Zm (m)"]) .. GENERATED FROM PYTHON SOURCE LINES 145-146 We create the joint input distribution. .. GENERATED FROM PYTHON SOURCE LINES 148-150 .. code-block:: Python inputRandomVector = ot.ComposedDistribution([Q, K_s, Z_v, Z_m]) .. GENERATED FROM PYTHON SOURCE LINES 151-152 Create a Monte-Carlo sample of the output :math:`H`. .. GENERATED FROM PYTHON SOURCE LINES 154-158 .. code-block:: Python nbobs = 20 inputSample = inputRandomVector.getSample(nbobs) outputH = g(inputSample) .. GENERATED FROM PYTHON SOURCE LINES 159-160 Generate the observation noise and add it to the output of the model. .. GENERATED FROM PYTHON SOURCE LINES 162-168 .. code-block:: Python sigmaObservationNoiseH = 0.1 # (m) noiseH = ot.Normal(0.0, sigmaObservationNoiseH) ot.RandomGenerator.SetSeed(0) sampleNoiseH = noiseH.getSample(nbobs) Hobs = outputH + sampleNoiseH .. GENERATED FROM PYTHON SOURCE LINES 169-170 Plot the Y observations versus the X observations. .. GENERATED FROM PYTHON SOURCE LINES 172-174 .. code-block:: Python Qobs = inputSample[:, 0] .. GENERATED FROM PYTHON SOURCE LINES 175-181 .. code-block:: Python graph = ot.Graph("Observations", "Q (m3/s)", "H (m)", True) cloud = ot.Cloud(Qobs, Hobs) graph.add(cloud) view = viewer.View(graph) .. image-sg:: /auto_calibration/bayesian_calibration/images/sphx_glr_plot_bayesian_calibration_flooding_001.png :alt: Observations :srcset: /auto_calibration/bayesian_calibration/images/sphx_glr_plot_bayesian_calibration_flooding_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 182-184 Setting the calibration parameters ---------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 186-191 Define the parametric model :math:`\vect z = f_Q(\vect\theta)` that associates each observation :math:`Q` and value of the parameters :math:`\vect \theta = (K_s, Z_v, Z_m)` to the parameters of the distribution of the corresponding observation: here :math:`\vect z=(\mu, \sigma)` with :math:`\mu = G(Q, K_s, Z_v, Z_m)` and :math:`\sigma = 0.5`. .. GENERATED FROM PYTHON SOURCE LINES 194-205 .. code-block:: Python def fullModelPy(X): Q, K_s, Z_v, Z_m = X mu = g(X)[0] sigma = 0.5 # (m^2) The standard deviation of the observation error. return [mu, sigma] fullModel = ot.PythonFunction(4, 2, fullModelPy) linkFunction = ot.ParametricFunction(fullModel, [0], [np.nan]) print(linkFunction) .. rst-class:: sphx-glr-script-out .. code-block:: none ParametricEvaluation(class=PythonEvaluation name=OpenTURNSPythonFunction, parameters positions=[0], parameters=[x0 : nan], input positions=[1,2,3]) .. GENERATED FROM PYTHON SOURCE LINES 206-208 Define the value of the reference values of the :math:`\vect\theta` parameter. In the Bayesian framework, this is called the mean of the *prior* Gaussian distribution. In the data assimilation framework, this is called the *background*. .. GENERATED FROM PYTHON SOURCE LINES 210-216 .. code-block:: Python KsInitial = 20.0 ZvInitial = 49.0 ZmInitial = 51.0 parameterPriorMean = [KsInitial, ZvInitial, ZmInitial] paramDim = len(parameterPriorMean) .. GENERATED FROM PYTHON SOURCE LINES 217-218 Define the covariance matrix of the parameters :math:`\vect\theta` to calibrate. .. GENERATED FROM PYTHON SOURCE LINES 220-224 .. code-block:: Python sigmaKs = 5.0 sigmaZv = 1.0 sigmaZm = 1.0 .. GENERATED FROM PYTHON SOURCE LINES 225-230 .. code-block:: Python parameterPriorCovariance = ot.CovarianceMatrix(paramDim) parameterPriorCovariance[0, 0] = sigmaKs**2 parameterPriorCovariance[1, 1] = sigmaZv**2 parameterPriorCovariance[2, 2] = sigmaZm**2 .. GENERATED FROM PYTHON SOURCE LINES 231-232 Define the prior distribution :math:`\pi(\vect\theta)` of the parameter :math:`\vect\theta` .. GENERATED FROM PYTHON SOURCE LINES 234-237 .. code-block:: Python prior = ot.Normal(parameterPriorMean, parameterPriorCovariance) prior.setDescription(["Ks", "Zv", "Zm"]) .. GENERATED FROM PYTHON SOURCE LINES 238-243 Define the distribution of observations :math:`\vect{y} | \vect{z}` conditional on model predictions. Note that its parameter dimension is the one of :math:`\vect{z}`, so the model must be adjusted accordingly. In other words, the input argument of the `setParameter` method of the conditional distribution must be equal to the dimension of the output of the `model`. Hence, we do not have to set the actual parameters: only the type of distribution is used. .. GENERATED FROM PYTHON SOURCE LINES 245-247 .. code-block:: Python conditional = ot.Normal() .. GENERATED FROM PYTHON SOURCE LINES 248-251 The proposed steps for :math:`K_s` :math:`Z_v` and :math:`Z_m` will all follow uniform distributions, but with different supports. .. GENERATED FROM PYTHON SOURCE LINES 253-255 .. code-block:: Python proposal = [ot.Uniform(-5.0, 5.0), ot.Uniform(-1.0, 1.0), ot.Uniform(-1.0, 1.0)] .. GENERATED FROM PYTHON SOURCE LINES 256-258 Build a Gibbs sampler --------------------- .. GENERATED FROM PYTHON SOURCE LINES 258-268 .. code-block:: Python initialState = parameterPriorMean mh_coll = [ ot.RandomWalkMetropolisHastings(prior, initialState, proposal[i], [i]) for i in range(paramDim) ] for mh in mh_coll: mh.setLikelihood(conditional, Hobs, linkFunction, Qobs) sampler = ot.Gibbs(mh_coll) .. GENERATED FROM PYTHON SOURCE LINES 269-270 Generate a sample from the posterior distribution of :math:`\vect \theta`. .. GENERATED FROM PYTHON SOURCE LINES 272-275 .. code-block:: Python sampleSize = 1000 sample = sampler.getSample(sampleSize) .. GENERATED FROM PYTHON SOURCE LINES 276-277 Look at the acceptance rates of the random walk Metropolis-Hastings samplers. .. GENERATED FROM PYTHON SOURCE LINES 277-280 .. code-block:: Python [mh.getAcceptanceRate() for mh in sampler.getMetropolisHastingsCollection()] .. rst-class:: sphx-glr-script-out .. code-block:: none [0.348, 0.362, 0.393] .. GENERATED FROM PYTHON SOURCE LINES 281-282 Build the distribution of the posterior by kernel smoothing. .. GENERATED FROM PYTHON SOURCE LINES 282-286 .. code-block:: Python kernel = ot.KernelSmoothing() posterior = kernel.build(sample) .. GENERATED FROM PYTHON SOURCE LINES 287-288 Display prior vs posterior for each parameter. .. GENERATED FROM PYTHON SOURCE LINES 288-338 .. code-block:: Python def plot_bayesian_prior_vs_posterior_pdf(prior, posterior): """ Plot the prior and posterior distribution of a Bayesian calibration Parameters ---------- prior : ot.Distribution(dimension) The prior. posterior : ot.Distribution(dimension) The posterior. Return ------ grid : ot.GridLayout(1, dimension) The prior and posterior PDF for each marginal. """ palette = ot.Drawable.BuildDefaultPalette(2) paramDim = prior.getDimension() grid = ot.GridLayout(1, paramDim) parameterNames = prior.getDescription() for parameter_index in range(paramDim): graph = ot.Graph("", parameterNames[parameter_index], "PDF", True) # Prior curve = prior.getMarginal(parameter_index).drawPDF().getDrawable(0) curve.setLineStyle( ot.ResourceMap.GetAsString("CalibrationResult-PriorLineStyle") ) curve.setLegend("Prior") graph.add(curve) # Posterior curve = posterior.getMarginal(parameter_index).drawPDF().getDrawable(0) curve.setLineStyle( ot.ResourceMap.GetAsString("CalibrationResult-PosteriorLineStyle") ) curve.setLegend("Posterior") graph.add(curve) # if parameter_index < paramDim - 1: graph.setLegends([""]) if parameter_index > 0: graph.setYTitle("") graph.setColors(palette) graph.setLegendPosition("upper right") grid.setGraph(0, parameter_index, graph) grid.setTitle("Bayesian calibration") return grid .. GENERATED FROM PYTHON SOURCE LINES 339-340 sphinx_gallery_thumbnail_number = 2 .. GENERATED FROM PYTHON SOURCE LINES 340-348 .. code-block:: Python grid = plot_bayesian_prior_vs_posterior_pdf(prior, posterior) viewer.View( grid, figure_kw={"figsize": (8.0, 3.0)}, legend_kw={"bbox_to_anchor": (1.0, 1.0), "loc": "upper left"}, ) pl.subplots_adjust(right=0.8, bottom=0.2, wspace=0.3) .. image-sg:: /auto_calibration/bayesian_calibration/images/sphx_glr_plot_bayesian_calibration_flooding_002.png :alt: Bayesian calibration :srcset: /auto_calibration/bayesian_calibration/images/sphx_glr_plot_bayesian_calibration_flooding_002.png :class: sphx-glr-single-img .. _sphx_glr_download_auto_calibration_bayesian_calibration_plot_bayesian_calibration_flooding.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_bayesian_calibration_flooding.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_bayesian_calibration_flooding.py `