.. 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_calibration_bayesian_calibration_plot_bayesian_calibration.py: Bayesian calibration of a computer code ======================================= In this example we are going to compute the parameters of a computer model thanks to Bayesian estimation. Let us denote :math:\underline y = (y_1, \dots, y_n) the observation sample, :math:\underline z = (f(x_1|\underline{\theta}), \ldots, f(x_n|\underline{\theta})) the model prediction, :math:p(y |z) the density function of observation :math:y conditional on model prediction :math:z, and :math:\underline{\theta} \in \mathbb{R}^p the calibration parameters we wish to estimate. The posterior distribution is given by Bayes theorem: .. math::\pi(\underline{\theta} | \underline y) \quad \propto \quad L\left(\underline y | \underline{\theta}\right) \times \pi(\underline{\theta}):math: where :math:\propto means "proportional to", regarded as a function of :math:\underline{\theta}. The posterior distribution is approximated here by the empirical distribution of the sample :math:\underline{\theta}^1, \ldots, \underline{\theta}^N generated by the Metropolis-Hastings algorithm. This means that any quantity characteristic of the posterior distribution (mean, variance, quantile, ...) is approximated by its empirical counterpart. Our model (i.e. the compute code to calibrate) is a standard normal linear regression, where .. math:: y_i = \theta_1 + x_i \theta_2 + x_i^2 \theta_3 + \varepsilon_i where :math:\varepsilon_i \stackrel{i.i.d.}{\sim} \mathcal N(0, 1). The "true" value of :math:\theta is: .. math:: \theta_{true} = (-4.5,4.8,2.2)^T. We use a normal prior on :math:\underline{\theta}: .. math:: \pi(\underline{\theta}) = \mathcal N(\mu_\theta, \Sigma_\theta) where .. math:: \mu_\theta = \begin{pmatrix} -3 \\ 4 \\ 1 \end{pmatrix} is the mean of the prior and .. math:: \Sigma_\theta = \begin{pmatrix} \sigma_{\theta_1}^2 & 0 & 0 \\ 0 & \sigma_{\theta_2}^2 & 0 \\ 0 & 0 & \sigma_{\theta_3}^2 \end{pmatrix} is the prior covariance matrix with .. math:: \sigma_{\theta_1} = 2, \qquad \sigma_{\theta_2} = 1, \qquad \sigma_{\theta_3} = 1.5. The following objects need to be defined in order to perform Bayesian calibration: - The conditional density :math:p(y|z) must be defined as a probability distribution - The computer model must be implemented thanks to the ParametricFunction class. This takes a value of :math:\underline{\theta} as input, and outputs the vector of model predictions :math:\underline z, as defined above (the vector of covariates :math:\underline x = (x_1, \ldots, x_n) is treated as a known constant). When doing that, we have to keep in mind that :math:z will be used as the vector of parameters corresponding to the distribution specified for :math:p(y |z). For instance, if :math:p(y|z) is normal, this means that :math:z must be a vector containing the mean and variance of :math:y - The prior density :math:\pi(\underline{\theta}) encoding the set of possible values for the calibration parameters, each value being weighted by its a priori probability, reflecting the beliefs about the possible values of :math:\underline{\theta} before consideration of the experimental data. Again, this is implemented as a probability distribution - The Metropolis-Hastings algorithm that samples from the posterior distribution of the calibration parameters requires a vector :math:\underline{\theta}_0 initial values for the calibration parameters, as well as the proposal laws used to update each parameter sequentially. .. code-block:: default import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) Dimension of the vector of parameters to calibrate .. code-block:: default paramDim = 3 # The number of obesrvations obsSize = 10 - Define the observed inputs :math:x_i .. code-block:: default xmin = -2. xmax = 3. step = (xmax-xmin)/(obsSize-1) rg = ot.RegularGrid(xmin, step, obsSize) x_obs = rg.getVertices() x_obs .. raw:: html
t -2 -1.444444 -0.8888889 -0.3333333 0.2222222 0.7777778 1.333333 1.888889 2.444444 3

- Define the parametric model :math:z = f(x,\theta) that associates each observation :math:x_i and values of the parameters :math:\theta_i to the parameters of the distribution of the corresponding observation: here :math:z=(\mu, \sigma) where :math:\mu, the first output of the model, is the mean and :math:\sigma, the second output of the model, is the standard deviation. .. code-block:: default fullModel = ot.SymbolicFunction( ['x1', 'theta1', 'theta2', 'theta3'], ['theta1+theta2*x1+theta3*x1^2','1.0']) model = ot.ParametricFunction(fullModel, , x_obs) model .. raw:: html

ParametricEvaluation([x1,theta1,theta2,theta3]->[theta1+theta2*x1+theta3*x1^2,1.0], parameters positions=, parameters=[x1 : -2], input positions=[1,2,3])

- Define the observation noise :math:\varepsilon {\sim} \mathcal N(0, 1) and create a sample from it. .. code-block:: default ot.RandomGenerator.SetSeed(0) noiseStandardDeviation = 1. noise = ot.Normal(0,noiseStandardDeviation) noiseSample = noise.getSample(obsSize) noiseSample .. raw:: html
X0 0.6082017 -1.266173 -0.4382656 1.205478 -2.181385 0.3500421 -0.355007 1.437249 0.810668 0.793156

- Define the vector of observations :math:y_i In this model, we use a constant value of the parameter. The "true" value of :math:\theta is used to compute the model outputs. .. code-block:: default thetaTrue = [-4.5,4.8,2.2] .. code-block:: default y_obs = ot.Sample(obsSize,1) for i in range(obsSize): model.setParameter(x_obs[i]) y_obs[i,0] = model(thetaTrue) + noiseSample[i,0] y_obs .. raw:: html
v0 -4.691798 -8.109383 -7.466661 -4.650077 -5.506077 0.9142396 5.456104 13.8533 21.18968 30.49316

- Draw the model vs the observations. .. code-block:: default functionnalModel = ot.ParametricFunction(fullModel, [1,2,3], thetaTrue) graphModel = functionnalModel.getMarginal(0).draw(xmin,xmax) observations = ot.Cloud(x_obs,y_obs) observations = ot.Cloud(x_obs,y_obs) observations.setColor("red") graphModel.add(observations) graphModel.setLegends(["Model","Observations"]) graphModel.setLegendPosition("topleft") view = viewer.View(graphModel) .. image:: /auto_calibration/bayesian_calibration/images/sphx_glr_plot_bayesian_calibration_001.png :alt: y0 as a function of x1 :class: sphx-glr-single-img - Define the distribution of observations :math:\underline{y} | \underline{z} conditional on model predictions Note that its parameter dimension is the one of :math:\underline{z}, so the model must be adjusted accordingly .. code-block:: default conditional = ot.Normal() conditional .. raw:: html

Normal(mu = 0, sigma = 1)

- Define the mean :math:\mu_\theta, the covariance matrix :math:\Sigma_\theta, then the prior distribution :math:\pi(\underline{\theta}) of the parameter :math:\underline{\theta}. .. code-block:: default thetaPriorMean = [-3.,4.,1.] .. code-block:: default sigma0 = ot.Point([2.,1.,1.5]) # standard deviations thetaPriorCovarianceMatrix = ot.CovarianceMatrix(paramDim) for i in range(paramDim): thetaPriorCovarianceMatrix[i, i] = sigma0[i]**2 prior = ot.Normal(thetaPriorMean, thetaPriorCovarianceMatrix) prior.setDescription(['theta1', 'theta2', 'theta3']) prior .. raw:: html

Normal(mu = [-3,4,1], sigma = [2,1,1.5], R = [[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]])

- Proposal distribution: uniform. .. code-block:: default proposal = [ot.Uniform(-1., 1.)] * paramDim proposal .. rst-class:: sphx-glr-script-out Out: .. code-block:: none [class=Uniform name=Uniform dimension=1 a=-1 b=1, class=Uniform name=Uniform dimension=1 a=-1 b=1, class=Uniform name=Uniform dimension=1 a=-1 b=1] Test the MCMC sampler --------------------- The MCMC sampler essentially computes the log-likelihood of the parameters. .. code-block:: default mymcmc = ot.MCMC(prior, conditional, model, x_obs, y_obs, thetaPriorMean) .. code-block:: default mymcmc.computeLogLikelihood(thetaPriorMean) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none -151.2962855240547 Test the Metropolis-Hastings sampler ------------------------------------ - Creation of the Random Walk Metropolis-Hastings (RWMH) sampler. .. code-block:: default initialState = thetaPriorMean .. code-block:: default RWMHsampler = ot.RandomWalkMetropolisHastings( prior, conditional, model, x_obs, y_obs, initialState, proposal) In order to check our model before simulating it, we compute the log-likelihood. .. code-block:: default RWMHsampler.computeLogLikelihood(initialState) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none -151.2962855240547 We observe that, as expected, the previous value is equal to the output of the same method in the MCMC object. Tuning of the RWMH algorithm. Strategy of calibration for the random walk (trivial example: default). .. code-block:: default strategy = ot.CalibrationStrategyCollection(paramDim) RWMHsampler.setCalibrationStrategyPerComponent(strategy) Other parameters. .. code-block:: default RWMHsampler.setVerbose(True) RWMHsampler.setThinning(1) RWMHsampler.setBurnIn(2000) Generate a sample from the posterior distribution of the parameters theta. .. code-block:: default sampleSize = 10000 sample = RWMHsampler.getSample(sampleSize) Look at the acceptance rate (basic checking of the efficiency of the tuning; value close to 0.2 usually recommended). .. code-block:: default RWMHsampler.getAcceptanceRate() .. raw:: html

[0.456667,0.2955,0.1305]

Build the distribution of the posterior by kernel smoothing. .. code-block:: default kernel = ot.KernelSmoothing() posterior = kernel.build(sample) Display prior vs posterior for each parameter. .. code-block:: default from openturns.viewer import View import pylab as pl fig = pl.figure(figsize=(12, 4)) for parameter_index in range(paramDim): graph = posterior.getMarginal(parameter_index).drawPDF() priorGraph = prior.getMarginal(parameter_index).drawPDF() priorGraph.setColors(['blue']) graph.add(priorGraph) graph.setLegends(['Posterior', 'Prior']) ax = fig.add_subplot(1, paramDim, parameter_index+1) _ = ot.viewer.View(graph, figure=fig, axes=[ax]) _ = fig.suptitle("Bayesian calibration") plt.show() .. image:: /auto_calibration/bayesian_calibration/images/sphx_glr_plot_bayesian_calibration_002.png :alt: Bayesian calibration :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 1.126 seconds) .. _sphx_glr_download_auto_calibration_bayesian_calibration_plot_bayesian_calibration.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_bayesian_calibration.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: plot_bayesian_calibration.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _