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.. _sphx_glr_auto_examples_inference_bayes_parameter_estimation_bayes_parameter_MCMC_regression.py:


Parameter estimation using MCMC - Regression Model
=============================================================================

Here a model is defined that is of the form

.. math:: y=f(θ) + \epsilon

where :math:`f` consists in running RunModel. In particular, here :math:`f(θ)=θ_{0} x + θ_{1} x^{2}` is a
regression model.

.. GENERATED FROM PYTHON SOURCE LINES 16-17

Initially we have to import the necessary modules.

.. GENERATED FROM PYTHON SOURCE LINES 20-39

.. code-block:: default

    import shutil

    import numpy as np
    import matplotlib.pyplot as plt

    from UQpy import PythonModel
    from UQpy.sampling.mcmc.MetropolisHastings import MetropolisHastings
    from UQpy.inference.inference_models.ComputationalModel import ComputationalModel
    from UQpy.run_model.RunModel import RunModel
    from UQpy.inference import BayesParameterEstimation
    from sklearn.neighbors import KernelDensity  # for the plots
    from UQpy.distributions import JointIndependent, Normal

    def pdf_from_kde(domain, samples1d):
        bandwidth = 1.06 * np.std(samples1d) * samples1d.size ** (-1 / 5)
        kde = KernelDensity(bandwidth=bandwidth).fit(samples1d.reshape((-1, 1)))
        log_dens = kde.score_samples(domain)
        return np.exp(log_dens)


.. GENERATED FROM PYTHON SOURCE LINES 40-41

First we generate synthetic data, and add some noise to it.

.. GENERATED FROM PYTHON SOURCE LINES 44-104

.. code-block:: default


    # Generate data
    param_true = np.array([1.0, 2.0]).reshape((1, -1))
    print('Shape of true parameter vector: {}'.format(param_true.shape))

    model = PythonModel(model_script='local_pfn_models.py', model_object_name='model_quadratic',
                        var_names=['theta_0', 'theta_1'])
    h_func = RunModel(model=model)
    h_func.run(samples=param_true)
    data_clean = np.array(h_func.qoi_list[0])
    print(data_clean.shape)

    # Add noise, use a RandomState for reproducible results
    error_covariance = 1.
    noise = Normal(loc=0., scale=np.sqrt(error_covariance)).rvs(nsamples=50, random_state=123).reshape((50,))
    data_3 = data_clean + noise
    print('Shape of data: {}'.format(data_3.shape))
    print(data_3[:4])

    p0 = Normal()
    p1 = Normal()
    prior = JointIndependent(marginals=[p0, p1])

    inference_model = ComputationalModel(n_parameters=2, runmodel_object=h_func, error_covariance=error_covariance,
                                         prior=prior)

    proposal = JointIndependent([Normal(scale=0.1), Normal(scale=0.05)])

    mh1 = MetropolisHastings(jump=10, burn_length=0, proposal=proposal, seed=[0.5, 2.5],
                             random_state=456)

    bayes_estimator = BayesParameterEstimation(inference_model=inference_model,
                                               data=data_3,
                                               sampling_class=mh1,
                                               nsamples=500)

    s = bayes_estimator.sampler.samples
    plt.scatter(s[:, 0], s[:, 1])
    plt.scatter(1.0, 2.0, label='true value')
    plt.title('MCMC samples')
    plt.show()

    fig, ax = plt.subplots(1, 2, figsize=(10, 4))

    domain = np.linspace(-4, 4, 200)[:, np.newaxis]
    pdf_ = pdf_from_kde(domain, s[:, 0])
    ax[0].plot(domain, pdf_, label='posterior')
    ax[0].plot(domain, p0.pdf(domain), label='prior')
    ax[0].set_title('posterior pdf of theta_{1}')

    domain = np.linspace(-4, 4, 200)[:, np.newaxis]
    pdf_ = pdf_from_kde(domain, s[:, 1])
    ax[1].plot(domain, pdf_, label='posterior')
    ax[1].plot(domain, p1.pdf(domain), label='prior')
    ax[1].set_title('posterior pdf of theta_{2}')

    plt.show()

    print(bayes_estimator.sampler.samples[:4])



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