.. DO NOT EDIT.
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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/inference/bayes_parameter_estimation/bayes_parameter_IS_regression.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_inference_bayes_parameter_estimation_bayes_parameter_IS_regression.py>`
        to download the full example code or to run this example in your browser via Binder

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.. _sphx_glr_auto_examples_inference_bayes_parameter_estimation_bayes_parameter_IS_regression.py:


Parameter estimation using Importance Sampling - 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 20-38

.. code-block:: default

    import shutil

    import numpy as np
    import matplotlib.pyplot as plt

    from UQpy import PythonModel
    from UQpy.sampling.ImportanceSampling import ImportanceSampling
    from UQpy.inference import BayesParameterEstimation, ComputationalModel
    from UQpy.run_model.RunModel import RunModel  # required to run the quadratic model
    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 39-40

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

.. GENERATED FROM PYTHON SOURCE LINES 43-85

.. code-block:: default


    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', delete_files=True,
                        var_names=['theta_0', 'theta_1'])
    h_func = RunModel(model=model)
    h_func.run(samples=param_true)

    # Add noise
    error_covariance = 1.
    data_clean = np.array(h_func.qoi_list[0])
    noise = Normal(loc=0., scale=np.sqrt(error_covariance)).rvs(nsamples=50).reshape((50,))
    data_3 = data_clean + noise
    print('Shape of data: {}'.format(data_3.shape))


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

    sampling = ImportanceSampling(proposal=JointIndependent([Normal(scale=2, )] * 2))
    bayes_estimator =\
        BayesParameterEstimation(inference_model=inference_model,
                                 data=data_3,
                                 sampling_class=sampling,
                                 nsamples=5000)

    s = bayes_estimator.sampler.samples
    w = bayes_estimator.sampler.weights
    print(sum(w))

    # print results
    fig, ax = plt.subplots(1, 2)
    for i in range(2):
        ax[i].hist(x=s[:, i], weights=None, density=True, range=(-4, 4), bins=20, color='blue', alpha=0.4,
                   label='prior')
        ax[i].hist(x=s[:, i], weights=w, density=True, range=(-4, 4), bins=20, color='orange', alpha=0.7,
                   label='posterior')
        ax[i].legend()
        ax[i].set_title('theta {}'.format(i + 1))

    plt.show()



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   **Total running time of the script:** ( 0 minutes  0.000 seconds)


.. _sphx_glr_download_auto_examples_inference_bayes_parameter_estimation_bayes_parameter_IS_regression.py:

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