.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_predictive_distribution.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_plot_predictive_distribution.py>`
        to download the full example code. or to run this example in your browser via Binder

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_plot_predictive_distribution.py:


Predictive distribution
=======================

Calculation of the predictive istribution of Bayesian linear regression model.

.. GENERATED FROM PYTHON SOURCE LINES 10-12

Import requirments
------------------

.. GENERATED FROM PYTHON SOURCE LINES 12-19

.. code-block:: Python


    import numpy as np
    import scipy.stats
    import plotly.graph_objects as go

    import bayesianLinearRegression








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Define a function to generate sinusoidal regression data
--------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-29

.. code-block:: Python


    def generateData(x, sigma=0.3):
        y = np.sin(2*np.pi*x)
        t = y + np.random.normal(loc=0, scale=sigma, size=len(y))
        return y, t









.. GENERATED FROM PYTHON SOURCE LINES 30-32

Define functions to generate the design matrix sinusoidal regression data
-------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-60

.. code-block:: Python


    def getGaussianBasisFunctions(mus, sigma):
        M = len(mus)
        basis_functions = [None for m in range(M)]
        for m in range(M):
            basis_functions[m] = lambda x, mu=mus[m], sigma=sigma: \
                np.exp(-(x-mu)**2/(2.0*sigma**2))
        return basis_functions


    def buildGaussianDesignMatrixRow(x, basis_functions):
        M = len(basis_functions)
        design_matrix_row = np.empty(shape=M, dtype=np.double)
        for m in range(M):
            design_matrix_row[m] = basis_functions[m](x)
        return design_matrix_row


    def buildGaussianDesignMatrix(x, basis_functions):
        M = len(basis_functions)
        N = len(x)
        design_matrix = np.empty(shape=(N, M), dtype=np.double)
        for n in range(N):
            design_matrix[n,:] = buildGaussianDesignMatrixRow(x=x[n],
                                                              basis_functions=basis_functions)
        return design_matrix









.. GENERATED FROM PYTHON SOURCE LINES 61-63

Generate train data
-------------------

.. GENERATED FROM PYTHON SOURCE LINES 63-72

.. code-block:: Python


    N = 10
    # N = 25
    # N = 4
    x = np.sort(np.random.uniform(size=N))
    _, t = generateData(x=x)
    x_dense = np.linspace(0, 1, 1000)
    y_dense, _ = generateData(x=x_dense)








.. GENERATED FROM PYTHON SOURCE LINES 73-75

Plot train data
---------------

.. GENERATED FROM PYTHON SOURCE LINES 75-86

.. code-block:: Python


    fig = go.Figure()
    trace_true = go.Scatter(x=x_dense, y=y_dense, mode="lines", line_color="green")
    trace_data = go.Scatter(x=x, y=t, mode="markers", marker_color="blue")
    fig.add_trace(trace_true)
    fig.add_trace(trace_data)
    fig.update_layout(xaxis_title="independent variable",
                      yaxis_title="dependent variable",
                      showlegend=False)
    fig






.. raw:: html

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    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 87-89

Set estimation parameters
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 89-96

.. code-block:: Python


    bf_mus = np.arange(0.1, 1.0, 0.1)
    bf_sigma = 1.0/(N-1)
    prior_precision = 2.0
    likelihood_precision = 25.0
    N_new = 100








.. GENERATED FROM PYTHON SOURCE LINES 97-99

Get and plot the basis functions
--------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 99-112

.. code-block:: Python


    basis_functions = getGaussianBasisFunctions(mus=bf_mus, sigma=bf_sigma)

    fig = go.Figure()
    for i in range(len(basis_functions)):
        basis_function_values = basis_functions[i](x_dense)
        trace = go.Scatter(x=x_dense, y=basis_function_values, mode="lines")
        fig.add_trace(trace)
    fig.update_layout(xaxis_title="x",
                      yaxis_title=r"$\phi_i(x)$",
                      showlegend=False)
    fig






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    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 113-115

Build design matrix
-------------------

.. GENERATED FROM PYTHON SOURCE LINES 115-118

.. code-block:: Python


    Phi = buildGaussianDesignMatrix(x=x, basis_functions=basis_functions)








.. GENERATED FROM PYTHON SOURCE LINES 119-121

Estimate posterior distribution
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 121-125

.. code-block:: Python


    mN, SN = bayesianLinearRegression.batchWithSimplePrior(
        Phi=Phi, y=t, alpha=prior_precision, beta=likelihood_precision)








.. GENERATED FROM PYTHON SOURCE LINES 126-128

Estimate predictive distribution
--------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 128-140

.. code-block:: Python


    new_x = np.sort(np.random.uniform(size=N_new))
    true_mean = np.empty(shape=N_new, dtype=np.double)
    new_mean = np.empty(shape=N_new, dtype=np.double)
    new_var = np.empty(shape=N_new, dtype=np.double)
    for n in range(N_new):
        true_mean[n] = np.sin(2*np.pi*new_x[n])
        phi = buildGaussianDesignMatrixRow(x=new_x[n],
                                           basis_functions=basis_functions)
        new_mean[n], new_var[n] = bayesianLinearRegression.predict(
            phi=phi, mn=mN, Sn=SN, beta=likelihood_precision)








.. GENERATED FROM PYTHON SOURCE LINES 141-143

Plot predictive distribution
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 143-170

.. code-block:: Python


    new_mean_upper = new_mean + 1.96*np.sqrt(new_var)
    new_mean_lower = new_mean - 1.96*np.sqrt(new_var)
    fig = go.Figure()
    trace_true = go.Scatter(x=new_x, y=true_mean, mode="lines", line_color="green")
    trace_mean = go.Scatter(x=new_x, y=new_mean, mode="lines", line_color="red")
    trace_mean_cb = go.Scatter(x=np.concatenate((new_x, new_x[::-1])),
                               y=np.concatenate((new_mean_upper,
                                                 new_mean_lower[::-1])),
                               fill="toself",
                               fillcolor="rgba(255,0,0,0.3)",
                               line=dict(color="rgba(255,255,255,0)"),
                               hoverinfo="skip",
                               showlegend=False,
                              )
    trace_data = go.Scatter(x=x, y=t, mode="markers", marker_color="blue",
                            marker_symbol="circle-open", marker_size=10)
    fig.add_trace(trace_true)
    fig.add_trace(trace_mean)
    fig.add_trace(trace_mean_cb)
    fig.add_trace(trace_data)
    fig.update_layout(xaxis_title="independent variable",
                      yaxis_title="dependent variable",
                      showlegend=False)
    fig

    # sphinx_gallery_thumbnail_path = '_static/predictive_distribution.png'





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.. _sphx_glr_download_auto_examples_plot_predictive_distribution.py:

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