Periodic Kernel Example
=======================

.. code:: python

    # load matplotlib
    %matplotlib inline
    
    # imports
    import numpy as np
    import scipy.stats
    
    # import the bayesian quadrature object
    from bayesian_quadrature import BQ
    from gp import PeriodicKernel
    
    # seed the numpy random generator, so we always get the same randomness
    np.random.seed(8706)
First, we need to define various parameters:

.. code:: python

    options = {
        'n_candidate': 5,
        'x_mean': 0.0,
        'x_var': 10.0,
        'candidate_thresh': 0.5,
        'kernel': PeriodicKernel,
        'optim_method': 'L-BFGS-B'
    }
Now, sample some random :math:`x` points and compute the :math:`y`
points from a standard normal distribution.

.. code:: python

    def vmpdf(x, mu, kappa):
        C = -np.log(2 * np.pi * scipy.special.iv(0, kappa))
        p = np.exp(C + (kappa * np.cos(x - mu)))
        return p
.. code:: python

    x = np.array([-1.75, -1, 1.25])
    f_y = lambda x: vmpdf(x, 0.1, 1.1)
    y = f_y(x)
Create the bayesian quadrature object, and fit its parameters.

.. code:: python

    bq = BQ(x, y, **options)
    bq.init(params_tl=(5, 2*np.pi, 1, 0), params_l=(0.2, np.pi / 2., 1, 0))
    bq.fit_hypers(['h', 'w'])
Plot the result.

.. code:: python

    fig, axes = bq.plot(f_y, xmin=-10, xmax=10)


.. image:: periodic-example_files/periodic-example_10_0.png


Print out the mean and variance of the integral, :math:`Z`:

.. code:: python

    bq._make_approx_x(xmin=0, xmax=2 * np.pi, n=1000)
    print "E[Z] = %f" % bq.Z_mean()
    print "V(Z) = %f" % bq.Z_var()

.. parsed-literal::

    E[Z] = 0.166155
    V(Z) = 0.000064


Compute the expected variance given a potential new observation, and
then plot the curve of the expected variances, along with the final
approximation.

.. code:: python

    fig, (ax1, ax2, ax3) = plt.subplots(3, 1, sharex=True)
    xmin, xmax = -10, 10
    bq.plot_l(ax1, f_l=f_y, xmin=xmin, xmax=xmax)
    bq.plot_expected_squared_mean(ax2, xmin=xmin, xmax=xmax)
    bq.plot_expected_variance(ax3, xmin=xmin, xmax=xmax)
    fig.set_figheight(8)
    plt.tight_layout()


.. image:: periodic-example_files/periodic-example_14_0.png