Durbin and Koopman: Box-Jenkins Examples
----------------------------------------

See Durbin and Koopman (2012), Chapter 8.4

.. code:: python

    %matplotlib inline

.. code:: python

    import numpy as np
    import pandas as pd
    from dismalpy import ssm
    import matplotlib.pyplot as plt

.. code:: python

    # Get the basic series
    dinternet = np.array(pd.read_csv('data/internet.csv').diff()[1:])
    
    # Remove datapoints
    missing = np.r_[6,16,26,36,46,56,66,72,73,74,75,76,86,96]-1
    dinternet[missing] = np.nan
    
    # Statespace
    mod = ssm.SARIMAX(dinternet, order=(1,0,1))
    res = mod.fit()
    print res.summary()


.. parsed-literal::

                               Statespace Model Results                           
    ==============================================================================
    Dep. Variable:                      y   No. Observations:                   99
    Model:               SARIMAX(1, 0, 1)   Log Likelihood                -225.770
    Date:                Mon, 21 Sep 2015   AIC                            457.541
    Time:                        14:49:44   BIC                            465.326
    Sample:                             0   HQIC                           460.691
                                     - 99                                         
    Covariance Type:                  opg                                         
    ==============================================================================
                     coef    std err          z      P>|z|      [0.025      0.975]
    ------------------------------------------------------------------------------
    ar.L1          0.6562      0.092      7.156      0.000       0.476       0.836
    ma.L1          0.4878      0.111      4.394      0.000       0.270       0.705
    sigma2        10.3402      1.569      6.590      0.000       7.265      13.416
    ===================================================================================
    Ljung-Box (Q):                         nan   Jarque-Bera (JB):                  nan
    Prob(Q):                               nan   Prob(JB):                          nan
    Heteroskedasticity (H):                nan   Skew:                              nan
    Prob(H) (two-sided):                   nan   Kurtosis:                          nan
    ===================================================================================
    
    Warnings:
    [1] Covariance matrix calculated using the outer product of gradients.


.. code:: python

    # In-sample one-step-ahead predictions, and out-of-sample forecasts
    nforecast = 20
    predict = res.get_prediction(end=mod.nobs + nforecast)
    idx = np.arange(len(predict.predicted_mean))
    predict_ci = predict.conf_int(alpha=0.5)
    
    # Graph
    fig, ax = plt.subplots(figsize=(12,6))
    ax.xaxis.grid()
    ax.plot(dinternet, 'k.')
    
    # Plot
    ax.plot(idx[:-nforecast], predict.predicted_mean[:-nforecast], 'gray')
    ax.plot(idx[-nforecast:], predict.predicted_mean[-nforecast:], 'k--', linestyle='--', linewidth=2)
    ax.fill_between(idx, predict_ci[:, 0], predict_ci[:, 1], alpha=0.15)
    
    ax.set(title='Figure 8.9 - Internet series');



.. image:: sarimax_internet.ipynb_4_0.png