.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/01_temporalTimeSeriesAnalysis/plot_autoregressiveAutocovariance.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_01_temporalTimeSeriesAnalysis_plot_autoregressiveAutocovariance.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_01_temporalTimeSeriesAnalysis_plot_autoregressiveAutocovariance.py:


AR(1) analytical and estimated autocovariance
=============================================

.. GENERATED FROM PYTHON SOURCE LINES 7-9

Import requirements
-------------------

.. GENERATED FROM PYTHON SOURCE LINES 9-14

.. code-block:: Python


    import os
    import numpy as np
    import plotly.graph_objects as go








.. GENERATED FROM PYTHON SOURCE LINES 15-17

Define variables
----------------

.. GENERATED FROM PYTHON SOURCE LINES 17-24

.. code-block:: Python


    srate = 1
    T = 10000
    sigma = 1.0
    phi = -0.9
    lags = np.arange(20)








.. GENERATED FROM PYTHON SOURCE LINES 25-28

Create white noise
------------------


.. GENERATED FROM PYTHON SOURCE LINES 28-33

.. code-block:: Python


    time = np.arange(0, T, 1.0/srate)
    N = len(time)
    w = np.random.normal(loc=0, scale=sigma, size=N)








.. GENERATED FROM PYTHON SOURCE LINES 34-37

Create autoregressive time series
---------------------------------


.. GENERATED FROM PYTHON SOURCE LINES 37-42

.. code-block:: Python

    ar = np.empty(len(w), dtype=np.double)
    ar[0] = w[0]
    for i in range(1, len(w)):
        ar[i] = phi * ar[i-1] + w[i]








.. GENERATED FROM PYTHON SOURCE LINES 43-46

Estimate autocovariance
-----------------------


.. GENERATED FROM PYTHON SOURCE LINES 46-60

.. code-block:: Python


    estAcov = np.zeros(len(lags))
    anaAcov = np.zeros(len(lags))
    arMu = np.mean(ar)
    for h in lags:
        if h > 0:
            xs = ar[h:] - arMu
            xt = ar[:-h] - arMu
        elif h == 0:
            xs = ar - arMu
            xt = ar - arMu
        estAcov[h] = np.mean(xs * xt)
        anaAcov[h] = phi**h * sigma**2 / (1 - phi**2)








.. GENERATED FROM PYTHON SOURCE LINES 61-64

Plot autoregressive time series, true and estimated autocovariance
------------------------------------------------------------------


.. GENERATED FROM PYTHON SOURCE LINES 64-91

.. code-block:: Python


    fig = go.Figure()
    trace = go.Scatter(x=time, y=ar, mode="lines+markers")
    fig.add_trace(trace)
    fig.update_layout(xaxis=dict(title="Time (sec)"), yaxis=dict(title="x"))

    if not os.path.exists("figures"):
        os.mkdir("figures")

    fig.write_html(f"figures/autoregressiveSamplesN{T}.html")
    fig.write_image(f"figures/autoregressiveSamplesN{T}.png")

    fig = go.Figure()
    trace = go.Scatter(x=lags, y=anaAcov, mode="lines+markers", name="analytical")
    fig.add_trace(trace)
    trace = go.Scatter(x=lags, y=estAcov, mode="lines+markers", name="estimated")
    fig.add_trace(trace)
    fig.update_layout(title=f"N={T}", xaxis=dict(title="Lag (samples)"),
                      yaxis=dict(title="Autocovariance"))

    if not os.path.exists("figures"):
        os.mkdir("figures")

    fig.write_html(f"figures/autoregressiveAutoCovN{T}.html")
    fig.write_image(f"figures/autoregressiveAutoCovN{T}.png")

    fig





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.. _sphx_glr_download_auto_examples_01_temporalTimeSeriesAnalysis_plot_autoregressiveAutocovariance.py:

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      :download:`Download zipped: plot_autoregressiveAutocovariance.zip <plot_autoregressiveAutocovariance.zip>`


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