Autoregressive Moving Average (ARMA): Artificial data
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.. _tsa_arma_1_notebook:
`Link to Notebook GitHub <https://github.com/statsmodels/statsmodels/blob/master/examples/notebooks/tsa_arma_1.ipynb>`_
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">print_function</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">from</span> <span class="nn">statsmodels.tsa.arima_process</span> <span class="k">import</span> <span class="n">arma_generate_sample</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">12345</span><span class="p">)</span>
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<p>Generate some data from an ARMA process:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">arparams</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">.</span><span class="mi">75</span><span class="p">,</span> <span class="o">-.</span><span class="mi">25</span><span class="p">])</span>
<span class="n">maparams</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">.</span><span class="mi">65</span><span class="p">,</span> <span class="o">.</span><span class="mi">35</span><span class="p">])</span>
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<p>The conventions of the arma_generate function require that we specify a 1 for the zero-lag of the AR and MA parameters and that the AR parameters be negated.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">arparams</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">arparams</span><span class="p">]</span>
<span class="n">maparam</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">maparams</span><span class="p">]</span>
<span class="n">nobs</span> <span class="o">=</span> <span class="mi">250</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">arma_generate_sample</span><span class="p">(</span><span class="n">arparams</span><span class="p">,</span> <span class="n">maparams</span><span class="p">,</span> <span class="n">nobs</span><span class="p">)</span>
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<p>Now, optionally, we can add some dates information. For this example, we'll use a pandas time series.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">dates</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">datetools</span><span class="o">.</span><span class="n">dates_from_range</span><span class="p">(</span><span class="s">'1980m1'</span><span class="p">,</span> <span class="n">length</span><span class="o">=</span><span class="n">nobs</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">TimeSeries</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">dates</span><span class="p">)</span>
<span class="n">arma_mod</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">ARMA</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
<span class="n">arma_res</span> <span class="o">=</span> <span class="n">arma_mod</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">trend</span><span class="o">=</span><span class="s">'nc'</span><span class="p">,</span> <span class="n">disp</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="n">arma_res</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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<div class=" highlight hl-ipython3"><pre><span class="n">y</span><span class="o">.</span><span class="n">tail</span><span class="p">()</span>
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<pre> ARMA Model Results
==============================================================================
Dep. Variable: y No. Observations: 250
Model: ARMA(2, 2) Log Likelihood -245.887
Method: css-mle S.D. of innovations 0.645
Date: Mon, 20 Jul 2015 AIC 501.773
Time: 17:45:06 BIC 519.381
Sample: 01-31-1980 HQIC 508.860
- 10-31-2000
==============================================================================
coef std err z P>|z| [95.0% Conf. Int.]
------------------------------------------------------------------------------
ar.L1.y 0.8411 0.403 2.089 0.038 0.052 1.630
ar.L2.y -0.2693 0.247 -1.092 0.276 -0.753 0.214
ma.L1.y 0.5352 0.412 1.299 0.195 -0.273 1.343
ma.L2.y 0.0157 0.306 0.051 0.959 -0.585 0.616
Roots
=============================================================================
Real Imaginary Modulus Frequency
-----------------------------------------------------------------------------
AR.1 1.5618 -1.1289j 1.9271 -0.0996
AR.2 1.5618 +1.1289j 1.9271 0.0996
MA.1 -1.9835 +0.0000j 1.9835 0.5000
MA.2 -32.1791 +0.0000j 32.1791 0.5000
-----------------------------------------------------------------------------
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">arma_res</span><span class="o">.</span><span class="n">plot_predict</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s">'1999m6'</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s">'2001m5'</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
<span class="n">legend</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">'upper left'</span><span class="p">)</span>
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