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1 Package:·python-statsmodels-doc1 Package:·python-statsmodels-doc
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Similarity: 0.9995828163771712% Differences: {"'cells'": "{22: {'outputs': {0: {'text': {insert: [(0, 'Optimization terminated " "successfully.\\n'), (1, ' Current function value: 1.548647\\n'), (2, " "' Iterations 7\\n')]}}, delete: [0]}}, 26: {'outputs': {1: {'text': {delete: " "[2, 1, 0]}}, insert: [(0, OrderedDict({'name': 'stdout', 'output_type': 'stream', " "'text': ['Optimization terminated successfully.\\n', ' Current function " "value: 3.091609\\n', ' […]
    
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384 ················{384 ················{
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389 ························"·········Current·function·value:·1.548647\n",389 ························"·········Current·function·value:·1.548647\n",
390 ························"·········Iterations·7\n"390 ························"·········Iterations·7\n",
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392 ················}, 
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400 ························"selfLR····0.297714··0.391669··0.573451··1.278772··1.346962···2.070080\n",394 ························"selfLR····0.297714··0.391669··0.573451··1.278772··1.346962···2.070080\n",
401 ························"age······-0.024945·-0.022898·-0.014851·-0.008681·-0.017904··-0.009433\n",395 ························"age······-0.024945·-0.022898·-0.014851·-0.008681·-0.017904··-0.009433\n",
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455 ················{449 ················{
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459 ························"Optimization·terminated·successfully.\n",453 ························"Optimization·terminated·successfully.\n",
460 ························"·········Current·function·value:·3.091609\n",454 ························"·········Current·function·value:·3.091609\n",
461 ························"·········Iterations·6\n",455 ························"·········Iterations·6\n"
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462 ························"··························Poisson·Regression·Results··························\n",462 ························"··························Poisson·Regression·Results··························\n",
463 ························"==============================================================================\n",463 ························"==============================================================================\n",
464 ························"Dep.·Variable:··················mdvis···No.·Observations:················20190\n",464 ························"Dep.·Variable:··················mdvis···No.·Observations:················20190\n",
465 ························"Model:························Poisson···Df·Residuals:····················20180\n",465 ························"Model:························Poisson···Df·Residuals:····················20180\n",
466 ························"Method:···························MLE···Df·Model:····························9\n",466 ························"Method:···························MLE···Df·Model:····························9\n",
467 ························"Date:················Sun,·10·Aug·2025···Pseudo·R-squ.:·················0.06343\n",467 ························"Date:················Sun,·10·Aug·2025···Pseudo·R-squ.:·················0.06343\n",
468 ························"Time:························13:13:47···Log-Likelihood:················-62420.\n",468 ························"Time:························13:13:47···Log-Likelihood:················-62420.\n",
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340 ························"method·RE:" 
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347 ························"·iterated\n",340 ························"method·RE:·iterated\n",
348 ························"························eff····sd_eff····ci_low····ci_upp······w_fe······w_re\n",341 ························"························eff····sd_eff····ci_low····ci_upp······w_fe······w_re\n",
349 ························"Carroll············0.094524··0.182680·-0.263521··0.452570··0.123885··0.152619\n",342 ························"Carroll············0.094524··0.182680·-0.263521··0.452570··0.123885··0.152619\n",
350 ························"Grant··············0.277356··0.176279·-0.068144··0.622857··0.133045··0.159157\n",343 ························"Grant··············0.277356··0.176279·-0.068144··0.622857··0.133045··0.159157\n",
351 ························"Peck···············0.366546··0.225573·-0.075569··0.808662··0.081250··0.116228\n",344 ························"Peck···············0.366546··0.225573·-0.075569··0.808662··0.081250··0.116228\n",
352 ························"Donat··············0.664385··0.102748··0.463002··0.865768··0.391606··0.257767\n",345 ························"Donat··············0.664385··0.102748··0.463002··0.865768··0.391606··0.257767\n",
353 ························"Stewart············0.461808··0.208310··0.053527··0.870089··0.095275··0.129428\n",346 ························"Stewart············0.461808··0.208310··0.053527··0.870089··0.095275··0.129428\n",
354 ························"Young··············0.185165··0.153729·-0.116139··0.486468··0.174939··0.184799\n",347 ························"Young··············0.185165··0.153729·-0.116139··0.486468··0.174939··0.184799\n",
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59 ··········<div·class="body"·role="main">59 ··········<div·class="body"·role="main">
60 ············60 ············
61 ··<section·id="Copula---Multivariate-joint-distribution">61 ··<section·id="Copula---Multivariate-joint-distribution">
62 <h1>Copula·-·Multivariate·joint·distribution<a·class="headerlink"·href="#Copula---Multivariate-joint-distribution"·title="Link·to·this·heading">¶</a></h1>62 <h1>Copula·-·Multivariate·joint·distribution<a·class="headerlink"·href="#Copula---Multivariate-joint-distribution"·title="Link·to·this·heading">¶</a></h1>
63 <div·class="nbinput·nblast·docutils·container">63 <div·class="nbinput·nblast·docutils·container">
64 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[1]:64 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:
65 </pre></div>65 </pre></div>
66 </div>66 </div>
67 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">import</span>·<span·class="nn">matplotlib.pyplot</span>·<span·class="k">as</span>·<span·class="nn">plt</span>67 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">import</span>·<span·class="nn">matplotlib.pyplot</span>·<span·class="k">as</span>·<span·class="nn">plt</span>
68 <span·class="kn">import</span>·<span·class="nn">numpy</span>·<span·class="k">as</span>·<span·class="nn">np</span>68 <span·class="kn">import</span>·<span·class="nn">numpy</span>·<span·class="k">as</span>·<span·class="nn">np</span>
69 <span·class="kn">import</span>·<span·class="nn">seaborn</span>·<span·class="k">as</span>·<span·class="nn">sns</span>69 <span·class="kn">import</span>·<span·class="nn">seaborn</span>·<span·class="k">as</span>·<span·class="nn">sns</span>
70 <span·class="kn">from</span>·<span·class="nn">scipy</span>·<span·class="kn">import</span>·<span·class="n">stats</span>70 <span·class="kn">from</span>·<span·class="nn">scipy</span>·<span·class="kn">import</span>·<span·class="n">stats</span>
  
71 <span·class="n">sns</span><span·class="o">.</span><span·class="n">set_style</span><span·class="p">(</span><span·class="s2">&quot;darkgrid&quot;</span><span·class="p">)</span>71 <span·class="n">sns</span><span·class="o">.</span><span·class="n">set_style</span><span·class="p">(</span><span·class="s2">&quot;darkgrid&quot;</span><span·class="p">)</span>
72 <span·class="n">sns</span><span·class="o">.</span><span·class="n">mpl</span><span·class="o">.</span><span·class="n">rc</span><span·class="p">(</span><span·class="s2">&quot;figure&quot;</span><span·class="p">,</span>·<span·class="n">figsize</span><span·class="o">=</span><span·class="p">(</span><span·class="mi">8</span><span·class="p">,</span>·<span·class="mi">8</span><span·class="p">))</span>72 <span·class="n">sns</span><span·class="o">.</span><span·class="n">mpl</span><span·class="o">.</span><span·class="n">rc</span><span·class="p">(</span><span·class="s2">&quot;figure&quot;</span><span·class="p">,</span>·<span·class="n">figsize</span><span·class="o">=</span><span·class="p">(</span><span·class="mi">8</span><span·class="p">,</span>·<span·class="mi">8</span><span·class="p">))</span>
73 <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">1234</span><span·class="p">)</span>·<span·class="c1">#·for·reproducibility</span>73 <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">1234</span><span·class="p">)</span>·<span·class="c1">#·for·reproducibility</span>
74 </pre></div>74 </pre></div>
75 </div>75 </div>
76 </div>76 </div>
77 <div·class="nbinput·docutils·container">77 <div·class="nbinput·nblast·docutils·container">
78 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[2]:78 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:
79 </pre></div>79 </pre></div>
80 </div>80 </div>
81 <div·class="input_area·highlight-javascript·notranslate"><div·class="highlight"><pre><span></span><span·class="o">%%</span><span·class="nx">javascript</span>81 <div·class="input_area·highlight-javascript·notranslate"><div·class="highlight"><pre><span></span><span·class="o">%%</span><span·class="nx">javascript</span>
82 <span·class="nx">IPython</span><span·class="p">.</span><span·class="nx">OutputArea</span><span·class="p">.</span><span·class="nx">prototype</span><span·class="p">.</span><span·class="nx">_should_scroll</span><span·class="w">·</span><span·class="o">=</span><span·class="w">·</span><span·class="kd">function</span><span·class="p">(</span><span·class="nx">lines</span><span·class="p">)</span><span·class="w">·</span><span·class="p">{</span>82 <span·class="nx">IPython</span><span·class="p">.</span><span·class="nx">OutputArea</span><span·class="p">.</span><span·class="nx">prototype</span><span·class="p">.</span><span·class="nx">_should_scroll</span><span·class="w">·</span><span·class="o">=</span><span·class="w">·</span><span·class="kd">function</span><span·class="p">(</span><span·class="nx">lines</span><span·class="p">)</span><span·class="w">·</span><span·class="p">{</span>
83 <span·class="w">····</span><span·class="k">return</span><span·class="w">·</span><span·class="kc">false</span><span·class="p">;</span>83 <span·class="w">····</span><span·class="k">return</span><span·class="w">·</span><span·class="kc">false</span><span·class="p">;</span>
84 <span·class="p">}</span>84 <span·class="p">}</span>
85 </pre></div>85 </pre></div>
86 </div>86 </div>
87 </div>87 </div>
88 <div·class="nboutput·nblast·docutils·container"> 
89 <div·class="prompt·empty·docutils·container"> 
90 </div> 
91 <div·class="output_area·docutils·container"> 
92 <div·class="output_javascript"></div> 
93 <script·type="text/javascript"> 
94 var·element·=·document.currentScript.previousSibling.previousSibling; 
95 IPython.OutputArea.prototype._should_scroll·=·function(lines)·{ 
96 ····return·false; 
97 } 
  
98 </script></div> 
99 </div> 
100 <p>When·modeling·a·system,·there·are·often·cases·where·multiple·parameters·are·involved.·Each·of·these·parameters·could·be·described·with·a·given·Probability·Density·Function·(PDF).·If·would·like·to·be·able·to·generate·a·new·set·of·parameter·values,·we·need·to·be·able·to·sample·from·these·distributions-also·called·marginals.·There·are·mainly·two·cases:·<em>(i)</em>·PDFs·are·independent;·<em>(ii)</em>·there·is·a·dependency.·One·way·to·model·the·dependency·it·to·use·a·<strong>copula</strong>.</p>88 <p>When·modeling·a·system,·there·are·often·cases·where·multiple·parameters·are·involved.·Each·of·these·parameters·could·be·described·with·a·given·Probability·Density·Function·(PDF).·If·would·like·to·be·able·to·generate·a·new·set·of·parameter·values,·we·need·to·be·able·to·sample·from·these·distributions-also·called·marginals.·There·are·mainly·two·cases:·<em>(i)</em>·PDFs·are·independent;·<em>(ii)</em>·there·is·a·dependency.·One·way·to·model·the·dependency·it·to·use·a·<strong>copula</strong>.</p>
101 <section·id="Sampling-from-a-copula">89 <section·id="Sampling-from-a-copula">
102 <h2>Sampling·from·a·copula<a·class="headerlink"·href="#Sampling-from-a-copula"·title="Link·to·this·heading">¶</a></h2>90 <h2>Sampling·from·a·copula<a·class="headerlink"·href="#Sampling-from-a-copula"·title="Link·to·this·heading">¶</a></h2>
103 <p>Let’s·use·a·bi-variate·example·and·assume·first·that·we·have·a·prior·and·know·how·to·model·the·dependence·between·our·2·variables.</p>91 <p>Let’s·use·a·bi-variate·example·and·assume·first·that·we·have·a·prior·and·know·how·to·model·the·dependence·between·our·2·variables.</p>
104 <p>In·this·case,·we·are·using·the·Gumbel·copula·and·fix·its·hyperparameter·<code·class="docutils·literal·notranslate"><span·class="pre">theta=2</span></code>.·We·can·visualize·it’s·2-dimensional·PDF.</p>92 <p>In·this·case,·we·are·using·the·Gumbel·copula·and·fix·its·hyperparameter·<code·class="docutils·literal·notranslate"><span·class="pre">theta=2</span></code>.·We·can·visualize·it’s·2-dimensional·PDF.</p>
105 <div·class="nbinput·docutils·container">93 <div·class="nbinput·nblast·docutils·container">
106 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[3]:94 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:
107 </pre></div>95 </pre></div>
108 </div>96 </div>
109 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">from</span>·<span·class="nn">statsmodels.distributions.copula.api</span>·<span·class="kn">import</span>·<span·class="p">(</span>97 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">from</span>·<span·class="nn">statsmodels.distributions.copula.api</span>·<span·class="kn">import</span>·<span·class="p">(</span>
110 ····<span·class="n">CopulaDistribution</span><span·class="p">,</span>·<span·class="n">GumbelCopula</span><span·class="p">,</span>·<span·class="n">IndependenceCopula</span><span·class="p">)</span>98 ····<span·class="n">CopulaDistribution</span><span·class="p">,</span>·<span·class="n">GumbelCopula</span><span·class="p">,</span>·<span·class="n">IndependenceCopula</span><span·class="p">)</span>
  
111 <span·class="n">copula</span>·<span·class="o">=</span>·<span·class="n">GumbelCopula</span><span·class="p">(</span><span·class="n">theta</span><span·class="o">=</span><span·class="mi">2</span><span·class="p">)</span>99 <span·class="n">copula</span>·<span·class="o">=</span>·<span·class="n">GumbelCopula</span><span·class="p">(</span><span·class="n">theta</span><span·class="o">=</span><span·class="mi">2</span><span·class="p">)</span>
112 <span·class="n">_</span>·<span·class="o">=</span>·<span·class="n">copula</span><span·class="o">.</span><span·class="n">plot_pdf</span><span·class="p">()</span>··<span·class="c1">#·returns·a·matplotlib·figure</span>100 <span·class="n">_</span>·<span·class="o">=</span>·<span·class="n">copula</span><span·class="o">.</span><span·class="n">plot_pdf</span><span·class="p">()</span>··<span·class="c1">#·returns·a·matplotlib·figure</span>
113 </pre></div>101 </pre></div>
114 </div>102 </div>
115 </div>103 </div>
116 <div·class="nboutput·nblast·docutils·container"> 
117 <div·class="prompt·empty·docutils·container"> 
118 </div> 
119 <div·class="output_area·docutils·container"> 
120 <img·alt="../../../_images/examples_notebooks_generated_copula_5_0.png"·src="../../../_images/examples_notebooks_generated_copula_5_0.png"·/> 
121 </div> 
122 </div> 
123 <p>And·we·can·sample·the·PDF.</p>104 <p>And·we·can·sample·the·PDF.</p>
124 <div·class="nbinput·docutils·container">105 <div·class="nbinput·nblast·docutils·container">
125 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[4]:106 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:
126 </pre></div>107 </pre></div>
127 </div>108 </div>
128 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">sample</span>·<span·class="o">=</span>·<span·class="n">copula</span><span·class="o">.</span><span·class="n">rvs</span><span·class="p">(</span><span·class="mi">10000</span><span·class="p">)</span>109 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">sample</span>·<span·class="o">=</span>·<span·class="n">copula</span><span·class="o">.</span><span·class="n">rvs</span><span·class="p">(</span><span·class="mi">10000</span><span·class="p">)</span>
129 <span·class="n">h</span>·<span·class="o">=</span>·<span·class="n">sns</span><span·class="o">.</span><span·class="n">jointplot</span><span·class="p">(</span><span·class="n">x</span><span·class="o">=</span><span·class="n">sample</span><span·class="p">[:,</span>·<span·class="mi">0</span><span·class="p">],</span>·<span·class="n">y</span><span·class="o">=</span><span·class="n">sample</span><span·class="p">[:,</span>·<span·class="mi">1</span><span·class="p">],</span>·<span·class="n">kind</span><span·class="o">=</span><span·class="s2">&quot;hex&quot;</span><span·class="p">)</span>110 <span·class="n">h</span>·<span·class="o">=</span>·<span·class="n">sns</span><span·class="o">.</span><span·class="n">jointplot</span><span·class="p">(</span><span·class="n">x</span><span·class="o">=</span><span·class="n">sample</span><span·class="p">[:,</span>·<span·class="mi">0</span><span·class="p">],</span>·<span·class="n">y</span><span·class="o">=</span><span·class="n">sample</span><span·class="p">[:,</span>·<span·class="mi">1</span><span·class="p">],</span>·<span·class="n">kind</span><span·class="o">=</span><span·class="s2">&quot;hex&quot;</span><span·class="p">)</span>
130 <span·class="n">_</span>·<span·class="o">=</span>·<span·class="n">h</span><span·class="o">.</span><span·class="n">set_axis_labels</span><span·class="p">(</span><span·class="s2">&quot;X1&quot;</span><span·class="p">,</span>·<span·class="s2">&quot;X2&quot;</span><span·class="p">,</span>·<span·class="n">fontsize</span><span·class="o">=</span><span·class="mi">16</span><span·class="p">)</span>111 <span·class="n">_</span>·<span·class="o">=</span>·<span·class="n">h</span><span·class="o">.</span><span·class="n">set_axis_labels</span><span·class="p">(</span><span·class="s2">&quot;X1&quot;</span><span·class="p">,</span>·<span·class="s2">&quot;X2&quot;</span><span·class="p">,</span>·<span·class="n">fontsize</span><span·class="o">=</span><span·class="mi">16</span><span·class="p">)</span>
131 </pre></div>112 </pre></div>
132 </div>113 </div>
133 </div>114 </div>
134 <div·class="nboutput·docutils·container"> 
135 <div·class="prompt·empty·docutils·container"> 
136 </div> 
137 <div·class="output_area·stderr·docutils·container"> 
138 <div·class="highlight"><pre> 
139 /usr/lib/python3/dist-packages/statsmodels/tools/rng_qrng.py:54:·FutureWarning:·Passing·`None`·as·the·seed·currently·return·the·NumPy·singleton·RandomState 
140 (np.random.mtrand._rand).·After·release·0.13·this·will·change·to·using·the 
141 default·generator·provided·by·NumPy·(np.random.default_rng()).·If·you·need 
142 reproducible·draws,·you·should·pass·a·seeded·np.random.Generator,·e.g., 
  
143 import·numpy·as·np 
144 seed·=·32839283923801 
145 rng·=·np.random.default_rng(seed)&#34; 
  
146 ··warnings.warn(_future_warn,·FutureWarning) 
147 </pre></div></div> 
148 </div> 
149 <div·class="nboutput·nblast·docutils·container"> 
150 <div·class="prompt·empty·docutils·container"> 
151 </div> 
152 <div·class="output_area·docutils·container"> 
153 <img·alt="../../../_images/examples_notebooks_generated_copula_7_1.png"·src="../../../_images/examples_notebooks_generated_copula_7_1.png"·/> 
154 </div> 
155 </div> 
156 <p>Let’s·come·back·to·our·2·variables·for·a·second.·In·this·case·we·consider·them·to·be·gamma·and·normally·distributed.·If·they·would·be·independent·from·each·other,·we·could·sample·from·each·PDF·individually.·Here·we·use·a·convenient·class·to·do·the·same·operation.</p>115 <p>Let’s·come·back·to·our·2·variables·for·a·second.·In·this·case·we·consider·them·to·be·gamma·and·normally·distributed.·If·they·would·be·independent·from·each·other,·we·could·sample·from·each·PDF·individually.·Here·we·use·a·convenient·class·to·do·the·same·operation.</p>
157 <section·id="Reproducibility">116 <section·id="Reproducibility">
158 <h3>Reproducibility<a·class="headerlink"·href="#Reproducibility"·title="Link·to·this·heading">¶</a></h3>117 <h3>Reproducibility<a·class="headerlink"·href="#Reproducibility"·title="Link·to·this·heading">¶</a></h3>
159 <p>Generating·reproducible·random·values·from·copulas·required·explicitly·setting·the·<code·class="docutils·literal·notranslate"><span·class="pre">seed</span></code>·argument.·<code·class="docutils·literal·notranslate"><span·class="pre">seed</span></code>·accepts·either·an·initialized·NumPy·<code·class="docutils·literal·notranslate"><span·class="pre">Generator</span></code>·or·<code·class="docutils·literal·notranslate"><span·class="pre">RandomState</span></code>,·or·any·argument·acceptable·to·<code·class="docutils·literal·notranslate"><span·class="pre">np.random.default_rng</span></code>,·e.g.,·an·integer·or·a·sequence·of·integers.·This·example·uses·an·integer.</p>118 <p>Generating·reproducible·random·values·from·copulas·required·explicitly·setting·the·<code·class="docutils·literal·notranslate"><span·class="pre">seed</span></code>·argument.·<code·class="docutils·literal·notranslate"><span·class="pre">seed</span></code>·accepts·either·an·initialized·NumPy·<code·class="docutils·literal·notranslate"><span·class="pre">Generator</span></code>·or·<code·class="docutils·literal·notranslate"><span·class="pre">RandomState</span></code>,·or·any·argument·acceptable·to·<code·class="docutils·literal·notranslate"><span·class="pre">np.random.default_rng</span></code>,·e.g.,·an·integer·or·a·sequence·of·integers.·This·example·uses·an·integer.</p>
160 <p>The·singleton·<code·class="docutils·literal·notranslate"><span·class="pre">RandomState</span></code>·that·is·directly·exposed·in·the·<code·class="docutils·literal·notranslate"><span·class="pre">np.random</span></code>·distributions·is·not·used,·and·setting·<code·class="docutils·literal·notranslate"><span·class="pre">np.random.seed</span></code>·has·no·effect·on·the·values·generated.</p>119 <p>The·singleton·<code·class="docutils·literal·notranslate"><span·class="pre">RandomState</span></code>·that·is·directly·exposed·in·the·<code·class="docutils·literal·notranslate"><span·class="pre">np.random</span></code>·distributions·is·not·used,·and·setting·<code·class="docutils·literal·notranslate"><span·class="pre">np.random.seed</span></code>·has·no·effect·on·the·values·generated.</p>
161 <div·class="nbinput·docutils·container">120 <div·class="nbinput·nblast·docutils·container">
162 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[5]:121 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:
163 </pre></div>122 </pre></div>
164 </div>123 </div>
165 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">marginals</span>·<span·class="o">=</span>·<span·class="p">[</span><span·class="n">stats</span><span·class="o">.</span><span·class="n">gamma</span><span·class="p">(</span><span·class="mi">2</span><span·class="p">),</span>·<span·class="n">stats</span><span·class="o">.</span><span·class="n">norm</span><span·class="p">]</span>124 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">marginals</span>·<span·class="o">=</span>·<span·class="p">[</span><span·class="n">stats</span><span·class="o">.</span><span·class="n">gamma</span><span·class="p">(</span><span·class="mi">2</span><span·class="p">),</span>·<span·class="n">stats</span><span·class="o">.</span><span·class="n">norm</span><span·class="p">]</span>
166 <span·class="n">joint_dist</span>·<span·class="o">=</span>·<span·class="n">CopulaDistribution</span><span·class="p">(</span><span·class="n">copula</span><span·class="o">=</span><span·class="n">IndependenceCopula</span><span·class="p">(),</span>·<span·class="n">marginals</span><span·class="o">=</span><span·class="n">marginals</span><span·class="p">)</span>125 <span·class="n">joint_dist</span>·<span·class="o">=</span>·<span·class="n">CopulaDistribution</span><span·class="p">(</span><span·class="n">copula</span><span·class="o">=</span><span·class="n">IndependenceCopula</span><span·class="p">(),</span>·<span·class="n">marginals</span><span·class="o">=</span><span·class="n">marginals</span><span·class="p">)</span>
167 <span·class="n">sample</span>·<span·class="o">=</span>·<span·class="n">joint_dist</span><span·class="o">.</span><span·class="n">rvs</span><span·class="p">(</span><span·class="mi">512</span><span·class="p">,</span>·<span·class="n">random_state</span><span·class="o">=</span><span·class="mi">20210801</span><span·class="p">)</span>126 <span·class="n">sample</span>·<span·class="o">=</span>·<span·class="n">joint_dist</span><span·class="o">.</span><span·class="n">rvs</span><span·class="p">(</span><span·class="mi">512</span><span·class="p">,</span>·<span·class="n">random_state</span><span·class="o">=</span><span·class="mi">20210801</span><span·class="p">)</span>
168 <span·class="n">h</span>·<span·class="o">=</span>·<span·class="n">sns</span><span·class="o">.</span><span·class="n">jointplot</span><span·class="p">(</span><span·class="n">x</span><span·class="o">=</span><span·class="n">sample</span><span·class="p">[:,</span>·<span·class="mi">0</span><span·class="p">],</span>·<span·class="n">y</span><span·class="o">=</span><span·class="n">sample</span><span·class="p">[:,</span>·<span·class="mi">1</span><span·class="p">],</span>·<span·class="n">kind</span><span·class="o">=</span><span·class="s2">&quot;scatter&quot;</span><span·class="p">)</span>127 <span·class="n">h</span>·<span·class="o">=</span>·<span·class="n">sns</span><span·class="o">.</span><span·class="n">jointplot</span><span·class="p">(</span><span·class="n">x</span><span·class="o">=</span><span·class="n">sample</span><span·class="p">[:,</span>·<span·class="mi">0</span><span·class="p">],</span>·<span·class="n">y</span><span·class="o">=</span><span·class="n">sample</span><span·class="p">[:,</span>·<span·class="mi">1</span><span·class="p">],</span>·<span·class="n">kind</span><span·class="o">=</span><span·class="s2">&quot;scatter&quot;</span><span·class="p">)</span>
169 <span·class="n">_</span>·<span·class="o">=</span>·<span·class="n">h</span><span·class="o">.</span><span·class="n">set_axis_labels</span><span·class="p">(</span><span·class="s2">&quot;X1&quot;</span><span·class="p">,</span>·<span·class="s2">&quot;X2&quot;</span><span·class="p">,</span>·<span·class="n">fontsize</span><span·class="o">=</span><span·class="mi">16</span><span·class="p">)</span>128 <span·class="n">_</span>·<span·class="o">=</span>·<span·class="n">h</span><span·class="o">.</span><span·class="n">set_axis_labels</span><span·class="p">(</span><span·class="s2">&quot;X1&quot;</span><span·class="p">,</span>·<span·class="s2">&quot;X2&quot;</span><span·class="p">,</span>·<span·class="n">fontsize</span><span·class="o">=</span><span·class="mi">16</span><span·class="p">)</span>
170 </pre></div>129 </pre></div>
171 </div>130 </div>
172 </div>131 </div>
173 <div·class="nboutput·nblast·docutils·container"> 
174 <div·class="prompt·empty·docutils·container"> 
175 </div> 
176 <div·class="output_area·docutils·container"> 
177 <img·alt="../../../_images/examples_notebooks_generated_copula_9_0.png"·src="../../../_images/examples_notebooks_generated_copula_9_0.png"·/> 
178 </div> 
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3 ····*·modules·|3 ····*·modules·|
4 ····*·next·|4 ····*·next·|
5 ····*·previous·|5 ····*·previous·|
6 ····*·statsmodels_0.14.5+dfsg·»6 ····*·statsmodels_0.14.5+dfsg·»
7 ····*·Examples·»7 ····*·Examples·»
8 ····*·Copula·-·Multivariate·joint·distribution8 ····*·Copula·-·Multivariate·joint·distribution
9 ******·Copula·-·Multivariate·joint·distribution¶·******9 ******·Copula·-·Multivariate·joint·distribution¶·******
10 [1]:10 [·]:
11 import·matplotlib.pyplot·as·plt11 import·matplotlib.pyplot·as·plt
12 import·numpy·as·np12 import·numpy·as·np
13 import·seaborn·as·sns13 import·seaborn·as·sns
14 from·scipy·import·stats14 from·scipy·import·stats
  
15 sns.set_style("darkgrid")15 sns.set_style("darkgrid")
16 sns.mpl.rc("figure",·figsize=(8,·8))16 sns.mpl.rc("figure",·figsize=(8,·8))
17 np.random.seed(1234)·#·for·reproducibility17 np.random.seed(1234)·#·for·reproducibility
18 [2]:18 [·]:
19 %%javascript19 %%javascript
20 IPython.OutputArea.prototype._should_scroll·=·function(lines)·{20 IPython.OutputArea.prototype._should_scroll·=·function(lines)·{
21 ····return·false;21 ····return·false;
22 }22 }
23 When·modeling·a·system,·there·are·often·cases·where·multiple·parameters·are23 When·modeling·a·system,·there·are·often·cases·where·multiple·parameters·are
24 involved.·Each·of·these·parameters·could·be·described·with·a·given·Probability24 involved.·Each·of·these·parameters·could·be·described·with·a·given·Probability
25 Density·Function·(PDF).·If·would·like·to·be·able·to·generate·a·new·set·of25 Density·Function·(PDF).·If·would·like·to·be·able·to·generate·a·new·set·of
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28 called·marginals.·There·are·mainly·two·cases:·(i)·PDFs·are·independent;·(ii)28 called·marginals.·There·are·mainly·two·cases:·(i)·PDFs·are·independent;·(ii)
29 there·is·a·dependency.·One·way·to·model·the·dependency·it·to·use·a·copula.29 there·is·a·dependency.·One·way·to·model·the·dependency·it·to·use·a·copula.
30 *****·Sampling·from·a·copula¶·*****30 *****·Sampling·from·a·copula¶·*****
31 Let’s·use·a·bi-variate·example·and·assume·first·that·we·have·a·prior·and·know31 Let’s·use·a·bi-variate·example·and·assume·first·that·we·have·a·prior·and·know
32 how·to·model·the·dependence·between·our·2·variables.32 how·to·model·the·dependence·between·our·2·variables.
33 In·this·case,·we·are·using·the·Gumbel·copula·and·fix·its·hyperparameter33 In·this·case,·we·are·using·the·Gumbel·copula·and·fix·its·hyperparameter
34 theta=2.·We·can·visualize·it’s·2-dimensional·PDF.34 theta=2.·We·can·visualize·it’s·2-dimensional·PDF.
35 [3]:35 [·]:
36 from·statsmodels.distributions.copula.api·import·(36 from·statsmodels.distributions.copula.api·import·(
37 ····CopulaDistribution,·GumbelCopula,·IndependenceCopula)37 ····CopulaDistribution,·GumbelCopula,·IndependenceCopula)
  
38 copula·=·GumbelCopula(theta=2)38 copula·=·GumbelCopula(theta=2)
39 _·=·copula.plot_pdf()··#·returns·a·matplotlib·figure39 _·=·copula.plot_pdf()··#·returns·a·matplotlib·figure
40 [../../../_images/examples_notebooks_generated_copula_5_0.png] 
41 And·we·can·sample·the·PDF.40 And·we·can·sample·the·PDF.
42 [4]:41 [·]:
43 sample·=·copula.rvs(10000)42 sample·=·copula.rvs(10000)
44 h·=·sns.jointplot(x=sample[:,·0],·y=sample[:,·1],·kind="hex")43 h·=·sns.jointplot(x=sample[:,·0],·y=sample[:,·1],·kind="hex")
45 _·=·h.set_axis_labels("X1",·"X2",·fontsize=16)44 _·=·h.set_axis_labels("X1",·"X2",·fontsize=16)
46 /usr/lib/python3/dist-packages/statsmodels/tools/rng_qrng.py:54:·FutureWarning: 
47 Passing·`None`·as·the·seed·currently·return·the·NumPy·singleton·RandomState 
48 (np.random.mtrand._rand).·After·release·0.13·this·will·change·to·using·the 
49 default·generator·provided·by·NumPy·(np.random.default_rng()).·If·you·need 
50 reproducible·draws,·you·should·pass·a·seeded·np.random.Generator,·e.g., 
  
51 import·numpy·as·np 
52 seed·=·32839283923801 
53 rng·=·np.random.default_rng(seed)" 
  
54 ··warnings.warn(_future_warn,·FutureWarning) 
55 [../../../_images/examples_notebooks_generated_copula_7_1.png] 
56 Let’s·come·back·to·our·2·variables·for·a·second.·In·this·case·we·consider·them45 Let’s·come·back·to·our·2·variables·for·a·second.·In·this·case·we·consider·them
57 to·be·gamma·and·normally·distributed.·If·they·would·be·independent·from·each46 to·be·gamma·and·normally·distributed.·If·they·would·be·independent·from·each
58 other,·we·could·sample·from·each·PDF·individually.·Here·we·use·a·convenient47 other,·we·could·sample·from·each·PDF·individually.·Here·we·use·a·convenient
59 class·to·do·the·same·operation.48 class·to·do·the·same·operation.
60 ****·Reproducibility¶·****49 ****·Reproducibility¶·****
61 Generating·reproducible·random·values·from·copulas·required·explicitly·setting50 Generating·reproducible·random·values·from·copulas·required·explicitly·setting
62 the·seed·argument.·seed·accepts·either·an·initialized·NumPy·Generator·or51 the·seed·argument.·seed·accepts·either·an·initialized·NumPy·Generator·or
63 RandomState,·or·any·argument·acceptable·to·np.random.default_rng,·e.g.,·an52 RandomState,·or·any·argument·acceptable·to·np.random.default_rng,·e.g.,·an
64 integer·or·a·sequence·of·integers.·This·example·uses·an·integer.53 integer·or·a·sequence·of·integers.·This·example·uses·an·integer.
65 The·singleton·RandomState·that·is·directly·exposed·in·the·np.random54 The·singleton·RandomState·that·is·directly·exposed·in·the·np.random
66 distributions·is·not·used,·and·setting·np.random.seed·has·no·effect·on·the55 distributions·is·not·used,·and·setting·np.random.seed·has·no·effect·on·the
67 values·generated.56 values·generated.
68 [5]:57 [·]:
69 marginals·=·[stats.gamma(2),·stats.norm]58 marginals·=·[stats.gamma(2),·stats.norm]
70 joint_dist·=·CopulaDistribution(copula=IndependenceCopula(),59 joint_dist·=·CopulaDistribution(copula=IndependenceCopula(),
71 marginals=marginals)60 marginals=marginals)
72 sample·=·joint_dist.rvs(512,·random_state=20210801)61 sample·=·joint_dist.rvs(512,·random_state=20210801)
73 h·=·sns.jointplot(x=sample[:,·0],·y=sample[:,·1],·kind="scatter")62 h·=·sns.jointplot(x=sample[:,·0],·y=sample[:,·1],·kind="scatter")
74 _·=·h.set_axis_labels("X1",·"X2",·fontsize=16)63 _·=·h.set_axis_labels("X1",·"X2",·fontsize=16)
75 [../../../_images/examples_notebooks_generated_copula_9_0.png] 
76 Now,·above·we·have·expressed·the·dependency·between·our·variables·using·a64 Now,·above·we·have·expressed·the·dependency·between·our·variables·using·a
77 copula,·we·can·use·this·copula·to·sample·a·new·set·of·observation·with·the·same65 copula,·we·can·use·this·copula·to·sample·a·new·set·of·observation·with·the·same
78 convenient·class.66 convenient·class.
79 [6]:67 [·]:
80 joint_dist·=·CopulaDistribution(copula,·marginals)68 joint_dist·=·CopulaDistribution(copula,·marginals)
81 #·Use·an·initialized·Generator·object69 #·Use·an·initialized·Generator·object
82 rng·=·np.random.default_rng([2,·0,·2,·1,·0,·8,·0,·1])70 rng·=·np.random.default_rng([2,·0,·2,·1,·0,·8,·0,·1])
83 sample·=·joint_dist.rvs(512,·random_state=rng)71 sample·=·joint_dist.rvs(512,·random_state=rng)
84 h·=·sns.jointplot(x=sample[:,·0],·y=sample[:,·1],·kind="scatter")72 h·=·sns.jointplot(x=sample[:,·0],·y=sample[:,·1],·kind="scatter")
85 _·=·h.set_axis_labels("X1",·"X2",·fontsize=16)73 _·=·h.set_axis_labels("X1",·"X2",·fontsize=16)
86 [../../../_images/examples_notebooks_generated_copula_11_0.png] 
87 There·are·two·things·to·note·here.·(i)·as·in·the·independent·case,·the74 There·are·two·things·to·note·here.·(i)·as·in·the·independent·case,·the
88 marginals·are·correctly·showing·a·gamma·and·normal·distribution;·(ii)·the75 marginals·are·correctly·showing·a·gamma·and·normal·distribution;·(ii)·the
89 dependence·is·visible·between·the·two·variables.76 dependence·is·visible·between·the·two·variables.
90 *****·Estimating·copula·parameters¶·*****77 *****·Estimating·copula·parameters¶·*****
91 Now,·imagine·we·already·have·experimental·data·and·we·know·that·there·is·a78 Now,·imagine·we·already·have·experimental·data·and·we·know·that·there·is·a
92 dependency·that·can·be·expressed·using·a·Gumbel·copula.·But·we·don’t·know·what79 dependency·that·can·be·expressed·using·a·Gumbel·copula.·But·we·don’t·know·what
93 is·the·hyperparameter·value·for·our·copula.·In·this·case,·we·can·estimate·the80 is·the·hyperparameter·value·for·our·copula.·In·this·case,·we·can·estimate·the
94 value.81 value.
95 We·are·going·to·use·the·sample·we·just·generated·as·we·already·know·the·value82 We·are·going·to·use·the·sample·we·just·generated·as·we·already·know·the·value
96 of·the·hyperparameter·we·should·get:·theta=2.83 of·the·hyperparameter·we·should·get:·theta=2.
97 [7]:84 [·]:
98 copula·=·GumbelCopula()85 copula·=·GumbelCopula()
99 theta·=·copula.fit_corr_param(sample)86 theta·=·copula.fit_corr_param(sample)
100 print(theta)87 print(theta)
101 2.049379621506455 
102 We·can·see·that·the·estimated·hyperparameter·value·is·close·to·the·value·set88 We·can·see·that·the·estimated·hyperparameter·value·is·close·to·the·value·set
103 previously.89 previously.
104 [Logo_of_statsmodels_0.14.5+dfsg]90 [Logo_of_statsmodels_0.14.5+dfsg]
105 ****·Table_of_Contents·****91 ****·Table_of_Contents·****
106 ····*·Installing_statsmodels92 ····*·Installing_statsmodels
107 ····*·Getting_started93 ····*·Getting_started
108 ····*·User_Guide94 ····*·User_Guide
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Similarity: 0.9995828163771712% Differences: {"'cells'": "{22: {'outputs': {0: {'text': {insert: [(0, 'Optimization terminated " "successfully.\\n'), (1, ' Current function value: 1.548647\\n'), (2, " "' Iterations 7\\n')]}}, delete: [0]}}, 26: {'outputs': {1: {'text': {delete: " "[2, 1, 0]}}, insert: [(0, OrderedDict({'name': 'stdout', 'output_type': 'stream', " "'text': ['Optimization terminated successfully.\\n', ' Current function " "value: 3.091609\\n', ' […]
    
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383 ············"outputs":·[383 ············"outputs":·[
384 ················{384 ················{
385 ····················"name":·"stdout",385 ····················"name":·"stdout",
386 ····················"output_type":·"stream",386 ····················"output_type":·"stream",
387 ····················"text":·[387 ····················"text":·[
388 ························"Optimization·terminated·successfully.\n",388 ························"Optimization·terminated·successfully.\n",
389 ························"·········Current·function·value:·1.548647\n",389 ························"·········Current·function·value:·1.548647\n",
390 ························"·········Iterations·7\n"390 ························"·········Iterations·7\n",
391 ····················] 
392 ················}, 
393 ················{ 
394 ····················"name":·"stdout", 
395 ····················"output_type":·"stream", 
396 ····················"text":·[ 
397 ························"·················0·········1·········2·········3·········4··········5\n",391 ························"·················0·········1·········2·········3·········4··········5\n",
398 ························"const····-0.373402·-2.250913·-3.665584·-7.613843·-7.060478·-12.105751\n",392 ························"const····-0.373402·-2.250913·-3.665584·-7.613843·-7.060478·-12.105751\n",
399 ························"logpopul·-0.011536·-0.088751·-0.105967·-0.091557·-0.093285··-0.140881\n",393 ························"logpopul·-0.011536·-0.088751·-0.105967·-0.091557·-0.093285··-0.140881\n",
400 ························"selfLR····0.297714··0.391669··0.573451··1.278772··1.346962···2.070080\n",394 ························"selfLR····0.297714··0.391669··0.573451··1.278772··1.346962···2.070080\n",
401 ························"age······-0.024945·-0.022898·-0.014851·-0.008681·-0.017904··-0.009433\n",395 ························"age······-0.024945·-0.022898·-0.014851·-0.008681·-0.017904··-0.009433\n",
402 ························"educ······0.082491··0.181043·-0.007152··0.199828··0.216939···0.321926\n",396 ························"educ······0.082491··0.181043·-0.007152··0.199828··0.216939···0.321926\n",
403 ························"income····0.005197··0.047874··0.057575··0.084498··0.080958···0.108894\n"397 ························"income····0.005197··0.047874··0.057575··0.084498··0.080958···0.108894\n"
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454 ············"outputs":·[448 ············"outputs":·[
455 ················{449 ················{
456 ····················"name":·"stdout",450 ····················"name":·"stdout",
457 ····················"output_type":·"stream",451 ····················"output_type":·"stream",
458 ····················"text":·[452 ····················"text":·[
459 ························"Optimization·terminated·successfully.\n",453 ························"Optimization·terminated·successfully.\n",
460 ························"·········Current·function·value:·3.091609\n",454 ························"·········Current·function·value:·3.091609\n",
461 ························"·········Iterations·6\n",455 ························"·········Iterations·6\n"
 456 ····················]
 457 ················},
 458 ················{
 459 ····················"name":·"stdout",
 460 ····················"output_type":·"stream",
 461 ····················"text":·[
462 ························"··························Poisson·Regression·Results··························\n",462 ························"··························Poisson·Regression·Results··························\n",
463 ························"==============================================================================\n",463 ························"==============================================================================\n",
464 ························"Dep.·Variable:··················mdvis···No.·Observations:················20190\n",464 ························"Dep.·Variable:··················mdvis···No.·Observations:················20190\n",
465 ························"Model:························Poisson···Df·Residuals:····················20180\n",465 ························"Model:························Poisson···Df·Residuals:····················20180\n",
466 ························"Method:···························MLE···Df·Model:····························9\n",466 ························"Method:···························MLE···Df·Model:····························9\n",
467 ························"Date:················Sun,·10·Aug·2025···Pseudo·R-squ.:·················0.06343\n",467 ························"Date:················Sun,·10·Aug·2025···Pseudo·R-squ.:·················0.06343\n",
468 ························"Time:························13:13:47···Log-Likelihood:················-62420.\n",468 ························"Time:························13:13:47···Log-Likelihood:················-62420.\n",
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333 ················"execution":·{}333 ················"execution":·{}
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335 ············"outputs":·[335 ············"outputs":·[
336 ················{336 ················{
337 ····················"name":·"stdout",337 ····················"name":·"stdout",
338 ····················"output_type":·"stream",338 ····················"output_type":·"stream",
339 ····················"text":·[339 ····················"text":·[
340 ························"method·RE:" 
341 ····················] 
342 ················}, 
343 ················{ 
344 ····················"name":·"stdout", 
345 ····················"output_type":·"stream", 
346 ····················"text":·[ 
347 ························"·iterated\n",340 ························"method·RE:·iterated\n",
348 ························"························eff····sd_eff····ci_low····ci_upp······w_fe······w_re\n",341 ························"························eff····sd_eff····ci_low····ci_upp······w_fe······w_re\n",
349 ························"Carroll············0.094524··0.182680·-0.263521··0.452570··0.123885··0.152619\n",342 ························"Carroll············0.094524··0.182680·-0.263521··0.452570··0.123885··0.152619\n",
350 ························"Grant··············0.277356··0.176279·-0.068144··0.622857··0.133045··0.159157\n",343 ························"Grant··············0.277356··0.176279·-0.068144··0.622857··0.133045··0.159157\n",
351 ························"Peck···············0.366546··0.225573·-0.075569··0.808662··0.081250··0.116228\n",344 ························"Peck···············0.366546··0.225573·-0.075569··0.808662··0.081250··0.116228\n",
352 ························"Donat··············0.664385··0.102748··0.463002··0.865768··0.391606··0.257767\n",345 ························"Donat··············0.664385··0.102748··0.463002··0.865768··0.391606··0.257767\n",
353 ························"Stewart············0.461808··0.208310··0.053527··0.870089··0.095275··0.129428\n",346 ························"Stewart············0.461808··0.208310··0.053527··0.870089··0.095275··0.129428\n",
354 ························"Young··············0.185165··0.153729·-0.116139··0.486468··0.174939··0.184799\n",347 ························"Young··············0.185165··0.153729·-0.116139··0.486468··0.174939··0.184799\n",
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57 ······<div·class="documentwrapper">57 ······<div·class="documentwrapper">
58 ········<div·class="bodywrapper">58 ········<div·class="bodywrapper">
59 ··········<div·class="body"·role="main">59 ··········<div·class="body"·role="main">
60 ············60 ············
61 ··<section·id="Autoregressive-Moving-Average-(ARMA):-Sunspots-data">61 ··<section·id="Autoregressive-Moving-Average-(ARMA):-Sunspots-data">
62 <h1>Autoregressive·Moving·Average·(ARMA):·Sunspots·data<a·class="headerlink"·href="#Autoregressive-Moving-Average-(ARMA):-Sunspots-data"·title="Link·to·this·heading">¶</a></h1>62 <h1>Autoregressive·Moving·Average·(ARMA):·Sunspots·data<a·class="headerlink"·href="#Autoregressive-Moving-Average-(ARMA):-Sunspots-data"·title="Link·to·this·heading">¶</a></h1>
63 <div·class="nbinput·nblast·docutils·container">63 <div·class="nbinput·nblast·docutils·container">
64 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:64 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[1]:
65 </pre></div>65 </pre></div>
66 </div>66 </div>
67 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="o">%</span><span·class="k">matplotlib</span>·inline67 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="o">%</span><span·class="k">matplotlib</span>·inline
68 </pre></div>68 </pre></div>
69 </div>69 </div>
70 </div>70 </div>
71 <div·class="nbinput·nblast·docutils·container">71 <div·class="nbinput·nblast·docutils·container">
72 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:72 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[2]:
73 </pre></div>73 </pre></div>
74 </div>74 </div>
75 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">import</span>·<span·class="nn">matplotlib.pyplot</span>·<span·class="k">as</span>·<span·class="nn">plt</span>75 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">import</span>·<span·class="nn">matplotlib.pyplot</span>·<span·class="k">as</span>·<span·class="nn">plt</span>
76 <span·class="kn">import</span>·<span·class="nn">numpy</span>·<span·class="k">as</span>·<span·class="nn">np</span>76 <span·class="kn">import</span>·<span·class="nn">numpy</span>·<span·class="k">as</span>·<span·class="nn">np</span>
77 <span·class="kn">import</span>·<span·class="nn">pandas</span>·<span·class="k">as</span>·<span·class="nn">pd</span>77 <span·class="kn">import</span>·<span·class="nn">pandas</span>·<span·class="k">as</span>·<span·class="nn">pd</span>
78 <span·class="kn">import</span>·<span·class="nn">statsmodels.api</span>·<span·class="k">as</span>·<span·class="nn">sm</span>78 <span·class="kn">import</span>·<span·class="nn">statsmodels.api</span>·<span·class="k">as</span>·<span·class="nn">sm</span>
79 <span·class="kn">from</span>·<span·class="nn">scipy</span>·<span·class="kn">import</span>·<span·class="n">stats</span>79 <span·class="kn">from</span>·<span·class="nn">scipy</span>·<span·class="kn">import</span>·<span·class="n">stats</span>
80 <span·class="kn">from</span>·<span·class="nn">statsmodels.tsa.arima.model</span>·<span·class="kn">import</span>·<span·class="n">ARIMA</span>80 <span·class="kn">from</span>·<span·class="nn">statsmodels.tsa.arima.model</span>·<span·class="kn">import</span>·<span·class="n">ARIMA</span>
81 </pre></div>81 </pre></div>
82 </div>82 </div>
83 </div>83 </div>
84 <div·class="nbinput·nblast·docutils·container">84 <div·class="nbinput·nblast·docutils·container">
85 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:85 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[3]:
86 </pre></div>86 </pre></div>
87 </div>87 </div>
88 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">from</span>·<span·class="nn">statsmodels.graphics.api</span>·<span·class="kn">import</span>·<span·class="n">qqplot</span>88 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">from</span>·<span·class="nn">statsmodels.graphics.api</span>·<span·class="kn">import</span>·<span·class="n">qqplot</span>
89 </pre></div>89 </pre></div>
90 </div>90 </div>
91 </div>91 </div>
92 <section·id="Sunspots-Data">92 <section·id="Sunspots-Data">
93 <h2>Sunspots·Data<a·class="headerlink"·href="#Sunspots-Data"·title="Link·to·this·heading">¶</a></h2>93 <h2>Sunspots·Data<a·class="headerlink"·href="#Sunspots-Data"·title="Link·to·this·heading">¶</a></h2>
94 <div·class="nbinput·nblast·docutils·container">94 <div·class="nbinput·docutils·container">
95 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:95 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[4]:
96 </pre></div>96 </pre></div>
97 </div>97 </div>
98 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="nb">print</span><span·class="p">(</span><span·class="n">sm</span><span·class="o">.</span><span·class="n">datasets</span><span·class="o">.</span><span·class="n">sunspots</span><span·class="o">.</span><span·class="n">NOTE</span><span·class="p">)</span>98 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="nb">print</span><span·class="p">(</span><span·class="n">sm</span><span·class="o">.</span><span·class="n">datasets</span><span·class="o">.</span><span·class="n">sunspots</span><span·class="o">.</span><span·class="n">NOTE</span><span·class="p">)</span>
99 </pre></div>99 </pre></div>
100 </div>100 </div>
101 </div>101 </div>
 102 <div·class="nboutput·nblast·docutils·container">
 103 <div·class="prompt·empty·docutils·container">
 104 </div>
 105 <div·class="output_area·docutils·container">
 106 <div·class="highlight"><pre>
 107 ::
  
 108 ····Number·of·Observations·-·309·(Annual·1700·-·2008)
 109 ····Number·of·Variables·-·1
 110 ····Variable·name·definitions::
  
 111 ········SUNACTIVITY·-·Number·of·sunspots·for·each·year
  
 112 ····The·data·file·contains·a·&#39;YEAR&#39;·variable·that·is·not·returned·by·load.
  
 113 </pre></div></div>
 114 </div>
102 <div·class="nbinput·nblast·docutils·container">115 <div·class="nbinput·nblast·docutils·container">
103 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:116 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[5]:
104 </pre></div>117 </pre></div>
105 </div>118 </div>
106 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">dta</span>·<span·class="o">=</span>·<span·class="n">sm</span><span·class="o">.</span><span·class="n">datasets</span><span·class="o">.</span><span·class="n">sunspots</span><span·class="o">.</span><span·class="n">load_pandas</span><span·class="p">()</span><span·class="o">.</span><span·class="n">data</span>119 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">dta</span>·<span·class="o">=</span>·<span·class="n">sm</span><span·class="o">.</span><span·class="n">datasets</span><span·class="o">.</span><span·class="n">sunspots</span><span·class="o">.</span><span·class="n">load_pandas</span><span·class="p">()</span><span·class="o">.</span><span·class="n">data</span>
107 </pre></div>120 </pre></div>
108 </div>121 </div>
109 </div>122 </div>
110 <div·class="nbinput·nblast·docutils·container">123 <div·class="nbinput·nblast·docutils·container">
111 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:124 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[6]:
112 </pre></div>125 </pre></div>
113 </div>126 </div>
114 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">dta</span><span·class="o">.</span><span·class="n">index</span>·<span·class="o">=</span>·<span·class="n">pd</span><span·class="o">.</span><span·class="n">Index</span><span·class="p">(</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="s2">&quot;1700&quot;</span><span·class="p">,</span>·<span·class="s2">&quot;2008&quot;</span><span·class="p">))</span>127 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">dta</span><span·class="o">.</span><span·class="n">index</span>·<span·class="o">=</span>·<span·class="n">pd</span><span·class="o">.</span><span·class="n">Index</span><span·class="p">(</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="s2">&quot;1700&quot;</span><span·class="p">,</span>·<span·class="s2">&quot;2008&quot;</span><span·class="p">))</span>
115 <span·class="n">dta</span><span·class="o">.</span><span·class="n">index</span><span·class="o">.</span><span·class="n">freq</span>·<span·class="o">=</span>·<span·class="n">dta</span><span·class="o">.</span><span·class="n">index</span><span·class="o">.</span><span·class="n">inferred_freq</span>128 <span·class="n">dta</span><span·class="o">.</span><span·class="n">index</span><span·class="o">.</span><span·class="n">freq</span>·<span·class="o">=</span>·<span·class="n">dta</span><span·class="o">.</span><span·class="n">index</span><span·class="o">.</span><span·class="n">inferred_freq</span>
116 <span·class="k">del</span>·<span·class="n">dta</span><span·class="p">[</span><span·class="s2">&quot;YEAR&quot;</span><span·class="p">]</span>129 <span·class="k">del</span>·<span·class="n">dta</span><span·class="p">[</span><span·class="s2">&quot;YEAR&quot;</span><span·class="p">]</span>
117 </pre></div>130 </pre></div>
118 </div>131 </div>
119 </div>132 </div>
120 <div·class="nbinput·nblast·docutils·container">133 <div·class="nbinput·docutils·container">
121 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:134 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[7]:
122 </pre></div>135 </pre></div>
123 </div>136 </div>
124 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">dta</span><span·class="o">.</span><span·class="n">plot</span><span·class="p">(</span><span·class="n">figsize</span><span·class="o">=</span><span·class="p">(</span><span·class="mi">12</span><span·class="p">,</span>·<span·class="mi">8</span><span·class="p">))</span>137 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">dta</span><span·class="o">.</span><span·class="n">plot</span><span·class="p">(</span><span·class="n">figsize</span><span·class="o">=</span><span·class="p">(</span><span·class="mi">12</span><span·class="p">,</span>·<span·class="mi">8</span><span·class="p">))</span>
125 </pre></div>138 </pre></div>
126 </div>139 </div>
127 </div>140 </div>
128 <div·class="nbinput·nblast·docutils·container">141 <div·class="nboutput·docutils·container">
129 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:142 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[7]:
 143 </pre></div>
 144 </div>
 145 <div·class="output_area·docutils·container">
 146 <div·class="highlight"><pre>
 147 &lt;Axes:·&gt;
 148 </pre></div></div>
 149 </div>
 150 <div·class="nboutput·nblast·docutils·container">
 151 <div·class="prompt·empty·docutils·container">
 152 </div>
 153 <div·class="output_area·docutils·container">
 154 <img·alt="../../../_images/examples_notebooks_generated_tsa_arma_0_8_1.png"·src="../../../_images/examples_notebooks_generated_tsa_arma_0_8_1.png"·/>
 155 </div>
 156 </div>
 157 <div·class="nbinput·docutils·container">
 158 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[8]:
130 </pre></div>159 </pre></div>
131 </div>160 </div>
132 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">fig</span>·<span·class="o">=</span>·<span·class="n">plt</span><span·class="o">.</span><span·class="n">figure</span><span·class="p">(</span><span·class="n">figsize</span><span·class="o">=</span><span·class="p">(</span><span·class="mi">12</span><span·class="p">,</span>·<span·class="mi">8</span><span·class="p">))</span>161 <div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">fig</span>·<span·class="o">=</span>·<span·class="n">plt</span><span·class="o">.</span><span·class="n">figure</span><span·class="p">(</span><span·class="n">figsize</span><span·class="o">=</span><span·class="p">(</span><span·class="mi">12</span><span·class="p">,</span>·<span·class="mi">8</span><span·class="p">))</span>
133 <span·class="n">ax1</span>·<span·class="o">=</span>·<span·class="n">fig</span><span·class="o">.</span><span·class="n">add_subplot</span><span·class="p">(</span><span·class="mi">211</span><span·class="p">)</span>162 <span·class="n">ax1</span>·<span·class="o">=</span>·<span·class="n">fig</span><span·class="o">.</span><span·class="n">add_subplot</span><span·class="p">(</span><span·class="mi">211</span><span·class="p">)</span>
134 <span·class="n">fig</span>·<span·class="o">=</span>·<span·class="n">sm</span><span·class="o">.</span><span·class="n">graphics</span><span·class="o">.</span><span·class="n">tsa</span><span·class="o">.</span><span·class="n">plot_acf</span><span·class="p">(</span><span·class="n">dta</span><span·class="o">.</span><span·class="n">values</span><span·class="o">.</span><span·class="n">squeeze</span><span·class="p">(),</span>·<span·class="n">lags</span><span·class="o">=</span><span·class="mi">40</span><span·class="p">,</span>·<span·class="n">ax</span><span·class="o">=</span><span·class="n">ax1</span><span·class="p">)</span>163 <span·class="n">fig</span>·<span·class="o">=</span>·<span·class="n">sm</span><span·class="o">.</span><span·class="n">graphics</span><span·class="o">.</span><span·class="n">tsa</span><span·class="o">.</span><span·class="n">plot_acf</span><span·class="p">(</span><span·class="n">dta</span><span·class="o">.</span><span·class="n">values</span><span·class="o">.</span><span·class="n">squeeze</span><span·class="p">(),</span>·<span·class="n">lags</span><span·class="o">=</span><span·class="mi">40</span><span·class="p">,</span>·<span·class="n">ax</span><span·class="o">=</span><span·class="n">ax1</span><span·class="p">)</span>
135 <span·class="n">ax2</span>·<span·class="o">=</span>·<span·class="n">fig</span><span·class="o">.</span><span·class="n">add_subplot</span><span·class="p">(</span><span·class="mi">212</span><span·class="p">)</span>164 <span·class="n">ax2</span>·<span·class="o">=</span>·<span·class="n">fig</span><span·class="o">.</span><span·class="n">add_subplot</span><span·class="p">(</span><span·class="mi">212</span><span·class="p">)</span>
136 <span·class="n">fig</span>·<span·class="o">=</span>·<span·class="n">sm</span><span·class="o">.</span><span·class="n">graphics</span><span·class="o">.</span><span·class="n">tsa</span><span·class="o">.</span><span·class="n">plot_pacf</span><span·class="p">(</span><span·class="n">dta</span><span·class="p">,</span>·<span·class="n">lags</span><span·class="o">=</span><span·class="mi">40</span><span·class="p">,</span>·<span·class="n">ax</span><span·class="o">=</span><span·class="n">ax2</span><span·class="p">)</span>165 <span·class="n">fig</span>·<span·class="o">=</span>·<span·class="n">sm</span><span·class="o">.</span><span·class="n">graphics</span><span·class="o">.</span><span·class="n">tsa</span><span·class="o">.</span><span·class="n">plot_pacf</span><span·class="p">(</span><span·class="n">dta</span><span·class="p">,</span>·<span·class="n">lags</span><span·class="o">=</span><span·class="mi">40</span><span·class="p">,</span>·<span·class="n">ax</span><span·class="o">=</span><span·class="n">ax2</span><span·class="p">)</span>
137 </pre></div>166 </pre></div>
138 </div>167 </div>
139 </div>168 </div>
 169 <div·class="nboutput·nblast·docutils·container">
 170 <div·class="prompt·empty·docutils·container">
 171 </div>
 172 <div·class="output_area·docutils·container">
 173 <img·alt="../../../_images/examples_notebooks_generated_tsa_arma_0_9_0.png"·src="../../../_images/examples_notebooks_generated_tsa_arma_0_9_0.png"·/>
 174 </div>
 175 </div>
140 <div·class="nbinput·nblast·docutils·container">176 <div·class="nbinput·nblast·docutils·container">
141 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:177 <div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:
142 </pre></div>178 </pre></div>
143 </div>179 </div>
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3 ····*·modules·|3 ····*·modules·|
4 ····*·next·|4 ····*·next·|
5 ····*·previous·|5 ····*·previous·|
6 ····*·statsmodels_0.14.5+dfsg·»6 ····*·statsmodels_0.14.5+dfsg·»
7 ····*·Examples·»7 ····*·Examples·»
8 ····*·Autoregressive·Moving·Average·(ARMA):·Sunspots·data8 ····*·Autoregressive·Moving·Average·(ARMA):·Sunspots·data
9 ******·Autoregressive·Moving·Average·(ARMA):·Sunspots·data¶·******9 ******·Autoregressive·Moving·Average·(ARMA):·Sunspots·data¶·******
10 [·]:10 [1]:
11 %matplotlib·inline11 %matplotlib·inline
12 [·]:12 [2]:
13 import·matplotlib.pyplot·as·plt13 import·matplotlib.pyplot·as·plt
14 import·numpy·as·np14 import·numpy·as·np
15 import·pandas·as·pd15 import·pandas·as·pd
16 import·statsmodels.api·as·sm16 import·statsmodels.api·as·sm
17 from·scipy·import·stats17 from·scipy·import·stats
18 from·statsmodels.tsa.arima.model·import·ARIMA18 from·statsmodels.tsa.arima.model·import·ARIMA
19 [·]:19 [3]:
20 from·statsmodels.graphics.api·import·qqplot20 from·statsmodels.graphics.api·import·qqplot
21 *****·Sunspots·Data¶·*****21 *****·Sunspots·Data¶·*****
22 [·]:22 [4]:
23 print(sm.datasets.sunspots.NOTE)23 print(sm.datasets.sunspots.NOTE)
 24 ::
  
 25 ····Number·of·Observations·-·309·(Annual·1700·-·2008)
 26 ····Number·of·Variables·-·1
 27 ····Variable·name·definitions::
  
 28 ········SUNACTIVITY·-·Number·of·sunspots·for·each·year
  
 29 ····The·data·file·contains·a·'YEAR'·variable·that·is·not·returned·by·load.
24 [·]:30 [5]:
25 dta·=·sm.datasets.sunspots.load_pandas().data31 dta·=·sm.datasets.sunspots.load_pandas().data
26 [·]:32 [6]:
27 dta.index·=·pd.Index(sm.tsa.datetools.dates_from_range("1700",·"2008"))33 dta.index·=·pd.Index(sm.tsa.datetools.dates_from_range("1700",·"2008"))
28 dta.index.freq·=·dta.index.inferred_freq34 dta.index.freq·=·dta.index.inferred_freq
29 del·dta["YEAR"]35 del·dta["YEAR"]
30 [·]:36 [7]:
31 dta.plot(figsize=(12,·8))37 dta.plot(figsize=(12,·8))
32 [·]:38 [7]:
 39 <Axes:·>
 40 [../../../_images/examples_notebooks_generated_tsa_arma_0_8_1.png]
 41 [8]:
33 fig·=·plt.figure(figsize=(12,·8))42 fig·=·plt.figure(figsize=(12,·8))
34 ax1·=·fig.add_subplot(211)43 ax1·=·fig.add_subplot(211)
35 fig·=·sm.graphics.tsa.plot_acf(dta.values.squeeze(),·lags=40,·ax=ax1)44 fig·=·sm.graphics.tsa.plot_acf(dta.values.squeeze(),·lags=40,·ax=ax1)
36 ax2·=·fig.add_subplot(212)45 ax2·=·fig.add_subplot(212)
37 fig·=·sm.graphics.tsa.plot_pacf(dta,·lags=40,·ax=ax2)46 fig·=·sm.graphics.tsa.plot_pacf(dta,·lags=40,·ax=ax2)
 47 [../../../_images/examples_notebooks_generated_tsa_arma_0_9_0.png]
38 [·]:48 [·]:
39 [·]:49 [9]:
40 arma_mod20·=·ARIMA(dta,·order=(2,·0,·0)).fit()50 arma_mod20·=·ARIMA(dta,·order=(2,·0,·0)).fit()
41 print(arma_mod20.params)51 print(arma_mod20.params)
42 [·]:52 const······49.746198
 53 ar.L1·······1.390633
 54 ar.L2······-0.688573
 55 sigma2····274.727183
 56 dtype:·float64
 57 [10]:
43 arma_mod30·=·ARIMA(dta,·order=(3,·0,·0)).fit()58 arma_mod30·=·ARIMA(dta,·order=(3,·0,·0)).fit()
44 [·]:59 [11]:
45 print(arma_mod20.aic,·arma_mod20.bic,·arma_mod20.hqic)60 print(arma_mod20.aic,·arma_mod20.bic,·arma_mod20.hqic)
46 [·]:61 2622.637093301387·2637.5704584089776·2628.607481146633
 62 [12]:
47 print(arma_mod30.params)63 print(arma_mod30.params)
48 [·]:64 const······49.751911
 65 ar.L1·······1.300818
 66 ar.L2······-0.508102
 67 ar.L3······-0.129644
 68 sigma2····270.101139
 69 dtype:·float64
 70 [13]:
49 print(arma_mod30.aic,·arma_mod30.bic,·arma_mod30.hqic)71 print(arma_mod30.aic,·arma_mod30.bic,·arma_mod30.hqic)
 72 2619.4036292456676·2638.0703356301565·2626.866614052225
50 ····*·Does·our·model·obey·the·theory?73 ····*·Does·our·model·obey·the·theory?
51 [·]:74 [14]:
52 sm.stats.durbin_watson(arma_mod30.resid.values)75 sm.stats.durbin_watson(arma_mod30.resid.values)
53 [·]:76 [14]:
 77 np.float64(1.9564953616072376)
 78 [15]:
54 fig·=·plt.figure(figsize=(12,·8))79 fig·=·plt.figure(figsize=(12,·8))
55 ax·=·fig.add_subplot(111)80 ax·=·fig.add_subplot(111)
56 ax·=·arma_mod30.resid.plot(ax=ax)81 ax·=·arma_mod30.resid.plot(ax=ax)
57 [·]:82 [../../../_images/examples_notebooks_generated_tsa_arma_0_18_0.png]
 83 [16]:
58 resid·=·arma_mod30.resid84 resid·=·arma_mod30.resid
59 [·]:85 [17]:
60 stats.normaltest(resid)86 stats.normaltest(resid)
61 [·]:87 [17]:
 88 NormaltestResult(statistic=np.float64(49.84393220876942),·pvalue=np.float64
 89 (1.5015079731491487e-11))
 90 [18]:
62 fig·=·plt.figure(figsize=(12,·8))91 fig·=·plt.figure(figsize=(12,·8))
63 ax·=·fig.add_subplot(111)92 ax·=·fig.add_subplot(111)
64 fig·=·qqplot(resid,·line="q",·ax=ax,·fit=True)93 fig·=·qqplot(resid,·line="q",·ax=ax,·fit=True)
65 [·]:94 [../../../_images/examples_notebooks_generated_tsa_arma_0_21_0.png]
 95 [19]:
66 fig·=·plt.figure(figsize=(12,·8))96 fig·=·plt.figure(figsize=(12,·8))
67 ax1·=·fig.add_subplot(211)97 ax1·=·fig.add_subplot(211)
68 fig·=·sm.graphics.tsa.plot_acf(resid.values.squeeze(),·lags=40,·ax=ax1)98 fig·=·sm.graphics.tsa.plot_acf(resid.values.squeeze(),·lags=40,·ax=ax1)
69 ax2·=·fig.add_subplot(212)99 ax2·=·fig.add_subplot(212)
70 fig·=·sm.graphics.tsa.plot_pacf(resid,·lags=40,·ax=ax2)100 fig·=·sm.graphics.tsa.plot_pacf(resid,·lags=40,·ax=ax2)
71 [·]:101 [../../../_images/examples_notebooks_generated_tsa_arma_0_22_0.png]
 102 [20]:
72 r,·q,·p·=·sm.tsa.acf(resid.values.squeeze(),·fft=True,·qstat=True)103 r,·q,·p·=·sm.tsa.acf(resid.values.squeeze(),·fft=True,·qstat=True)
73 data·=·np.c_[np.arange(1,·25),·r[1:],·q,·p]104 data·=·np.c_[np.arange(1,·25),·r[1:],·q,·p]
74 [·]:105 [21]:
75 table·=·pd.DataFrame(data,·columns=["lag",·"AC",·"Q",·"Prob(>Q)"])106 table·=·pd.DataFrame(data,·columns=["lag",·"AC",·"Q",·"Prob(>Q)"])
76 print(table.set_index("lag"))107 print(table.set_index("lag"))
 108 ············AC··········Q······Prob(>Q)
 109 lag
 110 1.0···0.009170···0.026239··8.713184e-01
 111 2.0···0.041793···0.572982··7.508939e-01
 112 3.0··-0.001338···0.573544··9.024612e-01
 113 4.0···0.136086···6.408642··1.706385e-01
 114 5.0···0.092465···9.111351··1.047043e-01
 115 6.0···0.091947··11.792661··6.675737e-02
 116 7.0···0.068747··13.296552··6.520425e-02
 117 8.0··-0.015022··13.368601··9.978086e-02
 118 9.0···0.187590··24.641072··3.394963e-03
 119 10.0··0.213715··39.320758··2.230588e-05
 120 11.0··0.201079··52.359565··2.346490e-07
 121 12.0··0.117180··56.802478··8.580351e-08
 122 13.0·-0.014057··56.866630··1.895209e-07
 123 14.0··0.015398··56.943864··4.000370e-07
 124 15.0·-0.024969··57.147642··7.746546e-07
 125 16.0··0.080916··59.295052··6.876728e-07
 126 17.0··0.041138··59.852008··1.111674e-06
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51582 ········"7":·"py:exception"51582 ········"7":·"py:exception"
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51590 ········"00000":·6466,51590 ········"00000":·6466,
51591 ········"000000":·[197,·198,·202,·207,·209,·219,·234,·254,·257,·258,·4692,·6466],51591 ········"000000":·[197,·198,·202,·207,·209,·219,·234,·254,·257,·258,·4692,·6466],
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2.31 MB
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