build based on cc56fb9

dev/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-09-21T17:53:36","documenter_version":"1.7.0"}}
+{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-09-21T18:08:57","documenter_version":"1.7.0"}}

dev/assets/references.bib

@@ -864,4 +864,14 @@ @incollection{lindskog2003kendall
   pages={149--156},
   year={2003},
   publisher={Springer}
+}
+
+@article{blier2022stochastic,
+  title={Stochastic representation of FGM copulas using multivariate Bernoulli random variables},
+  author={Blier-Wong, Christopher and Cossette, H{\'e}l{\`e}ne and Marceau, Etienne},
+  journal={Computational Statistics \& Data Analysis},
+  volume={173},
+  pages={107506},
+  year={2022},
+  publisher={Elsevier}
 }

dev/dependence_measures/index.html

@@ -3,4 +3,4 @@
    \lambda &= \lim\limits_{u \to 1} \frac{1 - 2u - C(u,..,u)}{1- u} \in [0,1]\\
    \chi(u) &= \frac{2 \ln(1-u)}{\ln(1-2u-C(u,...,u))} -1\\
    \chi &= \lim\limits_{u \to 1} \chi(u) \in [-1,1]
- \end{align}\]</p><p>When <span>$\lambda &gt; 0$</span>, we say that there is a strong upper tail dependency, and <span>$\chi = 1$</span>. When <span>$\lambda = 0$</span>, we say that there is no strong upper tail dependency, and if furthermore <span>$\chi \neq 0$</span> we say that there is weak upper tail dependency.</p></blockquote><p>The graph of <span>$u \to \chi(u)$</span> over <span>$[\frac{1}{2},1]$</span> is an interesting tool to assess the existence and strength of the tail dependency. The same kind of tools can be constructed for the lower tail. </p><p>All these coefficients quantify the behavior of the dependence structure, generally or in the extremes, and are therefore widely used in the literature either as verification tools to assess the quality of fits, or even as parameters. Many parametric copulas families have simple surjections, injections, or even bijections between these coefficients and their parametrization, allowing matching procedures of estimation (a lot like moments matching algorithm for fitting standard random variables).</p><div class="citation noncanonical"><dl><dt>[20]</dt><dd><div>A. J. McNeil and J. Nešlehová. <em>Multivariate Archimedean Copulas, d -Monotone Functions and L1 -Norm Symmetric Distributions</em>. The Annals of Statistics <strong>37</strong>, 3059–3097 (2009).</div></dd><dt>[46]</dt><dd><div>C. Genest, J. Nešlehová and N. Ben Ghorbal. <em>Estimators Based on Kendall&#39;s Tau in Multivariate Copula Models</em>. Australian &amp; New Zealand Journal of Statistics <strong>53</strong>, 157–177 (2011).</div></dd><dt>[47]</dt><dd><div>G. A. Fredricks and R. B. Nelsen. <em>On the Relationship between Spearman&#39;s Rho and Kendall&#39;s Tau for Pairs of Continuous Random Variables</em>. Journal of Statistical Planning and Inference <strong>137</strong>, 2143–2150 (2007).</div></dd><dt>[48]</dt><dd><div>A. Derumigny and J.-D. Fermanian. <em>À propos des tests de l&#39;hypothèse simplificatrice pour les copules conditionnelles</em>. JDS2017, 6 (2017).</div></dd><dt>[49]</dt><dd><div>H.-B. Fang, K.-T. Fang and S. Kotz. <em>The meta-elliptical distributions with given marginals</em>. Journal of multivariate analysis <strong>82</strong>, 1–16 (2002).</div></dd><dt>[50]</dt><dd><div>F. Lindskog, A. McNeil and U. Schmock. <em>Kendall’s tau for elliptical distributions</em>. In: <em>Credit risk: Measurement, evaluation and management</em> (Springer, 2003); pp. 149–156.</div></dd></dl></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../empirical/generalities/">« Empirical Copulas</a><a class="docs-footer-nextpage" href="../elliptical/available_models/">Elliptical Copulas »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Saturday 21 September 2024 17:53">Saturday 21 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+ \end{align}\]</p><p>When <span>$\lambda &gt; 0$</span>, we say that there is a strong upper tail dependency, and <span>$\chi = 1$</span>. When <span>$\lambda = 0$</span>, we say that there is no strong upper tail dependency, and if furthermore <span>$\chi \neq 0$</span> we say that there is weak upper tail dependency.</p></blockquote><p>The graph of <span>$u \to \chi(u)$</span> over <span>$[\frac{1}{2},1]$</span> is an interesting tool to assess the existence and strength of the tail dependency. The same kind of tools can be constructed for the lower tail. </p><p>All these coefficients quantify the behavior of the dependence structure, generally or in the extremes, and are therefore widely used in the literature either as verification tools to assess the quality of fits, or even as parameters. Many parametric copulas families have simple surjections, injections, or even bijections between these coefficients and their parametrization, allowing matching procedures of estimation (a lot like moments matching algorithm for fitting standard random variables).</p><div class="citation noncanonical"><dl><dt>[20]</dt><dd><div>A. J. McNeil and J. Nešlehová. <em>Multivariate Archimedean Copulas, d -Monotone Functions and L1 -Norm Symmetric Distributions</em>. The Annals of Statistics <strong>37</strong>, 3059–3097 (2009).</div></dd><dt>[46]</dt><dd><div>C. Genest, J. Nešlehová and N. Ben Ghorbal. <em>Estimators Based on Kendall&#39;s Tau in Multivariate Copula Models</em>. Australian &amp; New Zealand Journal of Statistics <strong>53</strong>, 157–177 (2011).</div></dd><dt>[47]</dt><dd><div>G. A. Fredricks and R. B. Nelsen. <em>On the Relationship between Spearman&#39;s Rho and Kendall&#39;s Tau for Pairs of Continuous Random Variables</em>. Journal of Statistical Planning and Inference <strong>137</strong>, 2143–2150 (2007).</div></dd><dt>[48]</dt><dd><div>A. Derumigny and J.-D. Fermanian. <em>À propos des tests de l&#39;hypothèse simplificatrice pour les copules conditionnelles</em>. JDS2017, 6 (2017).</div></dd><dt>[49]</dt><dd><div>H.-B. Fang, K.-T. Fang and S. Kotz. <em>The meta-elliptical distributions with given marginals</em>. Journal of multivariate analysis <strong>82</strong>, 1–16 (2002).</div></dd><dt>[50]</dt><dd><div>F. Lindskog, A. McNeil and U. Schmock. <em>Kendall’s tau for elliptical distributions</em>. In: <em>Credit risk: Measurement, evaluation and management</em> (Springer, 2003); pp. 149–156.</div></dd></dl></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../empirical/generalities/">« Empirical Copulas</a><a class="docs-footer-nextpage" href="../elliptical/available_models/">Elliptical Copulas »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Saturday 21 September 2024 18:08">Saturday 21 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/elliptical/available_models/index.html

@@ -3,8 +3,8 @@
 Random.rand!(C,u) # other calling syntax for rng.
 pdf(C,u) # to get the density
 cdf(C,u) # to get the distribution function 
-Ĉ = fit(GaussianCopula,u) # to fit on the sampled data. </code></pre><p>GaussianCopulas have a special case: </p><ul><li>When <code>isdiag(Σ)</code>, the constructor returns an <code>IndependentCopula(d)</code></li></ul><p>References:</p><ul><li>[<a href="../../references/#nelsen2006">3</a>] Nelsen, Roger B. An introduction to copulas. Springer, 2006.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/Copulas.jl/blob/5aa54b51066f1e16ed2852ff9613c6d6c4d0ac16/src/EllipticalCopulas/GaussianCopula.jl#L1-L38">source</a></section></article><h2 id="TCopula"><a class="docs-heading-anchor" href="#TCopula"><code>TCopula</code></a><a id="TCopula-1"></a><a class="docs-heading-anchor-permalink" href="#TCopula" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Copulas.TCopula" href="#Copulas.TCopula"><code>Copulas.TCopula</code></a> — <span class="docstring-category">Type</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">TCopula{d,MT}</code></pre><p>Fields:</p><ul><li>df::Int - number of degree of freedom</li><li>Σ::MT - covariance matrix</li></ul><p>Constructor</p><pre><code class="nohighlight hljs">TCopula(df,Σ)</code></pre><p>The Student&#39;s <a href="https://en.wikipedia.org/wiki/Multivariate_t-distribution#Copulas_based_on_the_multivariate_t">T Copula</a> is the  copula of a <a href="https://en.wikipedia.org/wiki/Multivariate_t-distribution">Multivariate Student distribution</a>. It is constructed as : </p><p class="math-container">\[C(\mathbf{x}; \boldsymbol{n,\Sigma}) = F_{n,\Sigma}(F_{n,\Sigma,i}^{-1}(x_i),i\in 1,...d)\]</p><p>where <span>$F_{n,\Sigma}$</span> is a cdf of a multivariate student random vector with covariance matrix <span>$\Sigma$</span> and <span>$n$</span> degrees of freedom. and <code>F_{n,\Sigma,i}</code> is the ith marignal cdf. </p><p>It can be constructed in Julia via:  </p><pre><code class="language-julia hljs">C = TCopula(2,Σ)</code></pre><p>You can sample it, compute pdf and cdf, or even fit the distribution via: </p><pre><code class="language-julia hljs">u = rand(C,1000)
+Ĉ = fit(GaussianCopula,u) # to fit on the sampled data. </code></pre><p>GaussianCopulas have a special case: </p><ul><li>When <code>isdiag(Σ)</code>, the constructor returns an <code>IndependentCopula(d)</code></li></ul><p>References:</p><ul><li>[<a href="../../references/#nelsen2006">3</a>] Nelsen, Roger B. An introduction to copulas. Springer, 2006.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/Copulas.jl/blob/cc56fb91514825ba69c42ff7c813af6832fe04b9/src/EllipticalCopulas/GaussianCopula.jl#L1-L38">source</a></section></article><h2 id="TCopula"><a class="docs-heading-anchor" href="#TCopula"><code>TCopula</code></a><a id="TCopula-1"></a><a class="docs-heading-anchor-permalink" href="#TCopula" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Copulas.TCopula" href="#Copulas.TCopula"><code>Copulas.TCopula</code></a> — <span class="docstring-category">Type</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">TCopula{d,MT}</code></pre><p>Fields:</p><ul><li>df::Int - number of degree of freedom</li><li>Σ::MT - covariance matrix</li></ul><p>Constructor</p><pre><code class="nohighlight hljs">TCopula(df,Σ)</code></pre><p>The Student&#39;s <a href="https://en.wikipedia.org/wiki/Multivariate_t-distribution#Copulas_based_on_the_multivariate_t">T Copula</a> is the  copula of a <a href="https://en.wikipedia.org/wiki/Multivariate_t-distribution">Multivariate Student distribution</a>. It is constructed as : </p><p class="math-container">\[C(\mathbf{x}; \boldsymbol{n,\Sigma}) = F_{n,\Sigma}(F_{n,\Sigma,i}^{-1}(x_i),i\in 1,...d)\]</p><p>where <span>$F_{n,\Sigma}$</span> is a cdf of a multivariate student random vector with covariance matrix <span>$\Sigma$</span> and <span>$n$</span> degrees of freedom. and <code>F_{n,\Sigma,i}</code> is the ith marignal cdf. </p><p>It can be constructed in Julia via:  </p><pre><code class="language-julia hljs">C = TCopula(2,Σ)</code></pre><p>You can sample it, compute pdf and cdf, or even fit the distribution via: </p><pre><code class="language-julia hljs">u = rand(C,1000)
 Random.rand!(C,u) # other calling syntax for rng.
 pdf(C,u) # to get the density
 cdf(C,u) # to get the distribution function 
-Ĉ = fit(TCopula,u) # to fit on the sampled data. </code></pre><p>Except that currently it does not work since <code>fit(Distributions.MvTDist,data)</code> does not dispatch. </p><p>References:</p><ul><li>[<a href="../../references/#nelsen2006">3</a>] Nelsen, Roger B. An introduction to copulas. Springer, 2006.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/Copulas.jl/blob/5aa54b51066f1e16ed2852ff9613c6d6c4d0ac16/src/EllipticalCopulas/TCopula.jl#L1-L39">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../dependence_measures/">« Dependence measures</a><a class="docs-footer-nextpage" href="../../archimedean/available_models/">Archimedean Generators »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Saturday 21 September 2024 17:53">Saturday 21 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+Ĉ = fit(TCopula,u) # to fit on the sampled data. </code></pre><p>Except that currently it does not work since <code>fit(Distributions.MvTDist,data)</code> does not dispatch. </p><p>References:</p><ul><li>[<a href="../../references/#nelsen2006">3</a>] Nelsen, Roger B. An introduction to copulas. Springer, 2006.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/Copulas.jl/blob/cc56fb91514825ba69c42ff7c813af6832fe04b9/src/EllipticalCopulas/TCopula.jl#L1-L39">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../dependence_measures/">« Dependence measures</a><a class="docs-footer-nextpage" href="../../archimedean/available_models/">Archimedean Generators »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Saturday 21 September 2024 18:08">Saturday 21 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>