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jorenham committed Nov 24, 2023
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2 changes: 1 addition & 1 deletion 404.html
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72 changes: 44 additions & 28 deletions distributions/index.html
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Expand Down Expand Up @@ -1188,12 +1188,14 @@ <h3 id="gev">GEV</h3>
distribution functions:</p>
<div class="arithmatex">\[
\begin{align*}
F(x) &amp;= e^{-\coxbox{-x}{\alpha}} \\
x(F) &amp;= -\boxcox{-\ln(F)}{\alpha}
F(x) &amp;= e^{-\qexp{1 - \alpha}{-x}} \\
x(F) &amp;= -\qlog{1 - \alpha}{-\ln(F)}
\end{align*}
\]</div>
<p>Here, <span class="arithmatex">\( \boxcox{\cdot}{\alpha} \)</span> is the
<a href="#def-boxcox">Box-Cox transformation function</a>.</p>
<p>Here, <span class="arithmatex">\( \qexp{q}{y} \)</span> and <span class="arithmatex">\( \qlog{q}{y} \)</span> are the
<a href="https://wikipedia.org/wiki/Tsallis_statistics">Tsallis</a>
<a href="#def-qexp"><span class="arithmatex">\( q \)</span>-exponential</a> and the <a href="#def-qlog"><span class="arithmatex">\( q \)</span>-logarithm</a>,
respectively.</p>
<p>An alternative parametrization is sometimes used, e.g. on
<a href="https://wikipedia.org/wiki/Generalized_extreme_value_distribution">Wikipedia</a>,
where <span class="arithmatex">\( \xi = -\alpha \)</span>.
Expand All @@ -1220,6 +1222,10 @@ <h3 id="gev">GEV</h3>
\label{eq:lr_gev}
\end{equation}
\]</div>
<p>Note that the GEV is effectively a reparametrized
<span class="arithmatex">\( q \)</span>-<a href="https://wikipedia.org/wiki/Gumbel_distribution">Gumbel</a>
<a href="https://wikipedia.org/wiki/Tsallis_distribution">Tsallis distribution</a>, with
<span class="arithmatex">\( q = 1 - \alpha \)</span>.</p>
<h3 id="glo">GLO</h3>
<p>The <em>generalized logistic distribution</em> (GLO), also known as the <a href="https://wikipedia.org/wiki/Shifted_log-logistic_distribution">shifted
log-logistic distribution
Expand All @@ -1228,8 +1234,8 @@ <h3 id="glo">GLO</h3>
distribution functions:</p>
<div class="arithmatex">\[
\begin{align*}
F(x) &amp;= \frac{1}{1 + \coxbox{x}{\alpha}} \\
x(F) &amp;= -\boxcox{\frac{1 - F}{F}}{\alpha}
F(x) &amp;= \frac{1}{1 + \qexp{1 - \alpha}{x}} \\
x(F) &amp;= -\qlog{1 - \alpha}{\frac{1 - F}{F}}
\end{align*}
\]</div>
<p>For <span class="arithmatex">\( -1 &lt; \alpha &lt; 1 \)</span>, the general trimmed L-moments of the GLO are:</p>
Expand Down Expand Up @@ -1259,15 +1265,19 @@ <h3 id="glo">GLO</h3>
<p>The corresponding <code>scipy.stats</code> implementation is
<a class="autorefs autorefs-external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kappa4.html#scipy.stats.kappa4"><code>kappa4</code></a>, with <code>h = -1</code> and <code>k</code> set to <span class="arithmatex">\( \alpha \)</span>;
<strong>not</strong> <a class="autorefs autorefs-external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.genlogistic.html#scipy.stats.genlogistic"><code>genlogistic</code></a>.</p>
<p>Note that the GLO is effectively a reparametrized
<span class="arithmatex">\( q \)</span>-<a href="https://wikipedia.org/wiki/Logistic_distribution">logistic</a>
<a href="https://wikipedia.org/wiki/Tsallis_distribution">Tsallis distribution</a>, with
<span class="arithmatex">\( q = 1 - \alpha \)</span>.</p>
<h3 id="gpd">GPD</h3>
<p>The <a href="https://wikipedia.org/wiki/Generalized_Pareto_distribution"><em>generalized Pareto distribution</em>
</a> (GPD), with
shape parameter <span class="arithmatex">\( \alpha \in \mathbb{R} \)</span>, has for <span class="arithmatex">\( x \ge 0 \)</span> the
distribution functions:</p>
<div class="arithmatex">\[
\begin{align*}
F(x) &amp;= 1 - 1 / \coxbox{x}{\alpha} \\
x(F) &amp;= \boxcox{1 / (1 - F)}{\alpha}
F(x) &amp;= 1 - 1 / \qexp{1 - \alpha}{x} \\
x(F) &amp;= \qlog{1 - \alpha}{1 / (1 - F)}
\end{align*}
\]</div>
<p>Note that this distribution is standard uniform if <span class="arithmatex">\( \alpha = 1 \)</span>, and
Expand Down Expand Up @@ -1312,6 +1322,10 @@ <h3 id="gpd">GPD</h3>
\]</div>
<p>See <a class="autorefs autorefs-external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.genpareto.html#scipy.stats.genpareto"><code>scipy.stats.genpareto</code></a> for the implementation of
the GPD.</p>
<p>Note that the GPD is a reparametrized <a href="https://wikipedia.org/wiki/Q-exponential_distribution"><span class="arithmatex">\( q \)</span>-exponential distribution
</a>, where
<span class="arithmatex">\( q = (2 \alpha + 1) / (\alpha + 1) \)</span> and <span class="arithmatex">\( \lambda = 1 / (2 - q) \)</span> s.t.
<span class="arithmatex">\( \alpha \neq -1 \)</span> and <span class="arithmatex">\( q &lt; 2 \)</span>.</p>
<h3 id="pareto-type-iv">Pareto Type IV</h3>
<p>The <a href="https://wikipedia.org/wiki/Pareto_distribution"><em>Pareto Type IV</em></a> has two
shape parameters <span class="arithmatex">\( \alpha \in \mathbb{R} \)</span> and
Expand Down Expand Up @@ -1498,9 +1512,7 @@ <h3 id="generalized-lambda">Generalized Lambda</h3>
<p>Like the Wakeby distribution, the generalized lambda has no closed-form PDF
or CDF. Instead, it is defined through its PPF:</p>
<div class="arithmatex">\[
x(F)
= \alpha \boxcox{F}{\beta}
- \gamma \boxcox{1 - F}{\delta}
x(F) = \alpha \qlog{1 - \beta}{F} - \gamma \qlog{1 - \delta}{1 - F}
\]</div>
<p>Although its central product moments have no closed-form expression, when
<span class="arithmatex">\( \beta &gt; -1 \)</span> and <span class="arithmatex">\( \delta &gt; -1 \)</span>, the general trimmed L-moments can be
Expand Down Expand Up @@ -1771,22 +1783,24 @@ <h2 id="constants-and-special-functions">Constants and special functions</h2>
</a>
</td>
</tr>
<tr id="def-boxcox" class="row-double-top">
<tr id="def-qexp" class="row-double-top">
<td>
<a
href="https://wikipedia.org/wiki/Power_transform#Box%E2%80%93Cox_transformation"
href="https://wikipedia.org/wiki/Tsallis_statistics"
target="_blank"
title="Box–Cox transformation - Power transform - Wikipedia"
title="Tsallis statistics"
>
Box–Cox transform
Tsallis \( q \)-exponential
</a>
</td>
<td>\[ \boxcox{y}{\lambda} \]</td>
<td>\[ \qexp{q}{x} \]</td>
<td>
\[
= \begin{cases}
(y^\lambda - 1) / \lambda & \text{if } \lambda \neq 0 \\
\ln(y) & \text{if } \lambda = 0
\displaystyle e^x
& \text{if } q = 1 \\
\displaystyle \sqrt[1 - q]{1 + (1 - q) \ x}
& \text{otherwise}
\end{cases}
\]
</td>
Expand All @@ -1795,26 +1809,28 @@ <h2 id="constants-and-special-functions">Constants and special functions</h2>
href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.boxcox.html"
target="_blank"
>
<code>scipy.special.boxcox</code>
<code>scipy.special.inv_boxcox(x, 1 - q)</code>
</a>
</td>
</tr>
<tr id="def-coxbox">
<tr id="def-qlog">
<td>
<a
href="https://wikipedia.org/wiki/Power_transform#Box%E2%80%93Cox_transformation"
href="https://wikipedia.org/wiki/Tsallis_statistics"
target="_blank"
title="Box–Cox transformation - Power transform - Wikipedia"
title="Tsallis statistics"
>
Inverse Box–Cox transform
Tsallis \( q \)-logarithm
</a>
</td>
<td>\[ \coxbox{x}{\lambda} \]</td>
<td>\[ \qlog{q}{x} \]</td>
<td>
\[
= \begin{cases}
(\lambda x + 1)^{1 / \lambda} & \text{if } \lambda \neq 0 \\
e^{x} & \text{if } \lambda = 0
\displaystyle \ln(x)
& \text{if } q = 1 \\
\displaystyle \frac{x^{1 - q} - 1}{1 - q}
& \text{otherwise} \\
\end{cases}
\]
</td>
Expand All @@ -1823,7 +1839,7 @@ <h2 id="constants-and-special-functions">Constants and special functions</h2>
href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.boxcox.html"
target="_blank"
>
<code>scipy.special.inv_boxcox</code>
<code>scipy.special.boxcox(x, 1 - q)</code>
</a>
</td>
</tr>
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<link rel="icon" href="img/favicon.ico">
<meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.10">
<meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.11">



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6 changes: 3 additions & 3 deletions scripts/mathjax.js
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Expand Up @@ -10,9 +10,9 @@ window.MathJax = {
E: "\\mathop{\\mathbb{E}}",
// Beta function
B: "\\mathop{\\mathrm{B}}",
// Box-Cox transformation, and its inverse
boxcox: ["\\rm \\mathcal{P}_{#2}\\left( #1 \\right)", 2],
coxbox: ["\\rm \\mathcal{P}^{-1}_{#2}\\left( #1 \\right)", 2],
// Tsallis' q-log and q-exp
qexp: ["e_{#1}^{#2}", 2],
qlog: ["\\ln_{#1}\\left({#2}\\right)", 2],
// Falling and rising factorials
ffact: ["\\left( #1 \\right)^{-}_{#2}", 2],
rfact: ["\\left( #1 \\right)^{+}_{#2}", 2],
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8 changes: 4 additions & 4 deletions sitemap.xml
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<urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
<url>
<loc>https://jorenham.github.io/Lmo/</loc>
<lastmod>2023-11-21</lastmod>
<lastmod>2023-11-24</lastmod>
<changefreq>daily</changefreq>
</url>
<url>
<loc>https://jorenham.github.io/Lmo/api/</loc>
<lastmod>2023-11-21</lastmod>
<lastmod>2023-11-24</lastmod>
<changefreq>daily</changefreq>
</url>
<url>
<loc>https://jorenham.github.io/Lmo/contributing/</loc>
<lastmod>2023-11-21</lastmod>
<lastmod>2023-11-24</lastmod>
<changefreq>daily</changefreq>
</url>
<url>
<loc>https://jorenham.github.io/Lmo/distributions/</loc>
<lastmod>2023-11-21</lastmod>
<lastmod>2023-11-24</lastmod>
<changefreq>daily</changefreq>
</url>
</urlset>
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