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Listing of first 4 GLD L-/LL-/LH-/TL-moments
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jorenham committed Dec 13, 2023
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208 changes: 160 additions & 48 deletions docs/distributions.md
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Expand Up @@ -1089,8 +1089,8 @@ general case.
Without loss of generality, Lmo uses a 3-parameter "standardized"
paremetrization, with shape parameters \( \beta,\ \delta,\ \phi \).

<img src="../gallery/wakeby.svg" width="100%"
style="height: auto; aspect-ratio: 16/9;" alt="Wakeby distribution PDF" />
See [`lmo.distributions.wakeby`][lmo.distributions.wakeby] for the
implementation.

Each of the following restrictions apply:

Expand Down Expand Up @@ -1127,6 +1127,9 @@ x(F) =
- \frac{1 - \phi}{\delta} (1 - (1 - F)^{-\delta})
\]

<img src="../gallery/wakeby.svg" width="100%"
style="height: auto; aspect-ratio: 16/9;" alt="Wakeby distribution PDF" />

/// note | Alternative parametrization
This 3-parameter Wakeby distribution is equivalent to the 5-parameter
variant that is generally used, after scaling by \( \sigma \) and shifting
Expand Down Expand Up @@ -1182,9 +1185,6 @@ L-moments can be expressed as

where \( H_n \) is a [harmonic number](#def-harmonic).

See [`lmo.distributions.wakeby`][lmo.distributions.wakeby] for the
implementation.

/// admonition | Special cases
type: info
There are several notable special cases of the Wakeby distribution:
Expand Down Expand Up @@ -1218,6 +1218,9 @@ paremetrization, with shape parameters \( \beta,\ \delta,\ \phi \), where
\( \phi \in [-1, 1] \) replaces the more commonly used shape parameters
\( \alpha \mapsto 1 + \phi \) and \( \gamma \mapsto 1 - \phi \).

The GLD is implemented as
[`lmo.distributions.genlamda`][lmo.distributions.genlambda].

As with the Wakeby distribution, the PDF and CDF of the GLD are not
analytically expressible. Instead, the GLD is defined through its PPF:

Expand Down Expand Up @@ -1290,52 +1293,161 @@ where
\end{cases}
\]

To illustrate; the first four L-moments of the GLD with
\( \beta > -1 \) and \( \delta > -1 \), are:
These equations look scarier that they actually are. To see why, take a look
at the first 4 L-moment, with 4 styles of trimming:

\[
\begin{align*}
\lmoment{1}
&= -(1 + \phi) \frac
{1}
{1 + \beta}
&&+ (1 - \phi) \frac
{1}
{1 + \delta}
\\
\lmoment{2}
&= \hphantom{-}(1 + \phi) \frac
{1}
{(1 + \beta)(2 + \beta)}
&&+ (1 - \phi) \frac
{1}
{(1 + \delta)(2 + \delta)}
\\
\lmoment{3}
&= -(1 + \phi) \frac
{1 - \beta}
{(1 + \beta)(2 + \beta)(3 + \beta)}
&&+ (1 - \phi) \frac
{(1 - \delta)}
{(1 + \delta)(2 + \delta)(3 + \delta)}
\\
\lmoment{4}
&= \hphantom{-}(1 + \phi) \frac
{(1 - \beta)(2 - \beta)}
{(1 + \beta)(2 + \beta)(3 + \beta)(4 + \beta)}
&&+ (1 - \phi) \frac
{(1 - \delta)(2 - \delta)}
{(1 + \delta)(2 + \delta)(3 + \delta)(4 + \delta)}
\end{align*}
\]
=== "L-moments"
If \( \beta > -1 \) and \( \delta > -1 \):

The GLD is implemented as
[`lmo.distributions.genlamda`][lmo.distributions.genlambda].
\[
\begin{align*}
\lmoment{1}
&= -(1 + \phi) \frac
{1}
{1 + \beta}
&&+ (1 - \phi) \frac
{1}
{1 + \delta}
\\
\lmoment{2}
&= \hphantom{-}(1 + \phi) \frac
{1}
{(1 + \beta)(2 + \beta)}
&&+ (1 - \phi) \frac
{1}
{(1 + \delta)(2 + \delta)}
\\
\lmoment{3}
&= -(1 + \phi) \frac
{1 - \beta}
{(1 + \beta)(2 + \beta)(3 + \beta)}
&&+ (1 - \phi) \frac
{1 - \delta}
{(1 + \delta)(2 + \delta)(3 + \delta)}
\\
\lmoment{4}
&= \hphantom{-}(1 + \phi) \frac
{(1 - \beta)(2 - \beta)}
{(1 + \beta)(2 + \beta)(3 + \beta)(4 + \beta)}
&&+ (1 - \phi) \frac
{(1 - \delta)(2 - \delta)}
{(1 + \delta)(2 + \delta)(3 + \delta)(4 + \delta)}
\end{align*}
\]
=== "LL-moments"
If \( \beta > -1 \) and \( \delta > -2 \):

\[
\begin{align*}
\tlmoment{0, 1}{1}
&= -(1 + \phi) \frac
{3 + \beta}
{(1 + \beta)(2 + \beta)}
&&+ (1 - \phi) \frac
{1}
{2 + \delta}
\\
\frac{1}{3}\tlmoment{0, 1}{2}
&= \hphantom{-} (1 + \phi) \frac
{1}
{(1 + \beta)(2 + \beta)(3 + \beta)}
&&+ \frac{1 - \phi}{2} \frac
{1}
{(2 + \delta)(3 + \delta)}
\\
\frac{1}{4} \tlmoment{0, 1}{3}
&= -(1 + \phi) \frac
{1 - \beta}
{(1 + \beta)(2 + \beta)(3 + \beta)(4 + \beta)}
&&+ \frac{1 - \phi}{3} \frac
{(1 - \delta)}
{(2 + \delta)(3 + \delta)(4 + \delta)}
\\
\frac{1}{5} \tlmoment{0, 1}{4}
&= \hphantom{-} (1 + \phi) \frac
{(1 - \beta)(2 - \beta)}
{(1 + \beta)(2 + \beta)(3 + \beta)(4 + \beta)(5 + \beta)}
&&+ \frac{1 - \phi}{4} \frac
{(1 - \delta)(2 - \delta)}
{(2 + \delta)(3 + \delta)(4 + \delta)(5 + \delta)}
\end{align*}
\]
=== "LH-moments"
If \( \beta > -2 \) and \( \delta > -1 \):

When \( \beta = \delta \) and \( \phi = 0 \), GLD is the
"regular" Tukey-lambda distribution with shape
\( \lambda \equiv \beta = \delta \), which is implemented as
[`scipy.stats.tukeylambda`][scipy.stats.tukeylambda].
\[
\begin{align*}
\tlmoment{1, 0}{1}
&= -(1 + \phi) \frac
{1}
{(2 + \beta)}
&&+ (1 - \phi) \frac
{3 + \beta}
{(1 + \delta)(2 + \delta)}
\\
\frac{1}{3}\tlmoment{1, 0}{2}
&= \hphantom{-} \frac{1 + \phi}{2} \frac
{1}
{(2 + \beta)(3 + \beta)}
&&+ (1 - \phi) \frac
{1}
{(1 + \delta)(2 + \delta)(3 + \delta)}
\\
\frac{1}{4} \tlmoment{1, 0}{3}
&= -\frac{1 + \phi}{3} \frac
{1 - \beta}
{(2 + \beta)(3 + \beta)(4 + \beta)}
&&+ (1 - \phi) \frac
{(1 - \delta)}
{(1 + \delta)(2 + \delta)(3 + \delta)(4 + \delta)}
\\
\frac{1}{5} \tlmoment{1, 0}{4}
&= \hphantom{-} \frac{1 + \phi}{4} \frac
{(1 - \beta)(2 - \beta)}
{(2 + \beta)(3 + \beta)(4 + \beta)(5 + \beta)}
&&+ (1 - \phi) \frac
{(1 - \delta)(2 - \delta)}
{(1 + \delta)(2 + \delta)(3 + \delta)(4 + \delta)(5 + \delta)}
\end{align*}
\]
=== "TL-moments"
If \( \beta > -2 \) and \( \delta > -2 \):

\[
\begin{align*}
\tlmoment{1}{1}
&= -(1 + \phi) \frac
{5 + \beta}
{(2 + \beta)(3 + \beta)}
&&+ (1 - \phi) \frac
{5 + \delta}
{(2 + \delta)(3 + \delta)}
\\
\frac{2}{3 \cdot 4} \tlmoment{1}{2}
&= \hphantom{-} (1 + \phi) \frac
{1}
{(2 + \beta)(3 + \beta)(4 + \beta)}
&&+ (1 - \phi) \frac
{1}
{(2 + \delta)(3 + \delta)(4 + \delta)}
\\
\frac{3}{4 \cdot 5} \tlmoment{1}{3}
&= -(1 + \phi) \frac
{(1 - \beta)}
{(2 + \beta)(3 + \beta)(4 + \beta)(5 + \beta)}
&&+ (1 - \phi) \frac
{(1 - \delta)}
{(2 + \delta)(3 + \delta)(4 + \delta)(5 + \delta)}
\\
\frac{4}{5 \cdot 6} \tlmoment{1}{4}
&= \hphantom{-} (1 + \phi) \frac
{(1 - \beta)(2 - \beta)}
{(2 + \beta)(3 + \beta)(4 + \beta)(5 + \beta)(6 + \beta)}
&&+ (1 - \phi) \frac
{(1 - \delta)(2 - \delta)}
{(2 + \delta)(3 + \delta)(4 + \delta)(5 + \delta)(6 + \delta)}
\end{align*}
\]

/// admonition | Special cases
type: info
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7 changes: 7 additions & 0 deletions mkdocs.yml
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Expand Up @@ -19,6 +19,7 @@ theme:
logo: img/logo.png
favicon: img/favicon.ico
features:
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- content.tooltips
- navigation.path
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Expand Down Expand Up @@ -97,6 +98,12 @@ markdown_extensions:
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slugify: !!python/object/apply:pymdownx.slugs.slugify
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