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187 changes: 183 additions & 4 deletions api/crps/index.html
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<title>Continuous Ranked Probability Score - Scoringrules</title>



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</span>
</a>

</li>

<li class="md-nav__item">
<a href="#scoringrules.crps_mixnorm" class="md-nav__link">
<span class="md-ellipsis">
crps_mixnorm
</span>
</a>

</li>

<li class="md-nav__item">
Expand Down Expand Up @@ -1168,6 +1177,15 @@
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#scoringrules.crps_mixnorm" class="md-nav__link">
<span class="md-ellipsis">
crps_mixnorm
</span>
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</li>

<li class="md-nav__item">
Expand Down Expand Up @@ -4731,6 +4749,167 @@ <h3 id="scoringrules.crps_lognormal" class="doc doc-heading">
<div class="doc doc-object doc-function">


<h3 id="scoringrules.crps_mixnorm" class="doc doc-heading">
<span class="doc doc-object-name doc-function-name">scoringrules.crps_mixnorm</span>


</h3>
<div class="language-python doc-signature highlight"><pre><span></span><code><span id="__span-0-1"><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nf">crps_mixnorm</span><span class="p">(</span>
</span><span id="__span-0-2"><a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">observation</span><span class="p">:</span> <span class="n">ArrayLike</span><span class="p">,</span>
</span><span id="__span-0-3"><a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="n">m</span><span class="p">:</span> <span class="n">ArrayLike</span><span class="p">,</span>
</span><span id="__span-0-4"><a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="n">s</span><span class="p">:</span> <span class="n">ArrayLike</span><span class="p">,</span>
</span><span id="__span-0-5"><a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="o">/</span><span class="p">,</span>
</span><span id="__span-0-6"><a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="n">w</span><span class="p">:</span> <span class="n">ArrayLike</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
</span><span id="__span-0-7"><a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="n">axis</span><span class="p">:</span> <span class="n">ArrayLike</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span>
</span><span id="__span-0-8"><a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="o">*</span><span class="p">,</span>
</span><span id="__span-0-9"><a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="n">backend</span><span class="p">:</span> <span class="n">Backend</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
</span><span id="__span-0-10"><a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">ArrayLike</span>
</span></code></pre></div>

<div class="doc doc-contents first">

<p>Compute the closed form of the CRPS for a mixture of normal distributions.</p>
<p>It is based on the following formulation from
<a href="https://doi.org/10.1256/qj.05.235">Grimit et al. (2006)</a>:</p>
<div class="arithmatex">\[ \mathrm{CRPS}(F, y) = \sum_{i=1}^{M} w_{i} A(y - \mu_{i}, \sigma_{i}^{2}) - \frac{1}{2} \sum_{i=1}^{M} \sum_{j=1}^{M} w_{i} w_{j} A(\mu_{i} - \mu_{j}, \sigma_{i}^{2} + \sigma_{j}^{2}), \]</div>
<p>where <span class="arithmatex">\(F(x) = \sum_{i=1}^{M} w_{i} \Phi \left( \frac{x - \mu_{i}}{\sigma_{i}} \right)\)</span>,
and <span class="arithmatex">\(A(\mu, \sigma^{2}) = \mu (2 \Phi(\frac{\mu}{\sigma}) - 1) + 2\sigma \phi(\frac{\mu}{\sigma}).\)</span></p>


<p><span class="doc-section-title">Parameters:</span></p>
<table>
<thead>
<tr>
<th>Name</th>
<th>Type</th>
<th>Description</th>
<th>Default</th>
</tr>
</thead>
<tbody>
<tr class="doc-section-item">
<td><code>observation</code></td>
<td>
<code><span title="scoringrules.core.typing.ArrayLike">ArrayLike</span></code>
</td>
<td>
<div class="doc-md-description">
<p>The observed values.</p>
</div>
</td>
<td>
<em>required</em>
</td>
</tr>
<tr class="doc-section-item">
<td><code>m</code></td>
<td>
<code><span title="scoringrules.core.typing.ArrayLike">ArrayLike</span></code>
</td>
<td>
<div class="doc-md-description">
<p>Means of the component normal distributions.</p>
</div>
</td>
<td>
<em>required</em>
</td>
</tr>
<tr class="doc-section-item">
<td><code>s</code></td>
<td>
<code><span title="scoringrules.core.typing.ArrayLike">ArrayLike</span></code>
</td>
<td>
<div class="doc-md-description">
<p>Standard deviations of the component normal distributions.</p>
</div>
</td>
<td>
<em>required</em>
</td>
</tr>
<tr class="doc-section-item">
<td><code>w</code></td>
<td>
<code><span title="scoringrules.core.typing.ArrayLike">ArrayLike</span></code>
</td>
<td>
<div class="doc-md-description">
<p>Non-negative weights assigned to each component.</p>
</div>
</td>
<td>
<code>None</code>
</td>
</tr>
<tr class="doc-section-item">
<td><code>axis</code></td>
<td>
<code><span title="scoringrules.core.typing.ArrayLike">ArrayLike</span></code>
</td>
<td>
<div class="doc-md-description">
<p>The axis corresponding to the mixture components. Default is the last axis.</p>
</div>
</td>
<td>
<code>-1</code>
</td>
</tr>
<tr class="doc-section-item">
<td><code>backend</code></td>
<td>
<code><span title="scoringrules.core.typing.Backend">Backend</span></code>
</td>
<td>
<div class="doc-md-description">
<p>The name of the backend used for computations. Defaults to 'numba' if available, else 'numpy'.</p>
</div>
</td>
<td>
<code>None</code>
</td>
</tr>
</tbody>
</table>


<p><span class="doc-section-title">Returns:</span></p>
<table>
<thead>
<tr>
<th>Name</th> <th>Type</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr class="doc-section-item">
<td><code>score</code></td> <td>
<code><span title="scoringrules.core.typing.ArrayLike">ArrayLike</span></code>
</td>
<td>
<div class="doc-md-description">
<p>The CRPS between MixNormal(m, s) and obs.</p>
</div>
</td>
</tr>
</tbody>
</table>


<p><span class="doc-section-title">Examples:</span></p>
<div class="language-pycon highlight"><pre><span></span><code><span id="__span-0-1"><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">scoringrules</span> <span class="k">as</span> <span class="nn">sr</span>
</span><span id="__span-0-2"><a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="gp">&gt;&gt;&gt; </span><span class="n">sr</span><span class="o">.</span><span class="n">crps_mixnormal</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
</span></code></pre></div>

</div>

</div>

<div class="doc doc-object doc-function">


<h3 id="scoringrules.crps_negbinom" class="doc doc-heading">
<span class="doc doc-object-name doc-function-name">scoringrules.crps_negbinom</span>

Expand Down Expand Up @@ -6474,10 +6653,10 @@ <h3 id="scoringrules.crps_quantile" class="doc doc-heading">
</div>


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Expand Down Expand Up @@ -726,8 +726,7 @@ <h3 id="scoringrules.energy_score" class="doc doc-heading">
<p>The Energy Score is a multivariate scoring rule expressed as</p>
<div class="arithmatex">\[\text{ES}(F_{ens}, \mathbf{y})= \frac{1}{M} \sum_{m=1}^{M} \| \mathbf{x}_{m} -
\mathbf{y} \| - \frac{1}{2 M^{2}} \sum_{m=1}^{M} \sum_{j=1}^{M} \| \mathbf{x}_{m} - \mathbf{x}_{j} \| \]</div>
<p>where <span class="arithmatex">\(\mathbf{X}\)</span> and <span class="arithmatex">\(\mathbf{X'}\)</span> are independent samples from <span class="arithmatex">\(F\)</span>
and <span class="arithmatex">\(||\cdot||\)</span> is the euclidean norm over the input dimensions (the variables).</p>
<p>where <span class="arithmatex">\(||\cdot||\)</span> is the euclidean norm over the input dimensions (the variables).</p>


<p><span class="doc-section-title">Parameters:</span></p>
Expand Down Expand Up @@ -1218,7 +1217,7 @@ <h3 id="scoringrules.vrenergy_score" class="doc doc-heading">
<div class="doc doc-contents first">

<p>Compute the Vertically Re-scaled Energy Score (vrES) for a finite multivariate ensemble.</p>
<p>Computation is performed using the ensemble representation of the twES in
<p>Computation is performed using the ensemble representation of the vrES in
<a href="https://arxiv.org/abs/2202.12732">Allen et al. (2022)</a>:</p>
<div class="arithmatex">\[
\begin{split}
Expand All @@ -1227,7 +1226,7 @@ <h3 id="scoringrules.vrenergy_score" class="doc doc-heading">
\end{split}
\]</div>
<p>where <span class="arithmatex">\(F_{ens}\)</span> is the ensemble forecast <span class="arithmatex">\(\mathbf{x}_{1}, \dots, \mathbf{x}_{M}\)</span> with
<span class="arithmatex">\(M\)</span> members, and <span class="arithmatex">\(v\)</span> is the chaining function used to target particular outcomes.</p>
<span class="arithmatex">\(M\)</span> members, and <span class="arithmatex">\(w\)</span> is the weight function used to target particular outcomes.</p>


<p><span class="doc-section-title">Parameters:</span></p>
Expand Down Expand Up @@ -1421,10 +1420,10 @@ <h3 id="scoringrules.vrenergy_score" class="doc doc-heading">
</div>


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