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<section id="priors-input-space-and-experimental-design">
<h1>Priors, input space and experimental design<a class="headerlink" href="#priors-input-space-and-experimental-design" title="Link to this heading">¶</a></h1>
<p>The surrogate models, as used in BayesValidRox, consider model formulations where at least one of the input parameters is associated with uncertainty.
This uncertainty can be described as probability distributions over possible values for the parameter.</p>
<div class="twocol docutils container">
<div class="leftside docutils container">
<p>Four classes contained in bayesvalidrox are associated with describing uncertain inputs: <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.inputs.Marginal.html#bayesvalidrox.surrogate_models.inputs.Marginal" title="bayesvalidrox.surrogate_models.inputs.Marginal"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.inputs.Marginal</span></code></a>, <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.inputs.Input.html#bayesvalidrox.surrogate_models.inputs.Input" title="bayesvalidrox.surrogate_models.inputs.Input"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.inputs.Input</span></code></a>, <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.input_space.InputSpace.html#bayesvalidrox.surrogate_models.input_space.InputSpace" title="bayesvalidrox.surrogate_models.input_space.InputSpace"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.input_space.InputSpace</span></code></a> and <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.exp_designs.ExpDesigns.html#bayesvalidrox.surrogate_models.exp_designs.ExpDesigns" title="bayesvalidrox.surrogate_models.exp_designs.ExpDesigns"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.exp_designs.ExpDesigns</span></code></a>.</p>
<p>Uncertain parameters are specified via their marginal distributions in <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.inputs.Marginal.html#bayesvalidrox.surrogate_models.inputs.Marginal" title="bayesvalidrox.surrogate_models.inputs.Marginal"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.inputs.Marginal</span></code></a> objects as either distribution types with associated parameters, or via a set of realizations.
Supported distribution types include <code class="docutils literal notranslate"><span class="pre">unif</span></code>, <code class="docutils literal notranslate"><span class="pre">norm</span></code>, <code class="docutils literal notranslate"><span class="pre">gamma</span></code>, <code class="docutils literal notranslate"><span class="pre">beta</span></code>, <code class="docutils literal notranslate"><span class="pre">lognorm</span></code>, <code class="docutils literal notranslate"><span class="pre">expon</span></code> and <code class="docutils literal notranslate"><span class="pre">weibull</span></code>.</p>
<p>All marginals contained in an <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.inputs.Input.html#bayesvalidrox.surrogate_models.inputs.Input" title="bayesvalidrox.surrogate_models.inputs.Input"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.inputs.Input</span></code></a> object should be defined in the same manner, mixing definitions via distribution type and sampels is not supported.</p>
<p>If <code class="docutils literal notranslate"><span class="pre">rosenblatt</span></code> is set as <code class="docutils literal notranslate"><span class="pre">True</span></code>, then a Rosenblatt transform will be applied for training the surrogate.</p>
</div>
<div class="rightside docutils container">
<a class="reference internal image-reference" href="_images/input_classes.png"><img alt="UML diagram for input-related classes in bayesvalidrox" src="_images/input_classes.png" style="width: 300px;" /></a>
</div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When using a polynomial-type surrogate setting <code class="docutils literal notranslate"><span class="pre">rosenblatt</span></code> to <code class="docutils literal notranslate"><span class="pre">True</span></code> results in all hermite polynomials.
If all uncertain parameters are specified as samples, the corresponding polynomials are automatically calculated following an aPCE.</p>
<p>#TODO Recheck this second point!</p>
</div>
<p>The <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.input_space.InputSpace.html#bayesvalidrox.surrogate_models.input_space.InputSpace" title="bayesvalidrox.surrogate_models.input_space.InputSpace"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.input_space.InputSpace</span></code></a> checks the validity of the given marginals and builds the input space.
This includes sampling from the distributions and applying the Rosenblatt transform or an isoprobabilistic transformation.</p>
<p>The <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.input_space.InputSpace.html#bayesvalidrox.surrogate_models.input_space.InputSpace" title="bayesvalidrox.surrogate_models.input_space.InputSpace"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.input_space.InputSpace</span></code></a> is used mainly in the surrogate model.
The class <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.exp_designs.ExpDesigns.html#bayesvalidrox.surrogate_models.exp_designs.ExpDesigns" title="bayesvalidrox.surrogate_models.exp_designs.ExpDesigns"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.exp_designs.ExpDesigns</span></code></a> additionally contains methods and attributes related to sampling from the input space for static and iterative training of the surrogate model.
Supported sampling methods include <code class="docutils literal notranslate"><span class="pre">random</span></code>, <code class="docutils literal notranslate"><span class="pre">latin-hypercube</span></code>, <code class="docutils literal notranslate"><span class="pre">sobol</span></code>, <code class="docutils literal notranslate"><span class="pre">halton</span></code>, <code class="docutils literal notranslate"><span class="pre">hammersley</span></code>, <code class="docutils literal notranslate"><span class="pre">chebyshev(FT)</span></code>, <code class="docutils literal notranslate"><span class="pre">grid(FT)</span></code> and <code class="docutils literal notranslate"><span class="pre">user</span></code> for user-defined sampling.</p>
<p>The options for iterative metamodel training are detailed in <a class="reference internal" href="surrogate_description.html"><span class="doc">Training surrogate models</span></a>.</p>
<section id="example">
<h2>Example<a class="headerlink" href="#example" title="Link to this heading">¶</a></h2>
<p>In practice, only the classes <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.inputs.Input.html#bayesvalidrox.surrogate_models.inputs.Input" title="bayesvalidrox.surrogate_models.inputs.Input"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.inputs.Input</span></code></a> and <a class="reference internal" href="_autosummary/bayesvalidrox.surrogate_models.exp_designs.ExpDesigns.html#bayesvalidrox.surrogate_models.exp_designs.ExpDesigns" title="bayesvalidrox.surrogate_models.exp_designs.ExpDesigns"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.surrogate_models.exp_designs.ExpDesigns</span></code></a> are directly used.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">bayesvalidrox</span> <span class="kn">import</span> <span class="n">Input</span><span class="p">,</span> <span class="n">ExpDesign</span>
</pre></div>
</div>
<p>Marginals of uncertain parameters can be directly added and defined on an <code class="docutils literal notranslate"><span class="pre">Input</span></code> object.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">Inputs</span> <span class="o">=</span> <span class="n">Input</span><span class="p">()</span>
</pre></div>
</div>
<p>If they are defined via distribution types, the <code class="docutils literal notranslate"><span class="pre">name</span></code>, <code class="docutils literal notranslate"><span class="pre">dist_type</span></code> and <code class="docutils literal notranslate"><span class="pre">parameters</span></code> for the distribution should be set.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">Inputs</span><span class="o">.</span><span class="n">add_marginals</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">Inputs</span><span class="o">.</span><span class="n">Marginals</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="s1">'$X$'</span>
<span class="gp">>>> </span><span class="n">Inputs</span><span class="o">.</span><span class="n">Marginals</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">dist_type</span> <span class="o">=</span> <span class="s1">'unif'</span>
<span class="gp">>>> </span><span class="n">Inputs</span><span class="o">.</span><span class="n">Marginals</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">parameters</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
</pre></div>
</div>
<p>If they are given via data, only <code class="docutils literal notranslate"><span class="pre">name</span></code> and <code class="docutils literal notranslate"><span class="pre">input_data</span></code> are relevant.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">inputParams</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">Inputs</span><span class="o">.</span><span class="n">add_marginals</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">Inputs</span><span class="o">.</span><span class="n">Marginals</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="s1">'$X$'</span>
<span class="gp">>>> </span><span class="n">Inputs</span><span class="o">.</span><span class="n">Marginals</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">input_data</span> <span class="o">=</span> <span class="n">inputParams</span>
</pre></div>
</div>
<p>An experimental design can be constructed based on these inputs.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">ExpDesign</span> <span class="o">=</span> <span class="n">ExpDesigns</span><span class="p">(</span><span class="n">Inputs</span><span class="p">)</span>
</pre></div>
</div>
<p>Samples of the marginals can be created by specifying a sampling method and generating the wanted number of samples.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">ExpDesign</span><span class="o">.</span><span class="n">sampling_method</span> <span class="o">=</span> <span class="s1">'latin_hypercube'</span>
<span class="gp">>>> </span><span class="n">samples</span> <span class="o">=</span> <span class="n">ExpDesign</span><span class="o">.</span><span class="n">generate_samples</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
</pre></div>
</div>
<p>The generated samples can be visualized against their marginal distributions.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">ExpDesign</span><span class="o">.</span><span class="n">plot_samples</span><span class="p">(</span><span class="n">samples</span><span class="p">)</span>
</pre></div>
</div>
<p>The results will be saved in the folder <code class="docutils literal notranslate"><span class="pre">Outputs_Priors</span></code>.</p>
<a class="reference internal image-reference" href="_images/prior_$X$.png"><img alt="Generated samples against their marginal distribution" src="_images/prior_%24X%24.png" style="width: 400px;" /></a>
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