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<section id="examples">
<h1>Examples<a class="headerlink" href="#examples" title="Link to this heading">#</a></h1>
<p>This is a first example of a real application of the differentiable programming paradigm in the context of mapping the brain’s functional connectivity. This example replicates the third experiment from our first preprint on this subject (“Covariance modelling”). The goal of this experiment is to create a simple, differentiable map of the dynamics of community structure in the brain’s functional connectome.</p>
<p>The connectome is a graph of the brain’s functional connectivity, where each node represents a brain region and each edge represents the strength of the connection between two regions. The community structure of the connectome is a partition of the nodes into groups that are more strongly connected to each other than to nodes in other groups. The dynamics of community structure is the evolution of the community structure over time. Here, we use a simple operationalisation of the dynamics that tracks only whether a community is present or absent at each time point.</p>
<p>This tutorial steps through the code for this experiment.</p>
<section id="loading-the-dataset">
<h2>Loading the dataset<a class="headerlink" href="#loading-the-dataset" title="Link to this heading">#</a></h2>
<p>In this tutorial, we’ll perform the experiment using a subset of the Midnight Scan Club (MSC) dataset. We select a subset of the dataset to expedite the processing steps, and because we can find a rich community structure even using only this subset. The MSC dataset is a collection of fMRI scans of 10 subjects, each performing a number of in-scanner tasks across 10 scanning sessions. Here, we use the first 3 resting-state scans from each subject.</p>
<p>The MSC dataset is available from the OpenNeuro website. Below, we’re using a utility function to retrieve a version of the dataset that has already been preprocessed. The preprocessing includes, among other standard steps, dimensionality reduction using a 400-region parcellation (brain atlas) and denoising using a 36-parameter model of motion estimates and nuisance signals.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pathlib</span>
<span class="kn">import</span> <span class="nn">jax.numpy</span> <span class="k">as</span> <span class="nn">jnp</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">from</span> <span class="nn">hypercoil.engine.axisutil</span> <span class="kn">import</span> <span class="n">extend_to_max_size</span>
<span class="kn">from</span> <span class="nn">hypercoil.neuro.data_msc</span> <span class="kn">import</span> <span class="n">minimal_msc_download</span>
<span class="n">dataset_root</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">minimal_msc_download</span><span class="p">()</span><span class="si">}</span><span class="s2">/data/ts/"</span>
<span class="n">paths</span> <span class="o">=</span> <span class="n">pathlib</span><span class="o">.</span><span class="n">Path</span><span class="p">(</span><span class="n">dataset_root</span><span class="p">)</span><span class="o">.</span><span class="n">glob</span><span class="p">(</span><span class="s2">"*ses-func0[1-3]*task-rest*ts.1D"</span><span class="p">)</span>
<span class="n">time_series</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s2">" "</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span> <span class="k">for</span> <span class="n">path</span> <span class="ow">in</span> <span class="n">paths</span><span class="p">)</span>
<span class="n">time_series</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">time_series</span><span class="p">)</span>
<span class="n">time_series</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="n">extend_to_max_size</span><span class="p">(</span><span class="n">time_series</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">))</span>
<span class="k">assert</span> <span class="n">time_series</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="mi">400</span><span class="p">,</span> <span class="mi">814</span><span class="p">)</span>
</pre></div>
</div>
</section>
<section id="defining-the-model">
<h2>Defining the model<a class="headerlink" href="#defining-the-model" title="Link to this heading">#</a></h2>
<p>Next, let’s define the model that will be learning the community structure. We need to define the model’s parameters, the model’s forward pass, and the model’s loss function. We begin here with the parameters. The model has two parameter tensors: one for the community structure and one for the dynamics of community structure. The community structure is a matrix of shape (n_nodes, n_communities), where each row represents a node and each column represents a community. The dynamics of community structure is a binary matrix of shape (n_time_points, n_communities), where each row represents a time point and each column represents a community.</p>
<p>We want to learn a community structure that is common to all subjects and all scans in the dataset. However, we also want the dynamics to be specific to each scan. To achieve this, we define the community structure as a parameter tensor that is shared across all scans, and we define the dynamics as a parameter tensor that is specific to each scan. We can achieve this by defining the community structure as a parameter tensor of shape (n_nodes, n_communities), and the dynamics as a parameter tensor of shape (n_scans, n_time_points, n_communities). The first dimension of the dynamics tensor will be used to index the dynamics of each scan.</p>
<p>We also want to impose some constraints on the values that the parameter tensors can take. For the community structure tensor, we’s ideally want each node to belong to exactly one community. But this would give us a combinatorial optimisation problem instead of a differentiable one, so we’ll relax this constraint to instead allow each node’s community assignment to be a categorical probability distribution over communities. We can achieve this by projecting the community structure tensor onto the probability simplex, which is the set of vectors whose elements are nonnegative and sum to 1.0. Behind the scenes, <code class="docutils literal notranslate"><span class="pre">hypercoil</span></code>’s <code class="docutils literal notranslate"><span class="pre">Probability</span> <span class="pre">SimplexParameter</span></code> implements this constraint using a softmax mapping.</p>
<p>For the dynamics tensor, we’d ideally want to impose the constraint that each community is either present or absent at each time point – but again, we’ll relax this constraint for the sake of differentiability. We’ll instead allow the presence of each community at each time point to vary continuously in (0, 1) using a <code class="docutils literal notranslate"><span class="pre">MappedLogits</span></code> parameter. Later, we’ll introduce some regularisations that encourage the dynamics to be close to binary.</p>
<p>The last model parameter is a scalar that sets the resolution of the community detection algorithm. This scalar, gamma, promotes discovery of more, smaller communities as it is increased. We won’t learn this parameter, but we’ll set it to a reasonable value for the purposes of this tutorial. In practice, we’ve found that a “default” value of 1 for gamma results in an unbalanced community structure that is dominated by a few large communities. We’ve found that a value of 5 for gamma results in a more balanced community structure.</p>
<p>With that said, let’s implement the model:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">jax</span>
<span class="kn">import</span> <span class="nn">equinox</span> <span class="k">as</span> <span class="nn">eqx</span>
<span class="kn">from</span> <span class="nn">hypercoil.engine</span> <span class="kn">import</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">PyTree</span>
<span class="kn">from</span> <span class="nn">hypercoil.init.mapparam</span> <span class="kn">import</span> <span class="p">(</span>
<span class="n">MappedLogits</span><span class="p">,</span>
<span class="n">ProbabilitySimplexParameter</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">class</span> <span class="nc">DynamicCommunityModel</span><span class="p">(</span><span class="n">eqx</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="n">n_nodes</span><span class="p">:</span> <span class="nb">int</span>
<span class="n">n_communities</span><span class="p">:</span> <span class="nb">int</span>
<span class="n">n_scans</span><span class="p">:</span> <span class="nb">int</span>
<span class="n">n_time_points</span><span class="p">:</span> <span class="nb">int</span>
<span class="n">gamma</span><span class="p">:</span> <span class="nb">float</span>
<span class="n">affiliation</span><span class="p">:</span> <span class="n">Tensor</span>
<span class="n">dynamics</span><span class="p">:</span> <span class="n">Tensor</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span>
<span class="bp">self</span><span class="p">,</span>
<span class="n">n_nodes</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
<span class="n">n_scans</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
<span class="n">n_communities</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
<span class="n">n_time_points</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
<span class="n">gamma</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">,</span>
<span class="n">init_scale_affiliation</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.01</span><span class="p">,</span>
<span class="n">init_scale_dynamics</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.001</span><span class="p">,</span>
<span class="o">*</span><span class="p">,</span>
<span class="n">key</span><span class="p">:</span> <span class="s1">'jax.random.PRNGKey'</span><span class="p">,</span>
<span class="p">):</span>
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_nodes</span> <span class="o">=</span> <span class="n">n_nodes</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_communities</span> <span class="o">=</span> <span class="n">n_communities</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_scans</span> <span class="o">=</span> <span class="n">n_scans</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_time_points</span> <span class="o">=</span> <span class="n">n_time_points</span>
<span class="bp">self</span><span class="o">.</span><span class="n">gamma</span> <span class="o">=</span> <span class="n">gamma</span>
<span class="bp">self</span><span class="o">.</span><span class="n">affiliation</span> <span class="o">=</span> <span class="n">init_scale_affiliation</span> <span class="o">*</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span>
<span class="n">key</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="n">n_communities</span><span class="p">))</span> <span class="o">+</span> <span class="mf">1.0</span>
<span class="bp">self</span><span class="o">.</span><span class="n">dynamics</span> <span class="o">=</span> <span class="n">init_scale_dynamics</span> <span class="o">*</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span>
<span class="n">key</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">n_scans</span><span class="p">,</span> <span class="n">n_communities</span><span class="p">,</span> <span class="n">n_time_points</span><span class="p">))</span> <span class="o">+</span> <span class="mf">0.5</span>
<span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">time_series</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-></span> <span class="n">Tensor</span><span class="p">:</span>
<span class="k">return</span> <span class="n">model_forward</span><span class="p">(</span>
<span class="n">time_series</span><span class="p">,</span>
<span class="bp">self</span><span class="o">.</span><span class="n">affiliation</span><span class="p">,</span>
<span class="bp">self</span><span class="o">.</span><span class="n">dynamics</span><span class="p">,</span>
<span class="bp">self</span><span class="o">.</span><span class="n">gamma</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">def</span> <span class="nf">parameterise_model</span><span class="p">(</span><span class="n">model</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">ProbabilitySimplexParameter</span><span class="o">.</span><span class="n">map</span><span class="p">(</span>
<span class="n">model</span><span class="p">,</span> <span class="n">where</span><span class="o">=</span><span class="s2">"affiliation"</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">MappedLogits</span><span class="o">.</span><span class="n">map</span><span class="p">(</span>
<span class="n">model</span><span class="p">,</span> <span class="n">where</span><span class="o">=</span><span class="s2">"dynamics"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">model</span>
</pre></div>
</div>
</section>
<section id="defining-the-loss-function">
<h2>Defining the loss function<a class="headerlink" href="#defining-the-loss-function" title="Link to this heading">#</a></h2>
<p>Next,</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">hypercoil.loss</span> <span class="kn">import</span> <span class="p">(</span>
<span class="n">LossScheme</span><span class="p">,</span>
<span class="n">LossApply</span><span class="p">,</span>
<span class="n">Loss</span><span class="p">,</span>
<span class="n">LossArgument</span><span class="p">,</span>
<span class="n">UnpackingLossArgument</span><span class="p">,</span>
<span class="n">ModularityLoss</span><span class="p">,</span>
<span class="n">SmoothnessLoss</span><span class="p">,</span>
<span class="n">BimodalSymmetricLoss</span><span class="p">,</span>
<span class="n">identity</span><span class="p">,</span>
<span class="n">sum_scalarise</span><span class="p">,</span>
<span class="n">mean_scalarise</span><span class="p">,</span>
<span class="n">vnorm_scalarise</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">def</span> <span class="nf">dynamic_community_loss</span><span class="p">(</span>
<span class="n">modularity_nu</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span>
<span class="n">smoothness_nu</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span>
<span class="n">dynamic_community_nu</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span>
<span class="n">bimodal_symmetric_nu</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span>
<span class="n">gamma</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span>
<span class="p">)</span> <span class="o">-></span> <span class="n">LossScheme</span><span class="p">:</span>
<span class="n">loss</span> <span class="o">=</span> <span class="n">LossScheme</span><span class="p">([</span>
<span class="n">LossApply</span><span class="p">(</span>
<span class="n">ModularityLoss</span><span class="p">(</span><span class="n">nu</span><span class="o">=</span><span class="n">modularity_nu</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">'Modularity'</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="n">gamma</span><span class="p">),</span>
<span class="n">apply</span><span class="o">=</span><span class="k">lambda</span> <span class="n">arg</span><span class="p">:</span> <span class="n">UnpackingLossArgument</span><span class="p">(</span>
<span class="n">A</span><span class="o">=</span><span class="n">arg</span><span class="o">.</span><span class="n">corr_unparam</span><span class="p">,</span>
<span class="n">Q</span><span class="o">=</span><span class="n">arg</span><span class="o">.</span><span class="n">affiliation</span><span class="p">,</span>
<span class="p">)),</span>
<span class="n">LossApply</span><span class="p">(</span>
<span class="n">Loss</span><span class="p">(</span>
<span class="n">nu</span><span class="o">=</span><span class="n">dynamic_community_nu</span><span class="p">,</span>
<span class="n">name</span><span class="o">=</span><span class="s1">'DynamicCommunities'</span><span class="p">,</span>
<span class="n">score</span><span class="o">=</span><span class="n">identity</span><span class="p">,</span>
<span class="n">scalarisation</span><span class="o">=</span><span class="n">mean_scalarise</span><span class="p">(</span>
<span class="n">axis</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">inner</span><span class="o">=</span><span class="n">sum_scalarise</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="p">),</span>
<span class="p">),</span>
<span class="n">apply</span><span class="o">=</span><span class="k">lambda</span> <span class="n">arg</span><span class="p">:</span> <span class="o">-</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">coaffiliation</span> <span class="o">*</span> <span class="n">arg</span><span class="o">.</span><span class="n">modularity</span><span class="p">)</span>
<span class="p">),</span>
<span class="n">LossScheme</span><span class="p">([</span>
<span class="n">SmoothnessLoss</span><span class="p">(</span>
<span class="n">nu</span><span class="o">=</span><span class="n">smoothness_nu</span><span class="p">,</span>
<span class="n">scalarisation</span><span class="o">=</span><span class="n">mean_scalarise</span><span class="p">(</span>
<span class="n">inner</span><span class="o">=</span><span class="n">vnorm_scalarise</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">))</span>
<span class="p">),</span>
<span class="n">BimodalSymmetricLoss</span><span class="p">(</span><span class="n">nu</span><span class="o">=</span><span class="n">bimodal_symmetric_nu</span><span class="p">,</span> <span class="n">modes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="p">],</span> <span class="n">apply</span><span class="o">=</span><span class="k">lambda</span> <span class="n">arg</span><span class="p">:</span> <span class="n">arg</span><span class="o">.</span><span class="n">dynamics</span><span class="p">)</span>
<span class="p">])</span>
<span class="k">return</span> <span class="n">loss</span>
</pre></div>
</div>
</section>
<section id="defining-the-forward-pass">
<h2>Defining the forward pass<a class="headerlink" href="#defining-the-forward-pass" title="Link to this heading">#</a></h2>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">hypercoil.engine</span> <span class="kn">import</span> <span class="n">_to_jax_array</span>
<span class="kn">from</span> <span class="nn">hypercoil.functional</span> <span class="kn">import</span> <span class="n">corr</span><span class="p">,</span> <span class="n">modularity_matrix</span><span class="p">,</span> <span class="n">coaffiliation</span>
<span class="k">def</span> <span class="nf">model_forward</span><span class="p">(</span>
<span class="n">time_series</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span>
<span class="n">affiliation</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span>
<span class="n">dynamics</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span>
<span class="n">gamma</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span>
<span class="p">)</span> <span class="o">-></span> <span class="n">Tensor</span><span class="p">:</span>
<span class="c1"># Ensure that all data tensors and parameters are JAX arrays.</span>
<span class="n">time_series</span> <span class="o">=</span> <span class="n">_to_jax_array</span><span class="p">(</span><span class="n">time_series</span><span class="p">)</span>
<span class="n">affiliation</span> <span class="o">=</span> <span class="n">_to_jax_array</span><span class="p">(</span><span class="n">affiliation</span><span class="p">)</span>
<span class="n">dynamics</span> <span class="o">=</span> <span class="n">_to_jax_array</span><span class="p">(</span><span class="n">dynamics</span><span class="p">)</span>
<span class="c1"># Compute the correlation matrix for each scan.</span>
<span class="n">corr_unparam</span> <span class="o">=</span> <span class="n">corr</span><span class="p">(</span><span class="n">time_series</span><span class="p">)</span>
<span class="n">corr_param</span> <span class="o">=</span> <span class="n">corr</span><span class="p">(</span><span class="n">time_series</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">,</span> <span class="o">...</span><span class="p">],</span> <span class="n">weight</span><span class="o">=</span><span class="n">dynamics</span><span class="p">)</span>
<span class="c1"># Compute the modularity matrix for each scan.</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">modularity_matrix</span><span class="p">(</span>
<span class="n">corr_param</span><span class="p">,</span>
<span class="n">normalise_modularity</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="n">gamma</span><span class="o">=</span><span class="n">gamma</span><span class="p">,</span>
<span class="p">)</span>
<span class="c1"># Compute the community co-affiliation matrix.</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">coaffiliation</span><span class="p">(</span>
<span class="n">affiliation</span><span class="o">.</span><span class="n">T</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">],</span>
<span class="n">normalise_coaffiliation</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="p">)</span>
<span class="c1"># Build arguments for the loss function.</span>
<span class="n">args</span> <span class="o">=</span> <span class="n">LossArgument</span><span class="p">(</span>
<span class="n">corr_unparam</span><span class="o">=</span><span class="n">corr_unparam</span><span class="p">,</span>
<span class="n">corr_param</span><span class="o">=</span><span class="n">corr_param</span><span class="p">,</span>
<span class="n">affiliation</span><span class="o">=</span><span class="n">affiliation</span><span class="p">,</span>
<span class="n">dynamics</span><span class="o">=</span><span class="n">dynamics</span><span class="p">,</span>
<span class="n">modularity</span><span class="o">=</span><span class="n">B</span><span class="p">,</span>
<span class="n">coaffiliation</span><span class="o">=</span><span class="n">H</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">return</span> <span class="n">args</span>
</pre></div>
</div>
</section>
<section id="defining-the-optimisation-loop">
<h2>Defining the optimisation loop<a class="headerlink" href="#defining-the-optimisation-loop" title="Link to this heading">#</a></h2>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Callable</span><span class="p">,</span> <span class="n">Tuple</span>
<span class="kn">import</span> <span class="nn">optax</span>
<span class="k">def</span> <span class="nf">init_optimiser</span><span class="p">(</span><span class="n">lr</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span> <span class="n">model</span><span class="p">:</span> <span class="n">PyTree</span><span class="p">)</span> <span class="o">-></span> <span class="n">optax</span><span class="o">.</span><span class="n">GradientTransformation</span><span class="p">:</span>
<span class="n">optim</span> <span class="o">=</span> <span class="n">optax</span><span class="o">.</span><span class="n">adam</span><span class="p">(</span><span class="n">lr</span><span class="p">)</span>
<span class="n">optim_state</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">init</span><span class="p">(</span><span class="n">eqx</span><span class="o">.</span><span class="n">filter</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">eqx</span><span class="o">.</span><span class="n">is_inexact_array</span><span class="p">))</span>
<span class="k">return</span> <span class="n">optim</span><span class="p">,</span> <span class="n">optim_state</span>
<span class="k">def</span> <span class="nf">update</span><span class="p">(</span>
<span class="n">model</span><span class="p">:</span> <span class="n">PyTree</span><span class="p">,</span>
<span class="nb">input</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span>
<span class="n">loss_scheme</span><span class="p">:</span> <span class="n">Callable</span><span class="p">,</span>
<span class="n">optim</span><span class="p">:</span> <span class="n">optax</span><span class="o">.</span><span class="n">GradientTransformation</span><span class="p">,</span>
<span class="n">optim_state</span><span class="p">:</span> <span class="n">PyTree</span><span class="p">,</span>
<span class="o">*</span><span class="p">,</span>
<span class="n">key</span><span class="p">:</span> <span class="s1">'jax.random.PRNGKey'</span><span class="p">,</span>
<span class="p">)</span> <span class="o">-></span> <span class="n">Tuple</span><span class="p">[</span><span class="n">PyTree</span><span class="p">,</span> <span class="n">optax</span><span class="o">.</span><span class="n">OptState</span><span class="p">]:</span>
<span class="k">def</span> <span class="nf">loss_fn</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
<span class="n">args</span> <span class="o">=</span> <span class="n">model_forward</span><span class="p">(</span>
<span class="nb">input</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">affiliation</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dynamics</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">gamma</span>
<span class="p">)</span>
<span class="k">return</span> <span class="n">loss_scheme</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">key</span><span class="p">)</span>
<span class="p">(</span><span class="n">loss</span><span class="p">,</span> <span class="n">meta</span><span class="p">),</span> <span class="n">grads</span> <span class="o">=</span> <span class="n">eqx</span><span class="o">.</span><span class="n">filter_value_and_grad</span><span class="p">(</span>
<span class="n">loss_fn</span><span class="p">,</span> <span class="n">has_aux</span><span class="o">=</span><span class="kc">True</span><span class="p">)(</span><span class="n">model</span><span class="p">,</span> <span class="nb">input</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">key</span><span class="p">)</span>
<span class="n">updates</span><span class="p">,</span> <span class="n">optim_state</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">update</span><span class="p">(</span>
<span class="n">eqx</span><span class="o">.</span><span class="n">filter</span><span class="p">(</span><span class="n">grads</span><span class="p">,</span> <span class="n">eqx</span><span class="o">.</span><span class="n">is_inexact_array</span><span class="p">),</span>
<span class="n">optim_state</span><span class="p">,</span>
<span class="n">eqx</span><span class="o">.</span><span class="n">filter</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">eqx</span><span class="o">.</span><span class="n">is_inexact_array</span><span class="p">),</span>
<span class="p">)</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">eqx</span><span class="o">.</span><span class="n">apply_updates</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">updates</span><span class="p">)</span>
<span class="k">return</span> <span class="n">model</span><span class="p">,</span> <span class="n">optim_state</span><span class="p">,</span> <span class="n">loss</span><span class="p">,</span> <span class="n">meta</span>
</pre></div>
</div>
</section>
<section id="train-the-model">
<h2>Train the model<a class="headerlink" href="#train-the-model" title="Link to this heading">#</a></h2>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Configure the hyperparameters.</span>
<span class="n">n_communities</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">n_time_points</span> <span class="o">=</span> <span class="mi">814</span>
<span class="n">n_nodes</span> <span class="o">=</span> <span class="mi">400</span>
<span class="n">n_scans</span> <span class="o">=</span> <span class="mi">30</span>
<span class="n">lr</span> <span class="o">=</span> <span class="mf">0.05</span>
<span class="n">modularity_nu</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">dynamic_community_nu</span> <span class="o">=</span> <span class="mf">2e-3</span>
<span class="n">smoothness_nu</span> <span class="o">=</span> <span class="mf">.2</span>
<span class="n">bimodal_symmetric_nu</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">max_epoch</span> <span class="o">=</span> <span class="mi">500</span>
<span class="n">gamma</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">key_model</span><span class="p">,</span> <span class="n">key_train</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
<span class="c1"># Initialise the model.</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">DynamicCommunityModel</span><span class="p">(</span>
<span class="n">n_nodes</span><span class="o">=</span><span class="n">n_nodes</span><span class="p">,</span>
<span class="n">n_scans</span><span class="o">=</span><span class="n">n_scans</span><span class="p">,</span>
<span class="n">n_communities</span><span class="o">=</span><span class="n">n_communities</span><span class="p">,</span>
<span class="n">n_time_points</span><span class="o">=</span><span class="n">n_time_points</span><span class="p">,</span>
<span class="n">gamma</span><span class="o">=</span><span class="n">gamma</span><span class="p">,</span>
<span class="n">key</span><span class="o">=</span><span class="n">key_model</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">parameterise_model</span><span class="p">(</span><span class="n">model</span><span class="p">)</span>
<span class="c1"># Initialise the loss function.</span>
<span class="n">loss_scheme</span> <span class="o">=</span> <span class="n">dynamic_community_loss</span><span class="p">(</span>
<span class="n">modularity_nu</span><span class="o">=</span><span class="n">modularity_nu</span><span class="p">,</span>
<span class="n">smoothness_nu</span><span class="o">=</span><span class="n">smoothness_nu</span><span class="p">,</span>
<span class="n">dynamic_community_nu</span><span class="o">=</span><span class="n">dynamic_community_nu</span><span class="p">,</span>
<span class="n">bimodal_symmetric_nu</span><span class="o">=</span><span class="n">bimodal_symmetric_nu</span><span class="p">,</span>
<span class="n">gamma</span><span class="o">=</span><span class="n">gamma</span><span class="p">,</span>
<span class="p">)</span>
<span class="c1"># Initialise the optimiser.</span>
<span class="n">optim</span><span class="p">,</span> <span class="n">optim_state</span> <span class="o">=</span> <span class="n">init_optimiser</span><span class="p">(</span><span class="n">lr</span><span class="p">,</span> <span class="n">model</span><span class="p">)</span>
<span class="c1"># Train the model.</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_epoch</span><span class="p">):</span>
<span class="n">key_epoch</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">fold_in</span><span class="p">(</span><span class="n">key_train</span><span class="p">,</span> <span class="n">epoch</span><span class="p">)</span>
<span class="n">model</span><span class="p">,</span> <span class="n">optim_state</span><span class="p">,</span> <span class="n">loss</span><span class="p">,</span> <span class="n">meta</span> <span class="o">=</span> <span class="n">eqx</span><span class="o">.</span><span class="n">filter_jit</span><span class="p">(</span><span class="n">update</span><span class="p">)(</span>
<span class="n">model</span><span class="p">,</span>
<span class="n">time_series</span><span class="p">,</span>
<span class="n">loss_scheme</span><span class="p">,</span>
<span class="n">optim</span><span class="p">,</span>
<span class="n">optim_state</span><span class="p">,</span>
<span class="n">key</span><span class="o">=</span><span class="n">key_epoch</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">epoch</span> <span class="o">%</span> <span class="mi">10</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'Epoch: </span><span class="si">{</span><span class="n">epoch</span><span class="si">}</span><span class="s1">, Loss: </span><span class="si">{</span><span class="n">loss</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
<span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">meta</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">k</span><span class="si">}</span><span class="s1">: </span><span class="si">{</span><span class="n">v</span><span class="o">.</span><span class="n">value</span><span class="si">:</span><span class="s1">.4f</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
</pre></div>
</div>
</section>
</section>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#loading-the-dataset">Loading the dataset</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-model">Defining the model</a></li>
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