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 <p>Now we can plot the landscape. This landscape show the values of HOI for
@@ -528,19 +528,19 @@ <h2>Multiplet isolation using the Topological Information<a class="headerlink" h
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 </pre></div>
 </div>
 <p>Again, we can plot the landscape. This time, as we can see, there’s no more
@@ -574,7 +574,7 @@ <h2>Multiplet isolation using the Topological Information<a class="headerlink" h
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diff --git a/auto_examples/metrics/plot_oinfo.html b/auto_examples/metrics/plot_oinfo.html
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    index  order       hoi  multiplet
 0      0      3  0.948908  [0, 1, 2]
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 0      0      3  1.698491  [0, 1, 2]
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@@ -789,14 +789,14 @@ <h3>Dynamic redundant and synergistic codings<a class="headerlink" href="#dynami
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diff --git a/auto_examples/metrics/plot_rsi.html b/auto_examples/metrics/plot_rsi.html
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 <img src="../../_images/sphx_glr_plot_rsi_001.png" srcset="../../_images/sphx_glr_plot_rsi_001.png" alt="plot rsi" class = "sphx-glr-single-img"/><div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/3 [00:00&lt;?,       ?it/s]
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@@ -644,7 +644,7 @@ <h2>Combining redundancy and synergy<a class="headerlink" href="#combining-redun
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diff --git a/auto_examples/metrics/plot_syn_phiID.html b/auto_examples/metrics/plot_syn_phiID.html
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- 33%|███▎      | RedMMI order 3:  1/3 [00:00&lt;00:00,    7.68it/s]
-100%|██████████| RedMMI order 5:  3/3 [00:00&lt;00:00,   19.18it/s]
+ 33%|███▎      | RedMMI order 3:  1/3 [00:00&lt;00:00,    7.34it/s]
+100%|██████████| RedMMI order 5:  3/3 [00:00&lt;00:00,   18.31it/s]
 
    index  order       hoi  multiplet
 0     20      3  0.392336  [1, 3, 5]
@@ -524,7 +524,7 @@ <h2>Simulate multivariate redundancy<a class="headerlink" href="#simulate-multiv
 </pre></div>
 </div>
 <div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/3 [00:00&lt;?,       ?it/s]
- 33%|███▎      | RedMMI order 3:  1/3 [00:00&lt;00:00,    7.36it/s]
+ 33%|███▎      | RedMMI order 3:  1/3 [00:00&lt;00:00,    7.07it/s]
 
    index  order       hoi  multiplet
 0     27      3  0.483097  [2, 3, 6]
@@ -562,9 +562,9 @@ <h2>Simulate univariate and multivariate synergy<a class="headerlink" href="#sim
 </pre></div>
 </div>
 <div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/3 [00:00&lt;?,       ?it/s]
- 33%|███▎      | SynMMI order 3:  1/3 [00:00&lt;00:00,    3.91it/s]
- 67%|██████▋   | SynMMI order 4:  2/3 [00:00&lt;00:00,    3.97it/s]
-100%|██████████| SynMMI order 5:  3/3 [00:00&lt;00:00,    3.97it/s]
+ 33%|███▎      | SynMMI order 3:  1/3 [00:00&lt;00:00,    3.63it/s]
+ 67%|██████▋   | SynMMI order 4:  2/3 [00:00&lt;00:00,    3.62it/s]
+100%|██████████| SynMMI order 5:  3/3 [00:00&lt;00:00,    3.64it/s]
 
    index  order           hoi  multiplet
 0     10      3  1.365911e+00  [0, 3, 5]
@@ -591,9 +591,9 @@ <h2>Simulate univariate and multivariate synergy<a class="headerlink" href="#sim
 </pre></div>
 </div>
 <div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/3 [00:00&lt;?,       ?it/s]
- 33%|███▎      | SynMMI order 3:  1/3 [00:00&lt;00:00,    3.68it/s]
- 67%|██████▋   | SynMMI order 4:  2/3 [00:00&lt;00:00,    3.64it/s]
-100%|██████████| SynMMI order 5:  3/3 [00:00&lt;00:00,    3.62it/s]
+ 33%|███▎      | SynMMI order 3:  1/3 [00:00&lt;00:00,    3.52it/s]
+ 67%|██████▋   | SynMMI order 4:  2/3 [00:00&lt;00:00,    3.39it/s]
+100%|██████████| SynMMI order 5:  3/3 [00:00&lt;00:00,    3.39it/s]
 
    index  order       hoi  multiplet
 0     18      3  1.600548  [1, 2, 6]
@@ -636,9 +636,9 @@ <h2>Combining redundancy and synergy<a class="headerlink" href="#combining-redun
 </pre></div>
 </div>
 <div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/3 [00:00&lt;?,       ?it/s]
- 33%|███▎      | SynMMI order 3:  1/3 [00:00&lt;00:00,    3.82it/s]
- 67%|██████▋   | SynMMI order 4:  2/3 [00:00&lt;00:00,    3.70it/s]
-100%|██████████| SynMMI order 5:  3/3 [00:00&lt;00:00,    3.66it/s]
+ 33%|███▎      | SynMMI order 3:  1/3 [00:00&lt;00:00,    3.61it/s]
+ 67%|██████▋   | SynMMI order 4:  2/3 [00:00&lt;00:00,    3.54it/s]
+100%|██████████| SynMMI order 5:  3/3 [00:00&lt;00:00,    3.45it/s]
 
 
 Synergy results
@@ -649,7 +649,7 @@ <h2>Combining redundancy and synergy<a class="headerlink" href="#combining-redun
 2      8      3  0.002230  [0, 2, 6]
 
   0%|          |  0/3 [00:00&lt;?,       ?it/s]
- 33%|███▎      | RedMMI order 3:  1/3 [00:00&lt;00:00,    7.44it/s]
+ 33%|███▎      | RedMMI order 3:  1/3 [00:00&lt;00:00,    7.01it/s]
 
 
 Redundancy results
@@ -684,7 +684,7 @@ <h2>Combining redundancy and synergy<a class="headerlink" href="#combining-redun
 <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
 </pre></div>
 </div>
-<img src="../../_images/sphx_glr_plot_syn_red_mmi_002.png" srcset="../../_images/sphx_glr_plot_syn_red_mmi_002.png" alt="plot syn red mmi" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 3.014 seconds)</p>
+<img src="../../_images/sphx_glr_plot_syn_red_mmi_002.png" srcset="../../_images/sphx_glr_plot_syn_red_mmi_002.png" alt="plot syn red mmi" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 3.228 seconds)</p>
 <div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-metrics-plot-syn-red-mmi-py">
 <div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
 <p><a class="reference download internal" download="" href="../../_downloads/506e66570f568588cfe575362572604a/plot_syn_red_mmi.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_syn_red_mmi.ipynb</span></code></a></p>
diff --git a/auto_examples/metrics/sg_execution_times.html b/auto_examples/metrics/sg_execution_times.html
index d12a485..1bde9d3 100644
--- a/auto_examples/metrics/sg_execution_times.html
+++ b/auto_examples/metrics/sg_execution_times.html
@@ -404,7 +404,7 @@ <h1>Computation times</h1>
                   
   <section id="computation-times">
 <span id="sphx-glr-auto-examples-metrics-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Link to this heading">#</a></h1>
-<p><strong>00:15.521</strong> total execution time for 5 files <strong>from auto_examples/metrics</strong>:</p>
+<p><strong>00:16.545</strong> total execution time for 5 files <strong>from auto_examples/metrics</strong>:</p>
 <div class="docutils container">
 <style scoped>
 <link href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/5.3.0/css/bootstrap.min.css" rel="stylesheet" />
@@ -426,23 +426,23 @@ <h1>Computation times</h1>
 </thead>
 <tbody>
 <tr class="row-even"><td><p><a class="reference internal" href="plot_rsi.html#sphx-glr-auto-examples-metrics-plot-rsi-py"><span class="std std-ref">Redundancy-Synergy Index</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_rsi.py</span></code>)</p></td>
-<td><p>00:03.954</p></td>
+<td><p>00:04.191</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="plot_oinfo.html#sphx-glr-auto-examples-metrics-plot-oinfo-py"><span class="std std-ref">O-information and its derivatives for network behavior and encoding</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_oinfo.py</span></code>)</p></td>
-<td><p>00:03.717</p></td>
+<td><p>00:03.844</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="reference internal" href="plot_infotopo.html#sphx-glr-auto-examples-metrics-plot-infotopo-py"><span class="std std-ref">Topological Information : conditioning on orders</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_infotopo.py</span></code>)</p></td>
-<td><p>00:03.311</p></td>
+<td><p>00:03.613</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="plot_syn_red_mmi.html#sphx-glr-auto-examples-metrics-plot-syn-red-mmi-py"><span class="std std-ref">Redundancy and Synergy MMI</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_syn_red_mmi.py</span></code>)</p></td>
-<td><p>00:03.014</p></td>
+<td><p>00:03.228</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="reference internal" href="plot_syn_phiID.html#sphx-glr-auto-examples-metrics-plot-syn-phiid-py"><span class="std std-ref">Integrated Information Decomposition</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_syn_phiID.py</span></code>)</p></td>
-<td><p>00:01.525</p></td>
+<td><p>00:01.668</p></td>
 <td><p>0.0</p></td>
 </tr>
 </tbody>
diff --git a/auto_examples/miscellaneous/plot_inspect_results.html b/auto_examples/miscellaneous/plot_inspect_results.html
index 7c1df95..a91dae5 100644
--- a/auto_examples/miscellaneous/plot_inspect_results.html
+++ b/auto_examples/miscellaneous/plot_inspect_results.html
@@ -466,18 +466,18 @@ <h2>Estimate HOI<a class="headerlink" href="#estimate-hoi" title="Link to this h
 </pre></div>
 </div>
 <div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/6 [00:00&lt;?,       ?it/s]
- 17%|█▋        | Entropy H(1):  1/6 [00:00&lt;00:00,    5.25it/s]
- 33%|███▎      | Entropy H(2):  2/6 [00:00&lt;00:00,    4.82it/s]
- 50%|█████     | Entropy H(3):  3/6 [00:00&lt;00:00,    5.44it/s]
- 67%|██████▋   | Entropy H(4):  4/6 [00:00&lt;00:00,    6.08it/s]
- 83%|████████▎ | Entropy H(5):  5/6 [00:00&lt;00:00,    6.49it/s]
-100%|██████████| Entropy H(6):  6/6 [00:00&lt;00:00,    6.82it/s]
+ 17%|█▋        | Entropy H(1):  1/6 [00:00&lt;00:01,    4.78it/s]
+ 33%|███▎      | Entropy H(2):  2/6 [00:00&lt;00:00,    4.72it/s]
+ 50%|█████     | Entropy H(3):  3/6 [00:00&lt;00:00,    5.31it/s]
+ 67%|██████▋   | Entropy H(4):  4/6 [00:00&lt;00:00,    5.97it/s]
+ 83%|████████▎ | Entropy H(5):  5/6 [00:00&lt;00:00,    6.38it/s]
+100%|██████████| Entropy H(6):  6/6 [00:00&lt;00:00,    6.59it/s]
 
 
 Mutual information:   0%|          | 0/42 [00:00&lt;?, ?it/s]
 
 
-Mutual information: 100%|██████████| 42/42 [00:00&lt;00:00, 16177.86it/s]
+Mutual information: 100%|██████████| 42/42 [00:00&lt;00:00, 16093.62it/s]
 </pre></div>
 </div>
 </section>
@@ -558,7 +558,7 @@ <h2>Plotting individual multiplets<a class="headerlink" href="#plotting-individu
 <span class="c1"># sphinx_gallery_thumbnail_number = 2</span>
 </pre></div>
 </div>
-<img src="../../_images/sphx_glr_plot_inspect_results_002.png" srcset="../../_images/sphx_glr_plot_inspect_results_002.png" alt="plot inspect results" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 1.665 seconds)</p>
+<img src="../../_images/sphx_glr_plot_inspect_results_002.png" srcset="../../_images/sphx_glr_plot_inspect_results_002.png" alt="plot inspect results" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 1.735 seconds)</p>
 <div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-miscellaneous-plot-inspect-results-py">
 <div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
 <p><a class="reference download internal" download="" href="../../_downloads/ad4b98878b000f24685a5165bd000d07/plot_inspect_results.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_inspect_results.ipynb</span></code></a></p>
diff --git a/auto_examples/miscellaneous/sg_execution_times.html b/auto_examples/miscellaneous/sg_execution_times.html
index ddb708d..3539e2d 100644
--- a/auto_examples/miscellaneous/sg_execution_times.html
+++ b/auto_examples/miscellaneous/sg_execution_times.html
@@ -404,7 +404,7 @@ <h1>Computation times</h1>
                   
   <section id="computation-times">
 <span id="sphx-glr-auto-examples-miscellaneous-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Link to this heading">#</a></h1>
-<p><strong>00:02.945</strong> total execution time for 2 files <strong>from auto_examples/miscellaneous</strong>:</p>
+<p><strong>00:03.043</strong> total execution time for 2 files <strong>from auto_examples/miscellaneous</strong>:</p>
 <div class="docutils container">
 <style scoped>
 <link href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/5.3.0/css/bootstrap.min.css" rel="stylesheet" />
@@ -426,11 +426,11 @@ <h1>Computation times</h1>
 </thead>
 <tbody>
 <tr class="row-even"><td><p><a class="reference internal" href="plot_inspect_results.html#sphx-glr-auto-examples-miscellaneous-plot-inspect-results-py"><span class="std std-ref">Inspect and plot Higher-Order Interactions</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_inspect_results.py</span></code>)</p></td>
-<td><p>00:01.665</p></td>
+<td><p>00:01.735</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="tuto_jax.html#sphx-glr-auto-examples-miscellaneous-tuto-jax-py"><span class="std std-ref">Quick introduction to jax</span></a> (<code class="docutils literal notranslate"><span class="pre">tuto_jax.py</span></code>)</p></td>
-<td><p>00:01.281</p></td>
+<td><p>00:01.308</p></td>
 <td><p>0.0</p></td>
 </tr>
 </tbody>
diff --git a/auto_examples/miscellaneous/tuto_jax.html b/auto_examples/miscellaneous/tuto_jax.html
index e606d88..6ceefdc 100644
--- a/auto_examples/miscellaneous/tuto_jax.html
+++ b/auto_examples/miscellaneous/tuto_jax.html
@@ -521,8 +521,8 @@ <h2>Compiling functions using jit<a class="headerlink" href="#compiling-function
 <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Time taken by jitted function: </span><span class="si">{</span><span class="n">t2</span><span class="si">}</span><span class="s2"> seconds&quot;</span><span class="p">)</span>
 </pre></div>
 </div>
-<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Time taken by pure NumPy function: 0.5754969579999738 seconds
-Time taken by jitted function: 0.2708683170000086 seconds
+<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Time taken by pure NumPy function: 0.5777375499999948 seconds
+Time taken by jitted function: 0.26260832099995923 seconds
 </pre></div>
 </div>
 <p>if the function takes optional argument, you can use the <cite>static_argnums</cite>
@@ -622,7 +622,7 @@ <h2>For loops with jax<a class="headerlink" href="#for-loops-with-jax" title="Li
 nice introduction to <cite>jax.lax.scan</cite></p></li>
 </ul>
 </div></blockquote>
-<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 1.281 seconds)</p>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 1.308 seconds)</p>
 <div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-miscellaneous-tuto-jax-py">
 <div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
 <p><a class="reference download internal" download="" href="../../_downloads/c9df0fdd61c061c32221dcc575a693a9/tuto_jax.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">tuto_jax.ipynb</span></code></a></p>
diff --git a/auto_examples/statistics/plot_bootstrapping.html b/auto_examples/statistics/plot_bootstrapping.html
index 5782b22..0a0e852 100644
--- a/auto_examples/statistics/plot_bootstrapping.html
+++ b/auto_examples/statistics/plot_bootstrapping.html
@@ -665,7 +665,7 @@ <h3>Plot the O-information with the confidence interval<a class="headerlink" hre
 <div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># sphinx_gallery_thumbnail_number = 2</span>
 </pre></div>
 </div>
-<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 15.466 seconds)</p>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 16.117 seconds)</p>
 <div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-statistics-plot-bootstrapping-py">
 <div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
 <p><a class="reference download internal" download="" href="../../_downloads/d2e94d8bac717c85e2a37d7d973930a2/plot_bootstrapping.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_bootstrapping.ipynb</span></code></a></p>
diff --git a/auto_examples/statistics/sg_execution_times.html b/auto_examples/statistics/sg_execution_times.html
index dd3292c..d157463 100644
--- a/auto_examples/statistics/sg_execution_times.html
+++ b/auto_examples/statistics/sg_execution_times.html
@@ -404,7 +404,7 @@ <h1>Computation times</h1>
                   
   <section id="computation-times">
 <span id="sphx-glr-auto-examples-statistics-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Link to this heading">#</a></h1>
-<p><strong>00:15.466</strong> total execution time for 1 file <strong>from auto_examples/statistics</strong>:</p>
+<p><strong>00:16.117</strong> total execution time for 1 file <strong>from auto_examples/statistics</strong>:</p>
 <div class="docutils container">
 <style scoped>
 <link href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/5.3.0/css/bootstrap.min.css" rel="stylesheet" />
@@ -426,7 +426,7 @@ <h1>Computation times</h1>
 </thead>
 <tbody>
 <tr class="row-even"><td><p><a class="reference internal" href="plot_bootstrapping.html#sphx-glr-auto-examples-statistics-plot-bootstrapping-py"><span class="std std-ref">Bootstrapping and confidence interval</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_bootstrapping.py</span></code>)</p></td>
-<td><p>00:15.466</p></td>
+<td><p>00:16.117</p></td>
 <td><p>0.0</p></td>
 </tr>
 </tbody>
diff --git a/auto_examples/tutorials/plot_ml_vs_it.html b/auto_examples/tutorials/plot_ml_vs_it.html
index adb284f..72dbb0e 100644
--- a/auto_examples/tutorials/plot_ml_vs_it.html
+++ b/auto_examples/tutorials/plot_ml_vs_it.html
@@ -602,7 +602,7 @@ <h2>Using information-theoretic approaches<a class="headerlink" href="#using-inf
 <img src="../../_images/sphx_glr_plot_ml_vs_it_003.png" srcset="../../_images/sphx_glr_plot_ml_vs_it_003.png" alt="Redundant interactions, Synergistic interactions" class = "sphx-glr-single-img"/><p>We retrieve the bump of information around sample 50 however this time, the
 bump is positive in case of redundant interactions and negative in case of
 synergistic interactions.</p>
-<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 9.166 seconds)</p>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 9.275 seconds)</p>
 <div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-tutorials-plot-ml-vs-it-py">
 <div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
 <p><a class="reference download internal" download="" href="../../_downloads/007eb65766f3b4811d99ff5b36a7b49b/plot_ml_vs_it.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_ml_vs_it.ipynb</span></code></a></p>
diff --git a/auto_examples/tutorials/plot_sim_red_syn.html b/auto_examples/tutorials/plot_sim_red_syn.html
index 91420db..870b4e1 100644
--- a/auto_examples/tutorials/plot_sim_red_syn.html
+++ b/auto_examples/tutorials/plot_sim_red_syn.html
@@ -490,7 +490,7 @@ <h3>Simulating synergistic behavior<a class="headerlink" href="#simulating-syner
 </pre></div>
 </div>
 <div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/1 [00:00&lt;?,       ?it/s]
-100%|██████████| Oinfo (3):  1/1 [00:00&lt;00:00,    1.71it/s]
+100%|██████████| Oinfo (3):  1/1 [00:00&lt;00:00,    1.56it/s]
 </pre></div>
 </div>
 <p>Print HOI value</p>
@@ -520,7 +520,7 @@ <h3>Simulating redundant behavior<a class="headerlink" href="#simulating-redunda
 </pre></div>
 </div>
 <div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/1 [00:00&lt;?,       ?it/s]
-100%|██████████| Oinfo (3):  1/1 [00:00&lt;00:00,    6.28it/s]
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 <p>Print HOI value</p>
@@ -552,7 +552,7 @@ <h3>Simulating synergistic behavior by sum operation<a class="headerlink" href="
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@@ -583,7 +583,7 @@ <h3>Simulating redundant behavior by copy operator<a class="headerlink" href="#s
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@@ -630,7 +630,7 @@ <h3>Simulating dynamic redundancy and synergy<a class="headerlink" href="#simula
 </pre></div>
 </div>
 <img src="../../_images/sphx_glr_plot_sim_red_syn_001.png" srcset="../../_images/sphx_glr_plot_sim_red_syn_001.png" alt="Dynamic HOI" class = "sphx-glr-single-img"/><div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/1 [00:00&lt;?,       ?it/s]
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 Text(0.5, 1.0, &#39;Dynamic HOI&#39;)
 </pre></div>
@@ -670,11 +670,11 @@ <h3>Simulating redundant encoding<a class="headerlink" href="#simulating-redunda
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@@ -767,11 +767,11 @@ <h3>Simulating synergistic encoding by sum operation<a class="headerlink" href="
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@@ -822,14 +822,14 @@ <h3>Simulating dynamic redundant and synergistic encoding<a class="headerlink" h
 </pre></div>
 </div>
 <img src="../../_images/sphx_glr_plot_sim_red_syn_002.png" srcset="../../_images/sphx_glr_plot_sim_red_syn_002.png" alt="Dynamic HOI" class = "sphx-glr-single-img"/><div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>  0%|          |  0/1 [00:00&lt;?,       ?it/s]
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-<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 4.698 seconds)</p>
+<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 4.841 seconds)</p>
 <div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-tutorials-plot-sim-red-syn-py">
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diff --git a/auto_examples/tutorials/sg_execution_times.html b/auto_examples/tutorials/sg_execution_times.html
index b04f9ab..6bbe02c 100644
--- a/auto_examples/tutorials/sg_execution_times.html
+++ b/auto_examples/tutorials/sg_execution_times.html
@@ -404,7 +404,7 @@ <h1>Computation times</h1>
                   
   <section id="computation-times">
 <span id="sphx-glr-auto-examples-tutorials-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Link to this heading">#</a></h1>
-<p><strong>00:13.863</strong> total execution time for 2 files <strong>from auto_examples/tutorials</strong>:</p>
+<p><strong>00:14.115</strong> total execution time for 2 files <strong>from auto_examples/tutorials</strong>:</p>
 <div class="docutils container">
 <style scoped>
 <link href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/5.3.0/css/bootstrap.min.css" rel="stylesheet" />
@@ -426,11 +426,11 @@ <h1>Computation times</h1>
 </thead>
 <tbody>
 <tr class="row-even"><td><p><a class="reference internal" href="plot_ml_vs_it.html#sphx-glr-auto-examples-tutorials-plot-ml-vs-it-py"><span class="std std-ref">Machine-learning vs. Information theoretic approaches for HOI</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_ml_vs_it.py</span></code>)</p></td>
-<td><p>00:09.166</p></td>
+<td><p>00:09.275</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="plot_sim_red_syn.html#sphx-glr-auto-examples-tutorials-plot-sim-red-syn-py"><span class="std std-ref">How to simulate redundancy and synergy</span></a> (<code class="docutils literal notranslate"><span class="pre">plot_sim_red_syn.py</span></code>)</p></td>
-<td><p>00:04.698</p></td>
+<td><p>00:04.841</p></td>
 <td><p>0.0</p></td>
 </tr>
 </tbody>
diff --git a/contributors.html b/contributors.html
index 72401c9..bbcdb6d 100644
--- a/contributors.html
+++ b/contributors.html
@@ -408,7 +408,7 @@ <h1>List of contributors<a class="headerlink" href="#list-of-contributors" title
 <li class="sphinx-contributors_list__item"><div class="sphinx-contributors_contributor docutils container">
 <img alt="https://avatars.githubusercontent.com/u/15892073?v=4" class="sphinx-contributors_contributor__image" src="https://avatars.githubusercontent.com/u/15892073?v=4" />
 <p class="sphinx-contributors_contributor__username"><a class="reference external" href="https://github.com/EtienneCmb">EtienneCmb</a></p>
-<p class="sphinx-contributors_contributor__contributions">295 contributions</p>
+<p class="sphinx-contributors_contributor__contributions">297 contributions</p>
 </div>
 </li>
 <li class="sphinx-contributors_list__item"><div class="sphinx-contributors_contributor docutils container">
diff --git a/index.html b/index.html
index 42118e2..a2ec0fa 100644
--- a/index.html
+++ b/index.html
@@ -418,7 +418,7 @@ <h2> Contents </h2>
 <div id="searchbox"></div>
                 <article class="bd-article">
                   
-  <p><a class="reference external" href="https://github.com/psf/black"><img alt="Black" src="https://img.shields.io/badge/code%20style-black-000000.svg" /></a> <a class="reference external" href="https://codecov.io/gh/brainets/hoi"><img alt="Codecov" src="https://codecov.io/gh/brainets/hoi/graph/badge.svg?token=7PNM2VD994" /></a></p>
+  <p><a class="reference external" href="https://github.com/psf/black"><img alt="Black" src="https://img.shields.io/badge/code%20style-black-000000.svg" /></a> <a class="reference external" href="https://codecov.io/gh/brainets/hoi"><img alt="Codecov" src="https://codecov.io/gh/brainets/hoi/graph/badge.svg?token=7PNM2VD994" /></a> <a class="reference external" href="https://doi.org/10.21105/joss.07360"><img alt="JOSS" src="https://joss.theoj.org/papers/10.21105/joss.07360/status.svg" /></a></p>
 <section id="hoi-high-performance-estimation-of-higher-order-interactions">
 <h1>HOI : High-Performance Estimation of Higher-Order Interactions<a class="headerlink" href="#hoi-high-performance-estimation-of-higher-order-interactions" title="Link to this heading">#</a></h1>
 <p><strong>HOI</strong> (Higher Order Interactions) is a Python package to go beyond pairwise interactions by quantifying the statistical dependencies between 2 or more units using information-theoretical metrics. The package is built on top of <a class="reference external" href="https://github.com/google/jax">Jax</a> allowing computations on CPU or GPU.</p>
diff --git a/searchindex.js b/searchindex.js
index 2ecfc58..03fe1cf 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
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"Compiling functions using jit": [[53, "compiling-functions-using-jit"]], "Computation times": [[41, null], [49, null], [52, null], [54, null], [57, null], [61, null], [69, null]], "Compute Higher-Order Interactions": [[68, "compute-higher-order-interactions"]], "Compute bootstrap": [[56, "compute-bootstrap"]], "Computing Higher-Order Interactions on large multiplets": [[67, "computing-higher-order-interactions-on-large-multiplets"]], "Computing entropy on large multi-dimensional arrays": [[67, "computing-entropy-on-large-multi-dimensional-arrays"]], "Contributing code using pull requests": [[62, "contributing-code-using-pull-requests"]], "Contributing to HOI": [[62, "contributing-to-hoi"]], "Core information theoretic measures": [[70, "core-information-theoretic-measures"]], "Data simulation": [[45, "data-simulation"], [59, "data-simulation"]], "Decoding Y using X": [[59, "decoding-y-using-x"]], "Definition": [[46, "definition"], [47, "definition"], [48, "definition"]], "Definition of MI estimators": [[39, "definition-of-mi-estimators"], [40, "definition-of-mi-estimators"]], "Definition of entropy estimators": [[38, "definition-of-entropy-estimators"]], "Definition of estimators of entropy": [[36, "definition-of-estimators-of-entropy"], [37, "definition-of-estimators-of-entropy"]], "Dependencies": [[66, "dependencies"]], "Developer Documentation": [[62, null]], "Dual Total Correlation (DTC)": [[45, "dual-total-correlation-dtc"]], "Dual Total correlation": [[70, "dual-total-correlation"]], "Dynamic redundant and synergistic codings": [[45, "dynamic-redundant-and-synergistic-codings"]], "Entropy": [[42, "entropy"]], "Entropy and mutual information": [[34, "entropy-and-mutual-information"], [35, null]], "Entropy of data sampled from a uniform distribution": [[36, "entropy-of-data-sampled-from-a-uniform-distribution"]], "Entropy of data sampled from an exponential distribution": [[36, "entropy-of-data-sampled-from-an-exponential-distribution"]], "Entropy of data sampled from normal distribution": [[36, "entropy-of-data-sampled-from-normal-distribution"]], "Estimate HOI": [[51, "estimate-hoi"]], "Estimation of the confidence interval using a bootstrapping approach": [[56, "estimation-of-the-confidence-interval-using-a-bootstrapping-approach"]], "Examples": [[34, null]], "Examples using hoi.core.get_entropy": [[12, "examples-using-hoi-core-get-entropy"]], "Examples using hoi.core.get_mi": [[13, "examples-using-hoi-core-get-mi"]], "Examples using hoi.metrics.DTC": [[15, "examples-using-hoi-metrics-dtc"]], "Examples using hoi.metrics.GradientOinfo": [[16, "examples-using-hoi-metrics-gradientoinfo"]], "Examples using hoi.metrics.InfoTopo": [[17, "examples-using-hoi-metrics-infotopo"]], "Examples using hoi.metrics.Oinfo": [[19, "examples-using-hoi-metrics-oinfo"]], "Examples using hoi.metrics.RSI": [[20, "examples-using-hoi-metrics-rsi"]], "Examples using hoi.metrics.RedundancyMMI": [[21, "examples-using-hoi-metrics-redundancymmi"]], "Examples using hoi.metrics.RedundancyphiID": [[22, "examples-using-hoi-metrics-redundancyphiid"]], "Examples using hoi.metrics.Sinfo": [[23, "examples-using-hoi-metrics-sinfo"]], "Examples using hoi.metrics.SynergyMMI": [[24, "examples-using-hoi-metrics-synergymmi"]], "Examples using hoi.metrics.SynergyphiID": [[25, "examples-using-hoi-metrics-synergyphiid"]], "Examples using hoi.metrics.TC": [[26, "examples-using-hoi-metrics-tc"]], "Examples using hoi.plot.plot_landscape": [[27, "examples-using-hoi-plot-plot-landscape"]], "Examples using hoi.simulation.simulate_hoi_gauss": [[28, "examples-using-hoi-simulation-simulate-hoi-gauss"]], "Examples using hoi.utils.digitize": [[29, "examples-using-hoi-utils-digitize"]], "Examples using hoi.utils.get_nbest_mult": [[30, "examples-using-hoi-utils-get-nbest-mult"]], "For loops with jax": [[53, "for-loops-with-jax"]], "From pairwise to higher-order interactions": [[70, "from-pairwise-to-higher-order-interactions"]], "Funding": [[65, "funding"]], "Get a summary of the results": [[51, "get-a-summary-of-the-results"]], "Get involved": [[65, "get-involved"]], "Glossary": [[64, null]], "Gradient of O-information": [[70, "gradient-of-o-information"]], "HOI : High-Performance Estimation of Higher-Order Interactions": [[65, null]], "How to simulate redundancy and synergy": [[60, null]], "Inspect and plot Higher-Order Interactions": [[51, null]], "Inspect the results": [[68, "inspect-the-results"]], "Installation": [[66, null], [68, "installation"]], "Installation for developers": [[62, "installation-for-developers"]], "Integrated Information Decomposition": [[47, null]], "Introduction to core information theoretical metrics": [[42, null]], "Jax.numpy module": [[53, "jax-numpy-module"]], "Jax: linear algebra backend": [[67, null]], "Landscape plot": [[51, "landscape-plot"]], "License": [[65, "license"]], "List of classes and functions": [[33, null]], "List of contributors": [[63, null]], "MI of data sampled from normal distribution": [[39, "mi-of-data-sampled-from-normal-distribution"]], "MI of data sampled from splitted multinormal distribution": [[40, "mi-of-data-sampled-from-splitted-multinormal-distribution"]], "Machine-learning vs. Information theoretic approaches for HOI": [[59, null]], "Measures of Entropy": [[0, "measures-of-entropy"]], "Measures of Mutual Information": [[0, "measures-of-mutual-information"]], "Measuring Entropy": [[70, "measuring-entropy"]], "Measuring Mutual Information (MI)": [[70, "measuring-mutual-information-mi"]], "Metrics of HOI": [[34, "metrics-of-hoi"], [43, null]], "Miscellaneous": [[34, "miscellaneous"], [50, null]], "Multiplet isolation using the Topological Information": [[44, "multiplet-isolation-using-the-topological-information"]], "Mutual information": [[42, "mutual-information"]], "Network behavior": [[1, "network-behavior"], [70, "network-behavior"]], "Network encoding": [[1, "network-encoding"], [70, "network-encoding"]], "O-information": [[45, "o-information"], [70, 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null]], "Redundancy-Synergy Index": [[46, null]], "Redundancy-Synergy index (RSI)": [[70, "redundancy-synergy-index-rsi"]], "Redundant and synergistic network behavior": [[60, "redundant-and-synergistic-network-behavior"]], "Redundant and synergistic network encoding": [[60, "redundant-and-synergistic-network-encoding"]], "S information": [[70, "s-information"]], "S-information": [[45, "s-information"]], "Simulate data": [[51, "simulate-data"], [68, "simulate-data"]], "Simulate multivariate redundancy": [[46, "simulate-multivariate-redundancy"], [48, "simulate-multivariate-redundancy"]], "Simulate redundancy": [[47, "simulate-redundancy"]], "Simulate redundant and synergistic interactions": [[44, "simulate-redundant-and-synergistic-interactions"]], "Simulate synergy": [[47, "simulate-synergy"]], "Simulate univariate and multivariate synergy": [[46, "simulate-univariate-and-multivariate-synergy"], [48, "simulate-univariate-and-multivariate-synergy"]], "Simulate univariate redundancy": 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\ No newline at end of file
diff --git a/sg_execution_times.html b/sg_execution_times.html
index 1215d38..2a0673e 100644
--- a/sg_execution_times.html
+++ b/sg_execution_times.html
@@ -404,7 +404,7 @@ <h1>Computation times</h1>
                   
   <section id="computation-times">
 <span id="sphx-glr-sg-execution-times"></span><h1>Computation times<a class="headerlink" href="#computation-times" title="Link to this heading">#</a></h1>
-<p><strong>02:55.794</strong> total execution time for 16 files <strong>from all galleries</strong>:</p>
+<p><strong>03:00.312</strong> total execution time for 16 files <strong>from all galleries</strong>:</p>
 <div class="docutils container">
 <style scoped>
 <link href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/5.3.0/css/bootstrap.min.css" rel="stylesheet" />
@@ -426,67 +426,67 @@ <h1>Computation times</h1>
 </thead>
 <tbody>
 <tr class="row-even"><td><p><a class="reference internal" href="auto_examples/it/plot_entropies_mvar.html#sphx-glr-auto-examples-it-plot-entropies-mvar-py"><span class="std std-ref">Comparison of entropy estimators for a multivariate normal</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/it/plot_entropies_mvar.py</span></code>)</p></td>
-<td><p>00:32.307</p></td>
+<td><p>00:32.953</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="auto_examples/it/plot_mi_high_dimensional.html#sphx-glr-auto-examples-it-plot-mi-high-dimensional-py"><span class="std std-ref">Comparison of MI estimators with high-dimensional data</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/it/plot_mi_high_dimensional.py</span></code>)</p></td>
-<td><p>00:30.210</p></td>
+<td><p>00:30.613</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="reference internal" href="auto_examples/it/plot_entropy_high_dimensional.html#sphx-glr-auto-examples-it-plot-entropy-high-dimensional-py"><span class="std std-ref">Comparison of entropy estimators with high-dimensional data</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/it/plot_entropy_high_dimensional.py</span></code>)</p></td>
-<td><p>00:24.120</p></td>
+<td><p>00:24.568</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="auto_examples/it/plot_mi.html#sphx-glr-auto-examples-it-plot-mi-py"><span class="std std-ref">Comparison of mutual-information estimators</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/it/plot_mi.py</span></code>)</p></td>
-<td><p>00:20.743</p></td>
+<td><p>00:21.318</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="reference internal" href="auto_examples/it/plot_entropies.html#sphx-glr-auto-examples-it-plot-entropies-py"><span class="std std-ref">Comparison of entropy estimators for various distributions</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/it/plot_entropies.py</span></code>)</p></td>
-<td><p>00:20.386</p></td>
+<td><p>00:20.804</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="auto_examples/statistics/plot_bootstrapping.html#sphx-glr-auto-examples-statistics-plot-bootstrapping-py"><span class="std std-ref">Bootstrapping and confidence interval</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/statistics/plot_bootstrapping.py</span></code>)</p></td>
-<td><p>00:15.466</p></td>
+<td><p>00:16.117</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="reference internal" href="auto_examples/tutorials/plot_ml_vs_it.html#sphx-glr-auto-examples-tutorials-plot-ml-vs-it-py"><span class="std std-ref">Machine-learning vs. Information theoretic approaches for HOI</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/tutorials/plot_ml_vs_it.py</span></code>)</p></td>
-<td><p>00:09.166</p></td>
+<td><p>00:09.275</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="auto_examples/tutorials/plot_sim_red_syn.html#sphx-glr-auto-examples-tutorials-plot-sim-red-syn-py"><span class="std std-ref">How to simulate redundancy and synergy</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/tutorials/plot_sim_red_syn.py</span></code>)</p></td>
-<td><p>00:04.698</p></td>
+<td><p>00:04.841</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="reference internal" href="auto_examples/metrics/plot_rsi.html#sphx-glr-auto-examples-metrics-plot-rsi-py"><span class="std std-ref">Redundancy-Synergy Index</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/metrics/plot_rsi.py</span></code>)</p></td>
-<td><p>00:03.954</p></td>
+<td><p>00:04.191</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="auto_examples/metrics/plot_oinfo.html#sphx-glr-auto-examples-metrics-plot-oinfo-py"><span class="std std-ref">O-information and its derivatives for network behavior and encoding</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/metrics/plot_oinfo.py</span></code>)</p></td>
-<td><p>00:03.717</p></td>
+<td><p>00:03.844</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="reference internal" href="auto_examples/metrics/plot_infotopo.html#sphx-glr-auto-examples-metrics-plot-infotopo-py"><span class="std std-ref">Topological Information : conditioning on orders</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/metrics/plot_infotopo.py</span></code>)</p></td>
-<td><p>00:03.311</p></td>
+<td><p>00:03.613</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="auto_examples/metrics/plot_syn_red_mmi.html#sphx-glr-auto-examples-metrics-plot-syn-red-mmi-py"><span class="std std-ref">Redundancy and Synergy MMI</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/metrics/plot_syn_red_mmi.py</span></code>)</p></td>
-<td><p>00:03.014</p></td>
+<td><p>00:03.228</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="reference internal" href="auto_examples/miscellaneous/plot_inspect_results.html#sphx-glr-auto-examples-miscellaneous-plot-inspect-results-py"><span class="std std-ref">Inspect and plot Higher-Order Interactions</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/miscellaneous/plot_inspect_results.py</span></code>)</p></td>
-<td><p>00:01.665</p></td>
+<td><p>00:01.735</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="auto_examples/metrics/plot_syn_phiID.html#sphx-glr-auto-examples-metrics-plot-syn-phiid-py"><span class="std std-ref">Integrated Information Decomposition</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/metrics/plot_syn_phiID.py</span></code>)</p></td>
-<td><p>00:01.525</p></td>
+<td><p>00:01.668</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="reference internal" href="auto_examples/miscellaneous/tuto_jax.html#sphx-glr-auto-examples-miscellaneous-tuto-jax-py"><span class="std std-ref">Quick introduction to jax</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/miscellaneous/tuto_jax.py</span></code>)</p></td>
-<td><p>00:01.281</p></td>
+<td><p>00:01.308</p></td>
 <td><p>0.0</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="reference internal" href="auto_examples/it/tuto_core.html#sphx-glr-auto-examples-it-tuto-core-py"><span class="std std-ref">Introduction to core information theoretical metrics</span></a> (<code class="docutils literal notranslate"><span class="pre">../examples/it/tuto_core.py</span></code>)</p></td>
-<td><p>00:00.232</p></td>
+<td><p>00:00.237</p></td>
 <td><p>0.0</p></td>
 </tr>
 </tbody>