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@@ -5,7 +5,7 @@ Welcome to the tutorials on the Voxelwise Encoding Model framework from the
 
 If you use these tutorials for your work, consider citing the corresponding paper:
 
-> T. Dupré La Tour, M. Visconti di Oleggio Castello, and J. L. Gallant. The Voxelwise Encoding Model framework: a tutorial introduction to fitting encoding models to fMRI data. PsyArXiv, 2024. [doi:10.31234/osf.io/t975e.](https://doi.org/10.31234/osf.io/t975e)
+> T. Dupré la Tour, M. Visconti di Oleggio Castello, and J. L. Gallant. The Voxelwise Encoding Model framework: a tutorial introduction to fitting encoding models to fMRI data. PsyArXiv, 2024. [doi:10.31234/osf.io/t975e.](https://doi.org/10.31234/osf.io/t975e)
 
 You can find a copy of the paper [here](https://github.com/gallantlab/voxelwise_tutorials/blob/main/paper/voxelwise_tutorials_paper.pdf).
 

notebooks/shortclips/03_compute_explainable_variance.html

@@ -437,8 +437,8 @@ <h1>Compute the explainable variance<a class="headerlink" href="#compute-the-exp
 across repetitions. For each repeat, we define the residual timeseries between
 brain response and average brain response as <span class="math notranslate nohighlight">\(r_i = y_i - \bar{y}\)</span>. The
 explainable variance (EV) is estimated as</p>
-<div class="amsmath math notranslate nohighlight" id="equation-6b5c66f3-6a1c-4d78-b878-859915a0fd50">
-<span class="eqno">(1)<a class="headerlink" href="#equation-6b5c66f3-6a1c-4d78-b878-859915a0fd50" title="Permalink to this equation">#</a></span>\[\begin{align}\text{EV} = \frac{1}{N}\sum_{i=1}^N\text{Var}(y_i) - \frac{N}{N-1}\sum_{i=1}^N\text{Var}(r_i)\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-8e371fe2-e3f6-4e66-94c0-74505eac3a2c">
+<span class="eqno">(1)<a class="headerlink" href="#equation-8e371fe2-e3f6-4e66-94c0-74505eac3a2c" title="Permalink to this equation">#</a></span>\[\begin{align}\text{EV} = \frac{1}{N}\sum_{i=1}^N\text{Var}(y_i) - \frac{N}{N-1}\sum_{i=1}^N\text{Var}(r_i)\end{align}\]</div>
 <p>In the literature, the explainable variance is also known as the <em>signal
 power</em>.</p>
 <p>For more information, see <span id="id1">Sahani and Linden [<a class="reference internal" href="vem_tutorials_merged_for_colab_model_fitting.html#id158" title="M. Sahani and J. Linden. How linear are auditory cortical responses? Adv. Neural Inf. Process. Syst., 2002.">2002</a>]</span>, <span id="id2">Hsu <em>et al.</em> [<a class="reference internal" href="vem_tutorials_merged_for_colab_model_fitting.html#id159" title="A. Hsu, A. Borst, and F. E. Theunissen. Quantifying variability in neural responses and its application for the validation of model predictions. Network, 2004.">2004</a>]</span>, and <span id="id3">Schoppe <em>et al.</em> [<a class="reference internal" href="vem_tutorials_merged_for_colab_model_fitting.html#id160" title="O. Schoppe, N. S. Harper, B. Willmore, A. King, and J. Schnupp. Measuring the performance of neural models. Front. Comput. Neurosci., 2016.">2016</a>]</span>.</p>

notebooks/shortclips/04_understand_ridge_regression.html

@@ -422,13 +422,13 @@ <h1>Understand ridge regression and hyperparameter selection<a class="headerlink
 variable <span class="math notranslate nohighlight">\(y \in \mathbb{R}^{n}\)</span> (the target). Specifically, linear
 regression uses a vector of coefficient <span class="math notranslate nohighlight">\(w \in \mathbb{R}^{p}\)</span> to
 predict the output</p>
-<div class="amsmath math notranslate nohighlight" id="equation-4d8e7a72-f95e-45b0-8300-c88feaa9beea">
-<span class="eqno">(2)<a class="headerlink" href="#equation-4d8e7a72-f95e-45b0-8300-c88feaa9beea" title="Permalink to this equation">#</a></span>\[\begin{align}\hat{y} = Xw\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-44016c9a-386e-4c5d-b726-0c593bd1d3d6">
+<span class="eqno">(2)<a class="headerlink" href="#equation-44016c9a-386e-4c5d-b726-0c593bd1d3d6" title="Permalink to this equation">#</a></span>\[\begin{align}\hat{y} = Xw\end{align}\]</div>
 <p>The model is considered accurate if the predictions <span class="math notranslate nohighlight">\(\hat{y}\)</span> are close
 to the true output values <span class="math notranslate nohighlight">\(y\)</span>. Therefore,  a good linear regression model
 is given by the vector <span class="math notranslate nohighlight">\(w\)</span> that minimizes the sum of squared errors:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-1a0b17c5-4527-49ed-82d6-6178a4173e16">
-<span class="eqno">(3)<a class="headerlink" href="#equation-1a0b17c5-4527-49ed-82d6-6178a4173e16" title="Permalink to this equation">#</a></span>\[\begin{align}w = \arg\min_w ||Xw - y||^2\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-164582e6-d0f8-4ec6-a812-94b6f84c155f">
+<span class="eqno">(3)<a class="headerlink" href="#equation-164582e6-d0f8-4ec6-a812-94b6f84c155f" title="Permalink to this equation">#</a></span>\[\begin{align}w = \arg\min_w ||Xw - y||^2\end{align}\]</div>
 <p>This is the simplest model for linear regression, and it is known as <em>ordinary
 least squares</em> (OLS).</p>
 <section id="ordinary-least-squares-ols">
@@ -480,8 +480,8 @@ <h2>Ordinary least squares (OLS)<a class="headerlink" href="#ordinary-least-squa
 </div>
 <p>The linear coefficient leading to the minimum squared loss can be found
 analytically with the formula:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-39886a10-bc4c-413a-96a6-a176b7120b6f">
-<span class="eqno">(4)<a class="headerlink" href="#equation-39886a10-bc4c-413a-96a6-a176b7120b6f" title="Permalink to this equation">#</a></span>\[\begin{align}w = (X^\top X)^{-1}  X^\top y\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-ec79a861-adc8-488d-b845-bd06194ee484">
+<span class="eqno">(4)<a class="headerlink" href="#equation-ec79a861-adc8-488d-b845-bd06194ee484" title="Permalink to this equation">#</a></span>\[\begin{align}w = (X^\top X)^{-1}  X^\top y\end{align}\]</div>
 <p>This is the OLS solution.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
@@ -621,8 +621,8 @@ <h2>Ridge regression<a class="headerlink" href="#ridge-regression" title="Link t
 <p>To solve the instability and under-determinacy issues of OLS, OLS can be
 extended to <em>ridge regression</em>. Ridge regression considers a different
 optimization problem:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-68b52747-55d0-4018-84bc-cf843e167110">
-<span class="eqno">(5)<a class="headerlink" href="#equation-68b52747-55d0-4018-84bc-cf843e167110" title="Permalink to this equation">#</a></span>\[\begin{align}w = \arg\min_w ||Xw - y||^2 + \alpha ||w||^2\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-f6de4535-4112-4241-8db4-1486e978a489">
+<span class="eqno">(5)<a class="headerlink" href="#equation-f6de4535-4112-4241-8db4-1486e978a489" title="Permalink to this equation">#</a></span>\[\begin{align}w = \arg\min_w ||Xw - y||^2 + \alpha ||w||^2\end{align}\]</div>
 <p>This optimization problem contains two terms: (i) a <em>data-fitting term</em>
 <span class="math notranslate nohighlight">\(||Xw - y||^2\)</span>, which ensures the regression correctly fits the
 training data; and (ii) a regularization term <span class="math notranslate nohighlight">\(\alpha||w||^2\)</span>, which
@@ -654,8 +654,8 @@ <h2>Ridge regression<a class="headerlink" href="#ridge-regression" title="Link t
 <p>To understand why the regularization term makes the solution more robust to
 noise, let’s consider the ridge solution. The ridge solution can be found
 analytically with the formula:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-1886f38e-bd35-4129-8cda-03e4abec2868">
-<span class="eqno">(6)<a class="headerlink" href="#equation-1886f38e-bd35-4129-8cda-03e4abec2868" title="Permalink to this equation">#</a></span>\[\begin{align}w = (X^\top X + \alpha I)^{-1}  X^\top y\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-ea124788-3420-4a26-b1fa-638985501426">
+<span class="eqno">(6)<a class="headerlink" href="#equation-ea124788-3420-4a26-b1fa-638985501426" title="Permalink to this equation">#</a></span>\[\begin{align}w = (X^\top X + \alpha I)^{-1}  X^\top y\end{align}\]</div>
 <p>where <code class="docutils literal notranslate"><span class="pre">I</span></code> is the identity matrix. In this formula, we can see that the
 inverted matrix is now <span class="math notranslate nohighlight">\((X^\top X + \alpha I)\)</span>. Compared to OLS, the
 additional term <span class="math notranslate nohighlight">\(\alpha I\)</span> adds a positive value <code class="docutils literal notranslate"><span class="pre">alpha</span></code> to all

notebooks/shortclips/06_visualize_hemodynamic_response.html

@@ -1660,8 +1660,8 @@ <h2>Visualize the HRF<a class="headerlink" href="#visualize-the-hrf" title="Link
 coefficients <span class="math notranslate nohighlight">\(\beta\)</span> obtained with a ridge regression, but the primal
 coefficients can be computed from the dual coefficients using the training
 features <span class="math notranslate nohighlight">\(X\)</span>:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-a4fc0a20-0616-4955-85a5-056abdb2afbb">
-<span class="eqno">(7)<a class="headerlink" href="#equation-a4fc0a20-0616-4955-85a5-056abdb2afbb" title="Permalink to this equation">#</a></span>\[\begin{align}\beta = X^\top w\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-647c6f7b-f485-4012-a2fa-40d3832f711b">
+<span class="eqno">(7)<a class="headerlink" href="#equation-647c6f7b-f485-4012-a2fa-40d3832f711b" title="Permalink to this equation">#</a></span>\[\begin{align}\beta = X^\top w\end{align}\]</div>
 <p>To better visualize the HRF, we will refit a model with more delays, but only
 on a selection of voxels to speed up the computations.</p>
 <div class="cell docutils container">

notebooks/shortclips/08_fit_motion_energy_model.html

@@ -1198,7 +1198,7 @@ <h2>Compare with the wordnet model<a class="headerlink" href="#compare-with-the-
 could predict responses in face-responsive areas without encoding any
 semantic information.</p>
 <p>To better disentangle the two feature spaces, we developed a joint model
-called <code class="docutils literal notranslate"><span class="pre">banded</span> <span class="pre">ridge</span> <span class="pre">regression</span></code> <span id="id4">[<a class="reference internal" href="../../pages/voxelwise_modeling.html#id25" title="T. Dupré La Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. NeuroImage, 267:119728, 2022. doi:10.1016/j.neuroimage.2022.119728.">Dupré La Tour <em>et al.</em>, 2022</a>, <a class="reference internal" href="../../pages/voxelwise_modeling.html#id22" title="A. O. Nunez-Elizalde, A. G. Huth, and J. L. Gallant. Voxelwise encoding models with non-spherical multivariate normal priors. Neuroimage, 197:482–492, 2019.">Nunez-Elizalde <em>et al.</em>, 2019</a>]</span>, which fits multiple feature spaces
+called <code class="docutils literal notranslate"><span class="pre">banded</span> <span class="pre">ridge</span> <span class="pre">regression</span></code> <span id="id4">[<a class="reference internal" href="../../pages/voxelwise_modeling.html#id25" title="T. Dupré la Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. NeuroImage, 267:119728, 2022. doi:10.1016/j.neuroimage.2022.119728.">Dupré la Tour <em>et al.</em>, 2022</a>, <a class="reference internal" href="../../pages/voxelwise_modeling.html#id22" title="A. O. Nunez-Elizalde, A. G. Huth, and J. L. Gallant. Voxelwise encoding models with non-spherical multivariate normal priors. Neuroimage, 197:482–492, 2019.">Nunez-Elizalde <em>et al.</em>, 2019</a>]</span>, which fits multiple feature spaces
 simultaneously with optimal regularization for each feature space. This model
 is described in the next example.</p>
 </section>
@@ -1207,8 +1207,8 @@ <h2>References<a class="headerlink" href="#references" title="Link to this headi
 <div class="docutils container" id="id5">
 <div role="list" class="citation-list">
 <div class="citation" id="id20" role="doc-biblioentry">
-<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">DLTENEG22</a><span class="fn-bracket">]</span></span>
-<p>T. Dupré La Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. <em>NeuroImage</em>, 267:119728, 2022. <a class="reference external" href="https://doi.org/10.1016/j.neuroimage.2022.119728">doi:10.1016/j.neuroimage.2022.119728</a>.</p>
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">DlTENEG22</a><span class="fn-bracket">]</span></span>
+<p>T. Dupré la Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. <em>NeuroImage</em>, 267:119728, 2022. <a class="reference external" href="https://doi.org/10.1016/j.neuroimage.2022.119728">doi:10.1016/j.neuroimage.2022.119728</a>.</p>
 </div>
 <div class="citation" id="id8" role="doc-biblioentry">
 <span class="label"><span class="fn-bracket">[</span>NVN+11<span class="fn-bracket">]</span></span>

notebooks/shortclips/09_fit_banded_ridge_model.html

@@ -422,7 +422,7 @@ <h1>Fit a voxelwise encoding model with both WordNet and motion-energy features<
 with two different feature spaces: motion energy and wordnet categories.</p>
 <p><em>Banded ridge regression:</em> Since the relative scaling of both feature spaces is
 unknown, we use two regularization hyperparameters (one per feature space) in a
-model called banded ridge regression <span id="id1">[<a class="reference internal" href="../../pages/voxelwise_modeling.html#id25" title="T. Dupré La Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. NeuroImage, 267:119728, 2022. doi:10.1016/j.neuroimage.2022.119728.">Dupré La Tour <em>et al.</em>, 2022</a>, <a class="reference internal" href="../../pages/voxelwise_modeling.html#id22" title="A. O. Nunez-Elizalde, A. G. Huth, and J. L. Gallant. Voxelwise encoding models with non-spherical multivariate normal priors. Neuroimage, 197:482–492, 2019.">Nunez-Elizalde <em>et al.</em>, 2019</a>]</span>.
+model called banded ridge regression <span id="id1">[<a class="reference internal" href="../../pages/voxelwise_modeling.html#id25" title="T. Dupré la Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. NeuroImage, 267:119728, 2022. doi:10.1016/j.neuroimage.2022.119728.">Dupré la Tour <em>et al.</em>, 2022</a>, <a class="reference internal" href="../../pages/voxelwise_modeling.html#id22" title="A. O. Nunez-Elizalde, A. G. Huth, and J. L. Gallant. Voxelwise encoding models with non-spherical multivariate normal priors. Neuroimage, 197:482–492, 2019.">Nunez-Elizalde <em>et al.</em>, 2019</a>]</span>.
 Just like with ridge regression, we optimize the hyperparameters over cross-validation.
 An efficient implementation of this model is available in the
 <a class="reference external" href="https://github.com/gallantlab/himalaya">himalaya</a> package.</p>
@@ -2827,8 +2827,8 @@ <h2>Plot the banded ridge split<a class="headerlink" href="#plot-the-banded-ridg
 take the kernel weights and the ridge (dual) weights corresponding to each
 feature space, and use them to compute the prediction from each feature space
 separately.</p>
-<div class="amsmath math notranslate nohighlight" id="equation-c515e9be-8919-4434-9ba6-8408cf6ab8bd">
-<span class="eqno">(8)<a class="headerlink" href="#equation-c515e9be-8919-4434-9ba6-8408cf6ab8bd" title="Permalink to this equation">#</a></span>\[\begin{align}\hat{y} = \sum_i^m \hat{y}_i = \sum_i^m \gamma_i K_i \hat{w}\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-eab8ce78-6b79-4fd7-ad7d-866dfd1987d3">
+<span class="eqno">(8)<a class="headerlink" href="#equation-eab8ce78-6b79-4fd7-ad7d-866dfd1987d3" title="Permalink to this equation">#</a></span>\[\begin{align}\hat{y} = \sum_i^m \hat{y}_i = \sum_i^m \gamma_i K_i \hat{w}\end{align}\]</div>
 <p>Then, we use these split predictions to compute split <span class="math notranslate nohighlight">\(\tilde{R}^2_i\)</span>
 scores. These scores are corrected so that their sum is equal to the
 <span class="math notranslate nohighlight">\(R^2\)</span> score of the full prediction <span class="math notranslate nohighlight">\(\hat{y}\)</span>.</p>
@@ -2887,16 +2887,16 @@ <h2>Plot the banded ridge split<a class="headerlink" href="#plot-the-banded-ridg
 motion-energy features predict brain activity in early visual cortex, while
 wordnet features predict in semantic visual areas. For more discussions about
 these results, we refer the reader to the publications describing the banded ridge
-regression approach <span id="id2">[<a class="reference internal" href="../../pages/voxelwise_modeling.html#id25" title="T. Dupré La Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. NeuroImage, 267:119728, 2022. doi:10.1016/j.neuroimage.2022.119728.">Dupré La Tour <em>et al.</em>, 2022</a>, <a class="reference internal" href="../../pages/voxelwise_modeling.html#id22" title="A. O. Nunez-Elizalde, A. G. Huth, and J. L. Gallant. Voxelwise encoding models with non-spherical multivariate normal priors. Neuroimage, 197:482–492, 2019.">Nunez-Elizalde <em>et al.</em>, 2019</a>]</span>.</p>
+regression approach <span id="id2">[<a class="reference internal" href="../../pages/voxelwise_modeling.html#id25" title="T. Dupré la Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. NeuroImage, 267:119728, 2022. doi:10.1016/j.neuroimage.2022.119728.">Dupré la Tour <em>et al.</em>, 2022</a>, <a class="reference internal" href="../../pages/voxelwise_modeling.html#id22" title="A. O. Nunez-Elizalde, A. G. Huth, and J. L. Gallant. Voxelwise encoding models with non-spherical multivariate normal priors. Neuroimage, 197:482–492, 2019.">Nunez-Elizalde <em>et al.</em>, 2019</a>]</span>.</p>
 </section>
 <section id="references">
 <h2>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h2>
 <div class="docutils container" id="id3">
 <div role="list" class="citation-list">
 <div class="citation" id="id18" role="doc-biblioentry">
-<span class="label"><span class="fn-bracket">[</span>DLTENEG22<span class="fn-bracket">]</span></span>
+<span class="label"><span class="fn-bracket">[</span>DlTENEG22<span class="fn-bracket">]</span></span>
 <span class="backrefs">(<a role="doc-backlink" href="#id1">1</a>,<a role="doc-backlink" href="#id2">2</a>)</span>
-<p>T. Dupré La Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. <em>NeuroImage</em>, 267:119728, 2022. <a class="reference external" href="https://doi.org/10.1016/j.neuroimage.2022.119728">doi:10.1016/j.neuroimage.2022.119728</a>.</p>
+<p>T. Dupré la Tour, M. Eickenberg, A.O. Nunez-Elizalde, and J. L. Gallant. Feature-space selection with banded ridge regression. <em>NeuroImage</em>, 267:119728, 2022. <a class="reference external" href="https://doi.org/10.1016/j.neuroimage.2022.119728">doi:10.1016/j.neuroimage.2022.119728</a>.</p>
 </div>
 <div class="citation" id="id15" role="doc-biblioentry">
 <span class="label"><span class="fn-bracket">[</span>NEHG19<span class="fn-bracket">]</span></span>