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Author: mvdoc

_sources/notebooks/shortclips/merged_for_colab.ipynb

@@ -5,7 +5,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 00_setup_colab.ipynb\n"
+    "# 01_setup_colab.ipynb\n"
    ]
   },
   {
@@ -16,8 +16,12 @@
     "# Setup Google Colab\n",
     "\n",
     "In this script, we setup a Google Colab environment. This script will only work\n",
-    "when run from [Google Colab](https://colab.research.google.com/)). You can\n",
-    "skip it if you run the tutorials on your machine.\n"
+    "when run from [Google Colab](https://colab.research.google.com/). You can\n",
+    "skip it if you run the tutorials on your machine.\n",
+    "\n",
+    "> **Note:** This script will install all the required dependencies and download the data.\n",
+    "> It will take around 10 minutes to run, but you need to run it only once in your Colab session.\n",
+    "> If your Colab session is disconnected, you will need to run this script again."
    ]
   },
   {
@@ -30,15 +34,15 @@
     "In Google Colab you can request access to a GPU by changing the runtime type.\n",
     "To do so, click the following menu options in Google Colab:\n",
     "\n",
-    "(Menu) \"Runtime\" -> \"Change runtime type\" -> \"Hardware accelerator\" -> \"GPU\".\n",
+    "> (Menu) \"Runtime\" -> \"Change runtime type\" -> \"Hardware accelerator\" -> \"GPU\".\n",
     "\n"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Download the data and install all required dependencies\n",
+    "## Install all required dependencies and download the data\n",
     "\n",
     "Uncomment and run the following cell to download the required packages.\n",
     "\n"
@@ -187,7 +191,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 01_plot_explainable_variance.ipynb\n"
+    "# 03_compute_explainable_variance.ipynb\n"
    ]
   },
   {
@@ -652,7 +656,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 02_plot_ridge_regression.ipynb\n"
+    "# 04_understand_ridge_regression.ipynb\n"
    ]
   },
   {
@@ -670,7 +674,7 @@
     "Linear regression is a method to model the relation between some input\n",
     "variables $X \\in \\mathbb{R}^{(n \\times p)}$ (the features) and an output\n",
     "variable $y \\in \\mathbb{R}^{n}$ (the target). Specifically, linear\n",
-    "regression uses a vector of coefficient $w \\in \\mathbb{R}^{p}$` to\n",
+    "regression uses a vector of coefficient $w \\in \\mathbb{R}^{p}$ to\n",
     "predict the output\n",
     "\n",
     "\\begin{align}\\hat{y} = Xw\\end{align}\n",
@@ -685,15 +689,6 @@
     "least squares* (OLS).\n"
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": []
-  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -1267,7 +1262,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 03_plot_wordnet_model.ipynb\n"
+    "# 05_fit_wordnet_model.ipynb\n"
    ]
   },
   {
@@ -2268,7 +2263,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 04_plot_hemodynamic_response.ipynb\n"
+    "# 06_visualize_hemodynamic_response.ipynb\n"
    ]
   },
   {
@@ -2855,7 +2850,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 05_plot_motion_energy_model.ipynb\n"
+    "# 08_fit_motion_energy_model.ipynb\n"
    ]
   },
   {
@@ -3368,7 +3363,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 06_plot_banded_ridge_model.ipynb\n"
+    "# 09_fit_banded_ridge_model.ipynb\n"
    ]
   },
   {

_sources/notebooks/shortclips/merged_for_colab_model_fitting.ipynb

@@ -5,7 +5,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 00_setup_colab.ipynb\n"
+    "# 01_setup_colab.ipynb\n"
    ]
   },
   {
@@ -16,8 +16,12 @@
     "# Setup Google Colab\n",
     "\n",
     "In this script, we setup a Google Colab environment. This script will only work\n",
-    "when run from [Google Colab](https://colab.research.google.com/)). You can\n",
-    "skip it if you run the tutorials on your machine.\n"
+    "when run from [Google Colab](https://colab.research.google.com/). You can\n",
+    "skip it if you run the tutorials on your machine.\n",
+    "\n",
+    "> **Note:** This script will install all the required dependencies and download the data.\n",
+    "> It will take around 10 minutes to run, but you need to run it only once in your Colab session.\n",
+    "> If your Colab session is disconnected, you will need to run this script again."
    ]
   },
   {
@@ -30,15 +34,15 @@
     "In Google Colab you can request access to a GPU by changing the runtime type.\n",
     "To do so, click the following menu options in Google Colab:\n",
     "\n",
-    "(Menu) \"Runtime\" -> \"Change runtime type\" -> \"Hardware accelerator\" -> \"GPU\".\n",
+    "> (Menu) \"Runtime\" -> \"Change runtime type\" -> \"Hardware accelerator\" -> \"GPU\".\n",
     "\n"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Download the data and install all required dependencies\n",
+    "## Install all required dependencies and download the data\n",
     "\n",
     "Uncomment and run the following cell to download the required packages.\n",
     "\n"
@@ -187,7 +191,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 01_plot_explainable_variance.ipynb\n"
+    "# 03_compute_explainable_variance.ipynb\n"
    ]
   },
   {
@@ -652,7 +656,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 03_plot_wordnet_model.ipynb\n"
+    "# 05_fit_wordnet_model.ipynb\n"
    ]
   },
   {
@@ -1653,7 +1657,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 05_plot_motion_energy_model.ipynb\n"
+    "# 08_fit_motion_energy_model.ipynb\n"
    ]
   },
   {
@@ -2166,7 +2170,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "# 06_plot_banded_ridge_model.ipynb\n"
+    "# 09_fit_banded_ridge_model.ipynb\n"
    ]
   },
   {

notebooks/shortclips/03_compute_explainable_variance.html

@@ -436,8 +436,8 @@ <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-7c4b943e-035d-41fd-9084-2bea6fe5d1be">
-<span class="eqno">(1)<a class="headerlink" href="#equation-7c4b943e-035d-41fd-9084-2bea6fe5d1be" 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-8571b8dd-458d-4804-9a62-21ea954a14fe">
+<span class="eqno">(1)<a class="headerlink" href="#equation-8571b8dd-458d-4804-9a62-21ea954a14fe" 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="merged_for_colab_model_fitting.html#id130" 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="merged_for_colab_model_fitting.html#id131" 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="merged_for_colab_model_fitting.html#id132" 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 cross-validation<a class="headerlink" href="
 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-253f3f88-d090-4f7f-9115-281a7a2e27fe">
-<span class="eqno">(2)<a class="headerlink" href="#equation-253f3f88-d090-4f7f-9115-281a7a2e27fe" title="Permalink to this equation">#</a></span>\[\begin{align}\hat{y} = Xw\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-010741a2-76d5-4c9c-9869-185c0d3cb2c4">
+<span class="eqno">(2)<a class="headerlink" href="#equation-010741a2-76d5-4c9c-9869-185c0d3cb2c4" 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-af5204d4-43c3-4e87-b187-c958e6c90d61">
-<span class="eqno">(3)<a class="headerlink" href="#equation-af5204d4-43c3-4e87-b187-c958e6c90d61" 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-04e7999a-297e-4b03-8416-a103d7ff49bf">
+<span class="eqno">(3)<a class="headerlink" href="#equation-04e7999a-297e-4b03-8416-a103d7ff49bf" 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-ca508ccb-fc89-4b5c-8482-e34fff729b99">
-<span class="eqno">(4)<a class="headerlink" href="#equation-ca508ccb-fc89-4b5c-8482-e34fff729b99" 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-08541323-7db5-4c17-bea4-760241648a56">
+<span class="eqno">(4)<a class="headerlink" href="#equation-08541323-7db5-4c17-bea4-760241648a56" 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-7ac6f82b-dd61-4d06-89f7-cf9ea8829f67">
-<span class="eqno">(5)<a class="headerlink" href="#equation-7ac6f82b-dd61-4d06-89f7-cf9ea8829f67" 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-152e4fe6-61c3-4b05-b6f4-2b165911ccca">
+<span class="eqno">(5)<a class="headerlink" href="#equation-152e4fe6-61c3-4b05-b6f4-2b165911ccca" 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-34f11fc4-5e65-4369-9500-5fc9adbf0f32">
-<span class="eqno">(6)<a class="headerlink" href="#equation-34f11fc4-5e65-4369-9500-5fc9adbf0f32" 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-492d00e7-ad9c-4c41-8a51-e718f0553dd4">
+<span class="eqno">(6)<a class="headerlink" href="#equation-492d00e7-ad9c-4c41-8a51-e718f0553dd4" 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-a3feabde-5706-4bdb-bb4b-47369e82c744">
-<span class="eqno">(7)<a class="headerlink" href="#equation-a3feabde-5706-4bdb-bb4b-47369e82c744" title="Permalink to this equation">#</a></span>\[\begin{align}\beta = X^\top w\end{align}\]</div>
+<div class="amsmath math notranslate nohighlight" id="equation-679d04fd-27f8-45fb-895d-39c2b830da09">
+<span class="eqno">(7)<a class="headerlink" href="#equation-679d04fd-27f8-45fb-895d-39c2b830da09" 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/09_fit_banded_ridge_model.html

@@ -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-c0687345-1407-48d0-8dd1-c430d921e59a">
-<span class="eqno">(8)<a class="headerlink" href="#equation-c0687345-1407-48d0-8dd1-c430d921e59a" 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-7b30bc1c-345f-4f87-b7b1-398ba820b074">
+<span class="eqno">(8)<a class="headerlink" href="#equation-7b30bc1c-345f-4f87-b7b1-398ba820b074" 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>