minor note 11 fix

constant_model_loss_transformations/loss_transformations.qmd

@@ -21,7 +21,7 @@ jupyter:
       format_version: '1.0'
       jupytext_version: 1.16.1
   kernelspec:
-    display_name: Python 3 (ipykernel)
+    display_name: ds100env
     language: python
     name: python3
 ---
@@ -116,7 +116,7 @@ $$
 \begin{align}
 0 &= {\frac{-2}{n}}\sum^{n}_{i=1} (y_i - \hat{\theta_0})
 \\ &= \sum^{n}_{i=1} (y_i - \hat{\theta_0}) \quad \quad \text{divide both sides by} \frac{-2}{n}
-\\ &= \left(\sum^{n}_{i=1} y_i\right) - \left(\sum^{n}_{i=1} \theta_0\right) \quad \quad \text{separate sums}
+\\ &= \left(\sum^{n}_{i=1} y_i\right) - \left(\sum^{n}_{i=1} \hat{\theta_0}\right) \quad \quad \text{separate sums}
 \\ &= \left(\sum^{n}_{i=1} y_i\right) - (n \cdot \hat{\theta_0}) \quad \quad  \text{c + c + … + c = nc}
 \\ n \cdot \hat{\theta_0} &= \sum^{n}_{i=1} y_i
 \\ \hat{\theta_0} &= \frac{1}{n} \sum^{n}_{i=1} y_i
@@ -159,7 +159,6 @@ The code for generating the graphs and models is included below, but we won't go
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 import numpy as np
 import pandas as pd
 import matplotlib.pyplot as plt
@@ -174,7 +173,6 @@ data_linear = dugongs[["Length", "Age"]]
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 # Big font helper
 def adjust_fontsize(size=None):
     SMALL_SIZE = 8
@@ -196,7 +194,6 @@ plt.style.use("default")  # Revert style to default mpl
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 # Constant Model + MSE
 plt.style.use('default') # Revert style to default mpl
 adjust_fontsize(size=16)
@@ -222,7 +219,6 @@ plt.legend();
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 # SLR + MSE
 def mse_linear(theta_0, theta_1, data_linear):
     data_x, data_y = data_linear.iloc[:, 0], data_linear.iloc[:, 1]
@@ -278,7 +274,6 @@ ax.set_zlabel("MSE");
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 # Predictions
 yobs = data_linear["Age"]  # The true observations y
 xs = data_linear["Length"]  # Needed for linear predictions
@@ -290,7 +285,6 @@ yhats_linear = [theta_0_hat + theta_1_hat * x for x in xs]
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 # Constant Model Rug Plot
 # In case we're in a weird style state
 sns.set_theme()
@@ -307,7 +301,6 @@ plt.yticks([]);
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 # SLR model scatter plot 
 # In case we're in a weird style state
 sns.set_theme()
@@ -421,7 +414,6 @@ Let's consider a dataset where each entry represents the number of drinks sold a
 
 ```{python}
 #| code-fold: false
-#| vscode: {languageId: python}
 drinks = np.array([20, 21, 22, 29, 33])
 drinks
 ```
@@ -430,7 +422,6 @@ From our derivations above, we know that the optimal model parameter under MSE c
 
 ```{python}
 #| code-fold: false
-#| vscode: {languageId: python}
 np.mean(drinks), np.median(drinks)
 ```
 
@@ -444,7 +435,6 @@ How do outliers affect each cost function? Imagine we replace the largest value
 
 ```{python}
 #| code-fold: false
-#| vscode: {languageId: python}
 drinks_with_outlier = np.append(drinks, 1033)
 display(drinks_with_outlier)
 np.mean(drinks_with_outlier), np.median(drinks_with_outlier)
@@ -458,7 +448,6 @@ Let's try another experiment. This time, we'll add an additional, non-outlying d
 
 ```{python}
 #| code-fold: false
-#| vscode: {languageId: python}
 drinks_with_additional_observation = np.append(drinks, 35)
 drinks_with_additional_observation
 ```
@@ -502,7 +491,6 @@ Let's revisit our dugongs example. The lengths and ages are plotted below:
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 # `corrcoef` computes the correlation coefficient between two variables
 # `std` finds the standard deviation
 x = dugongs["Length"]
@@ -530,7 +518,6 @@ An important word on $\log$: in Data 100 (and most upper-division STEM courses),
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 z = np.log(y)
 
 r = np.corrcoef(x, z)[0, 1]
@@ -568,7 +555,6 @@ $y$ is an *exponential* function of $x$. Applying an exponential fit to the untr
 
 ```{python}
 #| code-fold: true
-#| vscode: {languageId: python}
 plt.figure(dpi=120, figsize=(4, 3))
 
 plt.scatter(x, y)

docs/intro_to_modeling/intro_to_modeling.html

@@ -418,7 +418,7 @@ <h2 data-number="10.2" class="anchored" data-anchor-id="simple-linear-regression
 <li><span class="math inline">\(\text{regression estimate} = y\text{-intercept} + \text{slope}\cdot\text{}x\)</span></li>
 <li><span class="math inline">\(\text{residual} =\text{observed }y - \text{regression estimate}\)</span></li>
 </ul>
-<div id="1ab7ac5a" class="cell" data-execution_count="1">
+<div id="d6f418ab" class="cell" data-execution_count="1">
 <details class="code-fold">
 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> pandas <span class="im">as</span> pd</span>
@@ -465,7 +465,7 @@ <h4 data-number="10.2.1.2" class="anchored" data-anchor-id="correlation"><span c
 <li>Correlations range between -1 and 1: <span class="math inline">\(|r| \leq 1\)</span>, with <span class="math inline">\(r=1\)</span> indicating perfect positive linear association, and <span class="math inline">\(r=-1\)</span> indicating perfect negative association. The closer <span class="math inline">\(r\)</span> is to <span class="math inline">\(0\)</span>, the weaker the linear association is.</li>
 <li>Correlation says nothing about causation and non-linear association. Correlation does <strong>not</strong> imply causation. When <span class="math inline">\(r = 0\)</span>, the two variables are uncorrelated. However, they could still be related through some non-linear relationship.</li>
 </ol>
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 <details class="code-fold">
 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> plot_and_get_corr(ax, x, y, title):</span>
@@ -689,7 +689,7 @@ <h2 data-number="10.7" class="anchored" data-anchor-id="evaluating-the-slr-model
 <section id="four-mysterious-datasets-anscombes-quartet" class="level3" data-number="10.7.1">
 <h3 data-number="10.7.1" class="anchored" data-anchor-id="four-mysterious-datasets-anscombes-quartet"><span class="header-section-number">10.7.1</span> Four Mysterious Datasets (Anscombe’s quartet)</h3>
 <p>Let’s take a look at four different datasets.</p>
-<div id="f06c2852" class="cell" data-execution_count="3">
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 <details class="code-fold">
 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
@@ -701,7 +701,7 @@ <h3 data-number="10.7.1" class="anchored" data-anchor-id="four-mysterious-datase
 <span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> mpl_toolkits.mplot3d <span class="im">import</span> Axes3D</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </details>
 </div>
-<div id="0a263548" class="cell" data-execution_count="4">
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 <details class="code-fold">
 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Big font helper</span></span>
@@ -755,7 +755,7 @@ <h3 data-number="10.7.1" class="anchored" data-anchor-id="four-mysterious-datase
 <span id="cb4-49"><a href="#cb4-49" aria-hidden="true" tabindex="-1"></a>plt.style.use(<span class="st">"default"</span>)  <span class="co"># Revert style to default mpl</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </details>
 </div>
-<div id="7bec2a87" class="cell" data-execution_count="5">
+<div id="f0c62211" class="cell" data-execution_count="5">
 <details class="code-fold">
 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>plt.style.use(<span class="st">"default"</span>)  <span class="co"># Revert style to default mpl</span></span>
@@ -794,7 +794,7 @@ <h3 data-number="10.7.1" class="anchored" data-anchor-id="four-mysterious-datase
 <span id="cb5-34"><a href="#cb5-34" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> fig</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </details>
 </div>
-<div id="c6b48559" class="cell" data-execution_count="6">
+<div id="af3f30cd" class="cell" data-execution_count="6">
 <details class="code-fold">
 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb6"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Load in four different datasets: I, II, III, IV</span></span>
@@ -837,7 +837,7 @@ <h3 data-number="10.7.1" class="anchored" data-anchor-id="four-mysterious-datase
 </div>
 </div>
 <p>While these four sets of datapoints look very different, they actually all have identical means <span class="math inline">\(\bar x\)</span>, <span class="math inline">\(\bar y\)</span>, standard deviations <span class="math inline">\(\sigma_x\)</span>, <span class="math inline">\(\sigma_y\)</span>, correlation <span class="math inline">\(r\)</span>, and RMSE! If we only look at these statistics, we would probably be inclined to say that these datasets are similar.</p>
-<div id="2fbfddc5" class="cell" data-execution_count="7">
+<div id="d764b493" class="cell" data-execution_count="7">
 <details class="code-fold">
 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> dataset <span class="kw">in</span> [<span class="st">"I"</span>, <span class="st">"II"</span>, <span class="st">"III"</span>, <span class="st">"IV"</span>]:</span>
@@ -884,7 +884,7 @@ <h3 data-number="10.7.1" class="anchored" data-anchor-id="four-mysterious-datase
 </div>
 <p>We may also wish to visualize the model’s <strong>residuals</strong>, defined as the difference between the observed and predicted <span class="math inline">\(y_i\)</span> value (<span class="math inline">\(e_i = y_i - \hat{y}_i\)</span>). This gives a high-level view of how “off” each prediction is from the true observed value. Recall that you explored this concept in <a href="https://inferentialthinking.com/chapters/15/5/Visual_Diagnostics.html?highlight=heteroscedasticity#detecting-heteroscedasticity">Data 8</a>: a good regression fit should display no clear pattern in its plot of residuals. The residual plots for Anscombe’s quartet are displayed below. Note how only the first plot shows no clear pattern to the magnitude of residuals. This is an indication that SLR is not the best choice of model for the remaining three sets of points.</p>
 <!-- <img src="images/residual.png" alt='residual' width='600'> -->
-<div id="fd64711b" class="cell" data-execution_count="8">
+<div id="1d07f513" class="cell" data-execution_count="8">
 <details class="code-fold">
 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Residual visualization</span></span>

docs/ols/ols.html

@@ -356,15 +356,15 @@ <h3 data-number="12.1.1" class="anchored" data-anchor-id="multiple-linear-regres
 <p><span class="math display">\[\hat{y} = \theta_0\:+\:\theta_1x_{1}\:+\:\theta_2 x_{2}\:+\:...\:+\:\theta_p x_{p}\]</span></p>
 <p>Our predicted value of <span class="math inline">\(y\)</span>, <span class="math inline">\(\hat{y}\)</span>, is a linear combination of the single <strong>observations</strong> (features), <span class="math inline">\(x_i\)</span>, and the parameters, <span class="math inline">\(\theta_i\)</span>.</p>
 <p>We can explore this idea further by looking at a dataset containing aggregate per-player data from the 2018-19 NBA season, downloaded from <a href="https://www.kaggle.com/schmadam97/nba-regular-season-stats-20182019">Kaggle</a>.</p>
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 <details class="code-fold">
 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> pandas <span class="im">as</span> pd</span>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>nba <span class="op">=</span> pd.read_csv(<span class="st">'data/nba18-19.csv'</span>, index_col<span class="op">=</span><span class="dv">0</span>)</span>
 <span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>nba.index.name <span class="op">=</span> <span class="va">None</span> <span class="co"># Drops name of index (players are ordered by rank)</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </details>
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 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>nba.head(<span class="dv">5</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
@@ -535,7 +535,7 @@ <h3 data-number="12.1.1" class="anchored" data-anchor-id="multiple-linear-regres
 <li><code>AST</code>, the average number of assists per game</li>
 <li><code>3PA</code>, the average number of 3-point field goals attempted per game</li>
 </ul>
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 <summary>Code</summary>
 <div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>nba[[<span class="st">'FG'</span>, <span class="st">'AST'</span>, <span class="st">'3PA'</span>, <span class="st">'PTS'</span>]].head()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>