From 159d2441309a55c9d59ea1a74dc1544f611f7984 Mon Sep 17 00:00:00 2001
From: ArturoAmorQ <ArturoAmorQ@users.noreply.github.com>
Date: Tue, 17 Oct 2023 13:48:14 +0000
Subject: [PATCH] [ci skip] MAINT Fix broken sphinx-book-theme reference in CI
 (#732) 4b006d5eacab7802e4f9e6d2b15e8b6886bd9a81

---
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 ...ddfe7631a486b0cf3b5a2a87d34fac8c796537.png | Bin 0 -> 108107 bytes
 .../python_scripts/linear_models_sol_03.py    | 254 ++++++++++++---
 appendix/notebook_timings.html                |   4 +-
 python_scripts/linear_models_sol_03.html      | 308 ++++++++++++++----
 searchindex.js                                |   2 +-
 23 files changed, 460 insertions(+), 108 deletions(-)
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new file mode 100644
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new file mode 100644
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diff --git a/_sources/python_scripts/linear_models_sol_03.py b/_sources/python_scripts/linear_models_sol_03.py
index d789c8522..dc2a82f5c 100644
--- a/_sources/python_scripts/linear_models_sol_03.py
+++ b/_sources/python_scripts/linear_models_sol_03.py
@@ -8,14 +8,20 @@
 # %% [markdown]
 # # 📃 Solution for Exercise M4.03
 #
-# The parameter `penalty` can control the **type** of regularization to use,
-# whereas the regularization **strength** is set using the parameter `C`.
-# Setting`penalty="none"` is equivalent to an infinitely large value of `C`. In
-# this exercise, we ask you to train a logistic regression classifier using the
-# `penalty="l2"` regularization (which happens to be the default in
-# scikit-learn) to find by yourself the effect of the parameter `C`.
+# In the previous Module we tuned the hyperparameter `C` of the logistic
+# regression without mentioning that it controls the regularization strength.
+# Later, on the slides on 🎥 **Intuitions on regularized linear models** we
+# metioned that a small `C` provides a more regularized model, whereas a
+# non-regularized model is obtained with an infinitely large value of `C`.
+# Indeed, `C` behaves as the inverse of the `alpha` coefficient in the `Ridge`
+# model.
 #
-# We start by loading the dataset.
+# In this exercise, we ask you to train a logistic regression classifier using
+# different values of the parameter `C` to find its effects by yourself.
+#
+# We start by loading the dataset. We only keep the Adelie and Chinstrap classes
+# to keep the discussion simple.
+
 
 # %% [markdown]
 # ```{note}
@@ -27,7 +33,6 @@
 import pandas as pd
 
 penguins = pd.read_csv("../datasets/penguins_classification.csv")
-# only keep the Adelie and Chinstrap classes
 penguins = (
     penguins.set_index("Species").loc[["Adelie", "Chinstrap"]].reset_index()
 )
@@ -38,7 +43,9 @@
 # %%
 from sklearn.model_selection import train_test_split
 
-penguins_train, penguins_test = train_test_split(penguins, random_state=0)
+penguins_train, penguins_test = train_test_split(
+    penguins, random_state=0, test_size=0.4
+)
 
 data_train = penguins_train[culmen_columns]
 data_test = penguins_test[culmen_columns]
@@ -47,76 +54,227 @@
 target_test = penguins_test[target_column]
 
 # %% [markdown]
-# First, let's create our predictive model.
+# We define a function to help us fit a given `model` and plot its decision
+# boundary. We recall that by using a `DecisionBoundaryDisplay` with diverging
+# colormap, `vmin=0` and `vmax=1`, we ensure that the 0.5 probability is mapped
+# to the white color. Equivalently, the darker the color, the closer the
+# predicted probability is to 0 or 1 and the more confident the classifier is in
+# its predictions.
 
 # %%
-from sklearn.pipeline import make_pipeline
-from sklearn.preprocessing import StandardScaler
-from sklearn.linear_model import LogisticRegression
-
-logistic_regression = make_pipeline(
-    StandardScaler(), LogisticRegression(penalty="l2")
-)
-
-# %% [markdown]
-# Given the following candidates for the `C` parameter, find out the impact of
-# `C` on the classifier decision boundary. You can use
-# `sklearn.inspection.DecisionBoundaryDisplay.from_estimator` to plot the
-# decision function boundary.
-
-# %%
-Cs = [0.01, 0.1, 1, 10]
-
-# solution
 import matplotlib.pyplot as plt
 import seaborn as sns
 from sklearn.inspection import DecisionBoundaryDisplay
 
-for C in Cs:
-    logistic_regression.set_params(logisticregression__C=C)
-    logistic_regression.fit(data_train, target_train)
-    accuracy = logistic_regression.score(data_test, target_test)
 
-    DecisionBoundaryDisplay.from_estimator(
-        logistic_regression,
-        data_test,
-        response_method="predict",
+def plot_decision_boundary(model):
+    model.fit(data_train, target_train)
+    accuracy = model.score(data_test, target_test)
+
+    disp = DecisionBoundaryDisplay.from_estimator(
+        model,
+        data_train,
+        response_method="predict_proba",
+        plot_method="pcolormesh",
         cmap="RdBu_r",
-        alpha=0.5,
+        alpha=0.8,
+        vmin=0.0,
+        vmax=1.0,
+    )
+    DecisionBoundaryDisplay.from_estimator(
+        model,
+        data_train,
+        response_method="predict_proba",
+        plot_method="contour",
+        linestyles="--",
+        linewidths=1,
+        alpha=0.8,
+        levels=[0.5],
+        ax=disp.ax_,
     )
     sns.scatterplot(
-        data=penguins_test,
+        data=penguins_train,
         x=culmen_columns[0],
         y=culmen_columns[1],
         hue=target_column,
-        palette=["tab:red", "tab:blue"],
+        palette=["tab:blue", "tab:red"],
+        ax=disp.ax_,
     )
     plt.legend(bbox_to_anchor=(1.05, 0.8), loc="upper left")
     plt.title(f"C: {C} \n Accuracy on the test set: {accuracy:.2f}")
 
+
+# %% [markdown]
+# Let's now create our predictive model.
+
+# %%
+from sklearn.pipeline import make_pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.linear_model import LogisticRegression
+
+logistic_regression = make_pipeline(StandardScaler(), LogisticRegression())
+
+# %% [markdown]
+# ## Influence of the parameter `C` on the decision boundary
+#
+# Given the following candidates for the `C` parameter and the
+# `plot_decision_boundary` function, find out the impact of `C` on the
+# classifier's decision boundary.
+#
+# - How does the value of `C` impact the confidence on the predictions?
+# - How does it impact the underfit/overfit trade-off?
+# - How does it impact the position and orientation of the decision boundary?
+#
+# Try to give an interpretation on the reason for such behavior.
+
+# %%
+Cs = [1e-6, 0.01, 0.1, 1, 10, 100, 1e6]
+
+# solution
+for C in Cs:
+    logistic_regression.set_params(logisticregression__C=C)
+    plot_decision_boundary(logistic_regression)
+
+# %% [markdown] tags=["solution"]
+#
+# On this series of plots we can observe several important points. Regarding the
+# confidence on the predictions:
+#
+# - For low values of `C` (strong regularization), the classifier is less
+#   confident in its predictions. We are enforcing a **spread sigmoid**.
+# - For high values of `C` (weak regularization), the classifier is more
+#   confident: the areas with dark blue (very confident in predicting "Adelie")
+#   and dark red (very confident in predicting "Chinstrap") nearly cover the
+#   entire feature space. We are enforcing a **steep sigmoid**.
+#
+# To answer the next question, think that misclassified data points are more
+# costly when the classifier is more confident on the decision. Decision rules
+# are mostly driven by avoiding such cost. From the previous observations we can
+# then deduce that:
+#
+# - The smaller the `C` (the stronger the regularization), the lower the cost
+#   of a misclassification. As more data points lay in the low-confidence
+#   zone, the more the decision rules are influenced almost uniformly by all
+#   the data points. This leads to a less expressive model, which may underfit.
+# - The higher the value of `C` (the weaker the regularization), the more the
+#   decision is influenced by a few training points very close to the boundary,
+#   where decisions are costly. Remember that models may overfit if the number
+#   of samples in the training set is too small, as at least a minimum of
+#   samples is needed to average the noise out.
+#
+# The orientation is the result of two factors: minimizing the number of
+# misclassified training points with high confidence and their distance to the
+# decision boundary (notice how the contour line tries to align with the most
+# misclassified data points in the dark-colored zone). This is closely related
+# to the value of the weights of the model, which is explained in the next part
+# of the exercise.
+#
+# Finally, for small values of `C` the position of the decision boundary is
+# affected by the class imbalance: when `C` is near zero, the model predicts the
+# majority class (as seen in the training set) everywhere in the feature space.
+# In our case, there are approximately two times more "Adelie" than "Chinstrap"
+# penguins. This explains why the decision boundary is shifted to the right when
+# `C` gets smaller. Indeed, the most regularized model predicts light blue
+# almost everywhere in the feature space.
+
 # %% [markdown]
+# ## Impact of the regularization on the weights
+#
 # Look at the impact of the `C` hyperparameter on the magnitude of the weights.
+# **Hint**: You can [access pipeline
+# steps](https://scikit-learn.org/stable/modules/compose.html#access-pipeline-steps)
+# by name or position. Then you can query the attributes of that step such as
+# `coef_`.
 
 # %%
 # solution
-weights_ridge = []
+lr_weights = []
 for C in Cs:
     logistic_regression.set_params(logisticregression__C=C)
     logistic_regression.fit(data_train, target_train)
     coefs = logistic_regression[-1].coef_[0]
-    weights_ridge.append(pd.Series(coefs, index=culmen_columns))
+    lr_weights.append(pd.Series(coefs, index=culmen_columns))
 
 # %% tags=["solution"]
-weights_ridge = pd.concat(weights_ridge, axis=1, keys=[f"C: {C}" for C in Cs])
-weights_ridge.plot.barh()
+lr_weights = pd.concat(lr_weights, axis=1, keys=[f"C: {C}" for C in Cs])
+lr_weights.plot.barh()
 _ = plt.title("LogisticRegression weights depending of C")
 
 # %% [markdown] tags=["solution"]
-# We see that a small `C` will shrink the weights values toward zero. It means
-# that a small `C` provides a more regularized model. Thus, `C` is the inverse
-# of the `alpha` coefficient in the `Ridge` model.
 #
-# Besides, with a strong penalty (i.e. small `C` value), the weight of the
-# feature "Culmen Depth (mm)" is almost zero. It explains why the decision
+# As small `C` provides a more regularized model, it shrinks the weights values
+# toward zero, as in the `Ridge` model.
+#
+# In particular, with a strong penalty (e.g. `C = 0.01`), the weight of the feature
+# named "Culmen Depth (mm)" is almost zero. It explains why the decision
 # separation in the plot is almost perpendicular to the "Culmen Length (mm)"
 # feature.
+#
+# For even stronger penalty strengths (e.g. `C = 1e-6`), the weights of both
+# features are almost zero. It explains why the decision separation in the plot
+# is almost constant in the feature space: the predicted probability is only
+# based on the intercept parameter of the model (which is never regularized).
+
+# %% [markdown]
+# ## Impact of the regularization on with non-linear feature engineering
+#
+# Use the `plot_decision_boundary` function to repeat the experiment using a
+# non-linear feature engineering pipeline. For such purpose, insert
+# `Nystroem(kernel="rbf", gamma=1, n_components=100)` between the
+# `StandardScaler` and the `LogisticRegression` steps.
+#
+# - Does the value of `C` still impact the position of the decision boundary and
+#   the confidence of the model?
+# - What can you say about the impact of `C` on the underfitting vs overfitting
+#   trade-off?
+
+# %%
+from sklearn.kernel_approximation import Nystroem
+
+# solution
+classifier = make_pipeline(
+    StandardScaler(),
+    Nystroem(kernel="rbf", gamma=1.0, n_components=100, random_state=0),
+    LogisticRegression(penalty="l2", max_iter=1000),
+)
+
+for C in Cs:
+    classifier.set_params(logisticregression__C=C)
+    plot_decision_boundary(classifier)
+
+# %% [markdown] tags=["solution"]
+#
+# - For the lowest values of `C`, the overall pipeline underfits: it predicts
+#   the majority class everywhere, as previously.
+# - When `C` increases, the models starts to predict some datapoints from the
+#   "Chinstrap" class but the model is not very confident anywhere in the
+#   feature space.
+# - The decision boundary is no longer a straight line: the linear model is now
+#   classifying in the 100-dimensional feature space created by the `Nystroem`
+#   transformer. As are result, the decision boundary induced by the overall
+#   pipeline is now expressive enough to wrap around the minority class.
+# - For `C = 1` in particular, it finds a smooth red blob around most of the
+#   "Chinstrap" data points. When moving away from the data points, the model is
+#   less confident in its predictions and again tends to predict the majority
+#   class according to the proportion in the training set.
+# - For higher values of `C`, the model starts to overfit: it is very confident
+#   in its predictions almost everywhere, but it should not be trusted: the
+#   model also makes a larger number of mistakes on the test set (not shown in
+#   the plot) while adopting a very curvy decision boundary to attempt fitting
+#   all the training points, including the noisy ones at the frontier between
+#   the two classes. This makes the decision boundary very sensitive to the
+#   sampling of the training set and as a result, it does not generalize well in
+#   that region. This is confirmed by the (slightly) lower accuracy on the test
+#   set.
+#
+# Finally, we can also note that the linear model on the raw features was as
+# good or better than the best model using non-linear feature engineering. So in
+# this case, we did not really need this extra complexity in our pipeline.
+# **Simpler is better!**
+#
+# So to conclude, when using non-linear feature engineering, it is often
+# possible to make the pipeline overfit, even if the original feature space is
+# low-dimensional. As a result, it is important to tune the regularization
+# parameter in conjunction with the parameters of the transformers (e.g. tuning
+# `gamma` would be important here). This has a direct impact on the certainty of
+# the predictions.
diff --git a/appendix/notebook_timings.html b/appendix/notebook_timings.html
index 0bdb6d69d..94ea14b4c 100644
--- a/appendix/notebook_timings.html
+++ b/appendix/notebook_timings.html
@@ -1029,9 +1029,9 @@ <h1>Notebook timings<a class="headerlink" href="#notebook-timings" title="Permal
 <td><p>✅</p></td>
 </tr>
 <tr class="row-odd"><td><p><a class="xref doc reference internal" href="../python_scripts/linear_models_sol_03.html"><span class="doc">python_scripts/linear_models_sol_03</span></a></p></td>
-<td><p>2023-08-30 10:07</p></td>
+<td><p>2023-10-17 13:47</p></td>
 <td><p>cache</p></td>
-<td><p>3.97</p></td>
+<td><p>11.76</p></td>
 <td><p>✅</p></td>
 </tr>
 <tr class="row-even"><td><p><a class="xref doc reference internal" href="../python_scripts/linear_regression_in_sklearn.html"><span class="doc">python_scripts/linear_regression_in_sklearn</span></a></p></td>
diff --git a/python_scripts/linear_models_sol_03.html b/python_scripts/linear_models_sol_03.html
index c11870c99..7c020c1c9 100644
--- a/python_scripts/linear_models_sol_03.html
+++ b/python_scripts/linear_models_sol_03.html
@@ -656,7 +656,9 @@
   </button>
 `);
 </script>
-
+<label class="sidebar-toggle secondary-toggle btn btn-sm" for="__secondary"title="Toggle secondary sidebar" data-bs-placement="bottom" data-bs-toggle="tooltip">
+    <span class="fa-solid fa-list"></span>
+</label>
 </div></div>
       
     </div>
@@ -672,6 +674,16 @@ <h1>📃 Solution for Exercise M4.03</h1>
     <div id="print-main-content">
         <div id="jb-print-toc">
             
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#influence-of-the-parameter-c-on-the-decision-boundary">Influence of the parameter <code class="docutils literal notranslate"><span class="pre">C</span></code> on the decision boundary</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#impact-of-the-regularization-on-the-weights">Impact of the regularization on the weights</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#impact-of-the-regularization-on-with-non-linear-feature-engineering">Impact of the regularization on with non-linear feature engineering</a></li>
+</ul>
+            </nav>
         </div>
     </div>
 </div>
@@ -683,13 +695,17 @@ <h1>📃 Solution for Exercise M4.03</h1>
                   
   <section class="tex2jax_ignore mathjax_ignore" id="solution-for-exercise-m4-03">
 <h1>📃 Solution for Exercise M4.03<a class="headerlink" href="#solution-for-exercise-m4-03" title="Permalink to this heading">#</a></h1>
-<p>The parameter <code class="docutils literal notranslate"><span class="pre">penalty</span></code> can control the <strong>type</strong> of regularization to use,
-whereas the regularization <strong>strength</strong> is set using the parameter <code class="docutils literal notranslate"><span class="pre">C</span></code>.
-Setting<code class="docutils literal notranslate"><span class="pre">penalty=&quot;none&quot;</span></code> is equivalent to an infinitely large value of <code class="docutils literal notranslate"><span class="pre">C</span></code>. In
-this exercise, we ask you to train a logistic regression classifier using the
-<code class="docutils literal notranslate"><span class="pre">penalty=&quot;l2&quot;</span></code> regularization (which happens to be the default in
-scikit-learn) to find by yourself the effect of the parameter <code class="docutils literal notranslate"><span class="pre">C</span></code>.</p>
-<p>We start by loading the dataset.</p>
+<p>In the previous Module we tuned the hyperparameter <code class="docutils literal notranslate"><span class="pre">C</span></code> of the logistic
+regression without mentioning that it controls the regularization strength.
+Later, on the slides on 🎥 <strong>Intuitions on regularized linear models</strong> we
+metioned that a small <code class="docutils literal notranslate"><span class="pre">C</span></code> provides a more regularized model, whereas a
+non-regularized model is obtained with an infinitely large value of <code class="docutils literal notranslate"><span class="pre">C</span></code>.
+Indeed, <code class="docutils literal notranslate"><span class="pre">C</span></code> behaves as the inverse of the <code class="docutils literal notranslate"><span class="pre">alpha</span></code> coefficient in the <code class="docutils literal notranslate"><span class="pre">Ridge</span></code>
+model.</p>
+<p>In this exercise, we ask you to train a logistic regression classifier using
+different values of the parameter <code class="docutils literal notranslate"><span class="pre">C</span></code> to find its effects by yourself.</p>
+<p>We start by loading the dataset. We only keep the Adelie and Chinstrap classes
+to keep the discussion simple.</p>
 <div class="admonition note">
 <p class="admonition-title">Note</p>
 <p>If you want a deeper overview regarding this dataset, you can refer to the
@@ -700,7 +716,6 @@ <h1>📃 Solution for Exercise M4.03<a class="headerlink" href="#solution-for-ex
 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
 
 <span class="n">penguins</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s2">&quot;../datasets/penguins_classification.csv&quot;</span><span class="p">)</span>
-<span class="c1"># only keep the Adelie and Chinstrap classes</span>
 <span class="n">penguins</span> <span class="o">=</span> <span class="p">(</span>
     <span class="n">penguins</span><span class="o">.</span><span class="n">set_index</span><span class="p">(</span><span class="s2">&quot;Species&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">loc</span><span class="p">[[</span><span class="s2">&quot;Adelie&quot;</span><span class="p">,</span> <span class="s2">&quot;Chinstrap&quot;</span><span class="p">]]</span><span class="o">.</span><span class="n">reset_index</span><span class="p">()</span>
 <span class="p">)</span>
@@ -715,7 +730,9 @@ <h1>📃 Solution for Exercise M4.03<a class="headerlink" href="#solution-for-ex
 <div class="cell_input docutils container">
 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
 
-<span class="n">penguins_train</span><span class="p">,</span> <span class="n">penguins_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">penguins</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">penguins_train</span><span class="p">,</span> <span class="n">penguins_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span>
+    <span class="n">penguins</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.4</span>
+<span class="p">)</span>
 
 <span class="n">data_train</span> <span class="o">=</span> <span class="n">penguins_train</span><span class="p">[</span><span class="n">culmen_columns</span><span class="p">]</span>
 <span class="n">data_test</span> <span class="o">=</span> <span class="n">penguins_test</span><span class="p">[</span><span class="n">culmen_columns</span><span class="p">]</span>
@@ -726,97 +743,258 @@ <h1>📃 Solution for Exercise M4.03<a class="headerlink" href="#solution-for-ex
 </div>
 </div>
 </div>
-<p>First, let’s create our predictive model.</p>
+<p>We define a function to help us fit a given <code class="docutils literal notranslate"><span class="pre">model</span></code> and plot its decision
+boundary. We recall that by using a <code class="docutils literal notranslate"><span class="pre">DecisionBoundaryDisplay</span></code> with diverging
+colormap, <code class="docutils literal notranslate"><span class="pre">vmin=0</span></code> and <code class="docutils literal notranslate"><span class="pre">vmax=1</span></code>, we ensure that the 0.5 probability is mapped
+to the white color. Equivalently, the darker the color, the closer the
+predicted probability is to 0 or 1 and the more confident the classifier is in
+its predictions.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+<span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">DecisionBoundaryDisplay</span>
+
+
+<span class="k">def</span> <span class="nf">plot_decision_boundary</span><span class="p">(</span><span class="n">model</span><span class="p">):</span>
+    <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">data_train</span><span class="p">,</span> <span class="n">target_train</span><span class="p">)</span>
+    <span class="n">accuracy</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">data_test</span><span class="p">,</span> <span class="n">target_test</span><span class="p">)</span>
+
+    <span class="n">disp</span> <span class="o">=</span> <span class="n">DecisionBoundaryDisplay</span><span class="o">.</span><span class="n">from_estimator</span><span class="p">(</span>
+        <span class="n">model</span><span class="p">,</span>
+        <span class="n">data_train</span><span class="p">,</span>
+        <span class="n">response_method</span><span class="o">=</span><span class="s2">&quot;predict_proba&quot;</span><span class="p">,</span>
+        <span class="n">plot_method</span><span class="o">=</span><span class="s2">&quot;pcolormesh&quot;</span><span class="p">,</span>
+        <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;RdBu_r&quot;</span><span class="p">,</span>
+        <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span>
+        <span class="n">vmin</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span>
+        <span class="n">vmax</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span>
+    <span class="p">)</span>
+    <span class="n">DecisionBoundaryDisplay</span><span class="o">.</span><span class="n">from_estimator</span><span class="p">(</span>
+        <span class="n">model</span><span class="p">,</span>
+        <span class="n">data_train</span><span class="p">,</span>
+        <span class="n">response_method</span><span class="o">=</span><span class="s2">&quot;predict_proba&quot;</span><span class="p">,</span>
+        <span class="n">plot_method</span><span class="o">=</span><span class="s2">&quot;contour&quot;</span><span class="p">,</span>
+        <span class="n">linestyles</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">,</span>
+        <span class="n">linewidths</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+        <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span>
+        <span class="n">levels</span><span class="o">=</span><span class="p">[</span><span class="mf">0.5</span><span class="p">],</span>
+        <span class="n">ax</span><span class="o">=</span><span class="n">disp</span><span class="o">.</span><span class="n">ax_</span><span class="p">,</span>
+    <span class="p">)</span>
+    <span class="n">sns</span><span class="o">.</span><span class="n">scatterplot</span><span class="p">(</span>
+        <span class="n">data</span><span class="o">=</span><span class="n">penguins_train</span><span class="p">,</span>
+        <span class="n">x</span><span class="o">=</span><span class="n">culmen_columns</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
+        <span class="n">y</span><span class="o">=</span><span class="n">culmen_columns</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
+        <span class="n">hue</span><span class="o">=</span><span class="n">target_column</span><span class="p">,</span>
+        <span class="n">palette</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;tab:blue&quot;</span><span class="p">,</span> <span class="s2">&quot;tab:red&quot;</span><span class="p">],</span>
+        <span class="n">ax</span><span class="o">=</span><span class="n">disp</span><span class="o">.</span><span class="n">ax_</span><span class="p">,</span>
+    <span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">bbox_to_anchor</span><span class="o">=</span><span class="p">(</span><span class="mf">1.05</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">),</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper left&quot;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;C: </span><span class="si">{</span><span class="n">C</span><span class="si">}</span><span class="s2"> </span><span class="se">\n</span><span class="s2"> Accuracy on the test set: </span><span class="si">{</span><span class="n">accuracy</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Let’s now create our predictive model.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>
 <span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
 <span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
 
-<span class="n">logistic_regression</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span>
-    <span class="n">StandardScaler</span><span class="p">(),</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">penalty</span><span class="o">=</span><span class="s2">&quot;l2&quot;</span><span class="p">)</span>
-<span class="p">)</span>
+<span class="n">logistic_regression</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">StandardScaler</span><span class="p">(),</span> <span class="n">LogisticRegression</span><span class="p">())</span>
 </pre></div>
 </div>
 </div>
 </div>
-<p>Given the following candidates for the <code class="docutils literal notranslate"><span class="pre">C</span></code> parameter, find out the impact of
-<code class="docutils literal notranslate"><span class="pre">C</span></code> on the classifier decision boundary. You can use
-<code class="docutils literal notranslate"><span class="pre">sklearn.inspection.DecisionBoundaryDisplay.from_estimator</span></code> to plot the
-decision function boundary.</p>
+<section id="influence-of-the-parameter-c-on-the-decision-boundary">
+<h2>Influence of the parameter <code class="docutils literal notranslate"><span class="pre">C</span></code> on the decision boundary<a class="headerlink" href="#influence-of-the-parameter-c-on-the-decision-boundary" title="Permalink to this heading">#</a></h2>
+<p>Given the following candidates for the <code class="docutils literal notranslate"><span class="pre">C</span></code> parameter and the
+<code class="docutils literal notranslate"><span class="pre">plot_decision_boundary</span></code> function, find out the impact of <code class="docutils literal notranslate"><span class="pre">C</span></code> on the
+classifier’s decision boundary.</p>
+<ul class="simple">
+<li><p>How does the value of <code class="docutils literal notranslate"><span class="pre">C</span></code> impact the confidence on the predictions?</p></li>
+<li><p>How does it impact the underfit/overfit trade-off?</p></li>
+<li><p>How does it impact the position and orientation of the decision boundary?</p></li>
+</ul>
+<p>Try to give an interpretation on the reason for such behavior.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
-<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Cs</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">]</span>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Cs</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1e-6</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mf">1e6</span><span class="p">]</span>
 
 <span class="c1"># solution</span>
-<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
-<span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">DecisionBoundaryDisplay</span>
-
 <span class="k">for</span> <span class="n">C</span> <span class="ow">in</span> <span class="n">Cs</span><span class="p">:</span>
     <span class="n">logistic_regression</span><span class="o">.</span><span class="n">set_params</span><span class="p">(</span><span class="n">logisticregression__C</span><span class="o">=</span><span class="n">C</span><span class="p">)</span>
-    <span class="n">logistic_regression</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">data_train</span><span class="p">,</span> <span class="n">target_train</span><span class="p">)</span>
-    <span class="n">accuracy</span> <span class="o">=</span> <span class="n">logistic_regression</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">data_test</span><span class="p">,</span> <span class="n">target_test</span><span class="p">)</span>
-
-    <span class="n">DecisionBoundaryDisplay</span><span class="o">.</span><span class="n">from_estimator</span><span class="p">(</span>
-        <span class="n">logistic_regression</span><span class="p">,</span>
-        <span class="n">data_test</span><span class="p">,</span>
-        <span class="n">response_method</span><span class="o">=</span><span class="s2">&quot;predict&quot;</span><span class="p">,</span>
-        <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;RdBu_r&quot;</span><span class="p">,</span>
-        <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span>
-    <span class="p">)</span>
-    <span class="n">sns</span><span class="o">.</span><span class="n">scatterplot</span><span class="p">(</span>
-        <span class="n">data</span><span class="o">=</span><span class="n">penguins_test</span><span class="p">,</span>
-        <span class="n">x</span><span class="o">=</span><span class="n">culmen_columns</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
-        <span class="n">y</span><span class="o">=</span><span class="n">culmen_columns</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
-        <span class="n">hue</span><span class="o">=</span><span class="n">target_column</span><span class="p">,</span>
-        <span class="n">palette</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;tab:red&quot;</span><span class="p">,</span> <span class="s2">&quot;tab:blue&quot;</span><span class="p">],</span>
-    <span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">bbox_to_anchor</span><span class="o">=</span><span class="p">(</span><span class="mf">1.05</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">),</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper left&quot;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;C: </span><span class="si">{</span><span class="n">C</span><span class="si">}</span><span class="s2"> </span><span class="se">\n</span><span class="s2"> Accuracy on the test set: </span><span class="si">{</span><span class="n">accuracy</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+    <span class="n">plot_decision_boundary</span><span class="p">(</span><span class="n">logistic_regression</span><span class="p">)</span>
 </pre></div>
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/dd7f27dbcdf5564035e27c670f3123f844e7a981f8c63287750ece5b61a44928.png" src="../_images/dd7f27dbcdf5564035e27c670f3123f844e7a981f8c63287750ece5b61a44928.png" />
-<img alt="../_images/b292d3662c24fda8d7cbb974f613e06bc0eab2b7400074243c74184b04574c1c.png" src="../_images/b292d3662c24fda8d7cbb974f613e06bc0eab2b7400074243c74184b04574c1c.png" />
-<img alt="../_images/d6eafcd252a9a80ed6880db211eccc646407dd995a818a381022e7cbeba3db42.png" src="../_images/d6eafcd252a9a80ed6880db211eccc646407dd995a818a381022e7cbeba3db42.png" />
-<img alt="../_images/a4aec66658c53ca38b826f418b4b3db906c1a92ab89a9ddef7eb6c0c1867396d.png" src="../_images/a4aec66658c53ca38b826f418b4b3db906c1a92ab89a9ddef7eb6c0c1867396d.png" />
+<img alt="../_images/4995b02ae7b2d0dec5f4214107dd113c296209f435f295ac352e7de6fd7f5abb.png" src="../_images/4995b02ae7b2d0dec5f4214107dd113c296209f435f295ac352e7de6fd7f5abb.png" />
+<img alt="../_images/ca8118111e8a08b91b86eccb27d4c40542f7d4455efdbd2e0d985bf5615b293c.png" src="../_images/ca8118111e8a08b91b86eccb27d4c40542f7d4455efdbd2e0d985bf5615b293c.png" />
+<img alt="../_images/032d44039813a98a96b553b070215955b70d851f6f9fac169280d9be52105833.png" src="../_images/032d44039813a98a96b553b070215955b70d851f6f9fac169280d9be52105833.png" />
+<img alt="../_images/9e0634a92164f80e2d757d4a2bc574f1dea6cc857294526748584ac48c6c2b47.png" src="../_images/9e0634a92164f80e2d757d4a2bc574f1dea6cc857294526748584ac48c6c2b47.png" />
+<img alt="../_images/e0e8d7699e085bb07f9674f12dfeaa5dd14ee4b5dace4789a6d858e77b673d09.png" src="../_images/e0e8d7699e085bb07f9674f12dfeaa5dd14ee4b5dace4789a6d858e77b673d09.png" />
+<img alt="../_images/07c9345090d559dd29999ec0ee6c672b7ca65345dbe11259882d774345129b9b.png" src="../_images/07c9345090d559dd29999ec0ee6c672b7ca65345dbe11259882d774345129b9b.png" />
+<img alt="../_images/0cb9478c8ebafd1110a77a3591feb59f7009ca636e3eda3619943c8f950c57f3.png" src="../_images/0cb9478c8ebafd1110a77a3591feb59f7009ca636e3eda3619943c8f950c57f3.png" />
 </div>
 </div>
-<p>Look at the impact of the <code class="docutils literal notranslate"><span class="pre">C</span></code> hyperparameter on the magnitude of the weights.</p>
+<p>On this series of plots we can observe several important points. Regarding the
+confidence on the predictions:</p>
+<ul class="simple">
+<li><p>For low values of <code class="docutils literal notranslate"><span class="pre">C</span></code> (strong regularization), the classifier is less
+confident in its predictions. We are enforcing a <strong>spread sigmoid</strong>.</p></li>
+<li><p>For high values of <code class="docutils literal notranslate"><span class="pre">C</span></code> (weak regularization), the classifier is more
+confident: the areas with dark blue (very confident in predicting “Adelie”)
+and dark red (very confident in predicting “Chinstrap”) nearly cover the
+entire feature space. We are enforcing a <strong>steep sigmoid</strong>.</p></li>
+</ul>
+<p>To answer the next question, think that misclassified data points are more
+costly when the classifier is more confident on the decision. Decision rules
+are mostly driven by avoiding such cost. From the previous observations we can
+then deduce that:</p>
+<ul class="simple">
+<li><p>The smaller the <code class="docutils literal notranslate"><span class="pre">C</span></code> (the stronger the regularization), the lower the cost
+of a misclassification. As more data points lay in the low-confidence
+zone, the more the decision rules are influenced almost uniformly by all
+the data points. This leads to a less expressive model, which may underfit.</p></li>
+<li><p>The higher the value of <code class="docutils literal notranslate"><span class="pre">C</span></code> (the weaker the regularization), the more the
+decision is influenced by a few training points very close to the boundary,
+where decisions are costly. Remember that models may overfit if the number
+of samples in the training set is too small, as at least a minimum of
+samples is needed to average the noise out.</p></li>
+</ul>
+<p>The orientation is the result of two factors: minimizing the number of
+misclassified training points with high confidence and their distance to the
+decision boundary (notice how the contour line tries to align with the most
+misclassified data points in the dark-colored zone). This is closely related
+to the value of the weights of the model, which is explained in the next part
+of the exercise.</p>
+<p>Finally, for small values of <code class="docutils literal notranslate"><span class="pre">C</span></code> the position of the decision boundary is
+affected by the class imbalance: when <code class="docutils literal notranslate"><span class="pre">C</span></code> is near zero, the model predicts the
+majority class (as seen in the training set) everywhere in the feature space.
+In our case, there are approximately two times more “Adelie” than “Chinstrap”
+penguins. This explains why the decision boundary is shifted to the right when
+<code class="docutils literal notranslate"><span class="pre">C</span></code> gets smaller. Indeed, the most regularized model predicts light blue
+almost everywhere in the feature space.</p>
+</section>
+<section id="impact-of-the-regularization-on-the-weights">
+<h2>Impact of the regularization on the weights<a class="headerlink" href="#impact-of-the-regularization-on-the-weights" title="Permalink to this heading">#</a></h2>
+<p>Look at the impact of the <code class="docutils literal notranslate"><span class="pre">C</span></code> hyperparameter on the magnitude of the weights.
+<strong>Hint</strong>: You can <a class="reference external" href="https://scikit-learn.org/stable/modules/compose.html#access-pipeline-steps">access pipeline
+steps</a>
+by name or position. Then you can query the attributes of that step such as
+<code class="docutils literal notranslate"><span class="pre">coef_</span></code>.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># solution</span>
-<span class="n">weights_ridge</span> <span class="o">=</span> <span class="p">[]</span>
+<span class="n">lr_weights</span> <span class="o">=</span> <span class="p">[]</span>
 <span class="k">for</span> <span class="n">C</span> <span class="ow">in</span> <span class="n">Cs</span><span class="p">:</span>
     <span class="n">logistic_regression</span><span class="o">.</span><span class="n">set_params</span><span class="p">(</span><span class="n">logisticregression__C</span><span class="o">=</span><span class="n">C</span><span class="p">)</span>
     <span class="n">logistic_regression</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">data_train</span><span class="p">,</span> <span class="n">target_train</span><span class="p">)</span>
     <span class="n">coefs</span> <span class="o">=</span> <span class="n">logistic_regression</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">coef_</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">weights_ridge</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">culmen_columns</span><span class="p">))</span>
+    <span class="n">lr_weights</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">culmen_columns</span><span class="p">))</span>
 </pre></div>
 </div>
 </div>
 </div>
 <div class="cell tag_solution docutils container">
 <div class="cell_input docutils container">
-<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">weights_ridge</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">concat</span><span class="p">(</span><span class="n">weights_ridge</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keys</span><span class="o">=</span><span class="p">[</span><span class="sa">f</span><span class="s2">&quot;C: </span><span class="si">{</span><span class="n">C</span><span class="si">}</span><span class="s2">&quot;</span> <span class="k">for</span> <span class="n">C</span> <span class="ow">in</span> <span class="n">Cs</span><span class="p">])</span>
-<span class="n">weights_ridge</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">barh</span><span class="p">()</span>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">lr_weights</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">concat</span><span class="p">(</span><span class="n">lr_weights</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keys</span><span class="o">=</span><span class="p">[</span><span class="sa">f</span><span class="s2">&quot;C: </span><span class="si">{</span><span class="n">C</span><span class="si">}</span><span class="s2">&quot;</span> <span class="k">for</span> <span class="n">C</span> <span class="ow">in</span> <span class="n">Cs</span><span class="p">])</span>
+<span class="n">lr_weights</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">barh</span><span class="p">()</span>
 <span class="n">_</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;LogisticRegression weights depending of C&quot;</span><span class="p">)</span>
 </pre></div>
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/0e50749bd867085e6fd9c797f89a1870ff6983514e8822934f333abef22f2156.png" src="../_images/0e50749bd867085e6fd9c797f89a1870ff6983514e8822934f333abef22f2156.png" />
+<img alt="../_images/d29a899ee748eecaddc699f51059d814483a9f000303fa449509f98bdf524abe.png" src="../_images/d29a899ee748eecaddc699f51059d814483a9f000303fa449509f98bdf524abe.png" />
 </div>
 </div>
-<p>We see that a small <code class="docutils literal notranslate"><span class="pre">C</span></code> will shrink the weights values toward zero. It means
-that a small <code class="docutils literal notranslate"><span class="pre">C</span></code> provides a more regularized model. Thus, <code class="docutils literal notranslate"><span class="pre">C</span></code> is the inverse
-of the <code class="docutils literal notranslate"><span class="pre">alpha</span></code> coefficient in the <code class="docutils literal notranslate"><span class="pre">Ridge</span></code> model.</p>
-<p>Besides, with a strong penalty (i.e. small <code class="docutils literal notranslate"><span class="pre">C</span></code> value), the weight of the
-feature “Culmen Depth (mm)” is almost zero. It explains why the decision
+<p>As small <code class="docutils literal notranslate"><span class="pre">C</span></code> provides a more regularized model, it shrinks the weights values
+toward zero, as in the <code class="docutils literal notranslate"><span class="pre">Ridge</span></code> model.</p>
+<p>In particular, with a strong penalty (e.g. <code class="docutils literal notranslate"><span class="pre">C</span> <span class="pre">=</span> <span class="pre">0.01</span></code>), the weight of the feature
+named “Culmen Depth (mm)” is almost zero. It explains why the decision
 separation in the plot is almost perpendicular to the “Culmen Length (mm)”
 feature.</p>
+<p>For even stronger penalty strengths (e.g. <code class="docutils literal notranslate"><span class="pre">C</span> <span class="pre">=</span> <span class="pre">1e-6</span></code>), the weights of both
+features are almost zero. It explains why the decision separation in the plot
+is almost constant in the feature space: the predicted probability is only
+based on the intercept parameter of the model (which is never regularized).</p>
+</section>
+<section id="impact-of-the-regularization-on-with-non-linear-feature-engineering">
+<h2>Impact of the regularization on with non-linear feature engineering<a class="headerlink" href="#impact-of-the-regularization-on-with-non-linear-feature-engineering" title="Permalink to this heading">#</a></h2>
+<p>Use the <code class="docutils literal notranslate"><span class="pre">plot_decision_boundary</span></code> function to repeat the experiment using a
+non-linear feature engineering pipeline. For such purpose, insert
+<code class="docutils literal notranslate"><span class="pre">Nystroem(kernel=&quot;rbf&quot;,</span> <span class="pre">gamma=1,</span> <span class="pre">n_components=100)</span></code> between the
+<code class="docutils literal notranslate"><span class="pre">StandardScaler</span></code> and the <code class="docutils literal notranslate"><span class="pre">LogisticRegression</span></code> steps.</p>
+<ul class="simple">
+<li><p>Does the value of <code class="docutils literal notranslate"><span class="pre">C</span></code> still impact the position of the decision boundary and
+the confidence of the model?</p></li>
+<li><p>What can you say about the impact of <code class="docutils literal notranslate"><span class="pre">C</span></code> on the underfitting vs overfitting
+trade-off?</p></li>
+</ul>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.kernel_approximation</span> <span class="kn">import</span> <span class="n">Nystroem</span>
+
+<span class="c1"># solution</span>
+<span class="n">classifier</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span>
+    <span class="n">StandardScaler</span><span class="p">(),</span>
+    <span class="n">Nystroem</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s2">&quot;rbf&quot;</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">n_components</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">),</span>
+    <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">penalty</span><span class="o">=</span><span class="s2">&quot;l2&quot;</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">1000</span><span class="p">),</span>
+<span class="p">)</span>
+
+<span class="k">for</span> <span class="n">C</span> <span class="ow">in</span> <span class="n">Cs</span><span class="p">:</span>
+    <span class="n">classifier</span><span class="o">.</span><span class="n">set_params</span><span class="p">(</span><span class="n">logisticregression__C</span><span class="o">=</span><span class="n">C</span><span class="p">)</span>
+    <span class="n">plot_decision_boundary</span><span class="p">(</span><span class="n">classifier</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/4995b02ae7b2d0dec5f4214107dd113c296209f435f295ac352e7de6fd7f5abb.png" src="../_images/4995b02ae7b2d0dec5f4214107dd113c296209f435f295ac352e7de6fd7f5abb.png" />
+<img alt="../_images/4f064fcce2c9e74d5ca165c4b7a9d97adde6663fed9522e918e28a68073dd161.png" src="../_images/4f064fcce2c9e74d5ca165c4b7a9d97adde6663fed9522e918e28a68073dd161.png" />
+<img alt="../_images/7f99620cd8492571c1c6d809a38e7f691d3bffc6797b6c8f68bcca57101d2426.png" src="../_images/7f99620cd8492571c1c6d809a38e7f691d3bffc6797b6c8f68bcca57101d2426.png" />
+<img alt="../_images/eaf825e25e641c4d8d6a242333ddfe7631a486b0cf3b5a2a87d34fac8c796537.png" src="../_images/eaf825e25e641c4d8d6a242333ddfe7631a486b0cf3b5a2a87d34fac8c796537.png" />
+<img alt="../_images/453858549e380f9ecfac873f0dc202778dafe5323277d31a517a0d5f0437aa29.png" src="../_images/453858549e380f9ecfac873f0dc202778dafe5323277d31a517a0d5f0437aa29.png" />
+<img alt="../_images/e64c91b30862454474c755622a6a99fbf325cb90fde784598bc71a96daecd3ac.png" src="../_images/e64c91b30862454474c755622a6a99fbf325cb90fde784598bc71a96daecd3ac.png" />
+<img alt="../_images/4b8d72068b16a40033d1ae1c3e43e24f8a0695a944be1ef86aa1f8d61d157cc2.png" src="../_images/4b8d72068b16a40033d1ae1c3e43e24f8a0695a944be1ef86aa1f8d61d157cc2.png" />
+</div>
+</div>
+<ul class="simple">
+<li><p>For the lowest values of <code class="docutils literal notranslate"><span class="pre">C</span></code>, the overall pipeline underfits: it predicts
+the majority class everywhere, as previously.</p></li>
+<li><p>When <code class="docutils literal notranslate"><span class="pre">C</span></code> increases, the models starts to predict some datapoints from the
+“Chinstrap” class but the model is not very confident anywhere in the
+feature space.</p></li>
+<li><p>The decision boundary is no longer a straight line: the linear model is now
+classifying in the 100-dimensional feature space created by the <code class="docutils literal notranslate"><span class="pre">Nystroem</span></code>
+transformer. As are result, the decision boundary induced by the overall
+pipeline is now expressive enough to wrap around the minority class.</p></li>
+<li><p>For <code class="docutils literal notranslate"><span class="pre">C</span> <span class="pre">=</span> <span class="pre">1</span></code> in particular, it finds a smooth red blob around most of the
+“Chinstrap” data points. When moving away from the data points, the model is
+less confident in its predictions and again tends to predict the majority
+class according to the proportion in the training set.</p></li>
+<li><p>For higher values of <code class="docutils literal notranslate"><span class="pre">C</span></code>, the model starts to overfit: it is very confident
+in its predictions almost everywhere, but it should not be trusted: the
+model also makes a larger number of mistakes on the test set (not shown in
+the plot) while adopting a very curvy decision boundary to attempt fitting
+all the training points, including the noisy ones at the frontier between
+the two classes. This makes the decision boundary very sensitive to the
+sampling of the training set and as a result, it does not generalize well in
+that region. This is confirmed by the (slightly) lower accuracy on the test
+set.</p></li>
+</ul>
+<p>Finally, we can also note that the linear model on the raw features was as
+good or better than the best model using non-linear feature engineering. So in
+this case, we did not really need this extra complexity in our pipeline.
+<strong>Simpler is better!</strong></p>
+<p>So to conclude, when using non-linear feature engineering, it is often
+possible to make the pipeline overfit, even if the original feature space is
+low-dimensional. As a result, it is important to tune the regularization
+parameter in conjunction with the parameters of the transformers (e.g. tuning
+<code class="docutils literal notranslate"><span class="pre">gamma</span></code> would be important here). This has a direct impact on the certainty of
+the predictions.</p>
+</section>
 </section>
 
     <script type="text/x-thebe-config">
@@ -874,6 +1052,22 @@ <h1>📃 Solution for Exercise M4.03<a class="headerlink" href="#solution-for-ex
             
             
               
+                <div class="bd-sidebar-secondary bd-toc"><div class="sidebar-secondary-items sidebar-secondary__inner">
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+  <div class="sidebar-secondary-item">
+  <div class="page-toc tocsection onthispage">
+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#influence-of-the-parameter-c-on-the-decision-boundary">Influence of the parameter <code class="docutils literal notranslate"><span class="pre">C</span></code> on the decision boundary</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#impact-of-the-regularization-on-the-weights">Impact of the regularization on the weights</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#impact-of-the-regularization-on-with-non-linear-feature-engineering">Impact of the regularization on with non-linear feature engineering</a></li>
+</ul>
+  </nav></div>
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diff --git a/searchindex.js b/searchindex.js
index f3382abf1..d9df296b7 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
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157, 158, 162, 178, 184, 186], "befor": [3, 45, 49, 76, 80, 82, 84, 85, 87, 102, 104, 106, 110, 118, 121, 126, 129, 134, 140, 143, 146, 160, 164], "converg": [3, 68, 84, 87], "algorithm": [3, 13, 16, 18, 24, 35, 76, 83, 84, 87, 97, 110, 111, 115, 117, 118, 119, 121, 124, 126, 127, 129, 131, 136, 139, 141, 158, 159], "over": [3, 15, 24, 29, 39, 40, 76, 95, 97, 99, 100, 103, 105, 110, 114, 118, 120, 123, 146, 149, 151, 156, 158, 160, 162, 164, 171, 174], "done": [3, 6, 75, 83, 91, 118, 119, 121, 129, 131, 133, 134, 136, 152, 153, 155, 158, 162], "monitor": [3, 6, 29], "score": [3, 18, 27, 28, 29, 37, 49, 60, 62, 68, 75, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 104, 105, 109, 110, 115, 116, 117, 118, 119, 120, 122, 124, 125, 126, 127, 128, 129, 134, 136, 137, 139, 141, 143, 144, 145, 146, 147, 148, 149, 151, 152, 153, 154, 155, 156, 157, 158, 164, 176, 178, 180, 186], "jargon": [3, 83, 84], "onc": [3, 6, 18, 29, 82, 84, 92, 97, 102, 104, 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171, 180, 183, 186], "also": [3, 13, 22, 25, 27, 29, 33, 36, 46, 60, 62, 76, 82, 83, 84, 87, 88, 89, 90, 91, 92, 95, 98, 99, 100, 103, 104, 105, 106, 107, 110, 111, 112, 115, 116, 118, 119, 121, 124, 125, 127, 133, 134, 139, 140, 141, 142, 143, 146, 152, 153, 154, 155, 157, 158], "python": [3, 6, 29, 36, 68, 73, 76, 78, 79, 89, 91, 104, 107, 108, 109, 110, 112, 118, 134, 144, 147, 158, 159, 163, 164], "pass": [3, 18, 29, 49, 62, 68, 75, 79, 87, 88, 89, 91, 95, 99, 100, 104, 112, 140, 144, 145, 147, 148, 150, 151, 153, 157, 158, 186], "anoth": [3, 6, 18, 60, 76, 79, 99, 101, 103, 112, 121, 127, 128, 130, 134, 135, 139, 142, 143, 146, 158, 162], "includ": [3, 29, 34, 36, 66, 87, 99, 106, 127, 128, 131, 134, 136, 139, 152, 175], "gridsearchcv": [3, 6, 99, 120, 125, 150, 151, 153, 155, 157, 162, 178, 180, 186], "someth": [3, 92, 142, 152], "occur": [3, 87, 134], "your": [3, 6, 34, 39, 49, 62, 75, 76, 77, 80, 81, 85, 87, 88, 89, 90, 91, 92, 95, 96, 100, 102, 113, 114, 115, 116, 119, 126, 129, 130, 131, 132, 136, 144, 145, 149, 150, 156, 160, 161, 165, 178, 180], "stick": 3, "too": [3, 6, 63, 83, 93, 99, 104, 105, 115, 118, 119, 121, 124, 133, 134, 136, 151, 157, 162, 180], "so": [3, 6, 14, 29, 68, 76, 78, 84, 86, 87, 89, 91, 94, 95, 100, 101, 102, 103, 104, 106, 107, 110, 111, 116, 117, 119, 125, 131, 133, 134, 136, 138, 139, 143, 144, 147, 152, 153, 157, 178], "up": [3, 6, 10, 35, 45, 76, 79, 82, 87, 97, 99, 101, 104, 109, 110, 127, 130, 135, 139, 140, 143, 151, 163, 166, 183], "nois": [3, 49, 55, 60, 105, 112, 117, 133, 139, 142], "rather": [3, 29, 34, 79, 82, 107, 108, 141, 143, 148, 155, 157], "than": [3, 6, 18, 27, 28, 29, 34, 36, 44, 49, 55, 62, 75, 76, 79, 82, 83, 84, 85, 86, 87, 88, 91, 92, 94, 96, 97, 99, 100, 101, 103, 104, 105, 106, 107, 110, 111, 112, 118, 119, 120, 121, 124, 126, 128, 129, 130, 131, 133, 134, 135, 136, 141, 142, 143, 146, 148, 150, 151, 153, 154, 155, 157, 161, 162, 165, 175, 178, 180, 186], "relev": [3, 62, 82, 97, 110, 143], "pattern": [3, 23, 76, 97, 126, 129, 144, 147, 153], "tell": [3, 76, 105, 110], "great": [3, 6, 51], "poorli": [3, 91], "real": [3, 51, 76, 82, 83, 87, 88, 102, 104, 130, 135, 143, 146, 163], "world": [3, 163], "fit_predict": 3, "kneighborsclassifi": [3, 62, 80, 83, 85, 186], "decisiontreeregressor": [3, 18, 94, 98, 103, 104, 105, 112, 113, 117, 120, 121, 122, 139, 162, 163, 165, 178], "One": [3, 51, 62, 71, 75, 79, 80, 85, 87, 102, 103, 108, 110, 118, 143, 146, 153], "focu": [3, 18, 76, 82, 104, 106, 109, 111, 117, 134, 143, 146, 151, 152], "were": [3, 29, 68, 79, 82, 88, 97, 104, 105, 108, 117, 126, 129, 130, 133, 135, 143, 155, 157, 162, 165], "If": [3, 6, 28, 29, 37, 43, 55, 68, 76, 79, 83, 87, 94, 95, 97, 98, 99, 100, 103, 104, 105, 109, 110, 111, 113, 114, 115, 118, 119, 120, 121, 122, 123, 124, 127, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 152, 153, 157, 158, 160, 161, 162, 163, 164, 165, 176, 186], "do": [3, 6, 17, 18, 29, 60, 68, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 91, 92, 93, 96, 99, 101, 103, 104, 105, 106, 109, 115, 118, 119, 121, 124, 126, 127, 129, 131, 133, 134, 136, 138, 139, 143, 144, 147, 151, 152, 153, 154, 158, 163, 180, 182, 186], "1d": [3, 29, 139], "5": [3, 4, 18, 29, 49, 62, 68, 75, 76, 78, 79, 83, 84, 85, 86, 87, 88, 90, 92, 93, 94, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 119, 120, 121, 122, 123, 124, 125, 128, 131, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 146, 148, 151, 153, 154, 155, 157, 158, 159, 162, 163, 164, 165, 166, 182, 183, 186], "someon": [3, 76], "come": [3, 6, 29, 58, 76, 82, 117, 120, 127, 130, 135, 140, 152, 153, 163], "doe": [3, 6, 18, 24, 29, 36, 49, 76, 79, 84, 87, 88, 90, 91, 92, 94, 97, 98, 99, 100, 102, 103, 107, 110, 119, 122, 133, 134, 141, 142, 146, 148, 151, 155, 157, 158, 163, 180, 184], "15": [3, 4, 76, 82, 84, 87, 94, 105, 106, 116, 118, 119, 122, 125, 143, 155, 157, 158, 178], "continu": [3, 6, 37, 42, 44, 66, 76, 82, 104, 106, 110, 140, 141, 143, 146, 148, 159, 162], "price": [3, 51, 75, 93, 94, 104, 106, 109, 110, 145, 146, 148], "descript": [3, 29, 51, 75, 76, 80, 85, 94, 95, 98, 100, 104, 105, 109, 111, 113, 114, 115, 118, 120, 121, 122, 123, 124, 130, 131, 132, 134, 135, 136, 137, 138, 140, 141, 143, 144, 145, 146, 147, 148, 158, 160, 161, 162, 163, 164, 165, 166], "room": [3, 51, 104, 109, 110], "surfac": [3, 29], "locat": [3, 77, 78, 94, 106, 107, 109, 141, 154], "ag": [3, 49, 76, 79, 81, 82, 83, 84, 86, 87, 88, 104, 109, 110, 134, 151, 152, 155], "mri": 3, "scan": [3, 6, 152], "want": [3, 18, 76, 77, 78, 82, 83, 87, 90, 92, 93, 94, 95, 97, 98, 100, 102, 103, 104, 105, 107, 111, 113, 114, 115, 118, 120, 121, 122, 123, 124, 127, 128, 130, 131, 132, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 151, 152, 153, 158, 159, 160, 161, 162, 163, 164, 165, 178, 180, 186], "tree": [3, 6, 10, 13, 15, 16, 17, 18, 60, 76, 84, 87, 88, 90, 92, 94, 98, 103, 104, 105, 110, 111, 112, 114, 115, 116, 118, 120, 121, 122, 123, 124, 125, 128, 133, 139, 144, 147, 149, 151, 155, 156, 159, 160, 161, 164, 165, 171, 172, 173, 174, 175, 176, 178], "piecewis": [3, 133, 163, 175], "constant": [3, 24, 86, 110, 131, 133, 136, 139, 163, 175], "given": [3, 6, 18, 29, 43, 51, 60, 68, 82, 84, 87, 96, 99, 101, 102, 104, 106, 108, 109, 110, 112, 117, 121, 128, 132, 133, 134, 136, 137, 138, 140, 141, 143, 148, 153, 155, 158, 173, 180, 182, 183], "output": [3, 18, 42, 76, 79, 84, 87, 88, 103, 104, 107, 110, 112, 130, 131, 133, 135, 136, 141, 143, 148], "correspond": [3, 18, 29, 76, 79, 82, 87, 88, 89, 91, 97, 99, 104, 105, 106, 107, 108, 109, 110, 112, 114, 123, 136, 140, 141, 143, 151, 152, 154, 155, 163, 175, 178], "ridg": [3, 40, 45, 46, 49, 109, 110, 112, 134, 137], "order": [3, 6, 18, 29, 40, 49, 71, 76, 84, 88, 91, 92, 93, 97, 102, 104, 106, 116, 121, 125, 134, 140, 151, 154, 155, 186], "shrink": [3, 45, 46, 49, 134, 137], "constrain": [3, 40, 45, 63, 105, 133], "toward": [3, 45, 46, 49, 110, 127, 134, 137, 141], "zero": [3, 28, 29, 42, 44, 45, 46, 49, 55, 58, 76, 87, 105, 110, 134, 137, 139, 141], "2d": [3, 133, 139], "singl": [3, 15, 16, 18, 24, 25, 27, 28, 29, 35, 42, 43, 44, 45, 46, 49, 51, 55, 62, 66, 68, 71, 75, 76, 79, 82, 84, 87, 88, 93, 96, 99, 101, 102, 103, 104, 110, 112, 117, 120, 121, 122, 127, 128, 130, 131, 133, 135, 136, 139, 143, 144, 146, 147, 151, 152, 153, 158, 162, 173, 174, 175, 176, 178, 180, 182, 186], "orient": [3, 110], "clf": 3, "give": [3, 6, 14, 22, 24, 29, 33, 55, 58, 60, 73, 76, 79, 84, 86, 87, 88, 97, 98, 99, 102, 104, 107, 108, 110, 111, 112, 117, 118, 119, 120, 121, 127, 134, 138, 139, 141, 143, 146, 153, 157, 158], "concret": [3, 29, 60], "graphic": [3, 68, 88, 102, 109, 140], "plot": [3, 18, 29, 49, 62, 68, 76, 77, 78, 84, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 114, 116, 117, 123, 125, 127, 128, 130, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 144, 146, 147, 153, 154, 155, 157, 158, 159, 160, 161, 162, 163, 164, 165, 180], "compos": [3, 49, 75, 76, 83, 87, 88, 89, 90, 91, 92, 93, 96, 101, 106, 121, 146, 149, 151, 153, 155, 156, 162], "sinc": [3, 76, 79, 84, 95, 99, 100, 101, 102, 104, 107, 108, 109, 110, 111, 112, 117, 118, 119, 121, 127, 134, 138, 141, 142, 143, 146, 151, 158, 159, 163, 164], "potenti": [3, 46, 51, 76, 84, 88, 98, 104, 105, 118, 138, 142, 143, 153, 182], "choic": [3, 29, 36, 45, 55, 58, 60, 87, 100, 101, 104, 107, 109, 110, 112, 134, 138, 148, 151, 155, 157, 166, 186], "circl": [3, 76, 109, 112, 133, 142], "vs": [3, 84, 87, 143], "squar": [3, 28, 60, 107, 134, 135, 138, 139, 146, 148], "boil": 3, "down": [3, 6, 119], "fact": [3, 18, 84, 86, 92, 107, 117, 134, 155, 165, 178], "exactli": [3, 45, 58, 62, 82, 92, 97, 142, 186], "know": [3, 6, 88, 95, 100, 104, 105, 106, 109, 112, 117, 129, 134, 139, 143, 151], "frame": [3, 106, 107, 108, 109], "scienc": [3, 6, 36, 97, 107], "solv": [3, 6, 29, 45, 62, 76, 82, 88, 97, 100, 102, 104, 107, 108, 134, 138, 139, 140, 141, 146, 163, 186], "might": [3, 6, 29, 37, 51, 86, 87, 88, 90, 92, 97, 98, 102, 103, 104, 105, 107, 110, 119, 135, 143, 146, 151, 155], "speci": [3, 18, 77, 78, 111, 132, 137, 141, 158, 159, 160, 162, 164, 186], "commonli": [3, 76, 82, 83], "denot": 3, "eventu": 3, "ideal": [3, 104, 143, 146], "let": [3, 6, 18, 25, 44, 75, 76, 79, 80, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 117, 119, 127, 128, 131, 132, 133, 134, 135, 136, 137, 138, 139, 142, 143, 146, 148, 151, 152, 155, 158, 159, 162, 163, 186], "On": [3, 6, 29, 82, 84, 85, 88, 93, 99, 104, 110, 111, 112, 119, 124, 126, 129, 133, 134, 136, 138, 139, 142, 143, 146, 151, 153, 155, 158, 163, 164, 165], "figur": [3, 27, 76, 78, 79, 82, 102, 104, 107, 108, 109, 110, 111, 112, 117, 141, 153, 154, 155, 158, 159, 180], "mathemat": [3, 60, 95, 100, 133, 139, 140, 141, 146, 158], "b": [3, 15, 16, 17, 18, 24, 25, 26, 27, 28, 29, 35, 37, 42, 43, 44, 45, 46, 49, 51, 53, 55, 62, 63, 66, 68, 71, 75, 130, 135, 140, 155, 157, 173, 174, 175, 176, 178, 180, 182, 186], "creat": [3, 29, 45, 49, 62, 71, 73, 74, 75, 79, 80, 82, 84, 85, 87, 88, 95, 96, 98, 100, 101, 102, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 120, 122, 123, 124, 125, 126, 127, 129, 130, 131, 132, 134, 135, 136, 137, 139, 141, 142, 143, 144, 145, 147, 148, 151, 152, 153, 155, 158, 160, 161, 163, 164, 165, 174, 178, 183], "infin": 3, "vari": [3, 18, 82, 95, 98, 100, 102, 105, 107, 110, 125, 127, 134, 138, 140, 143, 152, 157, 178], "fulfil": 3, "requir": [3, 6, 13, 18, 22, 29, 33, 36, 39, 40, 44, 49, 58, 62, 73, 75, 76, 79, 82, 84, 88, 95, 100, 101, 107, 111, 118, 119, 134, 148, 153, 155, 157, 162, 171, 178, 183, 186], "minim": [3, 6, 40, 60, 66, 94, 99, 104, 105, 138, 146, 148, 158, 174], "sum": [3, 40, 42, 43, 60, 76, 83, 85, 117, 135, 141, 143], "error": [3, 6, 16, 28, 29, 40, 42, 53, 55, 58, 60, 61, 63, 76, 79, 83, 89, 91, 94, 97, 98, 99, 105, 111, 113, 114, 115, 117, 118, 119, 121, 122, 123, 124, 130, 131, 134, 135, 136, 138, 139, 142, 143, 145, 146, 148, 149, 156, 166, 174], "red": [3, 27, 79, 104, 111, 112, 117, 133, 137, 141, 142, 153, 158, 162, 164], "best": [3, 13, 18, 22, 33, 39, 55, 60, 62, 76, 93, 99, 100, 102, 105, 113, 115, 116, 118, 119, 121, 122, 124, 125, 130, 134, 135, 138, 139, 143, 146, 149, 150, 151, 152, 153, 155, 156, 157, 158, 171, 180, 183, 184, 186], "possibl": [3, 6, 15, 29, 36, 42, 44, 60, 62, 66, 71, 76, 84, 87, 94, 98, 99, 102, 104, 105, 110, 112, 118, 119, 121, 129, 130, 131, 133, 134, 135, 136, 139, 141, 142, 146, 148, 150, 151, 153, 154, 157, 161, 163, 165, 180, 186], "abstract": [3, 107], "manner": [3, 14, 29, 87, 97, 134, 148], "state": [3, 6, 14, 62, 76, 82, 83, 84, 87, 88, 109, 151, 155], "jockei": 3, "wheel": 3, "i": [3, 13, 18, 20, 22, 29, 49, 73, 74, 75, 76, 79, 82, 83, 84, 86, 87, 88, 91, 92, 98, 99, 102, 104, 110, 112, 117, 118, 119, 120, 130, 131, 133, 134, 135, 136, 137, 138, 141, 143, 146, 148, 151, 157, 166, 180, 186], "support": [3, 6, 84, 90, 92, 95, 100, 133, 139, 142, 144, 147], "standardscal": [3, 29, 49, 62, 68, 75, 79, 84, 88, 90, 92, 93, 95, 100, 101, 102, 109, 110, 132, 133, 137, 141, 142, 150, 152, 157, 178, 180, 182, 186], "columntransform": [3, 74, 88, 90, 92, 93, 133, 149, 151, 153, 155, 156], "enough": [3, 6, 91, 92, 101, 104, 105, 117, 119, 142, 146, 152, 158, 162, 164, 178], "flexibl": [3, 6, 46, 55, 58, 60, 63, 95, 100, 105, 121, 142], "opposit": 3, "cluster": [3, 104, 173], "whose": [3, 83, 118, 152], "group": [3, 6, 20, 25, 29, 76, 102, 103, 104, 109, 166], "subset": [3, 17, 42, 45, 73, 76, 79, 82, 83, 88, 93, 104, 107, 109, 119, 121, 124, 126, 127, 128, 129, 149, 156, 158, 159, 162, 178], "them": [3, 6, 13, 62, 76, 78, 79, 84, 87, 88, 96, 99, 101, 104, 110, 112, 114, 118, 119, 120, 123, 131, 133, 134, 136, 139, 141, 143, 146, 151, 152, 153, 157, 161, 164, 165, 178, 186], "broad": 3, "topic": [3, 97, 104, 109], "custom": [3, 139], "commerc": 3, "websit": [3, 36, 51, 80, 85], "although": 3, "mention": [3, 51, 79, 95, 100, 103, 106, 111, 118, 120, 133, 134, 138, 141, 143, 144, 146, 147, 152, 155, 165], "cover": [3, 52, 76, 79, 82, 87, 88], "impli": [3, 162], "fix": [3, 29, 49, 55, 60, 87, 95, 100, 119, 134, 150, 151, 157, 162, 178, 180, 184], "like": [3, 6, 18, 25, 29, 46, 55, 76, 82, 83, 87, 88, 89, 91, 99, 101, 104, 106, 108, 110, 111, 112, 119, 127, 134, 135, 139, 140, 141, 143, 144, 147, 148, 152], "necessari": [3, 6, 45, 76, 119, 133, 153], "subdivid": [3, 158], "select": [3, 6, 13, 15, 16, 17, 18, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 39, 42, 43, 44, 45, 46, 49, 51, 53, 55, 62, 63, 66, 68, 71, 75, 77, 78, 79, 81, 82, 84, 86, 89, 90, 91, 92, 99, 102, 104, 110, 112, 117, 119, 121, 125, 126, 129, 130, 135, 138, 139, 141, 151, 152, 153, 154, 157, 159, 171, 173, 174, 175, 176, 178, 180, 182, 186], "final": [3, 13, 23, 39, 45, 58, 62, 73, 83, 84, 87, 88, 99, 103, 104, 107, 109, 112, 116, 117, 119, 124, 125, 126, 128, 129, 134, 144, 145, 147, 148, 151, 152, 153, 161, 165, 183], "sometim": [3, 6, 60, 88, 101, 143, 146, 153, 155], "clear": [3, 76, 100, 105, 162], "mani": [3, 6, 29, 43, 49, 62, 75, 76, 77, 78, 83, 84, 87, 88, 99, 101, 102, 104, 105, 106, 109, 112, 119, 121, 131, 133, 134, 136, 143, 152, 157, 186], "need": [3, 6, 22, 24, 29, 37, 40, 62, 68, 76, 80, 82, 84, 85, 87, 88, 93, 95, 99, 100, 104, 105, 107, 110, 111, 112, 117, 119, 120, 121, 122, 133, 134, 138, 139, 142, 144, 145, 147, 148, 149, 151, 152, 153, 155, 156, 162, 176, 178], "criteria": [3, 119], "ml": [3, 6, 97], "cheatsheet": 3, "readthedoc": 3, "io": [3, 36], "en": 3, "latest": [3, 28], "googl": 3, "develop": [3, 6, 36, 73, 76, 89, 91, 107, 131, 136, 139], "com": [3, 6, 36], "advanc": [3, 6, 36, 73, 90, 92], "terminolog": 3, "modifi": [4, 6, 118, 120, 139], "run": [4, 6, 18, 45, 68, 80, 85, 94, 98, 99, 110, 116, 125, 134, 154, 155, 178, 180, 186], "statu": [4, 25, 76, 82, 87, 88, 151, 153, 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100, 101, 102, 108, 111, 117, 127, 128, 129, 130, 135, 137, 141, 144, 147, 148, 151, 152, 153, 154, 155, 156, 157, 158, 160, 164, 186], "imperfect": [6, 110], "estim": [6, 13, 17, 22, 29, 33, 39, 45, 49, 58, 62, 68, 73, 79, 83, 84, 87, 88, 93, 99, 101, 103, 109, 110, 111, 112, 113, 115, 116, 118, 119, 120, 121, 122, 124, 125, 126, 127, 129, 131, 134, 136, 139, 145, 146, 148, 151, 152, 153, 155, 171, 178, 180, 183, 186], "actual": [6, 76, 80, 83, 85, 94, 103, 104, 105, 119, 133, 143, 146, 153], "gener": [6, 17, 18, 22, 23, 28, 29, 49, 52, 58, 60, 61, 62, 68, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 108, 109, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 131, 134, 136, 139, 141, 142, 143, 145, 146, 148, 149, 150, 151, 152, 153, 155, 156, 157, 163, 166, 169, 178, 180, 186], "As": [6, 29, 49, 76, 79, 82, 83, 84, 87, 88, 90, 92, 95, 97, 99, 100, 101, 106, 108, 109, 110, 112, 117, 118, 133, 134, 141, 142, 143, 145, 146, 148, 152, 153, 154, 157, 162, 165], "narrow": 6, "spend": [6, 107, 119], "increasingli": 6, "effort": [6, 107], "split": [6, 15, 18, 23, 29, 37, 43, 45, 68, 76, 79, 81, 86, 87, 88, 94, 99, 102, 103, 104, 106, 110, 113, 117, 118, 119, 121, 122, 124, 126, 129, 142, 143, 146, 151, 153, 155, 158, 160, 162, 163, 164, 173, 174], "afford": 6, "trust": [6, 82, 83, 84, 85, 88, 93, 99, 104, 111, 133, 134, 138, 139, 142, 143, 151, 152, 153, 155, 158, 164, 165], "think": [6, 77, 78, 103, 107, 131, 136], "carefulli": [6, 97], "complet": [6, 15, 29, 36, 49, 99, 101, 126, 129, 133, 134, 150, 153, 157, 180], "futur": [6, 76, 79, 88, 93, 103, 104, 151, 158, 163, 164], "upon": [6, 93, 100, 101], "affect": [6, 103, 134, 141, 151, 163], "live": [6, 134], "sure": [6, 18, 62, 82, 84, 87, 92, 116, 125], "divers": [6, 158], "demograph": [6, 120], "increas": [6, 16, 18, 28, 29, 44, 45, 46, 49, 53, 55, 60, 78, 84, 87, 100, 104, 105, 110, 115, 118, 119, 124, 134, 140, 141, 142, 151, 155, 157, 158, 160, 162, 163, 164, 176, 184], "coverag": 6, "phrase": 6, "recommend": [6, 36, 73, 76, 87], "identifi": [6, 13, 73, 88, 97, 100, 105, 143, 180], "ani": [6, 15, 18, 29, 36, 44, 75, 76, 79, 82, 84, 86, 88, 91, 92, 94, 97, 100, 102, 103, 104, 105, 107, 108, 109, 110, 112, 119, 120, 121, 125, 126, 127, 128, 129, 133, 134, 138, 141, 146, 151, 153, 154, 157, 162, 175, 186], "bia": [6, 37, 53, 58, 60, 110, 131, 133, 136, 146, 166], "acquisit": 6, "full": [6, 8, 9, 18, 36, 48, 50, 51, 52, 54, 56, 64, 68, 69, 79, 80, 84, 85, 87, 99, 104, 119, 124, 126, 128, 129, 140, 149, 153, 156, 167, 180, 183], "chain": [6, 79, 84, 88], "acquir": [6, 13, 22, 33, 39, 58, 105, 107, 171, 183], "fanci": 6, "put": [6, 33, 37, 84, 105, 110, 134, 158], "product": [6, 99, 102, 104, 131, 134, 136, 151], "routin": [6, 18, 97, 186], "debt": 6, "simpler": [6, 14, 88], "easier": [6, 84, 87, 94, 139, 146], "maintain": 6, "less": [6, 29, 46, 88, 97, 103, 109, 110, 111, 112, 119, 120, 128, 133, 134, 136, 139, 151, 157], "power": [6, 29, 36, 107, 119, 120, 131, 134, 136, 139, 158], "drift": 6, "gave": [6, 143], "methodolog": [6, 36, 58, 104, 163], "element": [6, 22, 83, 87, 97, 104, 134, 135, 140, 143], "alwai": [6, 15, 18, 22, 24, 43, 45, 55, 81, 82, 83, 86, 88, 92, 94, 97, 99, 100, 101, 102, 104, 105, 108, 109, 121, 127, 142, 143, 152, 153, 154, 165, 178, 180, 186], "solid": 6, "conclus": [6, 76, 96, 97, 99, 101, 102, 103, 105, 108, 119, 127], "standpoint": 6, "biggest": 6, "shortcom": 6, "cannot": [6, 18, 29, 60, 76, 102, 103, 105, 110, 119, 126, 129, 133, 139, 142, 143, 146, 148, 151, 154, 165, 183], "autom": [6, 76, 166, 184], "domain": 6, "knowledg": [6, 36, 73, 99, 105, 112, 129, 139, 142, 153], "critic": [6, 36, 106], "thing": [6, 76, 87, 88, 93, 103, 139, 151], "oper": [6, 84, 112, 133, 143, 153], "risk": [6, 131, 136, 142], "advertis": 6, "individu": [6, 17, 29, 76, 84, 88, 108, 112, 114, 121, 123, 134, 141, 142, 158, 186], "caus": [6, 18, 53, 60, 76, 87, 91, 92, 119, 134, 151, 153, 180], "wast": [6, 119], "bit": [6, 29, 86, 104, 105, 110, 134, 139, 144, 147, 153], "monei": 6, "annoi": 6, "otherwis": [6, 18, 82, 91, 97, 99, 121, 133, 139, 140, 162], "mostli": [6, 157], "harmless": 6, "medicin": 6, "kill": 6, "logic": [6, 151], "fals": [6, 16, 55, 83, 87, 90, 92, 97, 99, 103, 107, 109, 125, 127, 128, 133, 134, 136, 139, 143, 147, 150, 151, 155, 157, 158, 164, 176], "brain": 6, "tumor": 6, "sent": 6, "surgeri": 6, "veri": [6, 18, 58, 62, 76, 82, 88, 90, 92, 99, 100, 102, 105, 108, 109, 110, 112, 117, 119, 121, 125, 133, 134, 139, 143, 151, 152, 153, 154, 155, 162, 180], "danger": [6, 134, 155], "mr": 6, "confirm": [6, 94, 102, 107, 109, 112, 126, 129, 133, 141, 146, 153, 154], "should": [6, 18, 23, 29, 34, 35, 45, 71, 75, 76, 83, 84, 88, 92, 97, 99, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 116, 117, 119, 120, 125, 126, 127, 129, 130, 134, 135, 139, 142, 143, 144, 147, 148, 149, 151, 152, 153, 156, 158, 160, 162, 163, 164, 176, 178, 180, 183, 184], "delai": 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76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 133, 134, 135, 136, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 155, 156, 157, 158, 159, 160, 164, 178, 186], "proxi": [6, 143], "reflect": [6, 104, 112, 121], "ground": [6, 117, 143], "truth": [6, 117, 143], "polici": [6, 25], "uneven": 6, "across": [6, 43, 49, 79, 84, 94, 110, 134, 141, 155, 186], "popul": [6, 76, 104, 109, 110, 143, 157], "eg": 6, "qualiti": [6, 110, 141, 146, 153], "affair": 6, "desir": [6, 81, 86, 121, 134], "qualif": 6, "respons": 6, "women": 6, "pai": [6, 87, 111], "men": 6, "pick": [6, 29, 93, 99, 106, 107, 110, 118, 152, 183], "amplifi": 6, "inequ": 6, "mechan": [6, 29, 83, 84, 153], "die": 6, "naiv": [6, 18, 29, 76, 99, 108, 118, 121], "bad": [6, 91, 100, 129, 134, 142, 143, 180], "health": [6, 36], "fallaci": 6, "compar": [6, 13, 17, 18, 22, 29, 37, 49, 51, 61, 62, 75, 76, 79, 81, 83, 84, 86, 87, 88, 89, 91, 99, 100, 101, 105, 109, 110, 112, 115, 117, 118, 122, 124, 127, 128, 129, 131, 134, 136, 143, 146, 155, 157, 158, 166, 178, 186], "wors": [6, 18, 28, 49, 75, 100, 101, 103, 138, 178], "baselin": [6, 22, 24, 81, 86, 88, 89, 91, 96, 97, 100, 101, 120, 166], "heart": [6, 29, 107, 131, 136], "pressur": 6, "greater": [6, 29, 75, 133], "trigger": 6, "care": [6, 34, 87, 99, 102, 108, 109, 126, 129, 134], "which": [6, 14, 16, 17, 18, 23, 25, 27, 29, 33, 34, 40, 43, 45, 49, 60, 68, 71, 74, 75, 76, 79, 82, 83, 84, 86, 87, 88, 92, 93, 94, 95, 98, 99, 100, 101, 103, 104, 105, 107, 108, 109, 110, 111, 112, 117, 118, 119, 121, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 143, 144, 146, 147, 148, 150, 151, 152, 154, 155, 157, 162, 163, 164, 171, 172, 173, 178, 180, 182, 184, 186], "learner": [6, 14, 111, 112, 117, 119, 120], "predictor": [6, 14, 15, 16, 17, 49, 68, 83, 84, 88, 101, 105, 122, 134, 151, 158], "pure": [6, 129, 164], "benefici": [6, 33, 84, 119, 127, 153, 155, 162], "intervent": [6, 76], "brittl": 6, "interpret": [6, 17, 36, 107, 109, 110, 136, 138, 139, 146, 154, 158], "subject": [6, 76, 143], "caution": [6, 92, 110], "feedback": 6, "loop": [6, 99, 110, 149, 151, 153, 156, 186], "todai": 6, "ai": 6, "alloc": 6, "loan": 6, "screen": [6, 8, 9, 48, 50, 52, 54, 56, 64, 69, 167], "job": [6, 84], "prioritis": 6, "treatement": 6, "law": [6, 29], "enforc": [6, 103, 107, 119, 134], "court": 6, "fairlearn": [6, 76], "assess": [6, 25, 49, 79, 83, 96, 98, 99, 101, 105, 115, 124, 130, 134, 135, 142, 143, 146, 153, 157], "shift": [6, 84], "technolog": [6, 97], "induc": [6, 110, 112], "societi": 6, "though": [6, 101, 118, 145, 148], "difficult": [6, 107, 119, 138, 146], "intersect": [6, 154, 157], "No": [6, 24], "found": [6, 62, 99, 108, 109, 116, 119, 125, 134, 138, 149, 151, 153, 156, 157, 158, 162, 163, 180, 186], "short": [6, 34, 82, 109, 120, 159], "move": [6, 88, 107, 154, 157], "choos": [6, 18, 36, 76, 89, 91, 104, 110, 126, 127, 129, 133, 134, 139, 141, 153, 176, 186], "revolut": 6, "fantast": [6, 129], "opportun": 6, "With": [6, 37, 91, 104, 107, 119, 127, 128, 143, 151, 155, 162], "lift": 6, "roadblock": 6, "hope": [6, 93, 133, 146], "empow": 6, "varieti": [6, 36, 79], "mindset": 6, "dream": 6, "being": [6, 43, 94, 141, 157], "adventur": 6, "navig": [8, 9, 48, 50, 52, 54, 56, 64, 69, 167], "slide": [8, 9, 48, 50, 52, 54, 56, 64, 69, 133, 141, 154, 157, 167], "click": [8, 9, 15, 48, 50, 52, 54, 56, 64, 69, 154, 157, 167, 180], "press": [8, 9, 48, 50, 52, 54, 56, 64, 69, 167], "arrow": [8, 9, 48, 50, 52, 54, 56, 64, 69, 167], "go": [8, 9, 13, 22, 29, 33, 36, 39, 48, 50, 52, 54, 56, 58, 64, 69, 73, 79, 80, 85, 87, 95, 97, 99, 100, 103, 104, 105, 107, 109, 110, 117, 120, 127, 141, 143, 148, 151, 158, 167, 171, 183], "next": [8, 9, 48, 50, 52, 54, 56, 60, 64, 69, 82, 83, 87, 93, 95, 96, 100, 101, 102, 103, 111, 117, 119, 120, 131, 133, 134, 136, 139, 152, 158, 167], "previou": [8, 9, 16, 18, 22, 29, 48, 49, 50, 52, 54, 56, 62, 64, 66, 68, 69, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 96, 97, 98, 99, 101, 102, 103, 104, 105, 112, 113, 116, 117, 118, 119, 121, 122, 125, 126, 129, 130, 131, 133, 134, 135, 136, 138, 139, 142, 144, 147, 149, 151, 152, 153, 154, 155, 156, 157, 158, 160, 161, 164, 165, 167, 178, 180, 183, 186], "p": [8, 9, 48, 50, 52, 54, 56, 64, 69, 97, 136, 144, 147, 167], "toggl": [8, 9, 48, 50, 52, 54, 56, 64, 69, 167], "mode": [8, 9, 36, 48, 50, 52, 54, 56, 64, 69, 167], "adapt": [10, 102, 120, 139, 142, 146, 155, 166], "adaboost": [10, 13, 117, 166], "gbdt": [10, 111, 124, 166], "exercis": [10, 11, 12, 19, 29, 30, 31, 32, 36, 38, 41, 47, 49, 61, 65, 67, 70, 82, 87, 119, 133, 138, 139, 151, 153, 158, 166, 168, 177, 178, 179, 181], "m6": [10, 11, 12, 119, 166], "speed": [10, 29, 34, 35, 45, 107, 110, 127, 153, 155, 166], "quiz": [10, 11, 12, 19, 20, 21, 30, 31, 38, 41, 47, 57, 59, 61, 65, 67, 70, 157, 166, 168, 169, 170, 177, 179, 181], "bag": [11, 13, 15, 16, 111, 113, 119, 120, 121, 122, 166], "introductori": [11, 141, 166], "forest": [11, 13, 14, 15, 17, 18, 37, 110, 114, 115, 117, 118, 120, 123, 124, 127, 128, 166], "togeth": [13, 14, 70, 75, 83, 84, 91, 96, 97, 99, 101, 111, 117, 142, 162, 166], "ensembl": [13, 14, 17, 18, 29, 60, 88, 90, 92, 110, 111, 112, 115, 117, 118, 119, 121, 122, 123, 124, 125, 127, 128, 148, 149, 151, 153, 155, 156], "famili": [13, 14, 39, 58, 60, 76, 82, 84, 88, 120, 152], "techniqu": [13, 33, 76, 110, 133], "bootstrap": [13, 14, 18, 119, 120, 121, 122, 166], "ii": [13, 22, 79, 83, 138], "belong": [13, 18, 49, 82, 86, 87, 97, 100, 108, 141, 148, 158, 178], "former": [13, 75, 76, 84, 127, 143], "strategi": [13, 14, 18, 22, 23, 24, 25, 29, 49, 62, 79, 81, 83, 86, 89, 91, 93, 94, 96, 97, 99, 101, 102, 103, 104, 106, 111, 112, 115, 118, 121, 124, 131, 135, 136, 138, 143, 146, 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133, 134, 136, 142, 153, 157, 162, 166, 171, 173, 176], "valid": [13, 18, 22, 23, 24, 25, 26, 29, 33, 34, 39, 40, 45, 49, 58, 59, 60, 62, 67, 68, 75, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 100, 101, 102, 103, 109, 110, 115, 116, 117, 118, 119, 120, 121, 124, 125, 126, 127, 128, 129, 131, 134, 136, 142, 143, 144, 145, 147, 148, 149, 151, 152, 153, 154, 155, 156, 162, 163, 166, 171, 176, 178, 180, 183, 184, 186], "principl": [13, 22, 34, 143], "through": [13, 22, 29, 33, 36, 39, 58, 73, 95, 100, 105, 110, 116, 125, 134, 140, 141, 144, 147, 151, 171, 180, 183], "hour": [13, 22, 39, 49, 58, 73, 76, 79, 81, 82, 83, 84, 86, 87, 88, 151, 152, 155, 171, 183], "saw": [14, 29, 40, 82, 84, 87, 88, 104, 105, 117, 119, 134, 138, 141, 142, 144, 147, 151, 153, 154, 155, 158, 162, 163, 164, 172], "parallel": [14, 16, 117, 153, 154, 157, 180], "sequenti": [14, 16, 55, 84, 118, 119, 154, 180], "intern": [14, 45, 68, 84, 88, 99, 104, 111, 112, 115, 116, 124, 125, 134, 146, 148, 151, 153, 180], 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156, 158, 162, 164, 167, 180], "iter": [3, 29, 68, 79, 84, 87, 102, 103, 104, 115, 119, 124, 134, 153, 154, 155], "optim": [3, 22, 23, 45, 66, 84, 95, 99, 100, 111, 116, 117, 118, 120, 125, 129, 131, 134, 136, 138, 143, 146, 148, 151, 152, 153, 155, 157, 158, 162, 178, 184, 186], "befor": [3, 45, 49, 76, 80, 82, 84, 85, 87, 102, 104, 106, 110, 118, 121, 126, 129, 134, 140, 143, 146, 160, 164], "converg": [3, 68, 84, 87], "algorithm": [3, 13, 16, 18, 24, 35, 76, 83, 84, 87, 97, 110, 111, 115, 117, 118, 119, 121, 124, 126, 127, 129, 131, 136, 139, 141, 158, 159], "over": [3, 15, 24, 29, 39, 40, 76, 95, 97, 99, 100, 103, 105, 110, 114, 118, 120, 123, 146, 149, 151, 156, 158, 160, 162, 164, 171, 174], "done": [3, 6, 75, 83, 91, 118, 119, 121, 129, 131, 133, 134, 136, 152, 153, 155, 158, 162], "monitor": [3, 6, 29], "score": [3, 18, 27, 28, 29, 37, 49, 60, 62, 68, 75, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 104, 105, 109, 110, 115, 116, 117, 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80, 82, 84, 85, 87, 88, 92, 101, 102, 105, 106, 109, 110, 112, 118, 119, 121, 133, 134, 139, 143, 146, 152, 158, 163, 165, 172, 178, 183], "avail": [3, 6, 29, 36, 51, 75, 76, 77, 78, 79, 82, 83, 93, 97, 98, 106, 108, 109, 114, 123, 127, 129, 134, 138, 140, 143, 152, 155, 178], "after": [3, 18, 28, 49, 79, 87, 104, 106, 112, 115, 118, 119, 122, 124, 138, 152, 156, 157], "been": [3, 76, 79, 82, 87, 94, 102, 104, 110, 111, 118, 124, 127, 136, 141, 152, 153, 158], "slope": [3, 29, 107, 140], "intercept": [3, 40, 130, 131, 135, 136, 137, 138, 139, 140, 141, 158], "one": [3, 6, 14, 18, 25, 37, 43, 44, 49, 60, 68, 71, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 88, 90, 92, 95, 96, 97, 99, 100, 101, 102, 104, 107, 109, 110, 111, 117, 118, 119, 121, 128, 133, 134, 136, 138, 139, 141, 143, 144, 146, 147, 151, 152, 153, 157, 158, 159, 162, 164, 174, 186], "section": [3, 79, 82, 87, 88, 94, 95, 97, 98, 100, 104, 105, 111, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 130, 131, 132, 134, 135, 136, 137, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 155, 158, 160, 161, 162, 163, 164, 165], "about": [3, 13, 17, 22, 24, 29, 33, 36, 39, 58, 62, 66, 73, 74, 76, 79, 83, 84, 87, 96, 98, 101, 104, 105, 106, 108, 110, 117, 120, 133, 137, 141, 142, 146, 151, 159, 163, 171, 180, 183, 186], "also": [3, 13, 22, 25, 27, 29, 33, 36, 46, 60, 62, 76, 82, 83, 84, 87, 88, 89, 90, 91, 92, 95, 98, 99, 100, 103, 104, 105, 106, 107, 110, 111, 112, 115, 116, 118, 119, 121, 124, 125, 127, 133, 134, 137, 139, 140, 141, 142, 143, 146, 152, 153, 154, 155, 157, 158], "python": [3, 6, 29, 36, 68, 73, 76, 78, 79, 89, 91, 104, 107, 108, 109, 110, 112, 118, 134, 144, 147, 158, 159, 163, 164], "pass": [3, 18, 29, 49, 62, 68, 75, 79, 87, 88, 89, 91, 95, 99, 100, 104, 112, 140, 144, 145, 147, 148, 150, 151, 153, 157, 158, 186], "anoth": [3, 6, 18, 60, 76, 79, 99, 101, 103, 112, 121, 127, 128, 130, 134, 135, 139, 142, 143, 146, 158, 162], "includ": [3, 29, 34, 36, 66, 87, 99, 106, 127, 128, 131, 134, 136, 137, 139, 152, 175], "gridsearchcv": [3, 6, 99, 120, 125, 150, 151, 153, 155, 157, 162, 178, 180, 186], "someth": [3, 92, 142, 152], "occur": [3, 87, 134], "your": [3, 6, 34, 39, 49, 62, 75, 76, 77, 80, 81, 85, 87, 88, 89, 90, 91, 92, 95, 96, 100, 102, 113, 114, 115, 116, 119, 126, 129, 130, 131, 132, 136, 144, 145, 149, 150, 156, 160, 161, 165, 178, 180], "stick": 3, "too": [3, 6, 63, 83, 93, 99, 104, 105, 115, 118, 119, 121, 124, 133, 134, 136, 137, 151, 157, 162, 180], "so": [3, 6, 14, 29, 68, 76, 78, 84, 86, 87, 89, 91, 94, 95, 100, 101, 102, 103, 104, 106, 107, 110, 111, 116, 117, 119, 125, 131, 133, 134, 136, 137, 138, 139, 143, 144, 147, 152, 153, 157, 178], "up": [3, 6, 10, 35, 45, 76, 79, 82, 87, 97, 99, 101, 104, 109, 110, 127, 130, 135, 139, 140, 143, 151, 163, 166, 183], "nois": [3, 49, 55, 60, 105, 112, 117, 133, 137, 139, 142], "rather": [3, 29, 34, 79, 82, 107, 108, 141, 143, 148, 155, 157], "than": [3, 6, 18, 27, 28, 29, 34, 36, 44, 49, 55, 62, 75, 76, 79, 82, 83, 84, 85, 86, 87, 88, 91, 92, 94, 96, 97, 99, 100, 101, 103, 104, 105, 106, 107, 110, 111, 112, 118, 119, 120, 121, 124, 126, 128, 129, 130, 131, 133, 134, 135, 136, 137, 141, 142, 143, 146, 148, 150, 151, 153, 154, 155, 157, 161, 162, 165, 175, 178, 180, 186], "relev": [3, 62, 82, 97, 110, 143], "pattern": [3, 23, 76, 97, 126, 129, 144, 147, 153], "tell": [3, 76, 105, 110], "great": [3, 6, 51], "poorli": [3, 91], "real": [3, 51, 76, 82, 83, 87, 88, 102, 104, 130, 135, 143, 146, 163], "world": [3, 163], "fit_predict": 3, "kneighborsclassifi": [3, 62, 80, 83, 85, 186], "decisiontreeregressor": [3, 18, 94, 98, 103, 104, 105, 112, 113, 117, 120, 121, 122, 139, 162, 163, 165, 178], "One": [3, 51, 62, 71, 75, 79, 80, 85, 87, 102, 103, 108, 110, 118, 143, 146, 153], "focu": [3, 18, 76, 82, 104, 106, 109, 111, 117, 134, 143, 146, 151, 152], "were": [3, 29, 68, 79, 82, 88, 97, 104, 105, 108, 117, 126, 129, 130, 133, 135, 143, 155, 157, 162, 165], "If": [3, 6, 28, 29, 37, 43, 55, 68, 76, 79, 83, 87, 94, 95, 97, 98, 99, 100, 103, 104, 105, 109, 110, 111, 113, 114, 115, 118, 119, 120, 121, 122, 123, 124, 127, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 152, 153, 157, 158, 160, 161, 162, 163, 164, 165, 176, 186], "do": [3, 6, 17, 18, 29, 60, 68, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 91, 92, 93, 96, 99, 101, 103, 104, 105, 106, 109, 115, 118, 119, 121, 124, 126, 127, 129, 131, 133, 134, 136, 138, 139, 143, 144, 147, 151, 152, 153, 154, 158, 163, 180, 182, 186], "1d": [3, 29, 139], "5": [3, 4, 18, 29, 49, 62, 68, 75, 76, 78, 79, 83, 84, 85, 86, 87, 88, 90, 92, 93, 94, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 119, 120, 121, 122, 123, 124, 125, 128, 131, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 146, 148, 151, 153, 154, 155, 157, 158, 159, 162, 163, 164, 165, 166, 182, 183, 186], "someon": [3, 76], "come": [3, 6, 29, 58, 76, 82, 117, 120, 127, 130, 135, 140, 152, 153, 163], "doe": [3, 6, 18, 24, 29, 36, 49, 76, 79, 84, 87, 88, 90, 91, 92, 94, 97, 98, 99, 100, 102, 103, 107, 110, 119, 122, 133, 134, 137, 141, 142, 146, 148, 151, 155, 157, 158, 163, 180, 184], "15": [3, 4, 76, 82, 84, 87, 94, 105, 106, 116, 118, 119, 122, 125, 143, 155, 157, 158, 178], "continu": [3, 6, 37, 42, 44, 66, 76, 82, 104, 106, 110, 140, 141, 143, 146, 148, 159, 162], "price": [3, 51, 75, 93, 94, 104, 106, 109, 110, 145, 146, 148], "descript": [3, 29, 51, 75, 76, 80, 85, 94, 95, 98, 100, 104, 105, 109, 111, 113, 114, 115, 118, 120, 121, 122, 123, 124, 130, 131, 132, 134, 135, 136, 137, 138, 140, 141, 143, 144, 145, 146, 147, 148, 158, 160, 161, 162, 163, 164, 165, 166], "room": [3, 51, 104, 109, 110], "surfac": [3, 29], "locat": [3, 77, 78, 94, 106, 107, 109, 141, 154], "ag": [3, 49, 76, 79, 81, 82, 83, 84, 86, 87, 88, 104, 109, 110, 134, 151, 152, 155], "mri": 3, "scan": [3, 6, 152], "want": [3, 18, 76, 77, 78, 82, 83, 87, 90, 92, 93, 94, 95, 97, 98, 100, 102, 103, 104, 105, 107, 111, 113, 114, 115, 118, 120, 121, 122, 123, 124, 127, 128, 130, 131, 132, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 151, 152, 153, 158, 159, 160, 161, 162, 163, 164, 165, 178, 180, 186], "tree": [3, 6, 10, 13, 15, 16, 17, 18, 60, 76, 84, 87, 88, 90, 92, 94, 98, 103, 104, 105, 110, 111, 112, 114, 115, 116, 118, 120, 121, 122, 123, 124, 125, 128, 133, 139, 144, 147, 149, 151, 155, 156, 159, 160, 161, 164, 165, 171, 172, 173, 174, 175, 176, 178], "piecewis": [3, 133, 163, 175], "constant": [3, 24, 86, 110, 131, 133, 136, 137, 139, 163, 175], "given": [3, 6, 18, 29, 43, 51, 60, 68, 82, 84, 87, 96, 99, 101, 102, 104, 106, 108, 109, 110, 112, 117, 121, 128, 132, 133, 134, 136, 137, 138, 140, 141, 143, 148, 153, 155, 158, 173, 180, 182, 183], "output": [3, 18, 42, 76, 79, 84, 87, 88, 103, 104, 107, 110, 112, 130, 131, 133, 135, 136, 141, 143, 148], "correspond": [3, 18, 29, 76, 79, 82, 87, 88, 89, 91, 97, 99, 104, 105, 106, 107, 108, 109, 110, 112, 114, 123, 136, 140, 141, 143, 151, 152, 154, 155, 163, 175, 178], "ridg": [3, 40, 45, 46, 49, 109, 110, 112, 134, 137], "order": [3, 6, 18, 29, 40, 49, 71, 76, 84, 88, 91, 92, 93, 97, 102, 104, 106, 116, 121, 125, 134, 140, 151, 154, 155, 186], "shrink": [3, 45, 46, 49, 134, 137], "constrain": [3, 40, 45, 63, 105, 133], "toward": [3, 45, 46, 49, 110, 127, 134, 137, 141], "zero": [3, 28, 29, 42, 44, 45, 46, 49, 55, 58, 76, 87, 105, 110, 134, 137, 139, 141], "2d": [3, 133, 139], "singl": [3, 15, 16, 18, 24, 25, 27, 28, 29, 35, 42, 43, 44, 45, 46, 49, 51, 55, 62, 66, 68, 71, 75, 76, 79, 82, 84, 87, 88, 93, 96, 99, 101, 102, 103, 104, 110, 112, 117, 120, 121, 122, 127, 128, 130, 131, 133, 135, 136, 139, 143, 144, 146, 147, 151, 152, 153, 158, 162, 173, 174, 175, 176, 178, 180, 182, 186], "orient": [3, 110, 137], "clf": 3, "give": [3, 6, 14, 22, 24, 29, 33, 55, 58, 60, 73, 76, 79, 84, 86, 87, 88, 97, 98, 99, 102, 104, 107, 108, 110, 111, 112, 117, 118, 119, 120, 121, 127, 134, 137, 138, 139, 141, 143, 146, 153, 157, 158], "concret": [3, 29, 60], "graphic": [3, 68, 88, 102, 109, 140], "plot": [3, 18, 29, 49, 62, 68, 76, 77, 78, 84, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 114, 116, 117, 123, 125, 127, 128, 130, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 144, 146, 147, 153, 154, 155, 157, 158, 159, 160, 161, 162, 163, 164, 165, 180], "compos": [3, 49, 75, 76, 83, 87, 88, 89, 90, 91, 92, 93, 96, 101, 106, 121, 146, 149, 151, 153, 155, 156, 162], "sinc": [3, 76, 79, 84, 95, 99, 100, 101, 102, 104, 107, 108, 109, 110, 111, 112, 117, 118, 119, 121, 127, 134, 138, 141, 142, 143, 146, 151, 158, 159, 163, 164], "potenti": [3, 46, 51, 76, 84, 88, 98, 104, 105, 118, 138, 142, 143, 153, 182], "choic": [3, 29, 36, 45, 55, 58, 60, 87, 100, 101, 104, 107, 109, 110, 112, 134, 138, 148, 151, 155, 157, 166, 186], "circl": [3, 76, 109, 112, 133, 142], "vs": [3, 84, 87, 137, 143], "squar": [3, 28, 60, 107, 134, 135, 138, 139, 146, 148], "boil": 3, "down": [3, 6, 119], "fact": [3, 18, 84, 86, 92, 107, 117, 134, 155, 165, 178], "exactli": [3, 45, 58, 62, 82, 92, 97, 142, 186], "know": [3, 6, 88, 95, 100, 104, 105, 106, 109, 112, 117, 129, 134, 139, 143, 151], "frame": [3, 106, 107, 108, 109], "scienc": [3, 6, 36, 97, 107], "solv": [3, 6, 29, 45, 62, 76, 82, 88, 97, 100, 102, 104, 107, 108, 134, 138, 139, 140, 141, 146, 163, 186], "might": [3, 6, 29, 37, 51, 86, 87, 88, 90, 92, 97, 98, 102, 103, 104, 105, 107, 110, 119, 135, 143, 146, 151, 155], "speci": [3, 18, 77, 78, 111, 132, 137, 141, 158, 159, 160, 162, 164, 186], "commonli": [3, 76, 82, 83], "denot": 3, "eventu": 3, "ideal": [3, 104, 143, 146], "let": [3, 6, 18, 25, 44, 75, 76, 79, 80, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 117, 119, 127, 128, 131, 132, 133, 134, 135, 136, 137, 138, 139, 142, 143, 146, 148, 151, 152, 155, 158, 159, 162, 163, 186], "On": [3, 6, 29, 82, 84, 85, 88, 93, 99, 104, 110, 111, 112, 119, 124, 126, 129, 133, 134, 136, 137, 138, 139, 142, 143, 146, 151, 153, 155, 158, 163, 164, 165], "figur": [3, 27, 76, 78, 79, 82, 102, 104, 107, 108, 109, 110, 111, 112, 117, 141, 153, 154, 155, 158, 159, 180], "mathemat": [3, 60, 95, 100, 133, 139, 140, 141, 146, 158], "b": [3, 15, 16, 17, 18, 24, 25, 26, 27, 28, 29, 35, 37, 42, 43, 44, 45, 46, 49, 51, 53, 55, 62, 63, 66, 68, 71, 75, 130, 135, 140, 155, 157, 173, 174, 175, 176, 178, 180, 182, 186], "creat": [3, 29, 45, 49, 62, 71, 73, 74, 75, 79, 80, 82, 84, 85, 87, 88, 95, 96, 98, 100, 101, 102, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 120, 122, 123, 124, 125, 126, 127, 129, 130, 131, 132, 134, 135, 136, 137, 139, 141, 142, 143, 144, 145, 147, 148, 151, 152, 153, 155, 158, 160, 161, 163, 164, 165, 174, 178, 183], "infin": 3, "vari": [3, 18, 82, 95, 98, 100, 102, 105, 107, 110, 125, 127, 134, 138, 140, 143, 152, 157, 178], "fulfil": 3, "requir": [3, 6, 13, 18, 22, 29, 33, 36, 39, 40, 44, 49, 58, 62, 73, 75, 76, 79, 82, 84, 88, 95, 100, 101, 107, 111, 118, 119, 134, 148, 153, 155, 157, 162, 171, 178, 183, 186], "minim": [3, 6, 40, 60, 66, 94, 99, 104, 105, 137, 138, 146, 148, 158, 174], "sum": [3, 40, 42, 43, 60, 76, 83, 85, 117, 135, 141, 143], "error": [3, 6, 16, 28, 29, 40, 42, 53, 55, 58, 60, 61, 63, 76, 79, 83, 89, 91, 94, 97, 98, 99, 105, 111, 113, 114, 115, 117, 118, 119, 121, 122, 123, 124, 130, 131, 134, 135, 136, 138, 139, 142, 143, 145, 146, 148, 149, 156, 166, 174], "red": [3, 27, 79, 104, 111, 112, 117, 133, 137, 141, 142, 153, 158, 162, 164], "best": [3, 13, 18, 22, 33, 39, 55, 60, 62, 76, 93, 99, 100, 102, 105, 113, 115, 116, 118, 119, 121, 122, 124, 125, 130, 134, 135, 137, 138, 139, 143, 146, 149, 150, 151, 152, 153, 155, 156, 157, 158, 171, 180, 183, 184, 186], "possibl": [3, 6, 15, 29, 36, 42, 44, 60, 62, 66, 71, 76, 84, 87, 94, 98, 99, 102, 104, 105, 110, 112, 118, 119, 121, 129, 130, 131, 133, 134, 135, 136, 137, 139, 141, 142, 146, 148, 150, 151, 153, 154, 157, 161, 163, 165, 180, 186], "abstract": [3, 107], "manner": [3, 14, 29, 87, 97, 134, 148], "state": [3, 6, 14, 62, 76, 82, 83, 84, 87, 88, 109, 151, 155], "jockei": 3, "wheel": 3, "i": [3, 13, 18, 20, 22, 29, 49, 73, 74, 75, 76, 79, 82, 83, 84, 86, 87, 88, 91, 92, 98, 99, 102, 104, 110, 112, 117, 118, 119, 120, 130, 131, 133, 134, 135, 136, 138, 141, 143, 146, 148, 151, 157, 166, 180, 186], "support": [3, 6, 84, 90, 92, 95, 100, 133, 139, 142, 144, 147], "standardscal": [3, 29, 49, 62, 68, 75, 79, 84, 88, 90, 92, 93, 95, 100, 101, 102, 109, 110, 132, 133, 137, 141, 142, 150, 152, 157, 178, 180, 182, 186], "columntransform": [3, 74, 88, 90, 92, 93, 133, 149, 151, 153, 155, 156], "enough": [3, 6, 91, 92, 101, 104, 105, 117, 119, 137, 142, 146, 152, 158, 162, 164, 178], "flexibl": [3, 6, 46, 55, 58, 60, 63, 95, 100, 105, 121, 142], "opposit": 3, "cluster": [3, 104, 173], "whose": [3, 83, 118, 152], "group": [3, 6, 20, 25, 29, 76, 102, 103, 104, 109, 166], "subset": [3, 17, 42, 45, 73, 76, 79, 82, 83, 88, 93, 104, 107, 109, 119, 121, 124, 126, 127, 128, 129, 149, 156, 158, 159, 162, 178], "them": [3, 6, 13, 62, 76, 78, 79, 84, 87, 88, 96, 99, 101, 104, 110, 112, 114, 118, 119, 120, 123, 131, 133, 134, 136, 139, 141, 143, 146, 151, 152, 153, 157, 161, 164, 165, 178, 186], "broad": 3, "topic": [3, 97, 104, 109], "custom": [3, 139], "commerc": 3, "websit": [3, 36, 51, 80, 85], "although": 3, "mention": [3, 51, 79, 95, 100, 103, 106, 111, 118, 120, 133, 134, 137, 138, 141, 143, 144, 146, 147, 152, 155, 165], "cover": [3, 52, 76, 79, 82, 87, 88, 137], "impli": [3, 162], "fix": [3, 29, 49, 55, 60, 87, 95, 100, 119, 134, 150, 151, 157, 162, 178, 180, 184], "like": [3, 6, 18, 25, 29, 46, 55, 76, 82, 83, 87, 88, 89, 91, 99, 101, 104, 106, 108, 110, 111, 112, 119, 127, 134, 135, 139, 140, 141, 143, 144, 147, 148, 152], "necessari": [3, 6, 45, 76, 119, 133, 153], "subdivid": [3, 158], "select": [3, 6, 13, 15, 16, 17, 18, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 39, 42, 43, 44, 45, 46, 49, 51, 53, 55, 62, 63, 66, 68, 71, 75, 77, 78, 79, 81, 82, 84, 86, 89, 90, 91, 92, 99, 102, 104, 110, 112, 117, 119, 121, 125, 126, 129, 130, 135, 138, 139, 141, 151, 152, 153, 154, 157, 159, 171, 173, 174, 175, 176, 178, 180, 182, 186], "final": [3, 13, 23, 39, 45, 58, 62, 73, 83, 84, 87, 88, 99, 103, 104, 107, 109, 112, 116, 117, 119, 124, 125, 126, 128, 129, 134, 137, 144, 145, 147, 148, 151, 152, 153, 161, 165, 183], "sometim": [3, 6, 60, 88, 101, 143, 146, 153, 155], "clear": [3, 76, 100, 105, 162], "mani": [3, 6, 29, 43, 49, 62, 75, 76, 77, 78, 83, 84, 87, 88, 99, 101, 102, 104, 105, 106, 109, 112, 119, 121, 131, 133, 134, 136, 143, 152, 157, 186], "need": [3, 6, 22, 24, 29, 37, 40, 62, 68, 76, 80, 82, 84, 85, 87, 88, 93, 95, 99, 100, 104, 105, 107, 110, 111, 112, 117, 119, 120, 121, 122, 133, 134, 137, 138, 139, 142, 144, 145, 147, 148, 149, 151, 152, 153, 155, 156, 162, 176, 178], "criteria": [3, 119], "ml": [3, 6, 97], "cheatsheet": 3, 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129, 131, 134, 136, 139, 145, 146, 148, 151, 152, 153, 155, 171, 178, 180, 183, 186], "actual": [6, 76, 80, 83, 85, 94, 103, 104, 105, 119, 133, 143, 146, 153], "gener": [6, 17, 18, 22, 23, 28, 29, 49, 52, 58, 60, 61, 62, 68, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 108, 109, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 131, 134, 136, 137, 139, 141, 142, 143, 145, 146, 148, 149, 150, 151, 152, 153, 155, 156, 157, 163, 166, 169, 178, 180, 186], "As": [6, 29, 49, 76, 79, 82, 83, 84, 87, 88, 90, 92, 95, 97, 99, 100, 101, 106, 108, 109, 110, 112, 117, 118, 133, 134, 137, 141, 142, 143, 145, 146, 148, 152, 153, 154, 157, 162, 165], "narrow": 6, "spend": [6, 107, 119], "increasingli": 6, "effort": [6, 107], "split": [6, 15, 18, 23, 29, 37, 43, 45, 68, 76, 79, 81, 86, 87, 88, 94, 99, 102, 103, 104, 106, 110, 113, 117, 118, 119, 121, 122, 124, 126, 129, 142, 143, 146, 151, 153, 155, 158, 160, 162, 163, 164, 173, 174], "afford": 6, "trust": [6, 82, 83, 84, 85, 88, 93, 99, 104, 111, 133, 134, 137, 138, 139, 142, 143, 151, 152, 153, 155, 158, 164, 165], "think": [6, 77, 78, 103, 107, 131, 136, 137], "carefulli": [6, 97], "complet": [6, 15, 29, 36, 49, 99, 101, 126, 129, 133, 134, 150, 153, 157, 180], "futur": [6, 76, 79, 88, 93, 103, 104, 151, 158, 163, 164], "upon": [6, 93, 100, 101], "affect": [6, 103, 134, 137, 141, 151, 163], "live": [6, 134], "sure": [6, 18, 62, 82, 84, 87, 92, 116, 125], "divers": [6, 158], "demograph": [6, 120], "increas": [6, 16, 18, 28, 29, 44, 45, 46, 49, 53, 55, 60, 78, 84, 87, 100, 104, 105, 110, 115, 118, 119, 124, 134, 137, 140, 141, 142, 151, 155, 157, 158, 160, 162, 163, 164, 176, 184], "coverag": 6, "phrase": 6, "recommend": [6, 36, 73, 76, 87], "identifi": [6, 13, 73, 88, 97, 100, 105, 143, 180], "ani": [6, 15, 18, 29, 36, 44, 75, 76, 79, 82, 84, 86, 88, 91, 92, 94, 97, 100, 102, 103, 104, 105, 107, 108, 109, 110, 112, 119, 120, 121, 125, 126, 127, 128, 129, 133, 134, 138, 141, 146, 151, 153, 154, 157, 162, 175, 186], "bia": [6, 37, 53, 58, 60, 110, 131, 133, 136, 146, 166], "acquisit": 6, "full": [6, 8, 9, 18, 36, 48, 50, 51, 52, 54, 56, 64, 68, 69, 79, 80, 84, 85, 87, 99, 104, 119, 124, 126, 128, 129, 140, 149, 153, 156, 167, 180, 183], "chain": [6, 79, 84, 88], "acquir": [6, 13, 22, 33, 39, 58, 105, 107, 171, 183], "fanci": 6, "put": [6, 33, 37, 84, 105, 110, 134, 158], "product": [6, 99, 102, 104, 131, 134, 136, 151], "routin": [6, 18, 97, 186], "debt": 6, "simpler": [6, 14, 88, 137], "easier": [6, 84, 87, 94, 139, 146], "maintain": 6, "less": [6, 29, 46, 88, 97, 103, 109, 110, 111, 112, 119, 120, 128, 133, 134, 136, 137, 139, 151, 157], "power": [6, 29, 36, 107, 119, 120, 131, 134, 136, 139, 158], "drift": 6, "gave": [6, 143], "methodolog": [6, 36, 58, 104, 163], "element": [6, 22, 83, 87, 97, 104, 134, 135, 140, 143], "alwai": [6, 15, 18, 22, 24, 43, 45, 55, 81, 82, 83, 86, 88, 92, 94, 97, 99, 100, 101, 102, 104, 105, 108, 109, 121, 127, 142, 143, 152, 153, 154, 165, 178, 180, 186], "solid": 6, "conclus": [6, 76, 96, 97, 99, 101, 102, 103, 105, 108, 119, 127], "standpoint": 6, "biggest": 6, "shortcom": 6, "cannot": [6, 18, 29, 60, 76, 102, 103, 105, 110, 119, 126, 129, 133, 139, 142, 143, 146, 148, 151, 154, 165, 183], "autom": [6, 76, 166, 184], "domain": 6, "knowledg": [6, 36, 73, 99, 105, 112, 129, 139, 142, 153], "critic": [6, 36, 106], "thing": [6, 76, 87, 88, 93, 103, 139, 151], "oper": [6, 84, 112, 133, 143, 153], "risk": [6, 131, 136, 142], "advertis": 6, "individu": [6, 17, 29, 76, 84, 88, 108, 112, 114, 121, 123, 134, 141, 142, 158, 186], "caus": [6, 18, 53, 60, 76, 87, 91, 92, 119, 134, 151, 153, 180], "wast": [6, 119], "bit": [6, 29, 86, 104, 105, 110, 134, 139, 144, 147, 153], "monei": 6, "annoi": 6, "otherwis": [6, 18, 82, 91, 97, 99, 121, 133, 139, 140, 162], "mostli": [6, 137, 157], "harmless": 6, "medicin": 6, "kill": 6, "logic": [6, 151], "fals": [6, 16, 55, 83, 87, 90, 92, 97, 99, 103, 107, 109, 125, 127, 128, 133, 134, 136, 139, 143, 147, 150, 151, 155, 157, 158, 164, 176], "brain": 6, "tumor": 6, "sent": 6, "surgeri": 6, "veri": [6, 18, 58, 62, 76, 82, 88, 90, 92, 99, 100, 102, 105, 108, 109, 110, 112, 117, 119, 121, 125, 133, 134, 137, 139, 143, 151, 152, 153, 154, 155, 162, 180], "danger": [6, 134, 155], "mr": 6, "confirm": [6, 94, 102, 107, 109, 112, 126, 129, 133, 137, 141, 146, 153, 154], "should": [6, 18, 23, 29, 34, 35, 45, 71, 75, 76, 83, 84, 88, 92, 97, 99, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 116, 117, 119, 120, 125, 126, 127, 129, 130, 134, 135, 137, 139, 142, 143, 144, 147, 148, 149, 151, 152, 153, 156, 158, 160, 162, 163, 164, 176, 178, 180, 183, 184], "delai": 6, "life": [6, 76, 126, 129, 146], "save": [6, 186], "treatment": [6, 60], "hospit": [6, 25], "stai": [6, 40, 45, 112, 134], "overcrowd": 6, "unit": [6, 76, 82, 84, 87, 88, 104, 107, 109, 110, 111, 134, 138, 140, 146, 151, 155], "chang": [6, 28, 29, 37, 49, 76, 78, 84, 96, 101, 107, 108, 109, 110, 111, 115, 124, 127, 134, 146, 149, 152, 156, 158, 159, 183, 186], "inpati": 6, "chose": [6, 55, 76, 117, 134], "load": [6, 18, 49, 66, 77, 78, 79, 80, 84, 85, 87, 88, 89, 91, 95, 98, 99, 100, 102, 103, 104, 105, 108, 109, 111, 113, 118, 119, 120, 122, 131, 132, 133, 134, 136, 137, 138, 139, 142, 143, 151, 152, 154, 155, 158, 159, 160, 161, 162, 163, 164, 165, 180, 186], "interest": [6, 51, 76, 77, 78, 94, 96, 101, 102, 104, 105, 107, 109, 112, 117, 131, 133, 136, 138, 139, 141, 143, 146, 148, 151, 152, 153, 154, 155, 157, 186], "focus": [6, 14, 63, 109, 111, 134, 143, 153], "easi": [6, 88, 103, 104, 107, 133, 162], "accumul": 6, "target": [6, 18, 29, 39, 40, 44, 49, 51, 60, 62, 66, 68, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 133, 134, 135, 136, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 155, 156, 157, 158, 159, 160, 164, 178, 186], "proxi": [6, 143], "reflect": [6, 104, 112, 121], "ground": [6, 117, 143], "truth": [6, 117, 143], "polici": [6, 25], "uneven": 6, "across": [6, 43, 49, 79, 84, 94, 110, 134, 141, 155, 186], "popul": [6, 76, 104, 109, 110, 143, 157], "eg": 6, "qualiti": [6, 110, 141, 146, 153], "affair": 6, "desir": [6, 81, 86, 121, 134], "qualif": 6, "respons": 6, "women": 6, "pai": [6, 87, 111], "men": 6, "pick": [6, 29, 93, 99, 106, 107, 110, 118, 152, 183], "amplifi": 6, "inequ": 6, "mechan": [6, 29, 83, 84, 153], "die": 6, "naiv": [6, 18, 29, 76, 99, 108, 118, 121], "bad": [6, 91, 100, 129, 134, 142, 143, 180], "health": [6, 36], "fallaci": 6, "compar": [6, 13, 17, 18, 22, 29, 37, 49, 51, 61, 62, 75, 76, 79, 81, 83, 84, 86, 87, 88, 89, 91, 99, 100, 101, 105, 109, 110, 112, 115, 117, 118, 122, 124, 127, 128, 129, 131, 134, 136, 143, 146, 155, 157, 158, 166, 178, 186], "wors": [6, 18, 28, 49, 75, 100, 101, 103, 138, 178], "baselin": [6, 22, 24, 81, 86, 88, 89, 91, 96, 97, 100, 101, 120, 166], "heart": [6, 29, 107, 131, 136], "pressur": 6, "greater": [6, 29, 75, 133], "trigger": 6, "care": [6, 34, 87, 99, 102, 108, 109, 126, 129, 134], "which": [6, 14, 16, 17, 18, 23, 25, 27, 29, 33, 34, 40, 43, 45, 49, 60, 68, 71, 74, 75, 76, 79, 82, 83, 84, 86, 87, 88, 92, 93, 94, 95, 98, 99, 100, 101, 103, 104, 105, 107, 108, 109, 110, 111, 112, 117, 118, 119, 121, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 143, 144, 146, 147, 148, 150, 151, 152, 154, 155, 157, 162, 163, 164, 171, 172, 173, 178, 180, 182, 184, 186], "learner": [6, 14, 111, 112, 117, 119, 120], "predictor": [6, 14, 15, 16, 17, 49, 68, 83, 84, 88, 101, 105, 122, 134, 151, 158], "pure": [6, 129, 164], "benefici": [6, 33, 84, 119, 127, 153, 155, 162], "intervent": [6, 76], "brittl": 6, "interpret": [6, 17, 36, 107, 109, 110, 136, 137, 138, 139, 146, 154, 158], "subject": [6, 76, 143], "caution": [6, 92, 110], "feedback": 6, "loop": [6, 99, 110, 149, 151, 153, 156, 186], "todai": 6, "ai": 6, "alloc": 6, "loan": 6, "screen": [6, 8, 9, 48, 50, 52, 54, 56, 64, 69, 167], "job": [6, 84], "prioritis": 6, "treatement": 6, "law": [6, 29], "enforc": [6, 103, 107, 119, 134, 137], "court": 6, "fairlearn": [6, 76], "assess": [6, 25, 49, 79, 83, 96, 98, 99, 101, 105, 115, 124, 130, 134, 135, 142, 143, 146, 153, 157], "shift": [6, 84, 137], "technolog": [6, 97], "induc": [6, 110, 112, 137], "societi": 6, "though": [6, 101, 118, 145, 148], "difficult": [6, 107, 119, 138, 146], "intersect": [6, 154, 157], "No": [6, 24], "found": [6, 62, 99, 108, 109, 116, 119, 125, 134, 138, 149, 151, 153, 156, 157, 158, 162, 163, 180, 186], "short": [6, 34, 82, 109, 120, 159], "move": [6, 88, 107, 137, 154, 157], "choos": [6, 18, 36, 76, 89, 91, 104, 110, 126, 127, 129, 133, 134, 139, 141, 153, 176, 186], "revolut": 6, "fantast": [6, 129], "opportun": 6, "With": [6, 37, 91, 104, 107, 119, 127, 128, 143, 151, 155, 162], "lift": 6, "roadblock": 6, "hope": [6, 93, 133, 146], "empow": 6, "varieti": [6, 36, 79], "mindset": 6, "dream": 6, "being": [6, 43, 94, 141, 157], "adventur": 6, "navig": [8, 9, 48, 50, 52, 54, 56, 64, 69, 167], "slide": [8, 9, 48, 50, 52, 54, 56, 64, 69, 133, 137, 141, 154, 157, 167], "click": [8, 9, 15, 48, 50, 52, 54, 56, 64, 69, 154, 157, 167, 180], "press": [8, 9, 48, 50, 52, 54, 56, 64, 69, 167], "arrow": [8, 9, 48, 50, 52, 54, 56, 64, 69, 167], "go": [8, 9, 13, 22, 29, 33, 36, 39, 48, 50, 52, 54, 56, 58, 64, 69, 73, 79, 80, 85, 87, 95, 97, 99, 100, 103, 104, 105, 107, 109, 110, 117, 120, 127, 141, 143, 148, 151, 158, 167, 171, 183], "next": [8, 9, 48, 50, 52, 54, 56, 60, 64, 69, 82, 83, 87, 93, 95, 96, 100, 101, 102, 103, 111, 117, 119, 120, 131, 133, 134, 136, 137, 139, 152, 158, 167], "previou": [8, 9, 16, 18, 22, 29, 48, 49, 50, 52, 54, 56, 62, 64, 66, 68, 69, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 96, 97, 98, 99, 101, 102, 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181, 182, 183, 184], "allow": [13, 14, 29, 45, 68, 76, 79, 82, 84, 87, 95, 100, 104, 106, 109, 110, 119, 126, 129, 130, 134, 135, 139, 140, 141, 144, 147, 153, 155, 162, 163, 171, 180, 183, 184], "skill": [13, 22, 33, 39, 58, 73, 171, 183], "carri": [13, 22, 33, 39, 49, 58, 73, 87, 98, 109, 128, 171, 183], "basic": [13, 22, 33, 36, 39, 52, 58, 73, 102, 104, 110, 122, 146, 171, 183], "usag": [13, 22, 33, 39, 58, 82, 106, 107, 108, 109, 121, 139, 153, 171, 183], "mainli": [13, 14, 22, 33, 39, 76, 106, 158, 171], "around": [13, 22, 33, 39, 76, 82, 93, 97, 100, 103, 104, 105, 107, 110, 136, 137, 142, 143, 146, 171], "overfit": [13, 16, 18, 22, 29, 33, 39, 45, 46, 53, 55, 58, 60, 61, 62, 63, 98, 99, 103, 110, 111, 112, 115, 119, 120, 121, 124, 128, 133, 134, 136, 137, 142, 153, 157, 162, 166, 171, 173, 176], "valid": [13, 18, 22, 23, 24, 25, 26, 29, 33, 34, 39, 40, 45, 49, 58, 59, 60, 62, 67, 68, 75, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 100, 101, 102, 103, 109, 110, 115, 116, 117, 118, 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66, 68, 71, 75, 76, 83, 97, 166, 173, 178, 180, 182, 186], "answer": [15, 16, 17, 18, 24, 25, 26, 27, 28, 29, 35, 42, 43, 44, 45, 46, 49, 51, 53, 55, 62, 63, 66, 68, 71, 75, 87, 106, 131, 136, 137, 157, 173, 174, 175, 176, 178, 180, 182, 186], "hint": [15, 28, 29, 49, 62, 75, 77, 78, 80, 81, 85, 86, 90, 92, 131, 134, 136, 137, 186], "base_estim": [15, 122], "decid": [15, 22, 76, 82, 104, 109, 126, 129, 131, 136, 153], "resampl": [15, 53, 99, 107, 111, 134], "perform": [15, 17, 18, 19, 22, 23, 24, 26, 29, 34, 35, 49, 62, 66, 68, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 108, 110, 111, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 131, 134, 136, 139, 142, 143, 145, 146, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 162, 171, 178, 180, 183, 184, 186], "correct": [16, 17, 18, 29, 62, 83, 85, 111, 117, 119, 131, 136, 143, 149, 155, 156, 162, 186], "statement": [16, 18, 27, 29, 43, 180, 186], "simultan": 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123, 124, 130, 131, 132, 134, 135, 136, 137, 138, 140, 141, 143, 144, 145, 146, 147, 148, 158, 160, 161, 162, 163, 164, 165], "exist": [17, 76, 79, 97], "maximum": [17, 29, 84, 87, 99, 105, 139, 143, 151, 155, 158, 160, 161, 163, 164, 165, 173, 176], "depth": [17, 18, 36, 77, 78, 105, 109, 111, 117, 118, 119, 120, 131, 132, 136, 137, 141, 149, 156, 158, 159, 160, 161, 162, 163, 164, 165, 173, 176, 178], "rate": [17, 29, 83, 94, 98, 104, 107, 119, 143, 151, 154, 156], "option": [17, 84, 87, 94, 102, 104, 107, 115, 124, 183, 186], "reduc": [17, 18, 40, 45, 49, 93, 97, 98, 118, 119, 120, 121, 127, 134, 135], "sensit": [17, 53, 137, 142, 143, 154, 157, 172], "notic": [18, 83, 84, 100, 104, 109, 110, 134, 136, 137, 139, 141, 143, 148, 150, 157], "tradit": 18, "panda": [18, 29, 36, 49, 62, 66, 73, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 114, 117, 119, 121, 122, 123, 125, 127, 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165, 186], "mass": [18, 29, 114, 123, 130, 131, 135, 136, 138, 140, 159, 161, 162, 163, 165, 186], "dropna": [18, 131, 136, 186], "frac": [18, 29], "random_st": [18, 37, 82, 84, 86, 88, 94, 95, 99, 100, 101, 102, 103, 104, 105, 110, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 128, 129, 132, 133, 134, 136, 137, 139, 141, 142, 143, 146, 149, 150, 151, 153, 155, 156, 157, 158, 160, 162, 164, 178], "reset_index": [18, 132, 137, 141], "drop": [18, 29, 37, 49, 62, 75, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 100, 101, 103, 106, 107, 108, 109, 110, 121, 143, 144, 145, 146, 147, 148, 149, 151, 153, 155, 156, 178], "therefor": [18, 29, 75, 76, 79, 84, 86, 88, 94, 97, 98, 99, 101, 104, 105, 109, 111, 117, 118, 119, 121, 127, 128, 129, 134, 135, 136, 139, 141, 142, 143, 146, 148, 153, 157, 162], "randomli": [18, 29, 99, 101, 104, 110, 112, 121, 155], "shuffl": [18, 37, 82, 97, 99, 102, 103, 104, 110, 116, 125, 143, 146], "break": [18, 97, 134], "spuriou": 18, "troubl": [18, 102], "outsid": [18, 151, 158, 161, 165], "scope": [18, 76, 139, 141, 146, 148], "regressor": [18, 24, 28, 29, 40, 46, 49, 83, 84, 94, 98, 103, 104, 105, 112, 113, 116, 117, 118, 119, 120, 121, 122, 124, 125, 134, 139, 146, 172, 175, 178], "sklearn": [18, 29, 45, 49, 62, 75, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 132, 133, 134, 136, 137, 138, 139, 141, 142, 143, 144, 146, 147, 148, 149, 150, 151, 152, 153, 155, 156, 157, 158, 160, 162, 163, 164, 165, 178, 180, 182, 186], "randomforestregressor": [18, 110, 117, 119, 121, 123, 124], "except": [18, 89, 91, 144, 147], "exact": [18, 29, 117], "fold": [18, 25, 29, 49, 62, 75, 79, 97, 102, 104, 110, 116, 125, 127, 128, 129, 130, 131, 134, 135, 136, 145, 148, 153, 155, 157, 178, 186], "model_select": [18, 29, 49, 62, 79, 81, 82, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 105, 109, 110, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 132, 134, 136, 137, 141, 143, 144, 146, 147, 148, 149, 150, 151, 152, 153, 155, 156, 157, 158, 160, 162, 164, 178, 180, 186], "cross_valid": [18, 29, 49, 62, 68, 75, 79, 87, 88, 89, 90, 91, 92, 93, 94, 95, 100, 101, 102, 105, 109, 110, 116, 117, 118, 120, 125, 127, 128, 134, 136, 144, 145, 147, 148, 151, 152, 153, 178, 186], "cv": [18, 29, 49, 68, 75, 79, 88, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 105, 110, 116, 120, 125, 128, 134, 136, 144, 147, 148, 151, 153, 155, 156, 162, 178, 180, 186], "store": [18, 29, 76, 82, 84, 94, 96, 101, 102, 104, 106, 107, 108, 112, 120, 127, 128, 133, 134, 138, 142, 150, 151, 157, 186], "return_train_scor": [18, 29, 62, 105, 134], "count": [18, 49, 62, 75, 76, 78, 82, 84, 86, 87, 94, 97, 102, 106, 107, 108, 109, 158, 178, 186], "rang": [18, 40, 49, 62, 68, 75, 82, 84, 97, 99, 102, 104, 107, 108, 109, 110, 112, 133, 134, 140, 153, 154, 155, 157, 159, 161, 165, 175, 178, 180, 186], "substanti": [18, 49, 75, 178, 186], "almost": [18, 49, 75, 83, 92, 104, 110, 137, 140, 146, 153, 164, 178], "100": [18, 49, 62, 82, 83, 94, 98, 101, 104, 105, 109, 110, 112, 113, 115, 116, 117, 118, 119, 120, 122, 124, 125, 126, 127, 128, 129, 131, 133, 134, 136, 137, 139, 142, 143, 146, 150, 155, 156, 157, 186], "again": [18, 95, 100, 110, 112, 115, 124, 137, 139, 143, 154], "curv": [18, 43, 60, 62, 95, 100, 115, 124, 136, 140, 141, 143, 166], "n_estim": [18, 111, 112, 115, 117, 118, 119, 120, 121, 122, 123, 124], "numpi": [18, 29, 36, 62, 66, 68, 73, 84, 94, 97, 98, 100, 101, 102, 105, 107, 109, 110, 111, 112, 117, 118, 123, 124, 126, 127, 129, 130, 133, 134, 135, 136, 138, 139, 140, 142, 143, 145, 146, 148, 154, 157, 162, 163, 165, 180], "np": [18, 29, 49, 62, 94, 95, 97, 98, 100, 101, 102, 105, 107, 109, 110, 111, 112, 115, 117, 118, 121, 123, 124, 126, 127, 129, 130, 131, 133, 134, 135, 136, 138, 139, 140, 142, 143, 145, 146, 148, 150, 154, 157, 162, 163, 165, 175, 180], "arrai": [18, 29, 43, 44, 62, 68, 79, 82, 83, 84, 85, 87, 88, 97, 98, 104, 105, 110, 111, 112, 115, 118, 123, 124, 131, 134, 136, 141, 143, 147, 151, 158], "200": [18, 62, 75, 79, 97, 107, 117, 118, 133, 155], "500": [18, 62, 87, 88, 91, 104, 107, 109, 146, 154, 155], "1_000": [18, 29, 97, 115, 124], "decreas": [18, 29, 49, 55, 60, 84, 110, 118, 127, 134, 146, 151, 157], "becom": [18, 95, 98, 100, 105, 110, 118, 119, 151, 155, 180], "reach": [18, 98, 105, 115, 124, 151, 154, 162, 180], "plateau": [18, 98, 115, 124], "experi": [18, 36, 49, 62, 73, 81, 86, 95, 97, 98, 99, 100, 104, 105, 109, 115, 116, 118, 124, 125, 137, 142, 144, 147, 157, 160, 163, 164], "instead": [18, 29, 49, 62, 75, 79, 83, 84, 87, 89, 90, 91, 92, 94, 95, 97, 98, 100, 101, 104, 112, 117, 119, 120, 121, 131, 134, 136, 139, 143, 144, 145, 146, 147, 148, 149, 150, 153, 154, 155, 156, 157, 158, 159, 163, 164, 178], "max_depth": [18, 105, 111, 112, 115, 116, 117, 119, 120, 124, 125, 139, 158, 163, 164, 165, 178], "gap": [18, 105, 134], "begin": [18, 83, 115, 124, 133, 134, 153], "consid": [18, 44, 62, 71, 75, 87, 103, 105, 106, 109, 110, 111, 117, 118, 119, 121, 133, 134, 139, 144, 147, 158, 164, 173, 182, 186], "none": [18, 49, 95, 97, 100, 104, 106, 107, 108, 109, 112, 115, 119, 120, 124, 132, 133, 139, 154, 186], "rf_1_tree": 18, "cv_results_tre": 18, "train_scor": [18, 29, 105, 134], "return": [18, 28, 29, 43, 68, 79, 82, 83, 87, 89, 91, 110, 112, 117, 130, 134, 135, 140, 141, 143, 144, 147, 151, 152, 154, 155, 180], "83120264": 18, "83309064": 18, "83195043": 18, "84834224": 18, "85790323": 18, "86235297": 18, "84791111": 18, "85183089": 18, "82241954": 18, "85045978": 18, "perfect": [18, 42, 58, 94, 104, 110, 117, 126, 128, 129, 142, 143, 162], "r2": [18, 28, 103, 109, 120, 125, 146, 148], "surpris": [18, 86, 97, 102, 103, 126, 129, 158, 164], "memor": [18, 83, 103, 104, 105], "expect": [18, 29, 68, 76, 78, 82, 87, 88, 99, 102, 103, 104, 109, 112, 119, 129, 131, 134, 136, 142, 146, 148, 153, 157, 162], "automat": [18, 68, 76, 82, 84, 87, 89, 91, 104, 152, 153, 178, 183], "prevent": [18, 44, 87, 91, 112, 119, 180], "max_it": [18, 29, 84, 87, 88, 91, 97, 118, 119, 125, 137], "recal": [18, 27, 62, 87, 99, 104, 118, 120, 137, 143, 152, 153, 155, 157, 186], "averag": [18, 29, 62, 83, 94, 97, 101, 102, 104, 107, 109, 110, 112, 117, 118, 119, 120, 121, 124, 127, 134, 137, 138, 143, 144, 146, 147, 148, 186], "small": [18, 40, 78, 79, 88, 97, 100, 102, 104, 105, 110, 112, 117, 118, 119, 131, 134, 136, 137, 139, 149, 151, 156, 180], "behav": [18, 95, 100, 101, 134, 137, 178], "high": [18, 29, 37, 53, 55, 60, 66, 76, 78, 82, 86, 87, 100, 105, 106, 107, 108, 109, 110, 133, 137, 141, 151, 152, 155], "optimum": 18, "m7": [19, 20, 21, 30, 31, 166], "stratif": [20, 166], "framework": [22, 23, 34, 58, 59, 98, 102, 105, 116, 125, 145, 148, 153, 166, 183], "keep": [22, 29, 76, 93, 97, 104, 106, 107, 109, 110, 127, 128, 129, 131, 132, 136, 137, 139, 141, 142, 153, 155], "mind": [22, 76, 104, 110, 119, 126, 127, 128, 129, 142, 155], "metric": [22, 27, 29, 68, 79, 83, 98, 103, 104, 108, 112, 113, 115, 122, 123, 124, 127, 131, 134, 135, 136, 138, 139, 144, 145, 146, 147, 148, 154, 163, 166, 186], "besid": [22, 23, 33, 39, 82, 88, 90, 92, 94, 97, 98, 116, 125, 127, 158, 171, 184], "insight": [22, 33, 36, 49, 66, 76, 103, 105, 112, 126, 129, 140, 141, 146, 155], "addit": [22, 29, 44, 71, 79, 83, 87, 104, 107, 109, 115, 116, 120, 121, 124, 125, 134, 136, 139, 140, 143, 146, 151, 152, 153, 155, 157, 178], "necess": [22, 104], "appropri": [22, 76], "nest": [22, 23, 26, 119, 134, 149, 151, 153, 156, 166, 178, 184, 186], "wise": [23, 133, 153, 154, 184], "encount": [23, 75, 87, 89, 91, 102], "show": [23, 39, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 93, 98, 99, 102, 104, 105, 106, 109, 111, 112, 117, 119, 120, 121, 128, 130, 131, 133, 134, 135, 136, 138, 139, 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