DOC Reorganize plot_nca_illustration example into subsections (#14795)

examples/neighbors/plot_nca_illustration.py

@@ -3,10 +3,10 @@
 Neighborhood Components Analysis Illustration
 =============================================
 
-An example illustrating the goal of learning a distance metric that maximizes
-the nearest neighbors classification accuracy. The example is solely for
-illustration purposes. Please refer to the :ref:`User Guide <nca>` for
-more information.
+This example illustrates a learned distance metric that maximizes
+the nearest neighbors classification accuracy. It provides a visual
+representation of this metric compared to the original point
+space. Please refer to the :ref:`User Guide <nca>` for more information.
 """
 
 # License: BSD 3 clause
@@ -20,23 +20,31 @@
 
 print(__doc__)
 
-random_state = 0
+##############################################################################
+# Original points
+# ---------------
+# First we create a data set of 9 samples from 3 classes, and plot the points
+# in the original space. For this example, we focus on the classification of
+# point no. 3. The thickness of a link between point no. 3 and another point
+# is proportional to their distance.
 
-# Create a tiny data set of 9 samples from 3 classes
 X, y = make_classification(n_samples=9, n_features=2, n_informative=2,
                            n_redundant=0, n_classes=3, n_clusters_per_class=1,
-                           class_sep=1.0, random_state=random_state)
+                           class_sep=1.0, random_state=0)
 
-# Plot the points in the original space
-plt.figure()
+plt.figure(1)
 ax = plt.gca()
-
-# Draw the graph nodes
 for i in range(X.shape[0]):
     ax.text(X[i, 0], X[i, 1], str(i), va='center', ha='center')
     ax.scatter(X[i, 0], X[i, 1], s=300, c=cm.Set1(y[[i]]), alpha=0.4)
 
-def p_i(X, i):
+ax.set_title("Original points")
+ax.axes.get_xaxis().set_visible(False)
+ax.axes.get_yaxis().set_visible(False)
+ax.axis('equal')  # so that boundaries are displayed correctly as circles
+
+
+def link_thickness_i(X, i):
     diff_embedded = X[i] - X
     dist_embedded = np.einsum('ij,ij->i', diff_embedded,
                               diff_embedded)
@@ -52,34 +60,30 @@ def p_i(X, i):
 def relate_point(X, i, ax):
     pt_i = X[i]
     for j, pt_j in enumerate(X):
-        thickness = p_i(X, i)
+        thickness = link_thickness_i(X, i)
         if i != j:
             line = ([pt_i[0], pt_j[0]], [pt_i[1], pt_j[1]])
             ax.plot(*line, c=cm.Set1(y[j]),
                     linewidth=5*thickness[j])
 
 
-# we consider only point 3
 i = 3
-
-# Plot bonds linked to sample i in the original space
 relate_point(X, i, ax)
-ax.set_title("Original points")
-ax.axes.get_xaxis().set_visible(False)
-ax.axes.get_yaxis().set_visible(False)
-ax.axis('equal')
+plt.show()
 
-# Learn an embedding with NeighborhoodComponentsAnalysis
-nca = NeighborhoodComponentsAnalysis(max_iter=30, random_state=random_state)
+##############################################################################
+# Learning an embedding
+# ---------------------
+# We use :class:`~sklearn.neighbors.NeighborhoodComponentsAnalysis` to learn an
+# embedding and plot the points after the transformation. We then take the
+# embedding and find the nearest neighbors.
+
+nca = NeighborhoodComponentsAnalysis(max_iter=30, random_state=0)
 nca = nca.fit(X, y)
 
-# Plot the points after transformation with NeighborhoodComponentsAnalysis
-plt.figure()
+plt.figure(2)
 ax2 = plt.gca()
-
-# Get the embedding and find the new nearest neighbors
 X_embedded = nca.transform(X)
-
 relate_point(X_embedded, i, ax2)
 
 for i in range(len(X)):
@@ -88,7 +92,6 @@ def relate_point(X, i, ax):
     ax2.scatter(X_embedded[i, 0], X_embedded[i, 1], s=300, c=cm.Set1(y[[i]]),
                 alpha=0.4)
 
-# Make axes equal so that boundaries are displayed correctly as circles
 ax2.set_title("NCA embedding")
 ax2.axes.get_xaxis().set_visible(False)
 ax2.axes.get_yaxis().set_visible(False)