Note

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# Neighborhood Components Analysis Illustration#

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 User Guide for more information.

```
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
import matplotlib.pyplot as plt
import numpy as np
from matplotlib import cm
from scipy.special import logsumexp
from sklearn.datasets import make_classification
from sklearn.neighbors import NeighborhoodComponentsAnalysis
```

## 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.

```
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=0,
)
plt.figure(1)
ax = plt.gca()
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)
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)
dist_embedded[i] = np.inf
# compute exponentiated distances (use the log-sum-exp trick to
# avoid numerical instabilities
exp_dist_embedded = np.exp(-dist_embedded - logsumexp(-dist_embedded))
return exp_dist_embedded
def relate_point(X, i, ax):
pt_i = X[i]
for j, pt_j in enumerate(X):
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])
i = 3
relate_point(X, i, ax)
plt.show()
```

## Learning an embedding#

We use `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)
plt.figure(2)
ax2 = plt.gca()
X_embedded = nca.transform(X)
relate_point(X_embedded, i, ax2)
for i in range(len(X)):
ax2.text(X_embedded[i, 0], X_embedded[i, 1], str(i), va="center", ha="center")
ax2.scatter(X_embedded[i, 0], X_embedded[i, 1], s=300, c=cm.Set1(y[[i]]), alpha=0.4)
ax2.set_title("NCA embedding")
ax2.axes.get_xaxis().set_visible(False)
ax2.axes.get_yaxis().set_visible(False)
ax2.axis("equal")
plt.show()
```

**Total running time of the script:** (0 minutes 0.137 seconds)

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