In [1]:
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LogisticRegression
from sklearn.svm import SVC
from sklearn.datasets import *
from ClassificationPlotter import plot_regions
from ipywidgets import *
Determining Features Based on the Radial Basis Function¶
Let L denote a "landmark" point. for a given observation $x_i$, we defined a new transformed feature z as follows: $\large z = e^{-\gamma \cdot d(L, x_i)}$.
If we have several landmarks, $L_1, L_2, ..., L_k$, then each landmark will determine a new feature, $z_1, z_2, ..., z_k$ in the transformed space.
In [2]:
np.random.seed(1)
L1 = np.array([2,2])
L2 = np.array([4,3])
L = np.vstack([L1,L2])
X1a, y1a = make_circles(n_samples=20, noise=0.1, factor=0.4)
X1b, y1b = make_circles(n_samples=20, noise=0.1, factor=0.4)
X1c, y1c = make_blobs(n_samples=10, centers=1, cluster_std=0.3)
X1d, y1d = make_blobs(n_samples=10, centers=1, cluster_std=0.3)
X1a = X1a + L1
X1b = X1b + L2
X1c = X1c + np.array([6.5,8.5])
X1d = X1d + np.array([7.5,7.5])
X1 = np.vstack([X1a, X1b, X1c, X1d])
y1 = np.hstack([y1a, y1b, y1c, y1d])
plt.figure(figsize = [8,6])
plt.scatter(X1[:, 0], X1[:, 1], c=y1, s=80, edgecolor='k', cmap='rainbow')
plt.scatter(L1[0], L1[1], c='darkorange', s=120, edgecolor='k', marker='D')
plt.scatter(L2[0], L2[1], c='darkgreen', s=120, edgecolor='k', marker='D')
plt.show()
In [3]:
%run -i Snippets/snippet13.py
Using Radial Basis Functions with Support Vector Machines¶
When using radial basis functions to create a higher dimensional feature space for linear classification, we will use each point in the training set as a separate landmark. The resulting transformed feature space will have n dimensions, where n is the number of training observations.
Example 1¶
In [4]:
%run -i Snippets/snippet14.py
In [5]:
mod_01 = SVC(kernel='rbf', C=20, gamma=0.5)
mod_01.fit(X1, y1)
print('Training Accuracy:', mod_01.score(X1, y1))
Example 2¶
In [6]:
np.random.seed(159)
X2, y2 = make_moons(n_samples=100, noise=0.1)
plt.figure(figsize = [8,6])
plt.scatter(X2[:, 0], X2[:, 1], marker='o', c=y2, s=50, edgecolor='k', cmap='rainbow')
plt.show()
In [7]:
%run -i Snippets/snippet15.py
In [8]:
mod_02 = SVC(kernel='rbf', C=20, gamma=1.6)
mod_02.fit(X2, y2)
print('Training Accuracy:', mod_02.score(X2, y2))
Example 3¶
In [9]:
np.random.seed(7530)
X3, y3 = make_blobs(n_samples=100, centers=6, n_features=2, cluster_std=2)
y3 = np.where(y3%2 == 0, 0, np.where(y3%2 == 1, 1, y3))
plt.figure(figsize = [8,6])
plt.scatter(X3[:, 0], X3[:, 1], marker='o', c=y3, s=50, edgecolor='k', cmap='rainbow')
plt.show()
In [10]:
%run -i Snippets/snippet16.py