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%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
# use seaborn plotting defaults
import seaborn as sns; sns.set()
SVMs operate by finding a seperation line that devides classes from each other.
As an example we consider the following dataset:
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from sklearn.datasets.samples_generator import make_blobs
X, y = make_blobs(n_samples=50, centers=2,
random_state=0, cluster_std=.6)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn');
Now the goal is to draw a straight line that seperates the two classes.
But there are several ways to do this. Which one is the best?
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xfit = np.linspace(-1, 3.5)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
for m, b in [(1, 0.65), (0.5, 1.6), (-0.2, 2.9)]:
plt.plot(xfit, m * xfit + b, '-k')
plt.xlim(-1, 3.5);
How should we classify new data? Depending on which line we choose, the label for the new data point varies.
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xfit = np.linspace(-1, 3.5)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plt.scatter([0.6], [2.1], color='blue')
for m, b in [(1, 0.65), (0.5, 1.6), (-0.2, 2.9)]:
plt.plot(xfit, m * xfit + b, '-k')
plt.xlim(-1, 3.5);
Support vector machines offer one way to improve on this.
Rather than simply drawing a line between the classes, we can draw around each line a margin of some width, up to the nearest point. Here is an example of how this might look:
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xfit = np.linspace(-1, 3.5)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
for m, b, d in [(1, 0.65, 0.33), (0.5, 1.6, 0.55), (-0.2, 2.9, 0.2)]:
yfit = m * xfit + b
plt.plot(xfit, yfit, '-k')
plt.fill_between(xfit, yfit - d, yfit + d, edgecolor='none',
color='#AAAAAA', alpha=0.4)
plt.xlim(-1, 3.5);
The line that maximizes this margin is the one we will choose as the optimal model.
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from sklearn.svm import SVC # "Support vector classifier"
model = SVC(kernel='linear', C=1E10)
model.fit(X, y)
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For the purpose of this lecture, let's write a function that prints the decision boundries of a SVM
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def plot_svc_decision_function(model, ax=None, plot_support=True):
"""Plot the decision function for a 2D SVC"""
if ax is None:
ax = plt.gca()
xlim = ax.get_xlim()
ylim = ax.get_ylim()
# create grid to evaluate model
x = np.linspace(xlim[0], xlim[1], 30)
y = np.linspace(ylim[0], ylim[1], 30)
Y, X = np.meshgrid(y, x)
xy = np.vstack([X.ravel(), Y.ravel()]).T
P = model.decision_function(xy).reshape(X.shape)
# plot decision boundary and margins
ax.contour(X, Y, P, colors='k',
levels=[-1, 0, 1], alpha=0.5,
linestyles=['--', '-', '--'])
# plot support vectors
if plot_support:
ax.scatter(model.support_vectors_[:, 0], model.support_vectors_[:, 1],
s=300, linewidth=1, facecolors='none', edgecolor='k');
ax.set_xlim(xlim)
ax.set_ylim(ylim)
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plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(model, plot_support=False);
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plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(model, plot_support=True);
The black circles indicate the only relevant datapoints for the model.
They are called support vectors. Hence, the name of the model. In Scikit-Learn these vectors are stored in:
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model.support_vectors_
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We can see this in the next example where similar datasets are plotted but with a varying number of datapoints.
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def plot_svm(N=10, ax=None):
X, y = make_blobs(n_samples=200, centers=2,
random_state=0, cluster_std=0.60)
X = X[:N]
y = y[:N]
model = SVC(kernel='linear', C=1E10)
model.fit(X, y)
ax = ax or plt.gca()
ax.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
ax.set_xlim(-1, 4)
ax.set_ylim(-1, 6)
plot_svc_decision_function(model, ax)
fig, ax = plt.subplots(1, 2, figsize=(16, 6))
fig.subplots_adjust(left=0.0625, right=0.95, wspace=0.1)
for axi, N in zip(ax, [60, 120]):
plot_svm(N, axi)
axi.set_title('N = {0}'.format(N))
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from ipywidgets import interact, fixed
interact(plot_svm, N=[10, 200], ax=fixed(None));
SVMs become really powerful when they are combined with another idea: kernels.
When a dataset is not linear seperable a kernel can transform the space to make linear seperable.
Consider the following dataset:
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from sklearn.datasets.samples_generator import make_circles
X, y = make_circles(100, factor=.1, noise=.1)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
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clf = SVC(kernel='linear').fit(X, y)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(clf, plot_support=False);
Obviously this dataset cannot be seperated by line, buth there is a obvious structure in the data.
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r = np.exp(-(X ** 2).sum(1))
We can visualize this extra data dimension using a three-dimensional plot:
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from mpl_toolkits import mplot3d
def plot_3D(elev=-90, azim=180, X=X, y=y):
ax = plt.subplot(projection='3d')
ax.scatter3D(X[:, 0], X[:, 1], r, c=y, s=50, cmap='autumn')
ax.view_init(elev=elev, azim=azim)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('r')
interact(plot_3D, elev=[-90, 90], azim=(-180, 180),
X=fixed(X), y=fixed(y));
kernel parameter to use the RBF kernel
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clf = SVC(kernel='rbf', C=1E6)
clf.fit(X, y)
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plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(clf)
plt.scatter(clf.support_vectors_[:, 0], clf.support_vectors_[:, 1],
s=300, lw=1, facecolors='none');
This trick no only applies to SVMs. It is used whenever a linear method must be transformed into a non-linear method.
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from IPython.display import Image
Image('images/machineLearning.png')
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Thus far we've only dealt with datasets where a perfect seperation could be found.
In practice, this never happens. So let's consider the following dataset:
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X, y = make_blobs(n_samples=100, centers=2,
random_state=0, cluster_std=1.2)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn');
Thus far we enforced a strict "no datapoint within the margins"-policy (by choosing a very high value $C$).
The small we choose the values for $C$ the more datapoints are allowed in the margin.
This plot depicts the influence of $C$:
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fig, ax = plt.subplots(1, 2, figsize=(16, 6))
fig.subplots_adjust(left=0.0625, right=0.95, wspace=0.1)
for axi, C in zip(ax, [10.0, 0.1]):
model = SVC(kernel='linear', C=C).fit(X, y)
axi.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(model, axi)
axi.scatter(model.support_vectors_[:, 0],
model.support_vectors_[:, 1],
s=300, lw=1, facecolors='none');
axi.set_title('C = {0:.1f}'.format(C), size=14)
The optimal value for $C$ depends on the dataset. Methods like grid-search or bayeasian optimization allow to automatically find well performing parameters.
We want to build a face recognition system.
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from sklearn.datasets import fetch_lfw_people
faces = fetch_lfw_people(min_faces_per_person=60)
print(faces.target_names)
print(faces.images.shape)
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fig, ax = plt.subplots(3, 5)
for i, axi in enumerate(ax.flat):
axi.imshow(faces.images[i], cmap='bone')
axi.set(xticks=[], yticks=[],
xlabel=faces.target_names[faces.target[i]])
Each sample is a 62 x 47 pixel image (~3000 features). We'll reduce that to 150 by applying a PCA dimensionality reduction (which will be subject to an upcoming lecture).
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from sklearn.svm import SVC
from sklearn.decomposition import PCA
from sklearn.pipeline import make_pipeline
pca = PCA(svd_solver='randomized', n_components=150, whiten=True, random_state=42)
svc = SVC(kernel='rbf', class_weight='balanced', C=10, gamma=0.0001, )
model = make_pipeline(pca, svc)
We split the data into a training and testing set:
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from sklearn.model_selection import train_test_split
Xtrain, Xtest, ytrain, ytest = train_test_split(faces.data, faces.target, random_state=42)
Then we fit the model:
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model.fit(Xtrain, ytrain)
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Predict the labels of the test set:
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yfit = model.predict(Xtest)
Let's examine the results.
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fig, ax = plt.subplots(4, 6)
for i, axi in enumerate(ax.flat):
axi.imshow(Xtest[i].reshape(62, 47), cmap='bone')
axi.set(xticks=[], yticks=[])
axi.set_ylabel(faces.target_names[yfit[i]].split()[-1],
color='black' if yfit[i] == ytest[i] else 'red')
fig.suptitle('Predicted Names; Incorrect Labels in Red', size=14);
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from sklearn.metrics import confusion_matrix
mat = confusion_matrix(ytest, yfit)
sns.heatmap(mat.T, square=True, annot=True, fmt='d', cbar=False,
xticklabels=faces.target_names,
yticklabels=faces.target_names)
plt.xlabel('true label')
plt.ylabel('predicted label');