This notebook was originally put together by [Jake Vanderplas](http://www.vanderplas.com) for PyCon 2014. [Peter Prettenhofer](https://github.com/pprett) adapted it for PyCon Ukraine 2014. Source and license info is on [GitHub](https://github.com/pprett/sklearn_pycon2014/).
Here we'll briefly go into a bit of depth on some important unsupervised learning techniques: Principal Component Analysis (PCA) and K-Means.
By the end of this section you should
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%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
Principal Component Analysis is a very powerful unsupervised method for dimensionality reduction in data. It's easiest to visualize by looking at a two-dimensional dataset:
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np.random.seed(1)
X = np.dot(np.random.random(size=(2, 2)), np.random.normal(size=(2, 200))).T
plt.plot(X[:, 0], X[:, 1], 'og')
plt.axis('equal')
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We can see that there is a definite trend in the data. What PCA seeks to do is to find the Principal Axes in the data, and explain how important those axes are in describing the data distribution:
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from sklearn.decomposition import PCA
pca = PCA(n_components=2)
pca.fit(X)
print(pca.explained_variance_)
print(pca.components_)
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plt.plot(X[:, 0], X[:, 1], 'og', alpha=0.3)
plt.axis('equal')
for length, vector in zip(pca.explained_variance_, pca.components_):
v = vector * 3 * np.sqrt(length)
plt.plot([0, v[0]], [0, v[1]], '-k', lw=3)
Notice that one direction is very important, while the other direction is not. This shows us that the second principal component could be completely ignored without much loss of information! Let's see what our data look like if we only keep 95% of the variance:
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clf = PCA(0.95)
X_trans = clf.fit_transform(X)
print(X.shape)
print(X_trans.shape)
By specifying that we want to throw away 5% of the variance, the data is now compressed by a factor of 50%! Let's see what the data look like after this compression:
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X_new = clf.inverse_transform(X_trans)
plt.plot(X[:, 0], X[:, 1], 'og', alpha=0.2)
plt.plot(X_new[:, 0], X_new[:, 1], 'og', alpha=0.8)
plt.axis('equal');
The light points are the original data, while the dark points are the projected version. We see that after truncating 5% of the variance of this dataset and then reprojecting it, the "most important" features of the data are maintained, and we've compressed the data by 50%!
This might seem a bit strange in two dimensions, but the projection and dimensionality reduction can be extremely useful when visualizing high-dimensional data. Let's take a quick look at the application of PCA to the digits data we looked at before:
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from sklearn.datasets import load_digits
digits = load_digits()
X = digits.data
y = digits.target
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pca = PCA(2) # project from 64 to 2 dimensions
Xproj = pca.fit_transform(X)
print(X.shape)
print(Xproj.shape)
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plt.scatter(Xproj[:, 0], Xproj[:, 1], c=y)
plt.colorbar();
This gives us an idea of the relationship between the digits. Essentially, we have found the optimal rotation in 64-dimensional space that allows us to see the layout of the digits, without reference to the labels.
But how much information have we thrown away? We can figure this out by looking at the variance:
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pca = PCA(64).fit(X)
plt.semilogx(np.cumsum(pca.explained_variance_ratio_))
plt.xlabel('number of components')
plt.ylabel('cumulative explained variance')
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Here we see that our two-dimensional projection loses a lot of information (as measured by the explained variance) and that we'd need about 20 components to retain 90% of the variance. Looking at this plot for a high-dimensional dataset can help you understand the level of redundancy present in multiple observations.
Note that scikit-learn contains many other unsupervised dimensionality reduction routines: some you might wish to try are Other dimensionality reduction techniques which are useful to know about:
In this exercise we will apply PCA to the Olivetti faces dataset we used in the second part of the tutorial. The idea is to reduce the dimensionality of face images by representing each image as a mixture over principal face images (so-called Eigen-Faces). Lets plot those principal components and see how they look like!
If you use a small number of components (e.g. 16) then we will perform lossy-compression -- if you plot the reconstruction we will see how much we loose. You can use PCA.transform to embed data into the space spanned by the principal components. If you compute the dot product between the embedding and the principal components pca.components_ you will get the reconstruction.
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from sklearn.datasets import fetch_olivetti_faces
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# %load solutions/05_eigen_faces.py
K Means is an algorithm for unsupervised clustering: that is, finding clusters in data based on the data attributes alone (not the labels).
K Means is a relatively easy-to-understand algorithm. It searches for cluster centers which are the mean of the points within them, such that every point is closest to the cluster center it is assigned to.
Let's look at how KMeans operates on the simple clusters we looked at previously. To emphasize that this is unsupervised, we'll not plot the colors of the clusters:
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from sklearn.datasets.samples_generator import make_blobs
X, y = make_blobs(n_samples=300, centers=4,
random_state=0, cluster_std=0.60)
plt.scatter(X[:, 0], X[:, 1], s=50);
By eye, it is relatively easy to pick out the four clusters. If you were to perform an exhaustive search for the different segmentations of the data, however, the search space would be exponential in the number of points. Fortunately, there is a well-known Expectation Maximization (EM) procedure which scikit-learn implements, so that KMeans can be solved relatively quickly.
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from sklearn.cluster import KMeans
est = KMeans(4) # 4 clusters
est.fit(X)
y_kmeans = est.predict(X)
plt.scatter(X[:, 0], X[:, 1], c=y_kmeans, s=50);
The algorithm identifies the four clusters of points in a manner very similar to what we would do by eye!
For a closer-to-real-world example, let's again take a look at the digits data. Here we'll use KMeans to automatically cluster the data in 64 dimensions, and then look at the cluster centers to see what the algorithm has found.
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est = KMeans(n_clusters=10)
clusters = est.fit_predict(digits.data)
est.cluster_centers_.shape
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We see ten clusters in 64 dimensions. Let's visualize each of these cluster centers to see what they represent:
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fig = plt.figure(figsize=(8, 3))
for i in range(10):
ax = fig.add_subplot(2, 5, 1 + i, xticks=[], yticks=[])
ax.imshow(est.cluster_centers_[i].reshape((8, 8)), cmap=plt.cm.binary)
We see that even without the labels, KMeans is able to find clusters whose means are recognizable digits (with apologies to the number 8).
For good measure, let's use our PCA visualization and look at the true cluster labels and K-means cluster labels:
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X = PCA(2).fit_transform(digits.data)
fig, ax = plt.subplots(1, 2, figsize=(8, 4))
ax[0].scatter(X[:, 0], X[:, 1], c=clusters)
ax[1].scatter(X[:, 0], X[:, 1], c=digits.target);
Though the colors are permuted, we see that in general (at least in a straightforward by-eye comparison) the KMeans clusters tend to reflect the true clustering.