Here we'll explore Principal Component Analysis, which is an extremely useful linear dimensionality reduction technique.
We'll start with our standard set of initial imports:
In [1]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
# use seaborn plotting defaults
# If this causes an error, you can comment it out.
import seaborn as sns
sns.set()
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:
In [2]:
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');
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:
In [3]:
from sklearn.decomposition import PCA
pca = PCA(n_components=2)
pca.fit(X)
print(pca.explained_variance_)
print(pca.components_)
In [4]:
plt.plot(X[:, 0], X[:, 1], 'og', alpha=0.3)
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)
plt.axis('equal');
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:
In [5]:
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:
In [7]:
from sklearn.datasets import load_digits
digits = load_digits()
X = digits.data
y = digits.target
In [8]:
pca = PCA(2) # project from 64 to 2 dimensions
Xproj = pca.fit_transform(X)
print(X.shape)
print(Xproj.shape)
In [9]:
plt.scatter(Xproj[:, 0], Xproj[:, 1], c=y, edgecolor='none', alpha=0.5,
cmap=plt.cm.get_cmap('nipy_spectral', 10))
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:
In [10]:
pca = PCA().fit(X)
plt.plot(np.cumsum(pca.explained_variance_ratio_))
plt.xlabel('number of components')
plt.ylabel('cumulative explained variance');
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.
In [11]:
fig, axes = plt.subplots(8, 8, figsize=(8, 8))
fig.subplots_adjust(hspace=0.1, wspace=0.1)
for i, ax in enumerate(axes.flat):
pca = PCA(i + 1).fit(X)
im = pca.inverse_transform(pca.transform(X[20:21]))
ax.imshow(im.reshape((8, 8)), cmap='binary')
ax.text(0.95, 0.05, 'n = {0}'.format(i + 1), ha='right',
transform=ax.transAxes, color='green')
ax.set_xticks([])
ax.set_yticks([])
Let's take another look at this by using IPython's interact functionality to view the reconstruction of several images at once:
In [12]:
from IPython.html.widgets import interact
def plot_digits(n_components):
plt.figure(figsize=(8, 8))
nside = 10
pca = PCA(n_components).fit(X)
Xproj = pca.inverse_transform(pca.transform(X[:nside ** 2]))
Xproj = np.reshape(Xproj, (nside, nside, 8, 8))
total_var = pca.explained_variance_ratio_.sum()
im = np.vstack([np.hstack([Xproj[i, j] for j in range(nside)])
for i in range(nside)])
plt.imshow(im)
plt.grid(False)
plt.title("n = {0}, variance = {1:.2f}".format(n_components, total_var),
size=18)
plt.clim(0, 16)
interact(plot_digits, n_components=[1, 64], nside=[1, 8]);
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: