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from __future__ import print_function, division
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
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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], 'o')
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:
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from sklearn.decomposition import PCA
pca = PCA(n_components=2)
pca.fit(X)
print(pca.explained_variance_ratio_)
print(pca.components_)
To see what these numbers mean, let's view them as vectors plotted on top of the data:
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plt.plot(X[:, 0], X[:, 1], 'o', alpha=0.5)
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 vector is longer than the other. In a sense, this tells us that that direction in the data is somehow more "important" than the other direction. The explained variance quantifies this measure of "importance" in direction.
Another way to think of it is 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) # keep 95% of variance
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], 'o', alpha=0.2)
plt.plot(X_new[:, 0], X_new[:, 1], 'ob', 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 is the sense in which "dimensionality reduction" works: if you can approximate a data set in a lower dimension, you can often have an easier time visualizing it or fitting complicated models to the data.
The dimensionality reduction might seem a bit abstract 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, 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 stretch and rotation in 64-dimensional space that allows us to see the layout of the digits, without reference to the labels.
PCA is a very useful dimensionality reduction algorithm, because it has a very intuitive interpretation via eigenvectors. The input data is represented as a vector: in the case of the digits, our data is
$$ x = [x_1, x_2, x_3 \cdots] $$but what this really means is
$$ image(x) = x_1 \cdot{\rm (pixel~1)} + x_2 \cdot{\rm (pixel~2)} + x_3 \cdot{\rm (pixel~3)} \cdots $$If we reduce the dimensionality in the pixel space to (say) 6, we recover only a partial image:
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from fig_code.figures import plot_image_components
sns.set_style('white')
plot_image_components(digits.data[0])
But the pixel-wise representation is not the only choice. We can also use other basis functions, and write something like
$$ image(x) = {\rm mean} + x_1 \cdot{\rm (basis~1)} + x_2 \cdot{\rm (basis~2)} + x_3 \cdot{\rm (basis~3)} \cdots $$What PCA does is to choose optimal basis functions so that only a few are needed to get a reasonable approximation. The low-dimensional representation of our data is the coefficients of this series, and the approximate reconstruction is the result of the sum:
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from fig_code.figures import plot_pca_interactive
plot_pca_interactive(digits.data)
Here we see that with only six PCA components, we recover a reasonable approximation of the input!
Thus we see that PCA can be viewed from two angles. It can be viewed as dimensionality reduction, or it can be viewed as a form of lossy data compression where the loss favors noise. In this way, PCA can be used as a filtering process as well.
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sns.set()
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.
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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:
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from IPython.html.widgets import interact
def plot_digits(n_components):
fig = plt.figure(figsize=(8, 8))
plt.subplot(1, 1, 1, frameon=False, xticks=[], yticks=[])
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:
Each of these has its own strengths & weaknesses, and areas of application. You can read about them on the scikit-learn website.