Chapter 8 – Dimensionality Reduction
This notebook contains all the sample code and solutions to the exercises in chapter 8.
First, let's import a few common modules, ensure MatplotLib plots figures inline and prepare a function to save the figures. We also check that Python 3.5 or later is installed (although Python 2.x may work, it is deprecated so we strongly recommend you use Python 3 instead), as well as Scikit-Learn ≥0.20.
# Python ≥3.5 is required
import sys
assert sys.version_info >= (3, 5)
# Scikit-Learn ≥0.20 is required
import sklearn
assert sklearn.__version__ >= "0.20"
# Common imports
import numpy as np
import os
# to make this notebook's output stable across runs
np.random.seed(42)
# To plot pretty figures
%matplotlib inline
import matplotlib as mpl
import matplotlib.pyplot as plt
mpl.rc('axes', labelsize=14)
mpl.rc('xtick', labelsize=12)
mpl.rc('ytick', labelsize=12)
# Where to save the figures
PROJECT_ROOT_DIR = "."
CHAPTER_ID = "dim_reduction"
IMAGES_PATH = os.path.join(PROJECT_ROOT_DIR, "images", CHAPTER_ID)
os.makedirs(IMAGES_PATH, exist_ok=True)
def save_fig(fig_id, tight_layout=True, fig_extension="png", resolution=300):
path = os.path.join(IMAGES_PATH, fig_id + "." + fig_extension)
print("Saving figure", fig_id)
if tight_layout:
plt.tight_layout()
plt.savefig(path, format=fig_extension, dpi=resolution)
Build 3D dataset:
np.random.seed(4)
m = 60
w1, w2 = 0.1, 0.3
noise = 0.1
angles = np.random.rand(m) * 3 * np.pi / 2 - 0.5
X = np.empty((m, 3))
X[:, 0] = np.cos(angles) + np.sin(angles)/2 + noise * np.random.randn(m) / 2
X[:, 1] = np.sin(angles) * 0.7 + noise * np.random.randn(m) / 2
X[:, 2] = X[:, 0] * w1 + X[:, 1] * w2 + noise * np.random.randn(m)
X_centered = X - X.mean(axis=0)
U, s, Vt = np.linalg.svd(X_centered)
c1 = Vt.T[:, 0]
c2 = Vt.T[:, 1]
m, n = X.shape
S = np.zeros(X_centered.shape)
S[:n, :n] = np.diag(s)
np.allclose(X_centered, U.dot(S).dot(Vt))
W2 = Vt.T[:, :2]
X2D = X_centered.dot(W2)
X2D_using_svd = X2D
With Scikit-Learn, PCA is really trivial. It even takes care of mean centering for you:
from sklearn.decomposition import PCA
pca = PCA(n_components = 2)
X2D = pca.fit_transform(X)
X2D[:5]
X2D_using_svd[:5]
Notice that running PCA multiple times on slightly different datasets may result in different results. In general the only difference is that some axes may be flipped. In this example, PCA using Scikit-Learn gives the same projection as the one given by the SVD approach, except both axes are flipped:
np.allclose(X2D, -X2D_using_svd)
Recover the 3D points projected on the plane (PCA 2D subspace).
X3D_inv = pca.inverse_transform(X2D)
Of course, there was some loss of information during the projection step, so the recovered 3D points are not exactly equal to the original 3D points:
np.allclose(X3D_inv, X)
We can compute the reconstruction error:
np.mean(np.sum(np.square(X3D_inv - X), axis=1))
The inverse transform in the SVD approach looks like this:
X3D_inv_using_svd = X2D_using_svd.dot(Vt[:2, :])
The reconstructions from both methods are not identical because Scikit-Learn's PCA
class automatically takes care of reversing the mean centering, but if we subtract the mean, we get the same reconstruction:
np.allclose(X3D_inv_using_svd, X3D_inv - pca.mean_)
The PCA
object gives access to the principal components that it computed:
pca.components_
Compare to the first two principal components computed using the SVD method:
Vt[:2]
Notice how the axes are flipped.
Now let's look at the explained variance ratio:
pca.explained_variance_ratio_
The first dimension explains 84.2% of the variance, while the second explains 14.6%.
By projecting down to 2D, we lost about 1.1% of the variance:
1 - pca.explained_variance_ratio_.sum()
Here is how to compute the explained variance ratio using the SVD approach (recall that s
is the diagonal of the matrix S
):
np.square(s) / np.square(s).sum()
Next, let's generate some nice figures! :)
Utility class to draw 3D arrows (copied from http://stackoverflow.com/questions/11140163)
from matplotlib.patches import FancyArrowPatch
from mpl_toolkits.mplot3d import proj3d
class Arrow3D(FancyArrowPatch):
def __init__(self, xs, ys, zs, *args, **kwargs):
FancyArrowPatch.__init__(self, (0,0), (0,0), *args, **kwargs)
self._verts3d = xs, ys, zs
def draw(self, renderer):
xs3d, ys3d, zs3d = self._verts3d
xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)
self.set_positions((xs[0],ys[0]),(xs[1],ys[1]))
FancyArrowPatch.draw(self, renderer)
Express the plane as a function of x and y.
axes = [-1.8, 1.8, -1.3, 1.3, -1.0, 1.0]
x1s = np.linspace(axes[0], axes[1], 10)
x2s = np.linspace(axes[2], axes[3], 10)
x1, x2 = np.meshgrid(x1s, x2s)
C = pca.components_
R = C.T.dot(C)
z = (R[0, 2] * x1 + R[1, 2] * x2) / (1 - R[2, 2])
Plot the 3D dataset, the plane and the projections on that plane.
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure(figsize=(6, 3.8))
ax = fig.add_subplot(111, projection='3d')
X3D_above = X[X[:, 2] > X3D_inv[:, 2]]
X3D_below = X[X[:, 2] <= X3D_inv[:, 2]]
ax.plot(X3D_below[:, 0], X3D_below[:, 1], X3D_below[:, 2], "bo", alpha=0.5)
ax.plot_surface(x1, x2, z, alpha=0.2, color="k")
np.linalg.norm(C, axis=0)
ax.add_artist(Arrow3D([0, C[0, 0]],[0, C[0, 1]],[0, C[0, 2]], mutation_scale=15, lw=1, arrowstyle="-|>", color="k"))
ax.add_artist(Arrow3D([0, C[1, 0]],[0, C[1, 1]],[0, C[1, 2]], mutation_scale=15, lw=1, arrowstyle="-|>", color="k"))
ax.plot([0], [0], [0], "k.")
for i in range(m):
if X[i, 2] > X3D_inv[i, 2]:
ax.plot([X[i][0], X3D_inv[i][0]], [X[i][1], X3D_inv[i][1]], [X[i][2], X3D_inv[i][2]], "k-")
else:
ax.plot([X[i][0], X3D_inv[i][0]], [X[i][1], X3D_inv[i][1]], [X[i][2], X3D_inv[i][2]], "k-", color="#505050")
ax.plot(X3D_inv[:, 0], X3D_inv[:, 1], X3D_inv[:, 2], "k+")
ax.plot(X3D_inv[:, 0], X3D_inv[:, 1], X3D_inv[:, 2], "k.")
ax.plot(X3D_above[:, 0], X3D_above[:, 1], X3D_above[:, 2], "bo")
ax.set_xlabel("$x_1$", fontsize=18, labelpad=10)
ax.set_ylabel("$x_2$", fontsize=18, labelpad=10)
ax.set_zlabel("$x_3$", fontsize=18, labelpad=10)
ax.set_xlim(axes[0:2])
ax.set_ylim(axes[2:4])
ax.set_zlim(axes[4:6])
# Note: If you are using Matplotlib 3.0.0, it has a bug and does not
# display 3D graphs properly.
# See https://github.com/matplotlib/matplotlib/issues/12239
# You should upgrade to a later version. If you cannot, then you can
# use the following workaround before displaying each 3D graph:
# for spine in ax.spines.values():
# spine.set_visible(False)
save_fig("dataset_3d_plot")
plt.show()
fig = plt.figure()
ax = fig.add_subplot(111, aspect='equal')
ax.plot(X2D[:, 0], X2D[:, 1], "k+")
ax.plot(X2D[:, 0], X2D[:, 1], "k.")
ax.plot([0], [0], "ko")
ax.arrow(0, 0, 0, 1, head_width=0.05, length_includes_head=True, head_length=0.1, fc='k', ec='k')
ax.arrow(0, 0, 1, 0, head_width=0.05, length_includes_head=True, head_length=0.1, fc='k', ec='k')
ax.set_xlabel("$z_1$", fontsize=18)
ax.set_ylabel("$z_2$", fontsize=18, rotation=0)
ax.axis([-1.5, 1.3, -1.2, 1.2])
ax.grid(True)
save_fig("dataset_2d_plot")
Swiss roll:
from sklearn.datasets import make_swiss_roll
X, t = make_swiss_roll(n_samples=1000, noise=0.2, random_state=42)
axes = [-11.5, 14, -2, 23, -12, 15]
fig = plt.figure(figsize=(6, 5))
ax = fig.add_subplot(111, projection='3d')
ax.scatter(X[:, 0], X[:, 1], X[:, 2], c=t, cmap=plt.cm.hot)
ax.view_init(10, -70)
ax.set_xlabel("$x_1$", fontsize=18)
ax.set_ylabel("$x_2$", fontsize=18)
ax.set_zlabel("$x_3$", fontsize=18)
ax.set_xlim(axes[0:2])
ax.set_ylim(axes[2:4])
ax.set_zlim(axes[4:6])
save_fig("swiss_roll_plot")
plt.show()
plt.figure(figsize=(11, 4))
plt.subplot(121)
plt.scatter(X[:, 0], X[:, 1], c=t, cmap=plt.cm.hot)
plt.axis(axes[:4])
plt.xlabel("$x_1$", fontsize=18)
plt.ylabel("$x_2$", fontsize=18, rotation=0)
plt.grid(True)
plt.subplot(122)
plt.scatter(t, X[:, 1], c=t, cmap=plt.cm.hot)
plt.axis([4, 15, axes[2], axes[3]])
plt.xlabel("$z_1$", fontsize=18)
plt.grid(True)
save_fig("squished_swiss_roll_plot")
plt.show()
from matplotlib import gridspec
axes = [-11.5, 14, -2, 23, -12, 15]
x2s = np.linspace(axes[2], axes[3], 10)
x3s = np.linspace(axes[4], axes[5], 10)
x2, x3 = np.meshgrid(x2s, x3s)
fig = plt.figure(figsize=(6, 5))
ax = plt.subplot(111, projection='3d')
positive_class = X[:, 0] > 5
X_pos = X[positive_class]
X_neg = X[~positive_class]
ax.view_init(10, -70)
ax.plot(X_neg[:, 0], X_neg[:, 1], X_neg[:, 2], "y^")
ax.plot_wireframe(5, x2, x3, alpha=0.5)
ax.plot(X_pos[:, 0], X_pos[:, 1], X_pos[:, 2], "gs")
ax.set_xlabel("$x_1$", fontsize=18)
ax.set_ylabel("$x_2$", fontsize=18)
ax.set_zlabel("$x_3$", fontsize=18)
ax.set_xlim(axes[0:2])
ax.set_ylim(axes[2:4])
ax.set_zlim(axes[4:6])
save_fig("manifold_decision_boundary_plot1")
plt.show()
fig = plt.figure(figsize=(5, 4))
ax = plt.subplot(111)
plt.plot(t[positive_class], X[positive_class, 1], "gs")
plt.plot(t[~positive_class], X[~positive_class, 1], "y^")
plt.axis([4, 15, axes[2], axes[3]])
plt.xlabel("$z_1$", fontsize=18)
plt.ylabel("$z_2$", fontsize=18, rotation=0)
plt.grid(True)
save_fig("manifold_decision_boundary_plot2")
plt.show()
fig = plt.figure(figsize=(6, 5))
ax = plt.subplot(111, projection='3d')
positive_class = 2 * (t[:] - 4) > X[:, 1]
X_pos = X[positive_class]
X_neg = X[~positive_class]
ax.view_init(10, -70)
ax.plot(X_neg[:, 0], X_neg[:, 1], X_neg[:, 2], "y^")
ax.plot(X_pos[:, 0], X_pos[:, 1], X_pos[:, 2], "gs")
ax.set_xlabel("$x_1$", fontsize=18)
ax.set_ylabel("$x_2$", fontsize=18)
ax.set_zlabel("$x_3$", fontsize=18)
ax.set_xlim(axes[0:2])
ax.set_ylim(axes[2:4])
ax.set_zlim(axes[4:6])
save_fig("manifold_decision_boundary_plot3")
plt.show()
fig = plt.figure(figsize=(5, 4))
ax = plt.subplot(111)
plt.plot(t[positive_class], X[positive_class, 1], "gs")
plt.plot(t[~positive_class], X[~positive_class, 1], "y^")
plt.plot([4, 15], [0, 22], "b-", linewidth=2)
plt.axis([4, 15, axes[2], axes[3]])
plt.xlabel("$z_1$", fontsize=18)
plt.ylabel("$z_2$", fontsize=18, rotation=0)
plt.grid(True)
save_fig("manifold_decision_boundary_plot4")
plt.show()
angle = np.pi / 5
stretch = 5
m = 200
np.random.seed(3)
X = np.random.randn(m, 2) / 10
X = X.dot(np.array([[stretch, 0],[0, 1]])) # stretch
X = X.dot([[np.cos(angle), np.sin(angle)], [-np.sin(angle), np.cos(angle)]]) # rotate
u1 = np.array([np.cos(angle), np.sin(angle)])
u2 = np.array([np.cos(angle - 2 * np.pi/6), np.sin(angle - 2 * np.pi/6)])
u3 = np.array([np.cos(angle - np.pi/2), np.sin(angle - np.pi/2)])
X_proj1 = X.dot(u1.reshape(-1, 1))
X_proj2 = X.dot(u2.reshape(-1, 1))
X_proj3 = X.dot(u3.reshape(-1, 1))
plt.figure(figsize=(8,4))
plt.subplot2grid((3,2), (0, 0), rowspan=3)
plt.plot([-1.4, 1.4], [-1.4*u1[1]/u1[0], 1.4*u1[1]/u1[0]], "k-", linewidth=1)
plt.plot([-1.4, 1.4], [-1.4*u2[1]/u2[0], 1.4*u2[1]/u2[0]], "k--", linewidth=1)
plt.plot([-1.4, 1.4], [-1.4*u3[1]/u3[0], 1.4*u3[1]/u3[0]], "k:", linewidth=2)
plt.plot(X[:, 0], X[:, 1], "bo", alpha=0.5)
plt.axis([-1.4, 1.4, -1.4, 1.4])
plt.arrow(0, 0, u1[0], u1[1], head_width=0.1, linewidth=5, length_includes_head=True, head_length=0.1, fc='k', ec='k')
plt.arrow(0, 0, u3[0], u3[1], head_width=0.1, linewidth=5, length_includes_head=True, head_length=0.1, fc='k', ec='k')
plt.text(u1[0] + 0.1, u1[1] - 0.05, r"$\mathbf{c_1}$", fontsize=22)
plt.text(u3[0] + 0.1, u3[1], r"$\mathbf{c_2}$", fontsize=22)
plt.xlabel("$x_1$", fontsize=18)
plt.ylabel("$x_2$", fontsize=18, rotation=0)
plt.grid(True)
plt.subplot2grid((3,2), (0, 1))
plt.plot([-2, 2], [0, 0], "k-", linewidth=1)
plt.plot(X_proj1[:, 0], np.zeros(m), "bo", alpha=0.3)
plt.gca().get_yaxis().set_ticks([])
plt.gca().get_xaxis().set_ticklabels([])
plt.axis([-2, 2, -1, 1])
plt.grid(True)
plt.subplot2grid((3,2), (1, 1))
plt.plot([-2, 2], [0, 0], "k--", linewidth=1)
plt.plot(X_proj2[:, 0], np.zeros(m), "bo", alpha=0.3)
plt.gca().get_yaxis().set_ticks([])
plt.gca().get_xaxis().set_ticklabels([])
plt.axis([-2, 2, -1, 1])
plt.grid(True)
plt.subplot2grid((3,2), (2, 1))
plt.plot([-2, 2], [0, 0], "k:", linewidth=2)
plt.plot(X_proj3[:, 0], np.zeros(m), "bo", alpha=0.3)
plt.gca().get_yaxis().set_ticks([])
plt.axis([-2, 2, -1, 1])
plt.xlabel("$z_1$", fontsize=18)
plt.grid(True)
save_fig("pca_best_projection_plot")
plt.show()
from sklearn.datasets import fetch_openml
mnist = fetch_openml('mnist_784', version=1)
mnist.target = mnist.target.astype(np.uint8)
from sklearn.model_selection import train_test_split
X = mnist["data"]
y = mnist["target"]
X_train, X_test, y_train, y_test = train_test_split(X, y)
pca = PCA()
pca.fit(X_train)
cumsum = np.cumsum(pca.explained_variance_ratio_)
d = np.argmax(cumsum >= 0.95) + 1
d
plt.figure(figsize=(6,4))
plt.plot(cumsum, linewidth=3)
plt.axis([0, 400, 0, 1])
plt.xlabel("Dimensions")
plt.ylabel("Explained Variance")
plt.plot([d, d], [0, 0.95], "k:")
plt.plot([0, d], [0.95, 0.95], "k:")
plt.plot(d, 0.95, "ko")
plt.annotate("Elbow", xy=(65, 0.85), xytext=(70, 0.7),
arrowprops=dict(arrowstyle="->"), fontsize=16)
plt.grid(True)
save_fig("explained_variance_plot")
plt.show()
pca = PCA(n_components=0.95)
X_reduced = pca.fit_transform(X_train)
pca.n_components_
np.sum(pca.explained_variance_ratio_)
pca = PCA(n_components = 154)
X_reduced = pca.fit_transform(X_train)
X_recovered = pca.inverse_transform(X_reduced)
def plot_digits(instances, images_per_row=5, **options):
size = 28
images_per_row = min(len(instances), images_per_row)
images = [instance.reshape(size,size) for instance in instances]
n_rows = (len(instances) - 1) // images_per_row + 1
row_images = []
n_empty = n_rows * images_per_row - len(instances)
images.append(np.zeros((size, size * n_empty)))
for row in range(n_rows):
rimages = images[row * images_per_row : (row + 1) * images_per_row]
row_images.append(np.concatenate(rimages, axis=1))
image = np.concatenate(row_images, axis=0)
plt.imshow(image, cmap = mpl.cm.binary, **options)
plt.axis("off")
plt.figure(figsize=(7, 4))
plt.subplot(121)
plot_digits(X_train[::2100])
plt.title("Original", fontsize=16)
plt.subplot(122)
plot_digits(X_recovered[::2100])
plt.title("Compressed", fontsize=16)
save_fig("mnist_compression_plot")
X_reduced_pca = X_reduced
from sklearn.decomposition import IncrementalPCA
n_batches = 100
inc_pca = IncrementalPCA(n_components=154)
for X_batch in np.array_split(X_train, n_batches):
print(".", end="") # not shown in the book
inc_pca.partial_fit(X_batch)
X_reduced = inc_pca.transform(X_train)
X_recovered_inc_pca = inc_pca.inverse_transform(X_reduced)
plt.figure(figsize=(7, 4))
plt.subplot(121)
plot_digits(X_train[::2100])
plt.subplot(122)
plot_digits(X_recovered_inc_pca[::2100])
plt.tight_layout()
X_reduced_inc_pca = X_reduced
Let's compare the results of transforming MNIST using regular PCA and incremental PCA. First, the means are equal:
np.allclose(pca.mean_, inc_pca.mean_)
But the results are not exactly identical. Incremental PCA gives a very good approximate solution, but it's not perfect:
np.allclose(X_reduced_pca, X_reduced_inc_pca)
memmap()
¶Let's create the memmap()
structure and copy the MNIST data into it. This would typically be done by a first program:
filename = "my_mnist.data"
m, n = X_train.shape
X_mm = np.memmap(filename, dtype='float32', mode='write', shape=(m, n))
X_mm[:] = X_train
Now deleting the memmap()
object will trigger its Python finalizer, which ensures that the data is saved to disk.
del X_mm
Next, another program would load the data and use it for training:
X_mm = np.memmap(filename, dtype="float32", mode="readonly", shape=(m, n))
batch_size = m // n_batches
inc_pca = IncrementalPCA(n_components=154, batch_size=batch_size)
inc_pca.fit(X_mm)
rnd_pca = PCA(n_components=154, svd_solver="randomized", random_state=42)
X_reduced = rnd_pca.fit_transform(X_train)
Let's time regular PCA against Incremental PCA and Randomized PCA, for various number of principal components:
import time
for n_components in (2, 10, 154):
print("n_components =", n_components)
regular_pca = PCA(n_components=n_components)
inc_pca = IncrementalPCA(n_components=n_components, batch_size=500)
rnd_pca = PCA(n_components=n_components, random_state=42, svd_solver="randomized")
for pca in (regular_pca, inc_pca, rnd_pca):
t1 = time.time()
pca.fit(X_train)
t2 = time.time()
print(" {}: {:.1f} seconds".format(pca.__class__.__name__, t2 - t1))
Now let's compare PCA and Randomized PCA for datasets of different sizes (number of instances):
times_rpca = []
times_pca = []
sizes = [1000, 10000, 20000, 30000, 40000, 50000, 70000, 100000, 200000, 500000]
for n_samples in sizes:
X = np.random.randn(n_samples, 5)
pca = PCA(n_components = 2, svd_solver="randomized", random_state=42)
t1 = time.time()
pca.fit(X)
t2 = time.time()
times_rpca.append(t2 - t1)
pca = PCA(n_components = 2)
t1 = time.time()
pca.fit(X)
t2 = time.time()
times_pca.append(t2 - t1)
plt.plot(sizes, times_rpca, "b-o", label="RPCA")
plt.plot(sizes, times_pca, "r-s", label="PCA")
plt.xlabel("n_samples")
plt.ylabel("Training time")
plt.legend(loc="upper left")
plt.title("PCA and Randomized PCA time complexity ")
And now let's compare their performance on datasets of 2,000 instances with various numbers of features:
times_rpca = []
times_pca = []
sizes = [1000, 2000, 3000, 4000, 5000, 6000]
for n_features in sizes:
X = np.random.randn(2000, n_features)
pca = PCA(n_components = 2, random_state=42, svd_solver="randomized")
t1 = time.time()
pca.fit(X)
t2 = time.time()
times_rpca.append(t2 - t1)
pca = PCA(n_components = 2)
t1 = time.time()
pca.fit(X)
t2 = time.time()
times_pca.append(t2 - t1)
plt.plot(sizes, times_rpca, "b-o", label="RPCA")
plt.plot(sizes, times_pca, "r-s", label="PCA")
plt.xlabel("n_features")
plt.ylabel("Training time")
plt.legend(loc="upper left")
plt.title("PCA and Randomized PCA time complexity ")
X, t = make_swiss_roll(n_samples=1000, noise=0.2, random_state=42)
from sklearn.decomposition import KernelPCA
rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
X_reduced = rbf_pca.fit_transform(X)
from sklearn.decomposition import KernelPCA
lin_pca = KernelPCA(n_components = 2, kernel="linear", fit_inverse_transform=True)
rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.0433, fit_inverse_transform=True)
sig_pca = KernelPCA(n_components = 2, kernel="sigmoid", gamma=0.001, coef0=1, fit_inverse_transform=True)
y = t > 6.9
plt.figure(figsize=(11, 4))
for subplot, pca, title in ((131, lin_pca, "Linear kernel"), (132, rbf_pca, "RBF kernel, $\gamma=0.04$"), (133, sig_pca, "Sigmoid kernel, $\gamma=10^{-3}, r=1$")):
X_reduced = pca.fit_transform(X)
if subplot == 132:
X_reduced_rbf = X_reduced
plt.subplot(subplot)
#plt.plot(X_reduced[y, 0], X_reduced[y, 1], "gs")
#plt.plot(X_reduced[~y, 0], X_reduced[~y, 1], "y^")
plt.title(title, fontsize=14)
plt.scatter(X_reduced[:, 0], X_reduced[:, 1], c=t, cmap=plt.cm.hot)
plt.xlabel("$z_1$", fontsize=18)
if subplot == 131:
plt.ylabel("$z_2$", fontsize=18, rotation=0)
plt.grid(True)
save_fig("kernel_pca_plot")
plt.show()