Chapter 10 - Unsupervised Learning¶
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# %load ../standard_import.txt
import pandas as pd
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.preprocessing import scale
from sklearn.decomposition import PCA
from sklearn.cluster import KMeans
from scipy.cluster import hierarchy
%matplotlib inline
plt.style.use('seaborn-white')
Lab 1: Principal Component Analysis¶
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# In R, I exported the dataset to a csv file. It is part of the base R distribution.
df = pd.read_csv('Data/USArrests.csv', index_col=0)
df.info()
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df.mean()
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df.var()
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X = pd.DataFrame(scale(df), index=df.index, columns=df.columns)
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# The loading vectors
pca_loadings = pd.DataFrame(PCA().fit(X).components_.T, index=df.columns, columns=['V1', 'V2', 'V3', 'V4'])
pca_loadings
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# Fit the PCA model and transform X to get the principal components
pca = PCA()
df_plot = pd.DataFrame(pca.fit_transform(X), columns=['PC1', 'PC2', 'PC3', 'PC4'], index=X.index)
df_plot
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fig , ax1 = plt.subplots(figsize=(9,7))
ax1.set_xlim(-3.5,3.5)
ax1.set_ylim(-3.5,3.5)
# Plot Principal Components 1 and 2
for i in df_plot.index:
ax1.annotate(i, (df_plot.PC1.loc[i], -df_plot.PC2.loc[i]), ha='center')
# Plot reference lines
ax1.hlines(0,-3.5,3.5, linestyles='dotted', colors='grey')
ax1.vlines(0,-3.5,3.5, linestyles='dotted', colors='grey')
ax1.set_xlabel('First Principal Component')
ax1.set_ylabel('Second Principal Component')
# Plot Principal Component loading vectors, using a second y-axis.
ax2 = ax1.twinx().twiny()
ax2.set_ylim(-1,1)
ax2.set_xlim(-1,1)
ax2.tick_params(axis='y', colors='orange')
ax2.set_xlabel('Principal Component loading vectors', color='orange')
# Plot labels for vectors. Variable 'a' is a small offset parameter to separate arrow tip and text.
a = 1.07
for i in pca_loadings[['V1', 'V2']].index:
ax2.annotate(i, (pca_loadings.V1.loc[i]*a, -pca_loadings.V2.loc[i]*a), color='orange')
# Plot vectors
ax2.arrow(0,0,pca_loadings.V1[0], -pca_loadings.V2[0])
ax2.arrow(0,0,pca_loadings.V1[1], -pca_loadings.V2[1])
ax2.arrow(0,0,pca_loadings.V1[2], -pca_loadings.V2[2])
ax2.arrow(0,0,pca_loadings.V1[3], -pca_loadings.V2[3]);
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# Standard deviation of the four principal components
np.sqrt(pca.explained_variance_)
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pca.explained_variance_
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pca.explained_variance_ratio_
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plt.figure(figsize=(7,5))
plt.plot([1,2,3,4], pca.explained_variance_ratio_, '-o', label='Individual component')
plt.plot([1,2,3,4], np.cumsum(pca.explained_variance_ratio_), '-s', label='Cumulative')
plt.ylabel('Proportion of Variance Explained')
plt.xlabel('Principal Component')
plt.xlim(0.75,4.25)
plt.ylim(0,1.05)
plt.xticks([1,2,3,4])
plt.legend(loc=2);
Lab 2: Clustering¶
10.5.1 K-Means Clustering¶
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# Generate data
np.random.seed(2)
X = np.random.standard_normal((50,2))
X[:25,0] = X[:25,0]+3
X[:25,1] = X[:25,1]-4
K = 2¶
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km1 = KMeans(n_clusters=2, n_init=20)
km1.fit(X)
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km1.labels_
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See plot for K=2 below.
K = 3¶
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np.random.seed(4)
km2 = KMeans(n_clusters=3, n_init=20)
km2.fit(X)
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pd.Series(km2.labels_).value_counts()
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km2.cluster_centers_
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km2.labels_
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# Sum of distances of samples to their closest cluster center.
km2.inertia_
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fig, (ax1, ax2) = plt.subplots(1,2, figsize=(14,5))
ax1.scatter(X[:,0], X[:,1], s=40, c=km1.labels_, cmap=plt.cm.prism)
ax1.set_title('K-Means Clustering Results with K=2')
ax1.scatter(km1.cluster_centers_[:,0], km1.cluster_centers_[:,1], marker='+', s=100, c='k', linewidth=2)
ax2.scatter(X[:,0], X[:,1], s=40, c=km2.labels_, cmap=plt.cm.prism)
ax2.set_title('K-Means Clustering Results with K=3')
ax2.scatter(km2.cluster_centers_[:,0], km2.cluster_centers_[:,1], marker='+', s=100, c='k', linewidth=2);
10.5.3 Hierarchical Clustering¶
scipy¶
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fig, (ax1,ax2,ax3) = plt.subplots(3,1, figsize=(15,18))
for linkage, cluster, ax in zip([hierarchy.complete(X), hierarchy.average(X), hierarchy.single(X)], ['c1','c2','c3'],
[ax1,ax2,ax3]):
cluster = hierarchy.dendrogram(linkage, ax=ax, color_threshold=0)
ax1.set_title('Complete Linkage')
ax2.set_title('Average Linkage')
ax3.set_title('Single Linkage');
Lab 3: NCI60 Data Example¶
§ 10.6.1 PCA¶
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# In R, I exported the two elements of this ISLR dataset to csv files.
# There is one file for the features and another file for the classes/types.
df2 = pd.read_csv('Data/NCI60_X.csv').drop('Unnamed: 0', axis=1)
df2.columns = np.arange(df2.columns.size)
df2.info()
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X = pd.DataFrame(scale(df2))
X.shape
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y = pd.read_csv('Data/NCI60_y.csv', usecols=[1], skiprows=1, names=['type'])
y.shape
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y.type.value_counts()
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# Fit the PCA model and transform X to get the principal components
pca2 = PCA()
df2_plot = pd.DataFrame(pca2.fit_transform(X))
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fig, (ax1, ax2) = plt.subplots(1,2, figsize=(15,6))
color_idx = pd.factorize(y.type)[0]
cmap = plt.cm.hsv
# Left plot
ax1.scatter(df2_plot.iloc[:,0], -df2_plot.iloc[:,1], c=color_idx, cmap=cmap, alpha=0.5, s=50)
ax1.set_ylabel('Principal Component 2')
# Right plot
ax2.scatter(df2_plot.iloc[:,0], df2_plot.iloc[:,2], c=color_idx, cmap=cmap, alpha=0.5, s=50)
ax2.set_ylabel('Principal Component 3')
# Custom legend for the classes (y) since we do not create scatter plots per class (which could have their own labels).
handles = []
labels = pd.factorize(y.type.unique())
norm = mpl.colors.Normalize(vmin=0.0, vmax=14.0)
for i, v in zip(labels[0], labels[1]):
handles.append(mpl.patches.Patch(color=cmap(norm(i)), label=v, alpha=0.5))
ax2.legend(handles=handles, bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)
# xlabel for both plots
for ax in fig.axes:
ax.set_xlabel('Principal Component 1')
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pd.DataFrame([df2_plot.iloc[:,:5].std(axis=0, ddof=0).as_matrix(),
pca2.explained_variance_ratio_[:5],
np.cumsum(pca2.explained_variance_ratio_[:5])],
index=['Standard Deviation', 'Proportion of Variance', 'Cumulative Proportion'],
columns=['PC1', 'PC2', 'PC3', 'PC4', 'PC5'])
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df2_plot.iloc[:,:10].var(axis=0, ddof=0).plot(kind='bar', rot=0)
plt.ylabel('Variances');
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fig , (ax1,ax2) = plt.subplots(1,2, figsize=(15,5))
# Left plot
ax1.plot(pca2.explained_variance_ratio_, '-o')
ax1.set_ylabel('Proportion of Variance Explained')
ax1.set_ylim(ymin=-0.01)
# Right plot
ax2.plot(np.cumsum(pca2.explained_variance_ratio_), '-ro')
ax2.set_ylabel('Cumulative Proportion of Variance Explained')
ax2.set_ylim(ymax=1.05)
for ax in fig.axes:
ax.set_xlabel('Principal Component')
ax.set_xlim(-1,65)
§ 10.6.2 Clustering¶
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X= pd.DataFrame(scale(df2), index=y.type, columns=df2.columns)
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fig, (ax1,ax2,ax3) = plt.subplots(1,3, figsize=(20,20))
for linkage, cluster, ax in zip([hierarchy.complete(X), hierarchy.average(X), hierarchy.single(X)],
['c1','c2','c3'],
[ax1,ax2,ax3]):
cluster = hierarchy.dendrogram(linkage, labels=X.index, orientation='right', color_threshold=0, leaf_font_size=10, ax=ax)
ax1.set_title('Complete Linkage')
ax2.set_title('Average Linkage')
ax3.set_title('Single Linkage');
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plt.figure(figsize=(10,20))
cut4 = hierarchy.dendrogram(hierarchy.complete(X),
labels=X.index, orientation='right', color_threshold=140, leaf_font_size=10)
plt.vlines(140,0,plt.gca().yaxis.get_data_interval()[1], colors='r', linestyles='dashed');
KMeans¶
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np.random.seed(2)
km4 = KMeans(n_clusters=4, n_init=50)
km4.fit(X)
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km4.labels_
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# Observations per KMeans cluster
pd.Series(km4.labels_).value_counts().sort_index()
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Hierarchical¶
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# Observations per Hierarchical cluster
cut4b = hierarchy.dendrogram(hierarchy.complete(X), truncate_mode='lastp', p=4, show_leaf_counts=True)
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# Hierarchy based on Principal Components 1 to 5
plt.figure(figsize=(10,20))
pca_cluster = hierarchy.dendrogram(hierarchy.complete(df2_plot.iloc[:,:5]), labels=y.type.values, orientation='right', color_threshold=100, leaf_font_size=10)
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cut4c = hierarchy.dendrogram(hierarchy.complete(df2_plot), truncate_mode='lastp', p=4,
show_leaf_counts=True)
# See also color coding in plot above.