Introduction to Sci-Kit Learn and Clustering¶
In this tutorial we will introduce the Sci-Kit Learn library:https://scikit-learn.org/stable/
This is a very important library with a huge toolkit for data processing, unsupervised and supervised learning. It is one of the core tools for data science.
We will see some of the capabilities of this toolkit and focus on clustering.
import numpy as np
import scipy as sp
import scipy.sparse as sp_sparse
import scipy.spatial.distance as sp_dist
import matplotlib.pyplot as plt
import sklearn as sk
import sklearn.datasets as sk_data
import sklearn.metrics as metrics
from sklearn import preprocessing
import sklearn.cluster as sk_cluster
import sklearn.feature_extraction.text as sk_text
import scipy.cluster.hierarchy as hr
import time
import seaborn as sns
%matplotlib inline
Computing distances¶
For the computation of distances there are libraries in Scipy
http://docs.scipy.org/doc/scipy-0.15.1/reference/spatial.distance.html#module-scipy.spatial.distance
but also in SciKit metrics library:
http://scikit-learn.org/stable/modules/generated/sklearn.metrics.pairwise.pairwise_distances.html
Compute distance between vectors
import scipy.spatial.distance as sp_dist
x = np.random.randint(2, size = 5)
y = np.random.randint(2, size = 5)
print (x)
print (y)
print (sp_dist.cosine(x,y))
print (sp_dist.euclidean(x,y))
print (sp_dist.jaccard(x,y))
print (sp_dist.hamming(x,y))
# When computing jaccard similarity of 0/1 matrices,
# 1 means that the element corresponding to the column is in the set,
# 0 that the element is not in the set
Compute pairwise distances in a table
A = np.random.randint(2, size = (5,3))
# computes the matrix of all pairwise distances of rows
# returns a vector with N(N-1)/2 entries (N number of rows)
D = sp_dist.pdist(A, 'jaccard')
print (A)
print('\n all row distances')
print (D)
print(sp_dist.squareform(D))
Compute distances using sklearn
import sklearn.metrics as metrics
#computes the matrix of all pairwise distances of rows
# returns a NxN matrix (N number of rows)
D2 = metrics.pairwise.pairwise_distances(A,metric = 'jaccard')
print('\n the matrix of row distances')
print(D2)
Compute distances between the rows of two tables
B = np.random.randint(2, size = (3,3))
print(A)
print (B)
#computes the matrix of all pairwise distances of rows of A with rows of B
# returns an NxM matrix (N rows of A, M rows of B)
D3 = metrics.pairwise.pairwise_distances(A,B,metric = 'jaccard')
print('\n the matrix of distances between the rows of A and B')
print(D3)
sp_dist.cdist(A,B,'jaccard')
We can apply everything to sparce matrices
d = np.array([[0, 0, 12],
[0, 1, 1],
[0, 5, 34],
[1, 3, 12],
[1, 2, 6],
[2, 0, 23],
[3, 4, 14],
])
s = sp_sparse.csr_matrix((d[:,2],(d[:,0],d[:,1])), shape=(4,6))
D4 = metrics.pairwise.pairwise_distances(s,metric = 'euclidean')
print(s.toarray())
print(D4)
Clustering¶
You can read more about clustering in SciKit here:
Generate data from Gaussian distributions.
More on data generation here: http://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_blobs.html
centers = [[1,1], [-1, -1], [1, -1]]
X, true_labels = sk_data.make_blobs(n_samples=500, centers=3, n_features=2,
center_box=(-10.0, 10.0),random_state=0)
plt.scatter(X[:,0], X[:,1])
print(type(X))
print(true_labels)
print(len(true_labels[true_labels==0]),len(true_labels[true_labels==1]),len(true_labels[true_labels==2]))
plt.scatter(X[true_labels==1,0], X[true_labels==1,1],c = 'r')
plt.scatter(X[true_labels==2,0], X[true_labels==2,1],c = 'b')
plt.scatter(X[true_labels==0,0], X[true_labels==0,1],c = 'g')
Useful command: We will create a colormap of the distance matrix using the pcolormesh method of matplotlib.pyplot
euclidean_dists = metrics.euclidean_distances(X)
plt.pcolormesh(euclidean_dists,cmap=plt.cm.coolwarm)
Clustering Algorithms¶
scikit-learn has a huge set of tools for unsupervised learning generally, and clustering specifically. These are in sklearn.cluster. http://scikit-learn.org/stable/modules/clustering.html
There are 3 functions in all the clustering classes,
- fit(): builds the model from the training data (e.g. for kmeans, it finds the centroids)
- predict(): assigns labels to the data after building the model
- fit_predict(): does both at the same data (e.g in kmeans, it finds the centroids and assigns the labels to the dataset)
K-means clustering¶
More on the k-means clustering here: http://scikit-learn.org/stable/modules/generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans
Important parameters
init: determines the way the initialization is done. kmeans++ is the default.
n_init: number of iterations
Important attributes:
labels_ the labels for each point
cluster_centers_: the cluster centroids
inertia_: the SSE value
import sklearn.cluster as sk_cluster
kmeans = sk_cluster.KMeans(init='k-means++', n_clusters=3, n_init=10)
kmeans.fit_predict(X)
centroids = kmeans.cluster_centers_
kmeans_labels = kmeans.labels_
error = kmeans.inertia_
print ("The total error of the clustering is: ", error)
print ('\nCluster labels')
print(kmeans_labels)
print ('\n Cluster Centroids')
print (centroids)
Useful command: numpy.argsort sorts a set of values and returns the sorted indices
idx = np.argsort(kmeans_labels) # returns the indices in sorted order
rX = X[idx,:]
r_euclid = metrics.euclidean_distances(rX)
#r_euclid = euclidean_dists[idx,:][:,idx]
plt.pcolormesh(r_euclid,cmap=plt.cm.coolwarm)
Confusion matrix: http://scikit-learn.org/stable/modules/generated/sklearn.metrics.confusion_matrix.html
Important: In the produced confusion matrix, the first list defines the rows and the second the columns. The matrix is always square, regarless if the number of classes and clusters are not the same. The extra rows or columns are filled with zeros.
Homogeneity and completeness: http://scikit-learn.org/stable/modules/clustering.html#homogeneity-completeness
Homogeneity and completeness are computed using the conditional entropy of the labels given the cluster, and the conditional entropy of the cluster labels given the class label. The V-measure combines these in a similar way like F-measure
Silhouette score: http://scikit-learn.org/stable/modules/generated/sklearn.metrics.silhouette_score.html
C= metrics.confusion_matrix(kmeans_labels,true_labels)
print (C)
plt.pcolormesh(C,cmap=plt.cm.Reds)
Compute precision and recall.
These metrics are for classification, so they assume that row i is mapped to column i
p = metrics.precision_score(true_labels,kmeans_labels, average=None)
print(p)
r = metrics.recall_score(true_labels,kmeans_labels, average = None)
print(r)
Create a function that maps each cluster to the class that has the most points.
You need to be careful if many clusters map to the same class. It will not work in this case
Useful command: numpy.argmax returns the index of the max element
def cluster_class_mapping(kmeans_labels,true_labels):
C= metrics.confusion_matrix(kmeans_labels,true_labels)
mapping = list(np.argmax(C,axis=1)) #for each row (cluster) find the best class in the confusion matrix
mapped_kmeans_labels = [mapping[l] for l in kmeans_labels]
C2= metrics.confusion_matrix(mapped_kmeans_labels,true_labels)
return mapped_kmeans_labels,C2
mapped_kmeans_labels,C = cluster_class_mapping(kmeans_labels,true_labels)
print(C)
plt.pcolormesh(C, cmap=plt.cm.Reds)
Compute different metrics for clustering quality
h = metrics.homogeneity_score(true_labels,mapped_kmeans_labels)
print(h)
c = metrics.completeness_score(true_labels,mapped_kmeans_labels)
print(c)
v = metrics.v_measure_score(true_labels,mapped_kmeans_labels)
print(v)
p = metrics.precision_score(true_labels,mapped_kmeans_labels, average=None)
print(p)
r = metrics.recall_score(true_labels,mapped_kmeans_labels, average = None)
print(r)
f = metrics.f1_score(true_labels,mapped_kmeans_labels, average = None)
print(f)
p = metrics.precision_score(true_labels,mapped_kmeans_labels, average='weighted')
print(p)
r = metrics.recall_score(true_labels,mapped_kmeans_labels, average = 'weighted')
print(r)
f = metrics.f1_score(true_labels,mapped_kmeans_labels, average = 'weighted')
print(f)
The SSE plot
error = np.zeros(11)
sh_score = np.zeros(11)
for k in range(1,11):
kmeans = sk_cluster.KMeans(init='k-means++', n_clusters=k, n_init=10)
kmeans.fit_predict(X)
error[k] = kmeans.inertia_
if k>1: sh_score[k]= metrics.silhouette_score(X, kmeans.labels_)
plt.plot(range(1,len(error)),error[1:])
plt.xlabel('Number of clusters')
plt.ylabel('Error')
The silhouette plot
We see a peak at k = 3 and k = 6 indicating that these may be good values for the cluster number
plt.plot(range(2,len(sh_score)),sh_score[2:])
plt.xlabel('Number of clusters')
plt.ylabel('silhouette score')
colors = np.array([x for x in 'bgrcmykbgrcmykbgrcmykbgrcmyk'])
colors = np.hstack([colors] * 20)
plt.scatter(X[:, 0], X[:, 1], color=colors[kmeans_labels].tolist(), s=10, alpha=0.8)
Agglomerative Clustering¶
More on Agglomerative Clustering here: http://scikit-learn.org/stable/modules/generated/sklearn.cluster.AgglomerativeClustering.html
agglo = sk_cluster.AgglomerativeClustering(linkage = 'complete', n_clusters = 3)
agglo_labels = agglo.fit_predict(X)
C_agglo= metrics.confusion_matrix(agglo_labels,true_labels)
print (C_agglo)
#plt.pcolor(C_agglo,cmap=plt.cm.coolwarm)
plt.pcolormesh(C_agglo,cmap=plt.cm.Reds)
mapped_agglo_labels,C_agglo = cluster_class_mapping(agglo_labels,true_labels)
print(C_agglo)
p = metrics.precision_score(true_labels,mapped_agglo_labels, average='weighted')
print(p)
r = metrics.recall_score(true_labels,mapped_agglo_labels, average = 'weighted')
print(r)
Another way to do agglomerative clustering using SciPy:
https://docs.scipy.org/doc/scipy/reference/cluster.hierarchy.html
import scipy.cluster.hierarchy as hr
Z = hr.linkage(X, method='complete', metric='euclidean')
print (Z.shape, X.shape)
import scipy.spatial.distance as sp_dist
D = sp_dist.pdist(X, 'euclidean')
Z = hr.linkage(D, method='complete')
print (Z.shape, X.shape)
Hierarchical clustering returns a 4 by (n-1) matrix Z. At the i-th iteration, clusters with indices Z[i, 0] and Z[i, 1] are combined to form cluster n + i. A cluster with an index less than n corresponds to one of the n original observations. The distance between clusters Z[i, 0] and Z[i, 1] is given by Z[i, 2]. The fourth value Z[i, 3] represents the number of original observations in the newly formed cluster.
fig = plt.figure(figsize=(10,10))
T = hr.dendrogram(Z,color_threshold=0.4, leaf_font_size=4)
fig.show()
Another way to do agglomerative clustering (and visualizing it): http://seaborn.pydata.org/generated/seaborn.clustermap.html
distances = metrics.euclidean_distances(X)
cg = sns.clustermap(distances, method="complete", figsize=(13,13), xticklabels=False)
print (cg.dendrogram_col.reordered_ind)
DBSCAN Algorithm¶
More on DBSCAN here: http://scikit-learn.org/stable/modules/generated/sklearn.cluster.DBSCAN.html
dbscan = sk_cluster.DBSCAN(eps=0.3)
dbscan_labels = dbscan.fit_predict(X)
print(dbscan_labels) #label -1 corresponds to noise
renamed_dbscan_labels = [x+1 for x in dbscan_labels]
C = metrics.confusion_matrix(renamed_dbscan_labels,true_labels)
#print(C)
print (C[:,:max(true_labels)+1])
#colors = np.array([x for x in 'bgrcmykbgrcmykbgrcmykbgrcmyk'])
#colors = np.hstack([colors] * 20)
colors = np.array([x for x in 'bgrcmywk'*10])
plt.scatter(X[:, 0], X[:, 1], color=colors[dbscan_labels].tolist(), s=10, alpha=0.8)
Processing Complex Data¶
So far we have assumed that the intput is in the form of numerical vectors to which we can apply directly the algorithms we have. Often the data will be more complex. For example what if we want to cluster categorical data, itemsets, or text? Python provides libraries for processing the data and transforming them to a format that we can use.
Python offers a set of tools for extracting features:http://scikit-learn.org/stable/modules/feature_extraction.html
DictVectorizer¶
The DictVectorizer feature extraction: http://scikit-learn.org/stable/modules/generated/sklearn.feature_extraction.DictVectorizer.html#sklearn.feature_extraction.DictVectorizer
The DictVectorizer takes a dictionary of attribute-value pairs and transforms them into numerical vectors. Real values are preserved, while categorical attributes are transformed into binary. The vectorizer produces a sparse representation.
from sklearn.feature_extraction import DictVectorizer
measurements = [
{'city': 'Dubai', 'temperature': 45},
{'city': 'London', 'temperature': 12},
{'city': 'San Fransisco', 'temperature': 23},
]
vec = DictVectorizer()
print(type(vec.fit_transform(measurements)))
print(vec.fit_transform(measurements).toarray())
vec.get_feature_names()
from sklearn.feature_extraction import DictVectorizer
measurements = [
{'city': 'Dubai', 'temperature': 45, 'dummy': 3},
{'city': 'London', 'temperature': 12},
{'city': 'San Fransisco', 'temperature': 23},
]
vec = DictVectorizer()
vec.fit(measurements)
print(vec.get_feature_names())
print(vec.transform(measurements).toarray())
x = {'city': 'Athens', 'temperature': 32, 'dummy2': 2}
print(vec.transform(x).toarray())
measurements = [
{'refund' : 'No','marital_status': 'married', 'income' : 100},
{'refund' : 'Yes','marital_status': 'single', 'income' : 120},
{'refund' : 'No','marital_status':'divorced', 'income' : 80},
]
vec = DictVectorizer()
print(vec.fit_transform(measurements))
vec.get_feature_names()
Text processing¶
Feature extraction from text: http://scikit-learn.org/stable/modules/classes.html#text-feature-extraction-ref
CountVectorizer¶
The CountVectorizer can be used to extract features in the form of bag of words. It is typically used for text, but you could use it to represent also a collection of itemsets (where each itemset will become a word).
import sklearn.feature_extraction.text as sk_text
corpus = ['This is the first document.',
'this is the second second document.',
'And the third one.',
'Is this the first document?',
]
vectorizer = sk_text.CountVectorizer(min_df=1)
X = vectorizer.fit_transform(corpus)
print(X.toarray())
vectorizer.get_feature_names()
vectorizer = sk_text.CountVectorizer(min_df=1,stop_words = 'english')
X2 = vectorizer.fit_transform(corpus)
print(X2.toarray())
vectorizer.get_feature_names()
TfIdfVectorizer¶
TfIdfVectorizer transforms text into a sparse matrix where rows are text and columns are words, and values are the tf-dif values. It performs tokenization, normalization, and removes stop-words. More here: http://scikit-learn.org/stable/modules/generated/sklearn.feature_extraction.text.TfidfVectorizer.html#sklearn.feature_extraction.text.TfidfVectorizer
vectorizer = sk_text.TfidfVectorizer(min_df=1)
X = vectorizer.fit_transform(corpus)
print(X.toarray())
print (vectorizer.get_feature_names())
Removing stop-words
vectorizer = sk_text.TfidfVectorizer(stop_words = 'english',min_df=1)
X = vectorizer.fit_transform(corpus)
print(X.toarray())
print (vectorizer.get_feature_names())
SciKit datasets: http://scikit-learn.org/stable/datasets/
We will use the 20-newsgroups datasets which consists of postings on 20 different newsgroups.
More information here: http://scikit-learn.org/stable/datasets/#the-20-newsgroups-text-dataset
from sklearn.datasets import fetch_20newsgroups
categories = ['comp.os.ms-windows.misc', 'sci.space','rec.sport.baseball']
#categories = ['alt.atheism', 'sci.space','rec.sport.baseball']
news_data = sk_data.fetch_20newsgroups(subset='train',
remove=('headers', 'footers', 'quotes'),
categories=categories)
print (news_data.target)
print (len(news_data.target))
print (type(news_data))
print (news_data.filenames)
print (news_data.target[:10])
print (news_data.data[1])
print (len(news_data.data))
vectorizer = sk_text.TfidfVectorizer(stop_words='english',
#max_features = 1000,
min_df=4, max_df=0.8)
data = vectorizer.fit_transform(news_data.data)
print(type(data))
Clustering text data¶
An example of what we want to do: http://scikit-learn.org/stable/auto_examples/text/document_clustering.html
import sklearn.cluster as sk_cluster
k=3
kmeans = sk_cluster.KMeans(n_clusters=k, init='k-means++', max_iter=100, n_init=1)
kmeans.fit_predict(data)
To understand the clusters we can print the words that have the highest values in the centroid
print("Top terms per cluster:")
asc_order_centroids = kmeans.cluster_centers_.argsort()#[:, ::-1]
order_centroids = asc_order_centroids[:,::-1]
terms = vectorizer.get_feature_names()
for i in range(k):
print ("Cluster %d:" % i)
for ind in order_centroids[i, :10]:
print (' %s' % terms[ind])
print
C = metrics.confusion_matrix(kmeans.labels_,news_data.target)
mapped_kmeans_labels,C = cluster_class_mapping(kmeans.labels_,news_data.target)
print (C)
p = metrics.precision_score(news_data.target,mapped_kmeans_labels, average=None)
print(p)
r = metrics.recall_score(news_data.target,mapped_kmeans_labels, average = None)
print(r)
agglo = sk_cluster.AgglomerativeClustering(linkage = 'complete', n_clusters = 3,)
dense = data.todense()
agglo_labels = agglo.fit_predict(dense) # agglomerative needs dense data
C_agglo= metrics.confusion_matrix(agglo_labels,news_data.target)
print (C_agglo)
dbscan = sk_cluster.DBSCAN(eps=0.1)
dbscan_labels = dbscan.fit_predict(data)
C = metrics.confusion_matrix(dbscan.labels_,news_data.target)
print (C)
Feature normalization¶
Python provides some functionality for normalizing and standardizing the data. Be careful though, some operations work only with dense data.
http://scikit-learn.org/stable/modules/preprocessing.html#preprocessing
Use the function preprocessing.scale to normalize by removing the mean and dividing by the standard deviation. This is done per feature, that is, per column of the dataset.
from sklearn import preprocessing
X = np.array([[ 1., -1., 2.],
[ 2., 0., 1.],
[ 0., 1., -1.]])
print("column means: ",X.mean(axis = 0))
print("column std: ",X.std(axis = 0))
X_scaled = preprocessing.scale(X)
print("after feature normalization")
print(X_scaled)
print("normalized column means: ",X_scaled.mean(axis=0))
print("normalized column std: ",X_scaled.var(axis = 0))
Feature normalization will not work with sparse data. In this case, the zeros are treated as values, so the sparse matrix will become non-sparse after normalization.
import scipy.sparse
cX = scipy.sparse.csc_matrix(X)
cX_scaled = preprocessing.scale(cX)
print(cX_scaled)
The same can be done with the StandardScaler from the preprocessing library of sklearn.
The function fit() computes the parameters for scaling, and transform() applies the scaling
from sklearn import preprocessing
std_scaler = preprocessing.StandardScaler()
std_scaler.fit(X)
print(std_scaler.mean_)
print(std_scaler.scale_)
X_std = std_scaler.transform(X)
print("scaled data:")
print(X_std)
The advantage is the we can now apply the transform to new data.
For example, we compute the parameters for the training data and we apply the scaling to the test data.
y = np.array([[2.,3.,1.],
[1.,2.,1.]])
print(std_scaler.transform(y))
The MinMaxScaler subbtracts from each column the minimum and then divides by the max-min.
min_max_scaler = preprocessing.MinMaxScaler()
X_minmax = min_max_scaler.fit_transform(X)
print(X_minmax)
print(min_max_scaler.transform(y))
The MaxAbsScaler divides with the maximum absolute value.
The MaxAbsScaler can work with sparse data, since it does not destroy the data sparseness. For the other datasets, removing the mean (or min) can destroy the sparseness of the data.
Sometimes we may choose to normalize only the non-zero values. This should be done manually.
max_abs_scaler = preprocessing.MaxAbsScaler()
X_maxabs = max_abs_scaler.fit_transform(X)
X_maxabs
# works with sparse data
cX_scaled = max_abs_scaler.transform(cX)
print(cX_scaled)
The normalize function normalizes the rows so that they become unit vectors in some norm that we specify. It can be applied to sparse matrices without destroying the sparsity.
#works with sparse data
X_normalized = preprocessing.normalize(X, norm='l2')
X_normalized
crX = scipy.sparse.csr_matrix(X)
crX_scaled = preprocessing.normalize(crX,norm='l1')
print(crX_scaled)
OneHotEncoder¶
The OneHotEncoder can be used for categorical data to transform them into binary, where for each attribute value we have 0 or 1 depending on whether this value appears in the feature vector. It works with numerical categorical values.
X = [[0,1,2],
[1,2,3],
[0,1,4]]
enc = preprocessing.OneHotEncoder(handle_unknown='ignore')
enc.fit(X)
enc.transform([[0,2,4],[1,1,2]]).toarray()
In this example every number in every column defines a separate feature
enc.categories_
We can also apply it selectively to some columns of the data
#works with sparse data
X = np.array([[0, 10, 45100],
[1, 20, 45221],
[0, 20, 45212]])
enc = preprocessing.OneHotEncoder(categorical_features=[2]) #only the third column is categorical
enc.fit(X)
enc.transform([[5,13,45212],[4,12,45221]]).toarray()