K-Nearest Neighbour Classifier

The Nearest Neighbour Classifier is one of the most straightforward classifier in the arsenal of machine learning techniques. It performs the classification by identifying the nearest neighbours to a query pattern and using those neighbors to determine the label of the query. The idea behind the algorithm is simple: Assign the query pattern to the class which occurs the most in the k nearest neighbors. In this post we'll use the function knn_search(...) that we have seen in the last post to implement a K-Nearest Neighbour Classifier. The implementation of the classifier is as follows:

from numpy import random,argsort,argmax,bincount,int_,array,vstack,round
from pylab import scatter,show

def knn_classifier(x, D, labels, K):
 """ Classify the vector x
     D - data matrix (each row is a pattern).
     labels - class of each pattern.
     K - number of neighbour to use.
     Returns the class label and the neighbors indexes.
 """
 neig_idx = knn_search(x,D,K)
 counts = bincount(labels[neig_idx]) # voting
 return argmax(counts),neig_idx

Let's test the classifier on some random data:

 # generating a random dataset with random labels
data = random.rand(2,150) # random points
labels = int_(round(random.rand(150)*1)) # random labels 0 or 1
x = random.rand(2,1) # random test point

# label assignment using k=5
result,neig_idx = knn_classifier(x,data,labels,5)
print 'Label assignment:', result

# plotting the data and the input pattern
# class 1, red points, class 0 blue points
scatter(data[0,:],data[1,:], c=labels,alpha=0.8)
scatter(x[0],x[1],marker='o',c='g',s=40)
# highlighting the neighbours
plot(data[0,neig_idx],data[1,neig_idx],'o',
  markerfacecolor='None',markersize=15,markeredgewidth=1)
show()

The script will show the following graph:

The query vector is represented with a green point and we can see that the 3 out of 5 nearest neighbors are red points (label 1) while the remaining 2 are blue (label 2).
The result of the classification will be printed on the console:

Label assignment: 1

As we expected, the green point have been assigned to the class with red markers.

k-nearest neighbor search

A k-nearest neighbor search identifies the top k nearest neighbors to a query. The problem is: given a dataset D of vectors in a d-dimensional space and a query point x in the same space, find the closest point in D to x. The following function performs a k-nearest neighbor search using the euclidean distance:

from numpy import random,argsort,sqrt
from pylab import plot,show

def knn_search(x, D, K):
 """ find K nearest neighbours of data among D """
 ndata = D.shape[1]
 K = K if K < ndata else ndata
 # euclidean distances from the other points
 sqd = sqrt(((D - x[:,:ndata])**2).sum(axis=0))
 idx = argsort(sqd) # sorting
 # return the indexes of K nearest neighbours
 return idx[:K]

The function computes the euclidean distance between every point of D and x then returns the indexes of the points for which the distance is smaller.
Now, we will test this function on a random bidimensional dataset:

# knn_search test
data = random.rand(2,200) # random dataset
x = random.rand(2,1) # query point

# performing the search
neig_idx = knn_search(x,data,10)

# plotting the data and the input point
plot(data[0,:],data[1,:],'ob',x[0,0],x[1,0],'or')
# highlighting the neighbours
plot(data[0,neig_idx],data[1,neig_idx],'o',
  markerfacecolor='None',markersize=15,markeredgewidth=1)
show()

The result is as follows:

The red point is the query vector and the blue ones represent the data. The blue points surrounded by a black circle are the nearest neighbors.

K- means clustering with scipy

K-means clustering is a method for finding clusters and cluster centers in a set of unlabeled data. Intuitively, we might think of a cluster as comprising a group of data points whose inter-point distances are small compared with the distances to points outside of the cluster. Given an initial set of K centers, the K-means algorithm alternates the two steps:

• for each center we identify the subset of training points (its cluster) that is closer to it than any other center;
• the means of each feature for the data points in each cluster are computed, and this mean vector becomes the new center for that cluster.

These two steps are iterated until the centers no longer move or the assignments no longer change. Then, a new point x can be assigned to the cluster of the closest prototype.
The Scipy library provides a good implementation of the K-Means algorithm. Let's see how to use it:

from pylab import plot,show
from numpy import vstack,array
from numpy.random import rand
from scipy.cluster.vq import kmeans,vq

# data generation
data = vstack((rand(150,2) + array([.5,.5]),rand(150,2)))

# computing K-Means with K = 2 (2 clusters)
centroids,_ = kmeans(data,2)
# assign each sample to a cluster
idx,_ = vq(data,centroids)

# some plotting using numpy's logical indexing
plot(data[idx==0,0],data[idx==0,1],'ob',
     data[idx==1,0],data[idx==1,1],'or')
plot(centroids[:,0],centroids[:,1],'sg',markersize=8)
show()

The result should be as follows:

In this case we splitted the data in 2 clusters, the blue points have been assigned to the first and the red ones to the second. The squares are the centers of the clusters.
Let's see try to split the data in 3 clusters:

# now with K = 3 (3 clusters)
centroids,_ = kmeans(data,3)
idx,_ = vq(data,centroids)

plot(data[idx==0,0],data[idx==0,1],'ob',
     data[idx==1,0],data[idx==1,1],'or',
     data[idx==2,0],data[idx==2,1],'og') # third cluster points
plot(centroids[:,0],centroids[:,1],'sm',markersize=8)
show()

This time the the result is as follows: