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Import necessary libraries

k-means clustering is a method of vector quantization, originally from signal processing, that aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean (cluster centers or cluster centroid), serving as a prototype of the cluster.

This results in a partitioning of the data space into Voronoi cells. k-means clustering minimizes within-cluster variances (squared Euclidean distances), but not regular Euclidean distances, which would be the more difficult Weber problem: the mean optimizes squared errors, whereas only the geometric median minimizes Euclidean distances. For instance, better Euclidean solutions can be found using k-medians and k-medoids.

import numpy as np

from scipy import ndimage

from time import time

from sklearn import datasets, manifold

from sklearn .cluster import KMeans, AgglomerativeClustering

from sklearn.mixture import GMM

from sklearn.cross_validation import StratifiedKFold

import matplotlib.pyplot as plt

import matplotlib as mpl

%matplotlib inline

k-Means Algorithm: Expectation–Maximization

Expectation–maximization (E–M) is a powerful algorithm that comes up in a variety of contexts within data science. k-means is a particularly simple and easy-to-understand application of the algorithm, and we will walk through it briefly here. In short, the expectation–maximization approach here consists of the following procedure:

Guess some cluster centers
Repeat until converged
E-Step: assign points to the nearest cluster center
M-Step: set the cluster centers to the mean

Here the "E-step" or "Expectation step" is so-named because it involves updating our expectation of which cluster each point belongs to. The "M-step" or "Maximization step" is so-named because it involves maximizing some fitness function that defines the location of the cluster centers—in this case, that maximization is accomplished by taking a simple mean of the data in each cluster.

The literature about this algorithm is vast, but can be summarized as follows: under typical circumstances, each repetition of the E-step and M-step will always result in a better estimate of the cluster characteristics.

We can visualize the algorithm as shown in the following figure. For the particular initialization shown here, the clusters converge in just three iterations. For an interactive version of this figure, refer to the code in the Appendix.

The k-Means algorithm is simple enough that we can write it in a few lines of code. The following is a very basic implementation:

Example 1: k-means on digits

To start, let's take a look at applying k-means on the same simple digits data that we saw in In-Depth: Decision Trees and Random Forests and In Depth: Principal Component Analysis. Here we will attempt to use k-means to try to identify similar digits without using the original label information; this might be similar to a first step in extracting meaning from a new dataset about which you don't have any a priori label information.

We will start by loading the digits and then finding the KMeans clusters. Recall that the digits consist of 1,797 samples with 64 features, where each of the 64 features is the brightness of one pixel in an 8×8 image: