k-means Partition Clustering

The simplest method of cluster analysis is partitioning which organizes the objects of a set into several exclusive groups. The partitioning algorithm organizes the objects into k partitions where each partition represents a cluster.

The well-known and commonly used partitioning methods are k-means and k-medoids. An objective function is used to assess the partitioning quality so that objects within a cluster are similar to one another but dissimilar to objects in other clusters. A centroid-based partitioning technique uses the centroid of a cluster, C_i , to represent that cluster. Conceptually, the centroid of a cluster is its center point.

The quality of cluster C_i can be measured by the within cluster variation which is the sum of squared error between all objects in C_i and the centroid c_i, defined as

where

E is the sum of the squared error for all objects in the data set

p is the point in space representing a given object

c_i is the centroid of cluster C_i

Optimizing the within-cluster variation is computationally challenging. The problem is NP-hard (non-deterministic polynomial-time hardness) for a general number of clusters k even in the 2-D Euclidean space.

If the number of clusters k and the dimensionality of the space d are fixed, the problem can be solved in time O(n^dk+1 log n), where n is the number of objects.

The k-means algorithm defines the centroid of a cluster as the mean value of the points within the cluster. The process of iteratively reassigning objects to clusters to improve the partitioning is referred to as iterative relocation.

The time complexity of the k-means algorithm is O(nkt)

where n is the total number of objects

k is the number of clusters.

t is the number of iterations.

The k-means method can be applied only when the mean of a set of objects is defined.

The k-modes method is a variant of k-means, which extends the k-means paradigm to cluster nominal data by replacing the means of clusters with modes.

Gaussian Elimination - Row reduction Algorithm

Gaussian elimination is a method for solving matrix equations of the form, Ax=b. This method is also known as the row reduction algorithm. Back Substitution Solving the last equation for the variable and then work backward into the first equation to solve it. The fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Pivot row The row that is used to perform elimination of a variable from other rows is called the pivot row. Example: Solving a linear equation The augmented matrix for the above equation shall be The equation shall be solved using back substitution. The eliminating the first variable (x1) in the first row (Pivot row) by carrying out the row operation. As the second row become zero, the row will be shifted to bottom by carrying out partial pivoting. Now, the second variable (x2) shall be eliminated by carrying out the row operation again. ...

Decision Tree - Gini Index

The Gini index is used in CART. The Gini index measures the impurity of the data set, where p i - probability that data in the data set, D belong to class, C i and pi = |C i,D |/|D| There are 2 v - 2 possible ways to form two partitions of the data set, D based on a binary split on a attribute. Each of the possible binary splits of the attribute is considered. The subset that gives the minimum Gini index is selected as the splitting subset for discrete valued attribute. The degree of Gini index varies between 0 and 1. The value 0 denotes that all elements belong to a certain class or if there exists only one class, and the value 1 denotes that the elements are randomly distributed across various classes. A Gini Index of 0.5 denotes equally distributed elements into some classes. The Gini index is biased toward multivalued attributes and has difficulty when the number of classes is large.

Clustering Analysis

Clustering is the process of grouping a set of data objects into multiple groups so that objects within a group have high similarity, but are very dissimilar to objects in other groups. The clustering would help to discover previously unknown groups within the data. The cluster analysis can be used to gain insight into the distribution of data, to observe the characteristics of each cluster, and to focus on a particular set of clusters for further analysis. The main focus has been given to distance-based cluster analysis . The Cluster analysis tools based on k-means, k-medoids, and several other methods also have been built into many statistical analysis software packages. The clustering is a form of learning by observation, rather than learning by examples. Requirement of Clustering Scalability Ability to deal various types of attributes Discovery of clusters with arbitrary shape Requirement for domain knowledge Ability to deal with noisy data Incremental clustering and insensitivity ...

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