K-Means Cluster Analysis

Chapter 3 PPDM Cl Class

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Introduction to Data Mining

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What is Cluster Analysis?
Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups
Intra-cluster distances are minimized Inter cluster Inter-cluster distances are maximized

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Applications of Cluster Analysis
Understanding
– Group related documents p for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations
Discovered Clusters Industry Group

1 2 3 4

Applied-Matl-DOWN,Bay-Network-Down,3-COM-DOWN, Cabletron-Sys-DOWN,CISCO-DOWN,HP-DOWN, DSC-Comm-DOWN,INTEL-DOWN,LSI-Logic-DOWN, Micron-Tech-DOWN,Texas-Inst-Down,Tellabs-Inc-Down, Natl-Semiconduct-DOWN,Oracl-DOWN,SGI-DOWN, Sun-DOWN Apple-Comp-DOWN,Autodesk-DOWN,DEC-DOWN, ADV-Micro-Device-DOWN,Andrew-Corp-DOWN, Computer-Assoc-DOWN,Circuit-City-DOWN, Compaq-DOWN, EMC-Corp-DOWN, Gen-Inst-DOWN, Motorola-DOWN,Microsoft-DOWN,Scientific-Atl-DOWN Fannie-Mae-DOWN,Fed-Home-Loan-DOWN, Fannie Mae DOWN Fed Home Loan DOWN MBNA-Corp-DOWN,Morgan-Stanley-DOWN Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP, Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP, Schlumberger-UP

Technology1-DOWN

Technology2-DOWN

Financial-DOWN Oil-UP

Summarization
– Reduce the size of large data sets
C uste g precipitation Clustering p ec p tat o in Australia
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What is not Cluster Analysis?
Supervised classification
– Have class label information

Simple segmentation
– Di idi students into different registration groups Dividing t d t i t diff t i t ti alphabetically, by last name

Results of a query
– Groupings are a result of an external specification

Graph partitioning
– Some mutual relevance and synergy, but areas are not identical
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Notion of a Cluster can be Ambiguous

How many clusters?

Six Clusters

Two Clusters

Four Clusters

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Types of Clusterings
A clustering is a set of clusters Important distinction between hierarchical and partitional sets of clusters Partitional Clustering
– A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset

Hierarchical clustering
– A set of nested clusters organized as a hierarchical tree

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Partitional Clustering

Original Points g

A Partitional Clustering g

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Hierarchical Clustering

p1 p3 p2 p4

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Traditional Hierarchical Clustering

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Traditional Dendrogram

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Non-traditional Hierarchical Clustering
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Non-traditional Dendrogram
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Introduction to Data Mining

Other Distinctions Between Sets of Clusters Exclusive versus non-exclusive
– In non-exclusive clusterings, points may belong to multiple clusters. clusters – Can represent multiple classes or ‘border’ points

Fuzzy versus non-fuzzy y y
– In fuzzy clustering, a point belongs to every cluster with some weight between 0 and 1 – Weights must sum to 1 – Probabilistic clustering has similar characteristics

Partial versus complete
– I some cases, we only want to cluster some of the data In l tt l t f th d t

Heterogeneous versus homogeneous
– Cluster of widely different sizes, shapes, and densities
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Types of Clusters
Well-separated clusters Center-based clusters Contiguous clusters Density-based clusters Property or Conceptual P t C t l Described by an Objective Function
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Types of Clusters: Well-Separated Well-Separated Clusters:
– A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster.

3 well-separated clusters
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Types of Clusters: Center-Based Center-based
– A cluster is a set of objects such that an object in a cluster is closer (more similar) to the “center” of a cluster, than to the center of any other cluster – The center of a cluster is often a centroid, the average of all the i t in the l t th points i th cluster, or a medoid, th most “representative” d id the t“ t ti ” point of a cluster

4 center-based clusters
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Types of Clusters: Contiguity-Based Contiguous Cluster (Nearest neighbor or Transitive)
– A cluster is a set of points such that a point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster.

8 contiguous clusters
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Types of Clusters: Density-Based Density-based
– A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density. – Used when the clusters are irregular or intertwined, and when noise and outliers are present.

6 density-based clusters
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Types of Clusters: Conceptual Clusters Shared Property or Conceptual Clusters
– Finds clusters that share some common property or represent a particular concept. .

2 Overlapping Circles
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Types of Clusters: Objective Function Clusters Defined by an Objective Function
– Finds clusters that minimize or maximize an objective function. – Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function (NP Hard) function. – Can have global or local objectives.
Hierarchical clustering algorithms typically have local objectives Partitional algorithms typically h P titi l l ith t i ll have global objectives l b l bj ti

– A variation of the global objective function approach is to fit the data to a parameterized model.
Parameters for the model are determined from the data. Mixture models assume that the data is a ‘mixture' of a number of statistical distributions.

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Types of Clusters: Objective Function … Map the clustering problem to a different domain and solve a related problem in that domain
– Proximity matrix defines a weighted graph, where the nodes are the points being clustered, and the weighted edges represent the proximities between points – Clustering is equivalent to breaking the graph into connected components, one for each cluster. – Want to minimize the edge weight between clusters and maximize the edge weight within clusters
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Characteristics of the Input Data Are Important
Type of proximity or density measure
– This is a derived measure, but central to clustering

Sparseness
– Dictates type of similarity – Adds to efficiency

Attribute type
– Dictates type of similarity

Type of Data
– Dictates type of similarity – Other characteristics, e.g., autocorrelation

Dimensionality Di i lit Noise and Outliers Type of Distribution
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Clustering Algorithms
K-means and its variants Hierarchical clustering Density-based clustering

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K-means Clustering
Partitional clustering approach Each cluster is associated with a centroid (center point) Each point is assigned to the cluster with the closest centroid Number of clusters, K must be specified clusters K, The basic algorithm is very simple

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K-means Clustering – Details
Initial centroids are often chosen randomly.
– Clusters produced vary from one run to another.

The centroid is (typically) the mean of the points in the cluster. ‘Closeness’ is measured by Euclidean distance, cosine similarity, correlation, etc. similarity correlation etc K-means will converge for common similarity measures mentioned above. Most of the convergence happens in the first few iterations.
– Often the stopping condition is changed to ‘Until relatively few pp g g y points change clusters’ n = number of points, K = number of clusters, I = number of iterations, d = number of attributes
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Complexity is O( n * K * I * d )
–

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Two different K-means Clusterings
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Optimal Clustering
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Sub-optimal Clustering
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Importance of Choosing Initial Centroids
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Importance of Choosing Initial Centroids
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Evaluating K-means Clusters
Most common measure is Sum of Squared Error (SSE)
– For each point, the error is the distance to the nearest cluster – To get SSE, we square these errors and sum them.

SSE = ∑ ∑ dist 2 ( mi , x ) i =1 x∈Ci

K

– x is a data point in cluster Ci and mi is the representative point for cluster Ci can show that mi corresponds to the center (mean) of the cluster

– Given two clusters, we can choose the one with the smallest error – One easy way to reduce SSE is to increase K, the number of clusters
A good clustering with smaller K can have a lower SSE than a poor clustering with hi h K l t i ith higher
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Importance of Choosing Initial Centroids …
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Importance of Choosing Initial Centroids …
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Problems with Selecting Initial Points
If there are K ‘real’ clusters then the chance of selecting one centroid from each cluster is small.
– – Chance is relatively small when K i l Ch i l ti l ll h is large If clusters are the same size, n, then

– – –

For example, if K = 10, then probability = 10!/1010 = 0.00036 Sometimes the initial centroids will readjust themselves in ‘right’ ‘ i ht’ way, and sometimes they d ’t d ti th don’t Consider an example of five pairs of clusters

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10 Clusters Example
Iteration 1 4 3 2
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x Starting with two initial centroids in one cluster of each pair of clusters
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10 Clusters Example
Iteration 1
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Starting with two initial centroids in one cluster of each pair of clusters
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10 Clusters Example
Iteration 1 4 3 2
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Starting with some pairs of clusters having three initial centroids, while other have only one.
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10 Clusters Example
Iteration 1
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Starting with some pairs of clusters having three initial centroids, while other have only one.
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Solutions to Initial Centroids Problem
Multiple runs
– Helps, but probability is not on y p , p y your side

Sample and use hierarchical clustering to determine initial centroids Select more than k initial centroids and then select among these initial centroids
– Select most widely separated

Postprocessing Bisecting K-means
– Not as susceptible to initialization issues

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Handling Empty Clusters
Basic K-means algorithm can yield empty clusters Several strategies
– Choose the point that contributes most to SSE – Choose a point from the cluster with the highest SSE – If there are several empty clusters, the above can be repeated several times.

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Updating Centers Incrementally
In the basic K-means algorithm, centroids are updated after all points are assigned to a centroid An alternative is to update the centroids after each assignment (incremental approach)
– – – – – Each assignment updates zero or two centroids More expensive Introduces an order dependency Never get an empty cluster N t t l t Can use “weights” to change the impact

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Pre-processing and Post-processing
Pre-processing
– Normalize the data – Eliminate outliers

Post-processing Post processing
– Eliminate small clusters that may represent outliers – Split ‘loose’ clusters i e clusters with relatively high loose clusters, i.e., SSE – Merge clusters that are ‘close’ and that have relatively g y low SSE – Can use these steps during the clustering process
ISODATA SO
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Bisecting K-means Bisecting K-means algorithm
– Variant of K-means that can produce a partitional or a hierarchical clustering

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Bisecting K-means Example

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Limitations of K-means
K-means has problems when clusters are of differing
– Sizes – Densities – Non-globular shapes

K-means has problems when the data contains outliers.

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Limitations of K-means: Differing Sizes

Original Points

K-means (3 Clusters)

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Limitations of K-means: Differing Density

Original Points

K-means (3 Clusters)

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Limitations of K-means: Non-globular Shapes

Original Points

K-means (2 Clusters)

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Overcoming K-means Limitations

Original Points

K-means Clusters

One solution is to use many clusters. y Find parts of clusters, but need to put together.
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Overcoming K-means Limitations

Original Points

K-means Clusters

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Overcoming K-means Limitations

Original Points

K-means Clusters

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