Cluster AnalysisCS240B Lecture notes

based on those by

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004

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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

Inter-cluster distances are maximized

Intra-cluster distances are

minimized

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Cluster Analysis: many & diverse applications

Understanding– Group related

documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations

Summarization– Reduce the size of large

data sets

Discovered Clusters Industry Group

1 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

Technology1-DOWN

2 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

Technology2-DOWN

3 Fannie-Mae-DOWN,Fed-Home-Loan-DOWN, MBNA-Corp-DOWN,Morgan-Stanley-DOWN

Financial-DOWN

4 Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP,

Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP, Schlumberger-UP

Oil-UP

Clustering precipitation in Australia

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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 A Partitional Clustering

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Hierarchical Clustering

p4p1

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Traditional Hierarchical Clustering

Non-traditional Hierarchical Clustering Non-traditional Dendrogram

Traditional Dendrogram

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Other Distinctions Between Sets of Clusters

Exclusive versus non-exclusive– In non-exclusive clusterings, points may belong to

multiple clusters.– Can represent multiple classes or ‘border’ points

Fuzzy versus non-fuzzy– 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– In some cases, we only want to cluster some of the

data

Heterogeneous versus homogeneous– Cluster of widely different sizes, shapes, and densities

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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 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. 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 points change clusters’

Complexity is O( n * K * I * d )– n = number of points, K = number of clusters,

I = number of iterations, d = number of attributes

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Two different K-means Clusterings

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K-means Clusterings-example

Initial centroids: Case 1

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids

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K-means Clusterings: Initial centers

Initial centers: Case 1

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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 2 ‘real’ clusters then any two centroids produces an optimal clustering—i.e., each point will become the centroid of cluster. But if we know that there are K>2 `real’ clusters the chance of selecting one centroid for each becomes small as k grows.

– For example, if K = 10, then probability = 10!/1010 = 0.00036

– Sometimes the initial centroids will readjust themselves in ‘right’ way, and sometimes they don’t

– An obvious limitation of random initial assignements (also we normally do not know K)

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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.

– x is a data point in cluster Ci and mi is the representative point for cluster Ci

– One easy way to reduce SSE is to increase K, the number of clusters

– But in general, the fewer the clusters the better. A good clustering with smaller K can have a lower SSE than a poor clustering with higher KHow do we minimize both K and SSE? For a given K we can try different sets of initial random points.

K

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10 Clusters Example (5 pairs)

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Starting with two initial centroids in one cluster of each pair of clusters

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10 Clusters Example

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Starting with two initial centroids in one cluster of each pair of clusters

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10 Clusters Example

Starting with some pairs of clusters having three initial centroids, while other have only one.

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10 Clusters Example

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Solutions to Initial Centroids Problem

Multiple runs– Helps, but probability is not on your side

Start with more than k initial centroids and then select k centroids from the most widely separated resulting clusters

Use hierarchical clustering to determine initial centroids on a small sample of data

Bisecting K-means– Not as susceptible to initialization issues

Postprocessing

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Improvements

Eliminate outliers (preprocessing) and very small clusters that may represent outliers

Outliers are defined as points that have large distances from every centroids (they are bad because they exercise an excessive—quadratic effect)

On the other hand many outliers in the same vicinity, could indicate the need to generate a new cluster

Split ‘loose’ clusters, i.e., clusters with relatively high SSE

Merge clusters that are ‘close’ and that have relatively low SSE

– Some clusters could turn out empty.

Can use these steps during the clustering process

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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.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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Hierarchical Clustering

Produces a set of nested clusters organized as a hierarchical tree

Can be visualized as a dendrogram– A tree like diagram that records the sequences

of merges or splits

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Strengths of Hierarchical Clustering

Do not have to assume any particular number of clusters– Any desired number of clusters can be

obtained by ‘cutting’ the dendogram at the proper level

They may correspond to meaningful taxonomies– Example in biological sciences (e.g., animal

kingdom, phylogeny reconstruction, …)

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Hierarchical Clustering

Two main types of hierarchical clustering– Agglomerative:

Start with the points as individual clusters At each step, merge the closest pair of clusters until only one cluster (or k clusters) left

– Divisive: Start with one, all-inclusive cluster At each step, split a cluster until each cluster contains a point (or there are k clusters)

Traditional hierarchical algorithms use a similarity or distance matrix

– Merge or split one cluster at a time

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Agglomerative Clustering Algorithm

More popular hierarchical clustering technique

Basic algorithm is straightforward1. Compute the proximity matrix

2. Let each data point be a cluster

3. Repeat

4. Merge the two closest clusters

5. Update the proximity matrix

6. Until only a single cluster remains

Key operation is the computation of the proximity of two clusters

– Different approaches to defining the distance between clusters distinguish the different algorithms

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How to Define Inter-Cluster Similarity

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MIN MAX Group Average Distance Between Centroids Objective function

– Ward’s Method

Proximity Matrix

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Cluster Similarity: Ward’s Method

Similarity of two clusters is based on the increase in squared error when two clusters are merged– Similar to group average if distance between

points is distance squared Less susceptible to noise and outliers Biased towards globular clusters Hierarchical analogue of K-means

– Can be used to initialize K-means

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Recent Hierarchical Clustering Methods

Major weakness of agglomerative clustering methods

– do not scale well: time complexity of at least O(n2), where n is the number of total objects

– can never undo what was done previously Integration of hierarchical with distance-based clustering

– BIRCH (1996): uses CF-tree and incrementally adjusts the quality of sub-clusters

– ROCK (1999): clustering categorical data by neighbor and link analysis

– CHAMELEON (1999): hierarchical clustering using dynamic modeling

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Cluster Validity

For supervised classification we have a variety of measures to evaluate how good our model is– Accuracy, precision, recall

For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters?

But “clusters are in the eye of the beholder”!

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Order the similarity matrix with respect to cluster labels and inspect visually.

Using Similarity Matrix for Cluster Validation

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Using Similarity Matrix for Cluster Validation

Clusters in random data are not so crisp

K-means

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K-means on data streams

Streams is partitioned into windows Initially: for each new point P a new centroid is

created with probability d/D where D is proportional to size of the domain, and d is the actual distance of P to the closest centroid [Callaghan ’02]

Later windows can use the centroids of the previous window---improved by

– Eliminating empty clusters– Splitting large sparse ones– Merging large clusters

Large changes in cluster position/population denote concept changes.

Maintenance on sliding windows, or window panes is harder.