Data Mining Introduction to Clustering Mauro Sozio

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Data Mining

Introduction to Clustering Mauro Sozio

some slides from Tan,Steinbach, Kumar, Introduction to Data Mining

Mauro Sozio Clustering 31/05/2013 Tuesday, June 25, 13

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What is Cluster Analysis?

l 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

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Mauro Sozio Clustering 31/05/2013

What is Cluster Analysis?

l 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

Tuesday, June 25, 13

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What is Cluster Analysis?

l 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

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What is Cluster Analysis?

l 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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Applications of Cluster Analysis

l Understanding

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

l Summarization

– Reduce the size of large data sets

Clustering precipitation

in Australia

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What is not Cluster Analysis?

l Supervised classification

– Have class label information

l Simple segmentation

– Dividing students into different registration groups alphabetically, by last name

l Results of a query

– Groupings are a result of an external specification

l 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?

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Notion of a Cluster can be Ambiguous

How many clusters?

Two Clusters

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Notion of a Cluster can be Ambiguous

How many clusters?

Two Clusters

Six Clusters

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Notion of a Cluster can be Ambiguous

How many clusters?

Four Clusters Two Clusters

Six Clusters

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Types of Clusterings

l A clustering is a set of clusters

l Important distinction between hierarchical and partitional sets of clusters

l Partitional Clustering

– A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset

l Hierarchical clustering

– A set of nested clusters organized as a hierarchical tree

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Partitional Clustering

Original Points

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Partitional Clustering

Original Points A Partitional Clustering

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Hierarchical Clustering

Traditional Hierarchical Clustering

Non-traditional Hierarchical Clustering Non-traditional Dendrogram Traditional Dendrogram

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Other Distinctions Between Sets of Clusters

l Exclusive versus non-exclusive

– In non-exclusive clusterings, points may belong to multiple clusters.

– Can represent multiple classes or ‘border’ points l 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 l Partial versus complete

– In some cases, we only want to cluster some of the data l Heterogeneous versus homogeneous

– Cluster of widely different sizes, shapes, and densities

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Types of Clusters

l Well-separated clusters

l Center-based clusters

l Contiguous clusters

l Density-based clusters

l Property or Conceptual

l Described by an Objective Function

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Types of Clusters: Well-Separated

l 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

l 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 points in the cluster, or a medoid, the most “representative”

point of a cluster

4 center-based clusters

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Types of Clusters: Contiguity-Based

l 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

l 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

l 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

l Clusters Defined by an Objective Function

– Finds clusters that minimize or maximize an objective function.

– Proximity matrix defines a weighted graph, where the nodes are the points being clustered, and the weighted edges represent the proximities between points

– Want to minimize the edge weight between clusters and maximize the edge weight within clusters (NP- Hard).

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Clustering Algorithms

l K-means and its variants

l Hierarchical clustering

l Density-based clustering

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K-means Clustering

l Partitional clustering approach

l Each cluster is associated with a centroid (center point)

l Each point is assigned to the cluster with the closest centroid

l Number of clusters, K, must be specified

l The basic algorithm is very simple

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K-means Clustering – Details

l Initial centroids are often chosen randomly.

– Clusters produced vary from one run to another.

l The centroid is (typically) the mean of the points in the cluster.

l ‘Closeness’ is measured by Euclidean distance, cosine similarity, correlation, etc.

l K-means will converge for common similarity measures

mentioned above.

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Two different K-means Clusterings

Original Points

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Two different K-means Clusterings

Optimal Clustering

Original Points

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Two different K-means Clusterings

Sub-optimal Clustering Optimal Clustering

Original Points

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids …

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Importance of Choosing Initial Centroids …

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Importance of Choosing Initial Centroids …

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Importance of Choosing Initial Centroids …

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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

l If there are K ‘real’ clusters then the chance of selecting one centroid from each cluster is small.

– Chance is relatively small when K is large – If clusters are the same size, n, then

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Evaluating K-means Clusters

l 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 C

_i

and m

_i

is the centroid of cluster C

_i

– Given two clusters, we can choose the one with smallest error – One easy way to reduce SSE is to increase K

 A good clustering with smaller K can have a lower SSE than a

poor clustering with higher K

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Solutions to Initial Centroids Problem

l Multiple runs

– Helps, but probability is not on your side

l Sample and use hierarchical clustering to determine initial centroids

l Select more than k initial centroids and then select among these initial centroids

– Select most widely separated

l Postprocessing

l Bisecting K-means

– Not as susceptible to initialization issues

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Handling Empty Clusters

l Basic K-means algorithm can yield empty clusters

l 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

l In the basic K-means algorithm, centroids are

updated after all points are assigned to a centroid

l 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

– Can use “weights” to change the impact

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Pre-processing and Post-processing

l Pre-processing

– Normalize the data – Eliminate outliers

l Post-processing

– Eliminate small clusters that may represent outliers – Split ‘loose’ clusters, i.e., clusters with relatively high

SSE

– Merge clusters that are ‘close’ and that have relatively

low SSE

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Limitations of K-means

l K-means has problems when clusters are of differing

– Sizes

– Densities

– Non-globular shapes

l K-means has problems when the data contains outliers.

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Limitations of K-means: Differing Sizes

Original Points

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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 Sizes

Original Points K-means (3 Clusters)

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Limitations of K-means: Differing Density

Original Points

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Limitations of K-means: Differing Density

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

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Limitations of K-means: Non-globular Shapes

Original Points K-means (2 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

l Produces a set of nested clusters organized as a hierarchical tree

l 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

l 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

l They may correspond to meaningful taxonomies

– Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, …)

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Hierarchical Clustering

l 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)

l Traditional hierarchical algorithms use a similarity or distance matrix

– Merge or split one cluster at a time

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Agglomerative Clustering Algorithm

l More popular hierarchical clustering technique

l Basic algorithm is straightforward

1. 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

l 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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Starting Situation

l Start with clusters of individual points and a proximity matrix

p1

p3

p5 p4 p2

p1 p2 p3 p4 p5 . . .

. .

.

Proximity Matrix

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Intermediate Situation

l After some merging steps, we have some clusters

C1

C4

C2 C5

C3

C2 C1

C1

C3

C5 C4 C2

C3 C4 C5

Proximity Matrix

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Intermediate Situation

l We want to merge the two closest clusters (C2 and C5) and update the proximity matrix.

C1

C4

C2 C5

C3

C2 C1

C1

C3

C5 C4 C2

C3 C4 C5

Proximity Matrix

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After Merging

l The question is “How do we update the proximity matrix?”

C1

C4

C2 U C5 C3

? ? ? ?

?

? C2 U C1 C5

C1

C3 C4 C2 U C5

C3 C4

Proximity Matrix

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How to Define Inter-Cluster Similarity

p1

p3

p5 p4 p2

p1 p2 p3 p4 p5 . . .

. . . Similarity?

l MIN

l MAX

l Group Average

l Distance Between Centroids

l Other methods driven by an objective function

– Ward’s Method uses squared error

Proximity Matrix

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How to Define Inter-Cluster Similarity

p1

p3

p5 p4 p2

p1 p2 p3 p4 p5 . . .

. .

. Proximity Matrix

l MIN

l MAX

l Group Average

l Distance Between Centroids

l Other methods driven by an objective function

– Ward’s Method uses squared error

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How to Define Inter-Cluster Similarity

p1

p3

p5 p4 p2

p1 p2 p3 p4 p5 . . .

. .

. Proximity Matrix

l MIN

l MAX

l Group Average

l Distance Between Centroids

l Other methods driven by an objective function

– Ward’s Method uses squared error

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How to Define Inter-Cluster Similarity

p1

p3

p5 p4 p2

p1 p2 p3 p4 p5 . . .

. .

. Proximity Matrix

l MIN

l MAX

l Group Average

l Distance Between Centroids

l Other methods driven by an objective function

– Ward’s Method uses squared error

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How to Define Inter-Cluster Similarity

p1

p3

p5 p4 p2

p1 p2 p3 p4 p5 . . .

. .

. Proximity Matrix

l MIN

l MAX

l Group Average

l Distance Between Centroids

l Other methods driven by an objective function

– Ward’s Method uses squared error

× ×

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Hierarchical Clustering: Problems and Limitations

l Once a decision is made to combine two clusters, it cannot be undone

l No objective function is directly minimized

l Different schemes have problems with one or more of the following:

– Sensitivity to noise and outliers

– Difficulty handling different sized clusters and convex shapes

– Breaking large clusters

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Cluster Validity

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

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

l Then why do we want to evaluate them?

– To avoid finding patterns in noise

– To compare clustering algorithms

– To compare two sets of clusters

– To compare two clusters

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Clusters found in Random Data

Random Points

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Clusters found in Random Data

Random Points

K-means

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Clusters found in Random Data

Random Points

K-means

DBSCAN

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Clusters found in Random Data

Random Points

K-means

DBSCAN

Complete

Link

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l Numerical measures that are applied to judge various aspects of cluster validity, are classified into the following three types.

– External Index: Used to measure the extent to which cluster labels match externally supplied class labels.



Entropy

– Internal Index: Used to measure the goodness of a clustering structure without respect to external information.



Sum of Squared Error (SSE)

– Relative Index: Used to compare two different clusterings or clusters.



Often an external or internal index is used for this function, e.g., SSE or entropy

l Sometimes these are referred to as criteria instead of indices

Measures of Cluster Validity

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l Clusters in more complicated figures aren’t well separated

l Internal Index: Used to measure the goodness of a clustering structure without respect to external information

– SSE

l SSE is good for comparing two clusterings or two clusters (average SSE).

l Can also be used to estimate the number of clusters

Internal Measures: SSE

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External Measures of Cluster Validity: Entropy and Purity

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“The validation of clustering structures is the most difficult and frustrating part of cluster analysis.

Without a strong effort in this direction, cluster

analysis will remain a black art accessible only to those true believers who have experience and

great courage.”

Algorithms for Clustering Data, Jain and Dubes

Final Comment on Cluster Validity

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K-means in MapReduce

l Iterate until clusters do not change:

– Map phase:

 partition data points;

 replicate centroids in each machine;

 for each vector v compute the closest centroid and output (v,clusterID).

– Reduce phase:

 group points with same cluster ID and output new centroid

l Problem: After the map phase we need to send all data points to the reducers. In practice, too much data has to be sent around!

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K-means in MapReduce: combiner

l Better: compute partial centroids on the map nodes themselves, need to aggregate them...

l Combiner: very similar to a reducer but it is

executed in a mapper (in main memory) on the data generated by one machine. E.g. useful to count number of occurrences of a word.

l Add this to conf.setCombinerClass(Reduce.class);

in the Driver to use the code of the reducer as a combiner.

combine(“computer”,1,1,1) -> (“computer”,3)

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K-means in MapReduce: more efficient

l Iterate until clusters do not change:

– Map phase

 partition data points and replicate centroids in each machine;

 for each v compute its closest centroid, output (v,clusterID).

– Combine:

 (partially) aggregates sum with a same ID and output (clusterID, “partial sum number of vectors #vectors”)

– Reduce phase

 group partial sums with a same cluster ID and compute average (centroids). output (centroid, cluster ID).