Data Mining Cluster Analysis: Basic Concepts and Algorithms. Clustering Algorithms. Lecture Notes for Chapter 8. Introduction to Data Mining

(1)

Data Mining

Cluster Analysis: Basic Concepts

and Algorithms

Lecture Notes for Chapter 8

Introduction to Data Mining

by

Tan, Steinbach, Kumar

Clustering Algorithms

l K-means and its variants

l Hierarchical clustering

l Density-based clustering

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

1 3 2 5 4 6

0 0.05 0.1 0.15 0.2

1 2

3 4

5 6

1 2

3 4

5

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, …)

Hierarchical Clustering

l Two main types of hierarchical clustering

– Agglomerative:

u

Start with the points as individual clusters

u

At each step, merge the closest pair of clusters until only one cluster

(or k clusters) left

– Divisive:

u

Start with one, all-inclusive cluster

u

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

Agglomerative Clustering Algorithm

l Most 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

(2)

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

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

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

After Merging

l

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

C1

C4

C2 U C5 C3

? ? ? ?

?

? C2 U C5 C1 C1

C3 C4 C2 U C5

C3 C4

Proximity Matrix

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

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

(3)

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

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

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

×××× ××××

Cluster Similarity: MIN or Single Link

l Similarity of two clusters is based on the two

most similar (closest) points in the different

clusters.

I1 I2 I3 I4 I5

I1 1.00 0.90 0.10 0.65 0.20

I2 0.90 1.00 0.70 0.60 0.50

I3 0.10 0.70 1.00 0.40 0.30

I4 0.65 0.60 0.40 1.00 0.80

I5 0.20 0.50 0.30 0.80 1.00 ₁ ₂ ₃ ₄ ₅

Hierarchical Clustering: MIN

Nested Clusters Dendrogram

1

2

3

4

5

6

1

2

3

4

5

3 6 2 5 4 1

0 0.05 0.1 0.15 0.2

Strength of MIN

Original Points Two Clusters

• Can handle non-elliptical shapes

(4)

Limitations of MIN

Original Points Two Clusters

• Sensitive to noise and outliers

Cluster Similarity: MAX or Complete Linkage

l Similarity of two clusters is based on the two least

similar (most distant) points in the different

clusters.

I1 I2 I3 I4 I5

I1 1.00 0.90 0.10 0.65 0.20

I2 0.90 1.00 0.70 0.60 0.50

I3 0.10 0.70 1.00 0.40 0.30

I4 0.65 0.60 0.40 1.00 0.80

I5 0.20 0.50 0.30 0.80 1.00 ₁ ₂ ₃ ₄ ₅

Hierarchical Clustering: MAX

Nested Clusters Dendrogram

3 6 4 1 2 5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

1

2

3

4

5

6

1 2 5

3

4 Strength of MAX

Original Points Two Clusters

• Less susceptible to noise and outliers

Limitations of MAX

Original Points Two Clusters

•Tends to break large clusters

•Biased towards globular clusters

Cluster Similarity: Group Average

l

Proximity of two clusters is the average of pairwise proximity

between points in the two clusters.

|

|Cluster

|

|Cluster ) p , p proximity(

) Cluster , Cluster proximity(

j i Cluster pCluster p

j i

j j

i i

= ∗

∑

∈∈

I1 I2 I3 I4 I5

I1 1.00 0.90 0.10 0.65 0.20

I2 0.90 1.00 0.70 0.60 0.50

I3 0.10 0.70 1.00 0.40 0.30

I4 0.65 0.60 0.40 1.00 0.80

I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5

(5)

Hierarchical Clustering: Group Average

Nested Clusters Dendrogram

3 6 4 1 2 5

0 0.05 0.1 0.15 0.2 0.25

1

2

3

4

5

6

1

2

5

3

4 Hierarchical Clustering: Group Average

l Compromise between Single and Complete

Linkage

l Strengths

– Less susceptible to noise and outliers

l Limitations

– Biased towards globular clusters

Cluster Similarity: Ward’s Method

l 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

squared euclidian distance

l Less susceptible to noise and outliers

l Biased towards globular clusters

l Hierarchical analogue of K-means

– Can be used to initialize K-means

Hierarchical Clustering: Comparison

Group Average Ward’s Method

1

2 3 4 5

6 1 2 5

3 4

MIN MAX

1

2 3 4 5

6 1 2

5

3 4

1

2 3 4 5

6 1

2 5

3 1 4

2 3 4 5

6 1 2

3

4 5

Hierarchical Clustering: Example

5.9

4.8

5

2.0

1.9

4

5.7

5.2

3

2.1

1.7

2

6.2

5.0

1 x ₂

x ₁

0

6.8

0.6

6.9

0.5

5

0

7.0

0.3

7.3

4

0

7.1

0.7

3

0

7.4

2

0

1

5

4

3

2

1 Use Manhattan distance as distance measure.

For example: d(2,1) = |1.7 - 5.0| + |2.1 - 6.2| = 7.4

Hierarchical Clustering: Example

5.9

4.8

5

2.0

1.9

4

5.7

5.2

3

2.1

1.7

2

6.2

5.0

1 x ₂

x ₁

0

6.8

0.6

6.9

0.5

5

0

7.0

0.3

7.3

4

0

7.1

0.7

3

0

7.4

2

0

1

5

4

3

2

1 Look for smallest non-diagonal entry in matrix.

Merge the corresponding clusters.

D ₁

(6)

Hierarchical Clustering: Example

l Objects 2 and 4 are most similar, since entry

(4,2) has the smallest value of the distance matrix

l Make a new cluster {4,2} and remove the single

point clusters {2} and {4}.

l Compute the distance between the newly formed

cluster and the remaining clusters {1}, {3} and {5}.

For example (single linkage):

d({1},{2,4}) = min{d(1,2),d(1,4)} = min{7.4,7.3} = 7.3

Hierarchical Clustering: Example

0

0.6

6.8

0.5

5

0

7.0

0.7

3

0

7.3 {2,4}

0

1

5

3 {2,4}

D ₂ 1

0

7.0

0.6

3

0

6.8 {2,4}

0 {1,5}

3 {2,4}

{1,5}

D ₃

0

6.8 {2,4}

0 {1,5,3}

{2,4}

{1,5,3}

D ₄

Example: data points

0 1 2 3 4 5 6

01234567

x1

x2

1

2

3

4

5 Example: Dendrogram

2 4 3 1 5

01234567

Cluster Dendrogram

hclust (*, "single") Single Linkage

Height

Hierarchical Clustering: Time and Space requirements

l O(N ² ) space since it uses the proximity matrix.

– N is the number of points.

l O(N ³ ) time in many cases

– There are N steps and at each step the size, N ² ,

proximity matrix must be updated and searched

– Complexity can be reduced to O(N ² log(N) ) time for

some approaches

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

(7)

DBSCAN

l DBSCAN is a density-based algorithm.

– Density = number of points within a specified radius (Eps)

– A point is a core point if it has at least a specified number of

points (MinPts) within Eps (including the point itself).

u

These are points that are at the interior of a cluster

– A border point has fewer than MinPts within Eps, but is in

the neighborhood of a core point

– A noise point is any point that is not a core point or a border

point.

DBSCAN: Core, Border, and Noise Points

5 DBSCAN Algorithm

l Label all points as core, border or noise.

l Eliminate noise points.

l Perform clustering on the remaining points

– Put an edge between all core points within Eps of

each other.

– Make each group of connected core points into a

separate cluster.

– Assign each border point to the cluster of one of its

associated core points.

DBSCAN: Core, Border and Noise Points

Original Points Point types: core,

border and noise

Eps = 10, MinPts = 4

When DBSCAN Works Well

Original Points Clusters

• Resistant to Noise

• Can handle clusters of different shapes and sizes

Potential Problem with DBSCAN

A B C D

noise noise

If we choose Minpts small enough that C and D

are found as separate clusters, than A+B+surrounding

noise will become a single cluster.

(8)

DBSCAN: Determining EPS and MinPts

l Idea is that for points in a cluster, their k ^th nearest

neighbors are at roughly the same distance

l Noise points have the k ^th nearest neighbor at farther

distance

l So, plot sorted distance of every point to its k ^th

nearest neighbor

Choose Eps = 10

because of sharp

increase in distance

to 4 ^th nearest

neighbour.

Cluster Validity

l For supervised classification we have a variety of

measures to evaluate how good our model is

– Accuracy, precision, recall

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

Clusters found in Random Data

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

Random Points

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

K-means

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

DBSCAN

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

Complete Link

1. Determining the clustering tendency of a set of data, i.e.,

distinguishing whether non-random structure actually exists in the

data.

2. Comparing the results of a cluster analysis to externally known

results, e.g., to externally given class labels.

3. Evaluating how well the results of a cluster analysis fit the data

without reference to external information.

- Use only the data

4. Comparing two sets of clusters to determine which is better.

5. Determining the ‘correct’ number of clusters.

For 2, 3, and 4, we can further distinguish whether we want to

evaluate the entire clustering or just individual clusters.

Different Aspects of Cluster Validation

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.

u

Entropy

– Internal Index: Used to measure the goodness of a

clustering structure without reference to external

information.

u

Sum of Squared Error (SSE)

– Relative Index: Used to compare two different

clusterings or clusters.

u

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

or entropy

Measures of Cluster Validity

l Two matrices

– Proximity Matrix

– “Incidence” Matrix

u

One row and one column for each data point

u

An entry is 1 if the associated pair of points belong to the same cluster

u

An entry is 0 if the associated pair of points belongs to different clusters

l Compute the correlation between the two matrices

– Since the matrices are symmetric, only the correlation between

n(n-1) / 2 entries needs to be calculated.

l Strong correlation indicates that points that belong to the

same cluster are close to each other.

l Not a good measure for some density or contiguity based

clusters.

Measuring Cluster Validity Via Correlation

(9)

Measuring Cluster Validity Via Correlation

l Correlation of incidence and distance matrices for

the K-means clusterings of the following two data

sets.

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

Corr = -0.9235 Corr = -0.5810

Correlation Example

0

6.8

0.6

6.9

0.5

5

0

7.0

0.3

7.3

4

0

7.1

0.7

3

0

7.4

2

0

1

5

4

3

2

1 D

1

0

1

0

1

5

1

0

1

0

4

1

0

1

3

1

0

2

1

5

4

3

2

1 …

…

(4,1)

0

7.1 (3,2)

1

0.7 (3,1)

0

7.4 (2,1)

corr = -0.998

C ₁ = {1,5,3}

C ₂ = {2,4}

l Order the similarity matrix with respect to cluster

labels and inspect visually.

Using Similarity Matrix for Cluster Validation

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

Points

20 40 60 80 100

10 20 30 40 50 60 70 80 90 100

Similarity0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

Using Similarity Matrix for Cluster Validation

l Clusters in random data are not so crisp

Points

20 40 60 80 100

10 20 30 40 50 60 70 80 90 100

Similarity0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DBSCAN

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

Points

20 40 60 80 100

10 20 30 40 50 60 70 80 90 100

Similarity0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Using Similarity Matrix for Cluster Validation

l Clusters in random data are not so crisp

K-means

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

Using Similarity Matrix for Cluster Validation

l Clusters in random data are not so crisp

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

Points

20 40 60 80 100

10 20 30 40 50 60 70 80 90 100

Similarity0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Complete Link

(10)

l Clusters in more complicated figures aren’t well separated

l

Internal Index: Used to measure the quality of a clustering

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

2 5 10 15 20 25 30

0 1 2 3 4 5 6 7 8 9 10

K

SSE

5 10 15

-6 -4 -2 0 2 4 6

Internal Measures: SSE

l SSE curve for a more complicated data set

1 2

3

5 6

4

7

SSE of clusters found using K-means

l Need a framework to interpret any measure.

– For example, if our measure of evaluation has the value, 10, is that

good, fair, or poor?

l Statistics provide a framework for cluster validity

– The more “atypical” a clustering result is, the more likely it represents

valid structure in the data

– Can compare the values of an index that result from random data or

clusterings to those of a clustering result.

u

If the value of the index is unlikely, then the cluster results are valid

– These approaches are more complicated and harder to understand.

l For comparing the results of two different sets of clusters,

a framework is less necessary.

– However, there is the question of whether the difference between two

index values is significant

Framework for Cluster Validity

l Example

– Compare SSE of 0.005 against three clusters in random data

– Histogram shows SSE of three clusters in 500 sets of random data

points of size 100 distributed over the range 0.2 – 0.8 for x and y

values

Statistical Framework for SSE

0.0160 0.0180.020.0220.0240.0260.0280.03 0.0320.034 5

10 15 20 25 30 35 40 45 50

SSE

Count

0 0. 2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

l Correlation of incidence and proximity matrices for the

K-means clusterings of the following two data sets.

Statistical Framework for Correlation

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

y

Corr = -0.9235 Corr = -0.5810

l Cluster Cohesion: Measures how closely related

are objects in a cluster

– Example: SSE

l Cluster Separation: Measure how distinct or well-

separated a cluster is from other clusters

l Example: Squared Error

– Cohesion is measured by the within cluster sum of squares (SSE)

– Separation is measured by the between cluster sum of squares

– Where |C

_i

| is the size of cluster i and m is the overall mean

Internal Measures: Cohesion and Separation

∑ ∑

∈ −

=

i x C i

i

m

x

WSS ( ) ²

∑ −

=

i

m m

C

BSS ( )

²

(11)

Internal Measures: Cohesion and Separation

l Example: SSE

– BSS + WSS = constant

1 ×××× 2 ×××× 3 4 ×××× 5

m

1

m

2

m

10

9

1

9 )

3

5 .

4 (

2 )

5 .

1

3 (

2

1 )

5 .

4

5 (

)

5 .

4

4 (

)

5 .

1

2 (

)

5 .

1

1 (

2 2

2 2 2 2

=

+

=

−

×

+

−

×

=

−

+

−

+

−

+

−

=

Total

BSS

K=2 clusters: WSS

10

0

10

0 )

3

3 (

4

10 )

3

5 (

)

3

4 (

)

3

2 (

)

3

1 (

2

2 2 2 2

=

+

=

−

×

=

−

+

−

+

−

+

−

=

Total

BSS

K=1 cluster: WSS

A proximity graph based approach can also be used for

cohesion and separation.

– Cluster cohesion is the sum of the weight of all links within a cluster.

– Cluster separation is the sum of the weights between nodes in the cluster

and nodes outside the cluster.

Internal Measures: Cohesion and Separation

cohesion separation

Silhouette Coefficient combine ideas of both cohesion and separation,

but for individual points, as well as clusters and clusterings

l

For an individual point, i

– Calculate a = average distance of i to the points in its cluster

– Calculate b = min (average distance of i to points in another cluster)

– The silhouette coefficient for a point is then given by

s = 1 – a/b if a < b,

(or s = b/a - 1 if a ≥b, not the usual case)

– Typically between 0 and 1.

– The closer to 1 the better.

l

Can calculate the Average Silhouette width for a cluster or a

clustering: does not necessarily increase with k.

Internal Measures: Silhouette Coefficient

a b