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Data Mining - Cluster Analysis - Types of Data, Study notes of Data Mining

Detailed informtion about Cluster Analysis, What is Cluster Analysis?, Types of Data in Cluster Analysis, Partitioning Methods, Hierarchical Methods, Density-Based Methods.

Typology: Study notes

2010/2011

Uploaded on 09/03/2011

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November 25, 2014 Data Mining: Concepts and
Techniques 1
Chapter 7. Cluster
Analysis
1. What is Cluster Analysis?
2. Types of Data in Cluster Analysis
3. A Categorization of Major Clustering Methods
4. Partitioning Methods
5. Hierarchical Methods
6. Density-Based Methods
7. Grid-Based Methods
8. Model-Based Methods
9. Clustering High-Dimensional Data
10.Constraint-Based Clustering
11.Outlier Analysis
12.Summary
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November 25, 2014

Data Mining: Concepts and 1

Chapter 7. Cluster

Analysis

  1. What is Cluster Analysis?
  2. Types of Data in Cluster Analysis
  3. A Categorization of Major Clustering Methods
  4. Partitioning Methods
  5. Hierarchical Methods
  6. Density-Based Methods
  7. Grid-Based Methods
  8. Model-Based Methods
  9. Clustering High-Dimensional Data

10.Constraint-Based Clustering

11.Outlier Analysis

12.Summary

November 25, 2014

Data Mining: Concepts and 2

Data Types and Distance

Metrics

Data Structures Data Structures

 Data Matrix (object-by-variable structure)

 n records, each with p attributes

 n-by-p matrix structure (two mode)

 (^) x ab

  • value for a

th record and b

th attribute

Attributes

np

... x nf

... x n

x

... ... ... ... ...

ip

... x if

... x i

x

... ... ... ... ...

1p

... x 1f

... x 11

x

record n

record i

record 1

November 25, 2014

Data Mining: Concepts and 4

Type of data in clustering analysis

Interval-Scaled Attributes

Binary Attributes

Nominal Attributes

Ordinal Attributes

Ratio-Scaled Attributes

Attributes of Mixed Type

November 25, 2014

Data Mining: Concepts and 5

Data Types and Distance

Metrics

Interval-Scaled Attributes Interval-Scaled Attributes

Continuous measurements on a

roughly linear scale Example

Height Scale Weight Scale

  1. Scale ranges over the

metre or foot scale

  1. Need to standardize

heights as different scale

can be used to express

same absolute

measurement

  1. Scale ranges over the

kilogram or pound

scale

20kg

40kg

60kg 100kg

80kg 120kg

November 25, 2014

Data Mining: Concepts and 7

Similarity and Dissimilarity

Between Objects

  • (^) Distances are normally used to measure the

similarity or dissimilarity between two data

objects

  • (^) Some popular ones include: Minkowski distance :

where i = ( x i

, x i

, …, x ip

) and j = ( x j

, x j

, …, x jp

)

are two p -dimensional data objects, and q is a

positive integer

  • (^) If q = 1 , d is Manhattan distance

q

q

p p

q q

j

x

i

x

j

x

i

x

j

x

i

d ( i , j ) (| x | | | ... | | )

1 1 2 2

( , ) | | | | ... | |

1 1 2 2 p jp

x

i

x

j

x

i

x

j

x

i

d i jx      

November 25, 2014

Data Mining: Concepts and 8

Similarity and Dissimilarity Between

Objects (Cont.)

  • (^) If q = 2 , d is Euclidean distance:
    • (^) Properties
      • (^) d(i,j)  0; Distance is nonnegative number
      • (^) d(i,i) = 0; The distance of an object to itself is 0
      • (^) d(i,j) = d(j,i) ; Distance is a symmetric Function
      • (^) d(i,j)d(i,k) + d(k,j)
  • (^) Also, one can use weighted distance, parametric

Pearson product moment correlation, or other

dissimilarity measures

2 2

2 2

2

1 1 p jp

x

i

x

j

x

i

x

j

x

i

d i jx      

November 25, 2014

Data Mining: Concepts and 10

Dissimilarity between Binary

Variables

  • (^) Example
    • (^) gender is a symmetric attribute
    • (^) the remaining attributes are asymmetric binary
    • (^) let the values Y and P be set to 1, and the value N be set

to 0

Name Gender Fever Cough Test-1 Test-2 Test-3 Test-

Jack M Y N P N N N

Mary F Y N P N P N

Jim M Y P N N N N

0. 75

1 1 2

1 2

( , )

0. 67

1 1 1

1 1 ( , )

0. 33

2 0 1

0 1 ( , )

 

 

 

 

 

d jim mary

d jack jim

d jack mary

November 25, 2014

Data Mining: Concepts and 11

Nominal / Categorical Variables

  • (^) A generalization of the binary variable in that it can

take more than 2 states, e.g., red, yellow, blue, green

  • (^) Method 1: Simple matching
    • (^) m : # of matches, p : total # of variables
  • (^) Method 2: use a large number of binary variables
    • (^) creating a new binary variable for each of the M

nominal states

p

p m

d i j

( , )

November 25, 2014

Data Mining: Concepts and 13

Nominal/ Categorical Example

  • (^) Consider object identifier and the variable( or

attribute) test-1 are available which is

categorical. Since here we have one categorical

variable, test-1, we set p= 1in eq so that d(i,j)

evaluates to 0 if objects I and j match, and 1 if

the objects differ. Thus we get 0

1 0

1 1 0

0 1 1 0

November 25, 2014

Data Mining: Concepts and 14

Ordinal Variables

  • (^) An ordinal variable can be discrete or continuous
  • (^) Order is important, e.g., rank
  • (^) Can be treated like interval-scaled
    • (^) replace x if by their rank
    • (^) map the range of each variable onto [0, 1] by

replacing i -th object in the f -th variable by

  • (^) compute the dissimilarity using methods for

interval-scaled variables

1

1

f

if

if M

r

z

{ 1 ,..., } if f

rM

November 25, 2014

Data Mining: Concepts and 16

Ratio-Scaled Variables

  • (^) Ratio-scaled variable: a positive measurement on a

nonlinear scale, approximately at exponential

scale, such as Ae

Bt or Ae

-Bt

  • (^) Methods:
    • (^) treat them like interval-scaled variables— not a

good choice! (why?—the scale can be distorted)

  • (^) apply logarithmic transformation

y if

= log(x if

)

  • (^) treat them as continuous ordinal data treat their

rank as interval-scaled

November 25, 2014

Data Mining: Concepts and 17

Ratio-Scaled Variables Example

  • (^) Consider object identifier and the variable( or attribute) test-3 are

available which is ratio-scaled variable. Let’s try a logarithmic

transformation. Take the log of test-3 results in the values 2.65,

1.34, 2.21 and 3.08 for the objects 1 to 4 respectively. Using

Euclidean distance on the transformed values, we obtain the

following dissimilarity matrix:

0

1.31 0

0.44 0.87 0

0.43 1.74 0.87 0

November 25, 2014

Data Mining: Concepts and 19

Vector Objects

  • (^) Vector objects: keywords in documents, gene features in micro-

arrays, etc.

  • (^) Broad applications: information retrieval, biologic taxonomy, etc.
  • (^) Cosine measure
  • (^) A variant: Tanimoto coefficient
  • (^) Suppose we are given two vectors, x =(1,1,0,0,) and y =(0,1,1,0).

By using above eq, the similarity between x and y is

  1. 5

2 2

( 0 1 0 0 )

( , ) 

  

s x y