1. What Is Good Clustering?
A good clustering method will produce high quality clusters with
high intra-class similarity
low inter-class similarity
The quality of a clustering result depends on the similarity measure used by the method.
The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns.

2. Vocabulary of Clustering
Records, data points, samples, items, objects, patterns…
Attributes, features, variables…
Similarity, dissimilarity, distances.
Centre, Centroid, Prototype.
Hard Clustering (Crisp Clustering)

3. Requirements of Clustering
Scalability
Ability to deal with different types of attributes
Discovery of clusters with arbitrary shape
Minimal requirements for domain knowledge to determine input parameters
Able to deal with noise and outliers
Insensitive to order of input records
Insensitive to the initial conditions
High dimensionality

4. Clustering Algorithms

5. Clustering Algorithms

6. Data Representation Data matrix (two mode) N objects with p attributes
Dissimilarity matrix (one mode)
d(i,j) : dissimilarity
between i and j
with p attributes

7. How to deal with missing values?

8. 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

9. Types of Clusters: 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

10. 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

11. Types of Clusters: 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

12. 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

13. 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.

14. Type of data in clustering analysis
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15. Symbol Table
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16. Symbol Table
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17. Frequency Table
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18. Frequency Table
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19. Frequency Table
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20. Frequency Table
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21. Type of data in clustering analysis
Binary variables
Nominal variables
Ordinal variables
Interval-scaled variables
Ratio variables
Variables of mixed types
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22. Binary variables
The binary variable is symmetric (Simple match coefficient)
The binary variable is asymmetric (Jaccard coefficient)
Object j
Object i
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23. Binary variables
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24. 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
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25. Nominal 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
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26. Nominal Variables Examples Eye Color Days of the week Religion Seasons
Job title
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27. Nominal Variables
Find the Proximity Matrix?
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28. Ordinal Variables Order is important, e.g., rank
Can be treated like interval-scaled
replacing xif 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
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29. Ordinal Variables
Find the Proximity Matrix?
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30. Interval-valued variables
Examples
Temperature
Weight
Time
Age
Length
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31. Interval-valued variables
Standardize data
Calculate the mean absolute deviation:
where
Calculate the standardized measurement (z-score)
Using mean absolute deviation is more robust than using standard deviation
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32. Ratio-Scaled Variables
Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as AeBt or Ae-Bt
Methods:
treat them like interval-scaled variables — not a good choice! (why?)
apply logarithmic transformation
yif = log(xif)
treat them as continuous ordinal data treat their rank as interval-scaled.
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33. Ratio-Scaled Variables
Find the Proximity Matrix?
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34. Variables of Mixed Types
A database may contain all the six types of variables
symmetric binary, asymmetric binary, nominal, ordinal, interval and ratio.
One may use a weighted formula to combine their effects.
f is binary or nominal:
dij(f) = 0 if xif = xjf , or dij(f) = 1 o.w.
f is interval-based: use the normalized distance
f is ordinal or ratio-scaled
compute ranks rif and
and treat zif as interval-scaled

35. Variables of Mixed Types
Find the Proximity Matrix?
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