Data Mining: Concepts and Techniques — Slides for Textbook — — Chapter 8 — ©Jiawei Han and Micheline Kamber Intelligent Database Systems Research Lab Simon Fraser University, Ari Visa, , Institute of Signal Processing Tampere University of Technology September 12, 2013
Data Mining: Concepts and Techniques
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Chapter 8. Cluster Analysis
What is Cluster Analysis?
Types of Data in Cluster Analysis
A Categorization of Major Clustering Methods
Partitioning Methods
Hierarchical Methods
Density-Based Methods
Grid-Based Methods
Model-Based Clustering Methods
Outlier Analysis
Summary
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General Applications of Clustering
Pattern Recognition Spatial Data Analysis create thematic maps in GIS by clustering feature spaces detect spatial clusters and explain them in spatial data mining Image Processing Economic Science (especially market research) WWW Document classification Cluster Weblog data to discover groups of similar access patterns
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Examples of Clustering Applications
Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs Land use: Identification of areas of similar land use in an earth observation database Insurance: Identifying groups of motor insurance policy holders with a high average claim cost City-planning: Identifying groups of houses according to their house type, value, and geographical location Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults
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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 both the similarity measure used by the method and its implementation. The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns.
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Requirements of Clustering in Data Mining
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
High dimensionality
Incorporation of user-specified constraints
Interpretability and usability
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Chapter 8. Cluster Analysis
What is Cluster Analysis?
Types of Data in Cluster Analysis
A Categorization of Major Clustering Methods
Partitioning Methods
Hierarchical Methods
Density-Based Methods
Grid-Based Methods
Model-Based Clustering Methods
Outlier Analysis
Summary
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Data Structures
Data matrix (two modes)
Dissimilarity matrix (one mode)
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x11 ... x i1 ... x n1
...
x1f
...
...
...
...
...
xif
...
...
...
...
... xnf
...
0 d(2,1) 0 d(3,1) d ( 3,2) 0 : : : d ( n,1) d ( n,2) ...
Data Mining: Concepts and Techniques
x1p ... xip ... xnp
... 0 9
Measure the Quality of Clustering
Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, which is typically metric: d(i, j) There is a separate “quality” function that measures the “goodness” of a cluster. The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio variables. Weights should be associated with different variables based on applications and data semantics. It is hard to define “similar enough” or “good enough” the answer is typically highly subjective.
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Type of data in clustering analysis
Interval-scaled variables:
Binary variables:
Nominal, ordinal, and ratio variables:
Variables of mixed types:
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Interval-valued variables
Standardize data
Calculate the mean absolute deviation: sf 1 n (| x1 f m f | | x2 f m f | ... | xnf m f |)
where
mf 1 n (x1 f x2 f
...
xnf )
.
Calculate the standardized measurement (z-score) xif m f zif sf
Using mean absolute deviation is more robust than using standard deviation
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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: d (i, j) q (| x x |q | x x |q ... | x x |q ) i1 j1 i2 j2 ip jp
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q is a positive integer
If q = 1, d is Manhattan distance
d (i, j) | x x | | x x | ... | x x | i1 j1 i2 j2 ip jp September 12, 2013
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Similarity and Dissimilarity Between Objects (Cont.)
If q = 2, d is Euclidean distance: d (i, j) (| x x |2 | x x |2 ... | x x |2 ) i1 j1 i2 j2 ip jp
Properties
d(i,j) 0 d(i,i) = 0 d(i,j) = d(j,i) d(i,j) d(i,k) + d(k,j)
Also one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures.
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Binary Variables
A contingency table for binary data Object j
Object i
1
0
sum
1
a
b
a b
0
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cd
sum a c b d
p
Simple matching coefficient (invariant, if the binary bc variable is symmetric): d (i, j) a bc d Jaccard coefficient (noninvariant if the binary variable is
asymmetric): September 12, 2013
d (i, j)
bc a bc
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Dissimilarity between Binary Variables
Example Name Jack Mary Jim
Gender M F M
Fever Y Y Y
Cough N N P
Test-1 P P N
Test-2 N N N
Test-3 N P N
Test-4 N N N
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 01 0.33 2 01 11 d ( jack , jim ) 0.67 111 1 2 d ( jim , mary ) 0.75 11 2 d ( jack , mary )
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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 m d (i, j) p p
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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Ordinal Variables
An ordinal variable can be discrete or continuous
order is important, e.g., rank
Can be treated like interval-scaled rif {1,..., M f } 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 rif 1 zif M f 1 compute the dissimilarity using methods for intervalscaled variables
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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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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. pf 1 ij( f ) dij( f ) d (i, j) pf 1 ij( f ) 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 r 1 z compute ranks rif and if M 1 and treat zif as interval-scaled if
f
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Chapter 8. Cluster Analysis
What is Cluster Analysis?
Types of Data in Cluster Analysis
A Categorization of Major Clustering Methods
Partitioning Methods
Hierarchical Methods
Density-Based Methods
Grid-Based Methods
Model-Based Clustering Methods
Outlier Analysis
Summary
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Major Clustering Approaches
Partitioning algorithms: Construct various partitions and
then evaluate them by some criterion
Hierarchy algorithms: Create a hierarchical decomposition of the set of data (or objects) using some criterion
Density-based: based on connectivity and density functions
Grid-based: based on a multiple-level granularity structure
Model-based: A model is hypothesized for each of the clusters and the idea is to find the best fit of that model to each other
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Chapter 8. Cluster Analysis
What is Cluster Analysis?
Types of Data in Cluster Analysis
A Categorization of Major Clustering Methods
Partitioning Methods
Hierarchical Methods
Density-Based Methods
Grid-Based Methods
Model-Based Clustering Methods
Outlier Analysis
Summary
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Partitioning Algorithms: Basic Concept
Partitioning method: Construct a partition of a database D of n objects into a set of k clusters Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion
Global optimal: exhaustively enumerate all partitions
Heuristic methods: k-means and k-medoids algorithms
k-means (MacQueen’67): Each cluster is represented by the center of the cluster
k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the cluster
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The K-Means Clustering Method
Given k, the k-means algorithm is implemented in 4 steps: Partition objects into k nonempty subsets Compute seed points as the centroids of the clusters of the current partition. The centroid is the center (mean point) of the cluster. Assign each object to the cluster with the nearest seed point. Go back to Step 2, stop when no more new assignment.
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The K-Means Clustering Method
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Comments on the K-Means Method
Strength
Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t
