4. Clustering Methods

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4. Clustering Methods

• Concepts
• Partitional (k-Means, k-Medoids)
• Hierarchical (Agglomerative Divisive, COBWEB)
• Density-based (DBSCAN, CLIQUE)
• Large size data (STING, BIRCH, CURE)

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Concepts of Clustering

• Clusters
• Different ways of representing clusters
• Division with boundaries
• Venn diagram or spheres
• Probabilistic
• Dendrograms
• Trees
• Rules

1 2 3
I1 I2 In
0.5 0.2 0.3
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• About clusters
• Inter-clusters distance ? maximization
• Intra-clusters distance ? minimization
• Clustering vs. classification
• Which one is more difficult? Why?
• Various possible ways of clustering, which way is
  the best?

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Major Categories

• Partitioning Divide into k partitions (k fixed)
  repartition to get better clustering.
• Hierarchical Divide into different number of
  partitions in layers - merge (bottom-up) or
  divide (top-down).
• Density-based Continue to grow a cluster as long
  as the density of the cluster exceeds a threshold
• Grid-based First divide space into grids, then
  perform clustering on the grids.

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k-Means

• Algorithm
• Given k
• Randomly pick k instances as the initial centers
• Assign the rest instances to closest one of k
  clusters
• Recalculate the mean of each cluster
• Repeat 3 4 until means dont change
• How good the clusters are
• Initial and final clusters
• Within-cluster variation ?diff(x,mean)2
• Why dont we consider inter-cluster distance?

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Example

• For simplicity, 1 dimensional objects and k2.
• Objects 1, 2, 5, 6,7
• K-means
• Randomly select 5 and 6 as initial centroids
• gt Two clusters 1,2,5 and 6,7 meanC18/3,
  meanC26.5
• gt 1,2, 5,6,7 meanC11.5, meanC26
• gt no change.
• Aggregate dissimilarity 0.52 0.52 12
  12 2.5

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Discussions

• Limitations
• Means cannot be defined for categorical
  attributes
• Choice of k
• Sensitive to outliers
• Crisp clustering
• Variants of k-means exist
• Using modes to deal with categorical attributes
• How about distance measures
• Is it similar to or different from k-NN?
• With and without learning

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k-Medoids

• k-Means algorithm is sensitive to outliers
• Is this true? How to prove it?
• Medoid the most centrally located point in a
  cluster, as a representative point of the
  cluster.
• In contrast, a centroid is not necessarily inside
  a cluster.
• An example

Initial Medoids
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Partition Around Medoids

• PAM
• Given k
• Randomly pick k instances as initial medoids
• Assign each instance to the nearest medoid x
• Calculate the objective function
• the sum of dissimilarities of all instances to
  their nearest medoids
• Randomly select an instance y
• Swap x by y if the swap reduces the objective
  function
• Repeat (3-6) until no change

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k-Means and k-Medoids

• The key difference lies in how they update means
  or medoids
• Both require distance calculation and
  reassignment of instances
• Time complexity
• Which one is more costly?
• Dealing with outliers

Outlier (100 unit away)
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EM (Expectation and Maximization)

• Moving away from crisp clusters as in k-Means by
  allowing an instance to belong to several
  clusters
• Finite mixtures a statistical clustering model
• A mixture is a set of k probability
  distributions, representing k clusters
• The simplest finite mixture one feature with a
  Gaussian
• When k2, we need to estimate 5 parameters 2
  pairs of µ and s and pA, where pB 1- pA
• EM
• Estimate using instances
• Maximize the overall likelihood that data came
  from this data set

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Agglomerative

• Each object is viewed as a cluster (bottom up).
• Repeat until the number of clusters is small
  enough
• Choose a closest pair of clusters
• Merge the two into one
• Defining closest Centroid (mean of cluster)
  distance, (average) sum of pairwise distance,
• Refer to the Evaluation part
• A dendrogram is a tree that shows clustering
  process.

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Dendrogram

• Cluster 1, 2, 4, 5, 6, 7 into two clusters
  (centriod distance)
• 1
• 2
• 4
• 5
• 6
• 7

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An example to show different Links

• Single link
• Merge the nearest clusters measured by the
  shortest edge between the two
• (((A B) (C D)) E)
• Complete link
• Merge the nearest clusters measured by the
  longest edge between the two
• (((A B) E) (C D))
• Average link
• Merge the nearest clusters measured by the
  average edge length between the two
• (((A B) (C D)) E)

B
A
E
C
D
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Divisive

• All instances belong to one cluster (top-down)
• To find an optimal division at each layer
  (especially the top one) is computationally
  prohibitive.
• One heuristic method is based on the Minimum
  Spanning Tree (MST) algorithm
• Connecting all instances with MST (O(N2))
• Repeatedly cut out the longest edges at each
  iteration until some stopping criterion is met or
  until one instance remains in each cluster.

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COBWEB

• Building a conceptual hierarchy incrementally
• Category Utility
• ?k?i?jP(fivij)P(fivijck)P(ckfivij)
• All categories ck, all features fi, all feature
  values vij
• It attempts to maximize both the probability that
  two objects in the same category have values in
  common and the probability that objects in
  different categories will have different property
  values

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• Processing one instance at a time by evaluating
• Placing the instance in the best existing
  category
• Adding a new category containing only the
  instance
• Merging of two existing categories into a new one
  and adding the instance to that category
• Splitting of an existing category into two and
  placing the instance in the best new resulting
  category

Grandparent
Grandparent
Split
Parent
Child 2
Child 1
Merge
Child 2
Child 1
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Density-based

• BBSCAN Density-Based Clustering of Applications
  with Noise
• It grows regions with sufficiently high density
  into clusters and can discover clusters of
  arbitrary shape in spatial databases with noise.
• Many existing clustering algorithms find
  spherical shapes of clusters
• DEBSCAN defines a cluster as a maximal set of
  density-connected points.

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• Defining density and connection
• ?-neighborhood of an object x (core object) (M,
  P, O)
• MinPts of objects within ?-neighborhood (say, 3)
• directly density-reachable (Q from M, M from P)
• density-reachable (Q from P, P not from Q)
  asymmetric
• density-connected (O, R, S) symmetric for
  border points
• What is the relationship between DR and DC?

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• Clustering with DBSCAN
• Search for clusters by checking the
  ?-neighborhood of each instance x
• If the ?-neighborhood of x contains more than
  MinPts, create a new cluster with x as a core
  object
• Iteratively collect directly density-reachable
  objects from these core object and merge
  density-reachable clusters
• Terminate when no new point can be add to any
  cluster
• DBSCAN is sensitive to the thresholds of density,
  but it is many folds faster than CLARANS
• Time complexity O(N log N) if a spatial index is
  used, O(N2) otherwise

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Dealing with Large Data

• Key ideas
• Reducing the number of instances yet to maintain
  the distribution
• Identifying relevant subspaces where clusters
  possibly exist
• Using summarized information to avoid repeated
  data access
• Sampling
• CLARA (Clustering LARge Applications) working on
  samples instead of the whole data
• CLARANS (Clustering Large Applications based on
  RANdomized Search)

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• Grid STING (STatistical INformation Grid)
• Statistical parameters of higher-level cells can
  easily be computed from those of lower-level
  cells
• Attribute-independent count
• Attribute-dependent mean, standard deviation,
  min, max
• Type of distribution normal, uniform,
  exponential, or unknown
• Irrelevant cells can be removed

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Representatives

• BIRCH using Clustering Feature (CF) and CF tree
• A cluster feature is a triplet about sub-clusters
  of instances (N, LS, SS)
• N - the number of instances, LS linear sum, SS
  square sum
• Two thresholds branching factor and the max
  number of children per non-leaf node
• Two phases
• Build an initial in-memory CF tree
• Apply a clustering algorithm to cluster the leaf
  nodes in CF tree
• CURE (Clustering Using REpresentitives) is
  another example

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• Taking advantage of the property of density
• If its dense in higher dimensional subspaces, it
  should be dense in some lower dimensional
  subspaces
• CLIQUE (CLustering In QUEst)
• With high dimensional data, there are many void
  subspaces
• Using the property identified, we can start with
  dense lower dimensional data
• CLIQUE is a density-based method that can
  automatically find subspaces of the highest
  dimensionality such that high-density clusters
  exist in those subspaces

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Chameleon

• A hierarchical Clustering Algorithm Using Dynamic
  Modeling
• Observations on the weakness of CURE and ROCK

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Summary

• There are many clustering algorithms
• Good clustering algorithms maximize inter-cluster
  dissimilarity and intra-cluster similarity
• Without prior knowledge, it is difficult to
  choose the best clustering algorithm.
• Clustering is an important tool for outlier
  analysis.

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Bibliography

• I.H. Witten and E. Frank. Data Mining Practical
  Machine Learning Tools and Techniques with Java
  Implementations. 2000. Morgan Kaufmann.
• M. Kantardzic. Data Mining Concepts, Models,
  Methods, and Algorithms. 2003. IEEE.
• J. Han and M. Kamber. Data Mining Concepts and
  Techniques. 2001. Morgan Kaufmann.
• M. H. Dunham. Data Mining Introductory and
  Advanced Topics.