Chapter 6 Cluster Analysis

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Chapter 6 Cluster Analysis

• By Jinn-Yi Yeh Ph.D.
• 4/21/2009

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Outline

• Chapter Objective
• 6.1 Clustering Concepts
• 6.2 Similarity Measures
• 6.3 Agglomerative Hierarchical Clustering
• 6.4 Partitional Clustering
• 6.5 Incremental Clustering

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Chapter Objectives

• Distinguish between different representations of
  clusters and different measures of similarities.
• Compare basic characteristics of agglomerative-
  and partitional-clustering algorithms.
• Implement agglomerative algorithms using
  single-link or complete-link measures of
  similarity.

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Chapter Objectives(cont.)

• Derive the K-means method for partitional
  clustering and analysis of its complexity.
• Explain the implementatation of
  incremental-clustering algorithms and its
  advantages and disadvantages.

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What is Cluster Analysis?

• Cluster analysis is a set of methodologies for
  automatic classification of samples into a number
  of group using a measure of association.
• Input
• A set of samples
• A measure of similarity(or dissmislarity) between
  two samples.
• Output
• A number of groups(cluster)
• A structure of partition
• A generalized description of every cluster

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6.1 Clustering Concepts

• Samples for clustering are represented as a
  vector of measurements, or more formally, as a
  point in a multidimensional space.
• Samples within a valid cluster are more similar
  to each other than they are to a sample belonging
  to a different cluster.

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Clustering Concepts(cont.)

• Clustering methodology is particularly
  appropriate for the exploration of
  interrelationships among samples to make a
  preliminary assessment of the sample structure.
• It is very difficult for humans to intuitively
  interpret data embedded in a high-dimensional
  space.

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Table 6.1
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Unsupervised Classification

• The samples in these data sets have only input
  dimensions, and the learning process is
  classified as unsupervised.
• The objective is to construct decision
  boundaries(classification surfaces).

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Problem of Clustering

• Data can reveal clusters with different shapes
  and sizes in an n-dimensional data space.
• Resolution(fine vs coarse)
• Euclidean 2D space

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Objective Criterion

• Input to a cluster analysis
• (X, s) or (X, d)
• X is a set of descriptions of samples s
  measures for similarity between samples
  dmeasures for dissimilarity (distance) between
  samples

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Objective Criterion(cont.)

• Output to a cluster analysis
• a partition ? G1, G2, , GN where Gk, k 1,
  , N is a crisp subset of X such that
• The members G1, G2, , GN of ? are called
  clusters.

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Formal Description of Discovered Clusters

• Represent a cluster of points in an n-dimensional
  space (samples) by their centroid or by a set of
  distant (border) points in a cluster.
• Represent a cluster graphically using nodes in a
  clustering tree.
• Represent clusters by using logical expression on
  sample attributes.

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Selection of Clustering Technique

• There is no clustering technique that is
  universally applicable in uncovering the variety
  of structures present in multidimensional data
  sets.
• The user's understanding of the problem and the,
  corresponding data types will be the best
  criteria to select the appropriate method.

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Selection of Clustering Technique(cont.)

• Most clustering algorithms are based on the
  following two popular approaches
• Hierarchical clustering
• organize data in a nested sequence of groups,
  which can be displayed In the form of a
  dendrogram or a tree structure.
• Iterative square-error partitional clustering
• attempt to obtain that partition which minimizes
  the within-cluster scatter or maximizes the
  between-cluster scatter.

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Selection of Clustering Technique(cont.)

• To guarantee that an optimum solution has been
  obtained, one has to examine all possible
  partitions of N samples of n-dimensions into K
  clusters (for a given K).
• Notice that the number of all possible partitions
  of a set of N objects into K clusters is given
  by

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6.2 Similarity Measures

• xi ? X, i 1, , n, is represented by a vector
  xi xi1, xi2, , xim.
• mthe number of dimensions (features) of samples
• nthe total number of samples

features
.
samples
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Describe Features

• These features can be either quantitative or
  qualitative descriptions of the object.
• Quantitative features can be subdivided as
• continuous valuesreal numbers where Pj ? R
• discrete valuesbinary numbers Pj 0, 1, or
  integers Pj ? Z
• interval valuesPj xij 20, 20 lt xij lt 40,
  xij 40

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Describe Features(cont.)

• Qualitative features can be
• nominal or unorderedcolor is "blue" or "red"
• ordinalmilitary rank with values "general",
  "colonel", etc.

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Similarity

• The word similarity in clustering means that
  the value of s(x, x) is large when x and x are
  two similar samples the value of s(x, x) is
  small when x and x are not similar.
• Similarity measure s is symmetric
• s(x, x) s(x, x) , ? x, x ? X
• Similarity measure is normalized
• 0 s(x, x) 1 , ? x, x ? X

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Dissimilarity

• Dissimilarity measure is denoted by d(x, x) ,
  ?x, x ? X, and it is frequently called a
  distance
• d(x, x) 0, ? x, x ? X
• d(x, x) d(x, x), ?x, x ? X
• if it is accepted as a metric distance measure,
  then a triangular inequality is required
• d(x, x) d(x, x) d(x, x), ?x, x, x?X
  (triangular inequality)

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Metric Distance Measure

• Euclidean distance in m-dimensional feature
  space
• L1 metric or city block distance

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Metric Distance Measure

• Minkowski metric (includes the Euclidean distance
  and city block distance as special cases)
• p 1, then d coincides with L1 distance
• p 2, d is identical with the Euclidean metric

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Example

• 4-dimensional vectors x1 l, 0, 1, 0 and x2
  2, 1, - 3, - 1 these distance measures are
  d1 1 1 4 1 7d2 (1 1 16 1)1/2
  4.36d3 (1 1 64 1)1/3 4.06

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Measures of Similarity

• Cosine-correlation
• It is easy to see that
• Example
• scos(xi,xj) (2030) / (2½.15½) -0.18

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Contingency Table

• athe number of binary attributes of samples xi
  and xj such that xik xjk 1.
• bthe number of binary attributes of samples xi
  and xj such that xik 1 and xjk 0.
• cthe number of binary attributes of samples xi
  and xj such that xik 0 and xjk 1.
• dthe number of binary attributes of samples xi
  and xj such that xik xjk 0

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Example

• if xi and xj are 8-dimensional vectors with
  binary feature values
• xi0,0,1,1,0,1,0,1
• xj0,1,1,0,0,1,0,0
• the values of the parameters introduced are
• a2,b2,c1,d3

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Similarity Measures with Binary Data

• Simple Matching Coeficient (SMC)
• Ssmc(xi, xj) (a d) / (a b c d)
• Jaccard Coefficient
• Sjc(xi, xj) a / (a b c )
• Raos Coefficient
• Src(xi, xj) a / (a b c d)
• Example
• Ssmc(xi, xj) 5/8, Sjc(xi, xj) 2/5, and
  Src(xi, xj) 2/8.

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Mutual Neighbor Distance

• MND(xi, xj) NN(xi, xj) NN(xj, xi)
• NN(xi, xj) is the neighbor number of xj with
  respect to xi.
• If xi is the closest point to xj then NN(xi, xj)
  is equal to 1
• if it is the second closest point to xjthen
  NN(xi, xj) is equal to 2

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Example

• NN(A, B) NN(B, A) 1 ? MND(A, B) 2
• NN(B, C) 1, NN(C, B) 2 ? MND(B, C) 3
• A and B are more similar than B and C using MND
  measure

Figure 6.3 A and B are more similar than B and C
using the MND measure
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Example

• NN(A, B) 1, NN(B, A) 4 ? MND(A, B) 5
• NN(B, C) 1, NN(C, B) 2 ? MND(B, C) 3
• After changes in the context, B and C are more
  similar than A and B using MND measure

Figure 6.4 After changes in the context, B and C
are more similar than A and B using the MND
measure
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Distance Measure Between Clusters

• These measures are an essential part in
  estimating the quality of a clustering process,
  and therefore they are part of clustering
  algorithms
• 1) Dmin(Ci,Cj)minpi-pjwhere pi?Ci and pj?Cj
• 2) Dmean(Ci,Cj)mi-mjwhere mi and mj are
  centriods of Ci and Cj
• 3) Davg(Ci,Cj)1/(ninj) ??pi-pjwhere pi?Ci and
  pj?Cj , and ni and nj are the numbers of samples
  in clusters Ci and Cj
• 4) Dmax(Ci,Cj) maxpi-pjwhere pi?Ci and pj?Cj

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6.3 Agglomerative Hierarchical Clustering

• Most procedures for hierarchical clustering are
  not based on the concept of optimization, and the
  goal is to find some approximate, suboptimal
  solution, using iterations for improvement of
  partitions until convergence.
• Algorithms of hierarchical cluster analysis are
  divided into the two categories
• divisible algorithms
• agglomerative algorithms.

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Divisible VS Agglomerative

• Divisible Algorithms
• Entire set of samples X ? subsets? smaller
  subsets ?
• Agglomerative Algorithms
• Bottom-up process
• Regards each object as an initial cluster ?Merged
  into a coarser partition ? One larger cluster
• Agglomerative algorithms are more frequently used
  in real-world applications than divisible methods
  

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Agglomerative Hierarchical Clustering Algorithms

• single-link complete-link
• These two basic algorithms differ only in the way
  they characterize the similarity between a pair
  of clusters.

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Steps

• 1. Place each sample in its own cluster.
  Construct the list of inter-cluster distances for
  all distinct unordered pairs of samples, and sort
  this list in ascending order.
• 2. Step through the sorted list of distances,
  forming for each distinct threshold value dk a
  graph of the samples where pairs of samples
  closer than dk are connected into a new cluster
  by a graph edge. If all the samples are members
  of a connected graph, stop. Otherwise, repeat
  this step.
• 3. The output of the algorithm is a nested
  hierarchy of graphs, which can be cut at a
  desired dissimilarity level forming a partition
  (clusters) identified by simple connected
  components in the corresponding sub-graph.

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Example

• x1(0,2) , x2 (0, 0) , x3 (1.5, 0) , x4 (5,
  0), x5 (5, 2)

Figure 6.6 Five two-dimensional samples for
clustering
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Example

• The distances between these points using the
  Euclidian measure
• d(x1,x2)2,d(x1,x3)2.5,d(x1,x4)5.39,d(x1,x5)5
• d(x2,x3)1.5,d(x2,x4)5,d(x2,x5)5.29
• d(x3,x4)3.5,d(x3,x5)4.03
• d(x4,x5)2

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Example-Single Link

• Final result x1,x2,x3 and x4,x5

Figure 6.7 Dendrogram by single-link method for
the data set in Figure 6.6
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Example-Complete Link

• Final result x1 and x2,x3 and x4,x5

Figure 6.8 Dendrogram by complete-link method
for the data set in Figure 6.6
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Chameleon Clustering Algorithm

• Unlike traditional agglomerative methods,
  Chameleon is a clustering algorithm that tries to
  improve the clustering quality by using a more
  elaborate criterion when merging two clusters.
• Two clusters will be merged if the
  interconnectivity and closeness of the merged
  clusters is very similar to the interconnectivity
  and closeness of the two individual clusters
  before merging.

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Chameleon Clustering Algorithm - Steps

• Step1creates a graph G (V, E)
• v ? V represents a data sample
• a weighted edge e(vi, vj)
• Graph G subgraphs
• Step2Chameleon determines the similarity between
  each pair of elementary clusters Ci and Cj
  according to their relative interconnectivity
  RI(Ci, Cj) and their relative closeness RC(Ci,
  Cj).

min-cut
min-cut
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Chameleon Clustering Algorithm - Steps(cont.)

• Interconnectivitythe total weight of edges that
  are removed when a min-cut is performed
• relative interconnectivity RI (Ci, Cj) the
  ratio between the interconnectivity of the merged
  cluster Ci and Cj to the average
  interconnectivity of Ci and Cj.
• closeness the average weight of the edges that
  are removed when a min-cut is performed on the
  cluster.
• relative closeness RC(Ci, Cj) the ratio between
  the closeness of the merged cluster of Ci and Cj
  to the average internal closeness of Ci and Cj

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Chameleon Clustering Algorithm - Steps(cont.)

• Step3 Compute similarity function
• a is a parameter between 0 and 1
• a 1, give equal weight to both measures alt1,
  place more emphasis on RI(Ci, Cj)
• Chameleon can automatically adapt to the internal
  characteristics of the clusters and it is
  effective in discovering arbitrarily-shaped
  clusters of varying density. However, algorithm
  is ineffective for high-dimensional data because
  its time complexity for n samples is O(n2).

RC(Ci,Cj) RI(Ci,Cj)a
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6.4 Partitional Clustering

• advantage in applications involving large data
  sets for which the construction of a dendrogram
  is computationally very complex.
• criterion function
• locallya subset of samples
• Minimal MND(Mutual Neighbor Distance)
• globallyall of the samples
• Euclidean square-error measure
• Therefore, identifying high-density regions in
  the data space is a basic criterion for forming
  clusters.

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Partitional Clustering(cont.)

• The most commonly used partitional-clustering
  strategy is based on the square-error criterion.
• objectiveobtain the partition that, for a fixed
  number of clusters, minimizes the total
  square-error.

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Partitional Clustering(cont.)

• Suppose that the given set of N samples in an
  n-dimensional space has somehow been partitioned
  into K clusters C1, C2, , Ck.
• Each Ck has nk samples and each sample is in
  exactly one cluster, so that ? nk N, where k
  1, , K.
• The mean vector Mk of cluster Ck is defined as
  the centroid of the cluster or

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Partitional Clustering(cont.)

• within-cluster variation
• The square-error for the entire clustering space

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K-means partitional-clustering algorithm

• employing a square-error criterion
• Steps
• select an initial partition with K clusters
  containing randomly chosen samples, and compute
  the centroids of the clusters,
• generate a new partition by assigning each sample
  to the closest cluster center,
• compute new cluster centers as the centroids of
  the clusters,
• repeat steps 2 and 3 until an optimum value of
  the criterion function is found (or until the
  cluster membership stabilizes).

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Example

• x1(0,2) , x2 (0, 0) , x3 (1.5, 0) , x4 (5,
  0), x5 (5, 2)
• Random distribution
• C1x1,x2,x4 and C2x3,x5
• The centriods for these two clusters are
• M1 (005)/3, (200)/3 1.66, 0.66
• M2 (1.55)/2, (02)/2 3.25, 1.00

Figure 6.6 Five two-dimensional samples for
clustering
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Example(cont.)

• Within-cluster variations
• e12 (0-1.66)2(2-0.66)2 (0-1.66)2
  (0-0.66)2(5-1.66)2 (0-0.66)2 19.36
• e22 (1.5-3.25)2 (0-1)2 (5-3.25)2
  (2-1)2 8.12
• Total square error
• E2 e12 e22 19.36 8.12 27.48

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Example(cont.)

• Reassign all samples
• d(M1, x1) 2.14 and d(M2, x1) 3.40 ? x1 ? C1
• d(M1, x2) 1.79 and d(M2, x2) 3.40 ? x2 ? C1
• d(M1, x3) 0.83 and d(M2, x3) 2.01 ? x3 ? C1
• d(M1, x4) 3.41 and d(M2, x4) 2.01 ? x4 ? C2
• d(M1, x5) 3.60 and d(M2, x5) 2.01 ? x5 ? C2
• New clusters C1 x1, x2, x3 and C2 x4, x5
• New centroids M1 0.5, 0.67 and M2 5.0,
  1.0
• Errors
• e12 4.17 and e22 2.00
• E2 6.17

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Why K-means is so popular?

• Its time complexity is O(nkl), the algorithm has
  linear time complexity in the size of the data
  set.
• n is the number of samples
• k is the number of clusters
• l is the number of iterations taken by the
  algorithm to converge
• k and l are fixed
• Its space complexity is O(k n).
• It is an order-independent algorithm.

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Disadvantages of K-means algorithm

• A big frustration in using iterative
  partitional-clustering programs is the lack of
  guidelines available for choosing K-number of
  clusters.
• The K-means algorithm is very sensitive to noise
  and outlier data points
• K-mediodsuses the most centrally located object
  (mediods) in a cluster to be the cluster
  representative.

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6.5 Incremental Clustering

• There are more and more applications where it is
  necessary to cluster a large collection of data.
• large
• 1960sseveral thousand samples for clustering
• Nowmillions of samples of high dimensionality
• ProblemThere are applications where the entire
  data set cannot be stored in the main memory
  because of its size.

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Possible Approaches

• divide-and-conquer approach
• The data set can be stored in a secondary memory
  and subsets of this data are clustered
  independently, followed by a merging step to
  yield a clustering of the entire set.
• Incremental-clustering algorithm
• Data are stored in the secondary memory and data
  items are transferred to the main memory one at a
  time for clustering. Only the cluster
  representations are stored permanently in the
  main memory to alleviate space limitations.
• A parallel implementation of a clustering
  algorithm
• The advantages of parallel computers increase the
  efficiency of the divide-and-conquer approach.

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Incremental-Clustering Algorithm - Steps

• Assign the first data item to the first cluster.
• Consider the next data item. Either assign this
  item to one of the existing clusters or assign it
  to a new cluster. This assignment is done based
  on some criterion, e.g., the distance between the
  new item and the existing cluster centroids.
  After every addition of a new item to an existing
  cluster, recompute a new value for the centroid.
• Repeat step 2 till all the data samples are
  clustered.

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Features of Incremental-Clustering Algorithm

• Advantages
• The space requirements of the incremental
  algorithm are very small.
• centroids of the clusters
• Their time requirements are also small.
• algorithms are noniterative
• Disadvantages
• Not order-independence

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Example - Figure6.6

• x1(0,2) , x2 (0, 0) , x3 (1.5, 0) , x4 (5,
  0), x5 (5, 2)
• Inputx1?x2?x3?x4?x5
• the threshold level of similarity between
  clusters is d 3.
• Steps
• The first sample x1 will become the first cluster
  C1 x1. The coordinates of x1 will be the
  coordinates of the centroid M1 0, 2.

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Example(cont.)

• Start analysis of the other samples.
• Second sample x2 is compared with M1d(x2, M1)
  (02 22)1/2 2.0 lt 3 Therefore, x2 ? C1. New
  centroid will be M10, 1
• Third sample x3 is compared with the centroid M1
  (still the only centroid!) d(x3, M1) (1.52
  12) 1/2 1.8 lt 3 x3? C1 ? C1 x1, x2, x3 ?
  M1 0.5, 0.66
• Fourth sample x4 is compared with the centroid
  M1 d(x4, M1) (4.52 0.662)1/2 4.55 gt 3 C2
  x4 with the new centroid M2 5, 0.

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Example(cont.)

• Fifth example x5 is comparing with both cluster
  centroids d(x5, M1) (4.52 1.442) 1/2 4.72
  gt 3 d(x5, M2) (02 22)1/2 2 lt 3 C2 x4,
  x5 ? M2 5, 1
• All samples are analyzed and a final clustering
  solution C1 x1, x2, x3 and C2 x4, x5
• The reader may check that the result of the
  incremental-clustering process will not be the
  same if the order of the samples is different.

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Cluster Feature Vector

• Components of CF
• the number of points (samples) of the cluster
• the centroid of the cluster
• the radius of the cluster
• the square root of the average mean-squared
  distance from the centroid to the points in the
  cluster
• It is very important that we do not need the set
  of points in the cluster to compute a new CF.

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Birch clustering algorithm

• We have to mention that this technique is very
  efficient for two reasons
• CFs occupy less space than any other
  representation of clusters.
• CFs are sufficient for calculating all the values
  involved in making clustering decisions.

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K-nearest neighbor Algorithm

• If samples are with categorical data, then we do
  not have a method to calculate centroids as
  representatives of the clusters.
• K-nearest neighbor may be used to estimate
  distances (or similarities) between samples and
  existing clusters.

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K-nearest neighbor Algorithm -Steps

• The distances between new sample and all previous
  samples, already classified into clusters, are
  computed.
• The distances are sorted in increasing order, and
  K samples with smallest distance values are
  selected.
• Voting principle is applied New sample will be
  added (classified) to the cluster that belongs to
  the largest number out of K selected samples.

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Example

• Given six 6-dimensional categorical samples
• X1 A, B, A, B, C, B
• X2 A, A, A, B, A, B
• X3 B, B, A, B, A, B
• X4 B, C, A, B, B, A
• X5 B, A, B, A, C, A
• X6 A, C, B, A, B, B
• Clustered into two clustersC1 X1, X2, X3
  and C2 X4, X5, X6
• Classify the new sample Y A, C, A, B, C, A

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Example(cont.)

• Using SMC measure
• Using 1-nearest neighbor rule (K 1) new sample
  cannot be classified because there are two
  samples (X1 and X4) with the same, highest
  similarity (smallest distances), and one of them
  is in the class C1 and the other in the class C2.
  

Similarities with elements in C1 SMC(Y, X1)
4/6 0.66 SMC(Y, X2) 3/6 0.50
SMC(Y, X3) 2/6 0.33
Similarities with elements in C2 SMC(Y, X4)
4/6 0.66 SMC(Y, X5) 2/6 0.33
SMC(Y, X6) 2/6 0.33
similarity 0.66?0.66?0.50?0.33?0.33?0.33
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Example(cont.)

• Using 3-nearest neighbor rule (K 3), and
  selecting three largest similarities in the set,
  we can see that two samples (X1 and X2) belong to
  class C1, and only one sample to class C2.
• Using simple voting system Y ? C1 class.

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How to evaluate a clustering algorithm ?

• The first step in evaluation is actually an
  assessment of the data domain rather than the
  clustering algorithm itself .
• Cluster validity is the second step, when we
  expect to have our data clusters.
• It is subjective .

70
Validation Studies for Clustering Algorithms

• External assessment
• Compares the discovered structure to an a priori
  structure.
• Internal examination
• Try to determine if the discovered structure is
  intrinsically appropriate for the data.
• Both assessments are subjective and
  domain-dependent.
• Relative test
• Compares the two structures obtained either from
  different cluster methodologies or by using the
  same methodology but with different clustering
  parameters, such as the order of input samples.
• We still need to resolve the question how to
  select the structures for comparison.

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Keep in Your Mind

• Every clustering algorithm will find clusters in
  a given data set whether they exist or not. the
  data should, therefore, be subjected to tests for
  clustering tendency before applying a clustering
  algorithm, followed by a validation of the
  clusters generated by the algorithm.
• There is no best clustering algorithm therefore
  a user is advised to try several algorithms on a
  given data set.

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• THE END