CLUSTERING

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CLUSTERING
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Overview

• Definition of Clustering
• Existing clustering methods
• Clustering examples

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Definition

• Clustering can be considered the most important
  unsupervised learning technique so, as every
  other problem of this kind, it deals with finding
  a structure in a collection of unlabeled data.
• Clustering is the process of organizing objects
  into groups whose members are similar in some
  way.
• A cluster is therefore a collection of objects
  which are similar between them and are
  dissimilar to the objects belonging to other
  clusters.

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Why clustering?

• A few good reasons ...
• Simplifications
• Pattern detection
• Useful in data concept construction
• Unsupervised learning process

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Where to use clustering?

• Data mining
• Information retrieval
• text mining
• Web analysis
• medical diagnostic

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Major Existing clustering methods

• Distance-based
• Hierarchical
• Partitioning
• Probabilistic

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Measuring Similarity

• 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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Hierarchical clustering

• Agglomerative (bottom up)
• start with 1 point (singleton)
• recursively add two or more appropriate clusters
• Stop when k number of clusters is achieved.

• Divisive (top down)
• Start with a big cluster
• Recursively divide into smaller clusters
• Stop when k number of clusters is achieved.

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general steps of hierarchical clustering

• Given a set of N items to be clustered, and an
  NN distance (or similarity) matrix, the basic
  process of hierarchical clustering (defined by
  S.C. Johnson in 1967) is this
• Start by assigning each item to a cluster, so
  that if you have N items, you now have N
  clusters, each containing just one item. Let the
  distances (similarities) between the clusters the
  same as the distances (similarities) between the
  items they contain.
• Find the closest (most similar) pair of clusters
  and merge them into a single cluster, so that now
  you have one cluster less.
• Compute distances (similarities) between the new
  cluster and each of the old clusters.
• Repeat steps 2 and 3 until all items are
  clustered into K number of clusters

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• Exclusive vs. non exclusive clustering
• In the first case data are grouped in an
  exclusive way, so that if a certain datum belongs
  to a definite cluster then it could not be
  included in another cluster. A simple example of
  that is shown in the figure below, where the
  separation of points is achieved by a straight
  line on a bi-dimensional plane.
• On the contrary the second type, the overlapping
  clustering, uses fuzzy sets to cluster data, so
  that each point may belong to two or more
  clusters with different degrees of membership.

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Partitioning clustering

• Divide data into proper subset
• recursively go through each subset and relocate
  points between clusters (opposite to visit-once
  approach in Hierarchical approach)
• This recursive relocation higher quality cluster

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Probabilistic clustering

1. Data are picked from mixture of probability
   distribution.
2. Use the mean, variance of each distribution as
   parameters for cluster
3. Single cluster membership

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Single-Linkage Clustering(hierarchical)

• The NN proximity matrix is D d(i,j)
• The clusterings are assigned sequence numbers
  0,1,......, (n-1)
• L(k) is the level of the kth clustering
• A cluster with sequence number m is denoted (m)
• The proximity between clusters (r) and (s) is
  denoted d (r),(s)

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The algorithm is composed of the following steps

• Begin with the disjoint clustering having level
  L(0) 0 and sequence number m 0.
• Find the least dissimilar pair of clusters in the
  current clustering, say pair (r), (s), according
  tod(r),(s) min d(i),(j)where the
  minimum is over all pairs of clusters in the
  current clustering.

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The algorithm is composed of the following
steps(cont.)

• Increment the sequence number m m 1. Merge
  clusters (r) and (s) into a single cluster to
  form the next clustering m. Set the level of this
  clustering toL(m) d(r),(s)
• Update the proximity matrix, D, by deleting the
  rows and columns corresponding to clusters (r)
  and (s) and adding a row and column corresponding
  to the newly formed cluster. The proximity
  between the new cluster, denoted (r,s) and old
  cluster (k) is defined in this wayd(k),
  (r,s) min d(k),(r), d(k),(s)
• If all objects are in one cluster, stop. Else, go
  to step 2.

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Hierarchical clustering example

• Lets now see a simple example a hierarchical
  clustering of distances in kilometers between
  some Italian cities. The method used is
  single-linkage.
• Input distance matrix (L 0 for all the
  clusters)

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• The nearest pair of cities is MI and TO, at
  distance 138. These are merged into a single
  cluster called "MI/TO". The level of the new
  cluster is L(MI/TO) 138 and the new sequence
  number is m 1.Then we compute the distance
  from this new compound object to all other
  objects. In single link clustering the rule is
  that the distance from the compound object to
  another object is equal to the shortest distance
  from any member of the cluster to the outside
  object. So the distance from "MI/TO" to RM is
  chosen to be 564, which is the distance from MI
  to RM, and so on.

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• After merging MI with TO we obtain the following
  matrix

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• min d(i,j) d(NA,RM) 219 gt merge NA and RM
  into a new cluster called NA/RML(NA/RM) 219m
  2

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• min d(i,j) d(BA,NA/RM) 255 gt merge BA and
  NA/RM into a new cluster called
  BA/NA/RML(BA/NA/RM) 255m 3

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• min d(i,j) d(BA/NA/RM,FI) 268 gt merge
  BA/NA/RM and FI into a new cluster called
  BA/FI/NA/RML(BA/FI/NA/RM) 268m 4

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• Finally, we merge the last two clusters at level
  295.
• The process is summarized by the following
  hierarchical tree

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K-mean algorithm

• It accepts the number of clusters to group data
  into, and the dataset to cluster as input values.
  
• It then creates the first K initial clusters (K
  number of clusters needed) from the dataset by
  choosing K rows of data randomly from the
  dataset. For Example, if there are 10,000 rows of
  data in the dataset and 3 clusters need to be
  formed, then the first K3 initial clusters will
  be created by selecting 3 records randomly from
  the dataset as the initial clusters. Each of the
  3 initial clusters formed will have just one row
  of data.

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3. The K-Means algorithm calculates the
Arithmetic Mean of each cluster formed in the
dataset. The Arithmetic Mean of a cluster is the
mean of all the individual records in the
cluster. In each of the first K initial
clusters, their is only one record. The
Arithmetic Mean of a cluster with one record is
the set of values that make up that record. For
Example if the dataset we are discussing is a set
of Height, Weight and Age measurements for
students in a University, where a record P in the
dataset S is represented by a Height, Weight and
Age measurement, then P Age, Height,
Weight. Then a record containing
the measurements of a student John, would be
represented as John 20, 170, 80 where John's
Age 20 years, Height 1.70 meters and Weight
80 Pounds. Since there is only one record in each
initial cluster then the Arithmetic Mean of a
cluster with only the record for John as a member
20, 170, 80.
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• Next, K-Means assigns each record in the dataset
  to only one of the initial clusters. Each record
  is assigned to the nearest cluster (the cluster
  which it is most similar to) using a measure of
  distance or similarity like the Euclidean
  Distance Measure or Manhattan/City-Block Distance
  Measure.
• K-Means re-assigns each record in
  the dataset to the most similar cluster
  and re-calculates the arithmetic mean of all the
  clusters in the dataset. The arithmetic mean of a
  cluster is the arithmetic mean of all the records
  in that cluster. For Example, if a cluster
  contains two records where the record of the set
  of measurements for John 20, 170, 80 and
  Henry 30, 160, 120, then the arithmetic mean
  Pmean is represented as Pmean Agemean,
  Heightmean, Weightmean).  Agemean (20 30)/2,
  Heightmean (170 160)/2 and Weightmean (80
  120)/2. The arithmetic mean of this cluster
  25, 165, 100. This new arithmetic mean becomes
  the center of this new cluster. Following the
  same procedure, new cluster centers are formed
  for all the existing clusters.

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• K-Means re-assigns each record in the dataset to
  only one of the new clusters formed. A record or
  data point is assigned to the nearest cluster
  (the cluster which it is most similar to) using a
  measure of distance or similarity 
• The preceding steps are repeated until stable
  clusters are formed and the K-Means clustering
  procedure is completed. Stable clusters are
  formed when new iterations or repetitions of the
  K-Means clustering algorithm does not create new
  clusters as the cluster center or Arithmetic Mean
  of each cluster formed is the same as the old
  cluster center. There are different techniques
  for determining when a stable cluster is formed
  or when the k-means clustering algorithm
  procedure is completed.