Clustering Algorithms BIRCH and CURE

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Clustering AlgorithmsBIRCH and CURE

• Anna Putnam
• Diane Xu

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What is Cluster Analysis?

• Cluster Analysis is like Classification, but the
  class label of each object is not known.
• Clustering is the process of grouping the data
  into classes or clusters so that objects within a
  cluster have high similarity in comparison to one
  another, but are very dissimilar to objects in
  other clusters.

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Applications for Cluster Analysis

• Marketing discover distinct groups in customer
  bases, and develop targeted marketing programs.
• Land use Identify areas of similar land use.
• Insurance Identify groups of motor insurance
  policy holders with a high average claim cost.
• City-planning Identify 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.
• Biology plant and animal taxonomies, genes
  functionality
• Also used for Pattern recognition, data analysis,
  and image processing.
• Clustering is Studied in Data mining, statistics,
  machine learning, spatial database technology,
  biology, and marketing.

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Some 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.
• Ability to deal with noisy data
• Insensitivity to the order of input records
• High dimensionality
• Constraint-based clustering
• Interpretability and usability

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Categorization of Major Clustering Methods

• Partitioning Methods
• Density Based Methods
• Grid Based Methods
• Model Based Methods
• Hierarchical Methods

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Partitioning Methods

• Given a database of n objects or data tuples, a
  partitioning method constructs k partitions of
  the data, where each partition represents a
  cluster and kn.
• It classifies the data into k groups, which
  together satisfy the following requirements
• each group must contain at least one object, and
• each object must belong to exactly one group.
• Given k, the number of partitions to construct a
  partitioning method creates an initial
  partitioning.
• It then uses an iterative relocation technique
  that attempts to improve the partitioning by
  moving objects from one group to another.

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Density Based Methods

• Most partitioning methods cluster objects based
  on the distance between objects.
• Such methods can find only spherical-shaped
  clusters and encounter difficulty at discovering
  clusters of arbitrary density.
• The idea is to continue growing the given cluster
  as long as the density (number of objects or data
  points) in the neighborhood exceeds some
  threshold.
• DBSCAN and OPTICS are two density based methods.

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Grid-Based Methods

• Grid-based methods quantize the object space into
  a finite number of cells that form a gird
  structure.
• All the clustering operations are performed on
  the grid structure.
• The advantage of this approach is fast processing
  time
• STING, CLIQUE, and Wave-Cluster are examples of
  grid-based clustering algorithms.

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Model-based methods

• Hypothesize a model for each of the clusters and
  find the best fit of the data to the given model.
  
• A model-based algorithm may locate clusters by
  constructing a density function that reflects the
  spatial distribution of the data points.

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Hierarchical Methods

• Creates a hierarchical decomposition of the given
  set of data objects.
• Two approaches
• agglomerative (bottom-up) starts with each
  object forming a separate group. Merges the
  objects or groups close to one another until all
  of the groups are merged into one, or until a
  termination condition holds.
• divisive (top-down) starts with all the objects
  in the same cluster. In each iteration, a
  cluster is split up into smaller clusters, until
  eventually each object is one cluster, or until a
  termination condition holds.
• This type of method suffers from the fact that
  once a step (merge or split) is done, it can
  never be undone.
• BIRCH and CURE are examples of hierarchical
  methods.

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BIRCH

• Balanced Iterative Reducing and Clustering Using
  Hierarchies
• Begins by partitioning objects hierarchically
  using tree structures, and then applies other
  clustering algorithms to refine the clusters.

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Clustering Problem

• Given the desired number of cluster K and a
  dataset of N points, and a distance-based
  measurement function, we are asked to find a
  partition of the dataset that minimizes the value
  of the measurement function.
• Due to an abundance of local minima, there is
  typically no way to find a global minimal
  solution without trying all possible partitions.
• Constraint The amount of memory available is
  limited (typically much smaller than the data set
  size) and we want to minimize the time required
  for I/O.

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Previous Work

• Probability-based approaches
• Typically make the assumption that probability
  distributions on separate attributes are
  statistically independent of each other.
• Makes updating and storing the clusters very
  expensive.
• Distance-based approaches
• Assume that all data points are given in advance
  and can be scanned frequently.
• Ignore the fact that not all data points in the
  dataset are equally important with respect to the
  clustering purpose.
• Are global methods. They inspect all data points
  or all current clusters equally no matter how
  close or far away they are.

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Contributions of BIRCH

• BIRCH is local (instead of global). Each
  clustering decision is made without scanning all
  data points or currently existing clusters.
• BIRCH exploits the observation that the data
  space is usually not uniformly occupied, and
  therefore not every data point is equally
  important for clustering purposes.
• BIRCH makes full use of available memory to
  derive the finest possible subclusters while
  minimizing I/O costs.

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Definitions

• Centroid
• Diameter D

• Radius R

• Given the centroids of two clusters and
  
• -The centroid Euclidian distance D0 -The
  centroid Manhattan distance D1

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Definitions (cont.)

• Given N1 d-dimensional data points in a
  cluster where i1,2,,N1, and N2
  data points in another cluster where j
  N11, N12,,N1N2
• -Average inter-cluster distance D2 -Average
  intra-cluster distance D3
• -Variance increase distance D4

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Clustering Feature

• A Clustering Feature (CF) is a triple summarizing
  the information that we maintain about a cluster.
• N is the number of data points in the cluster
• is the linear sum of the N data points
• SS is the square sum of the N data points

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CF Additivity Theorem

• Assume that and
• are the CF vectors of two disjoint clusters.
  Then the CF vector of th cluster that is formed
  by merging the two disjoint clusters is
• From the CF definition and additivity theorem, we
  know that the CF vectors of clusters can be
  stored and calculated incrementally as clusters
  are merged.
• We can think of a cluster as a set of data
  points, but only the CF vector is stored as
  summary.

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CF Tree

• A CF tree is a height-balanced tree with two
  parameters
• Branching factor B
• Threshold T
• Each nonleaf node contains at most B entries of
  the form CFi,childi where i1,2,B.
• childi is a pointer to its i-th child node
• CFi is the CF of the sub-cluster represented by
  this child.
• A leaf node contains at most L entries, each of
  the form CFi where i1,2,,L.
• A leaf node also represents a cluster made up of
  all the subclusters represented by its entries.
• All entries in a leaf node must satisfy a
  threshold value T the diameter (or radius) has
  to be less than T.

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CF Tree
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Insertion into a CF Tree

• Given entry Ent
• Identify the appropriate leaf Starting from the
  root, it recursively descends the CF tree by
  choosing the closest child node according to a
  chosen distance metric D0,D1,D2,D3 or D4
• Modify the leaf When it reaches a leaf node, it
  finds the closest leaf entry, say Li and then
  tests whether Li can absorb Ent without
  violating the threshold condition. If so, the CF
  vector for Li is updated to reflect this. If
  not, an new entry for Ent is added to the leaf.
  If there is space on the leaf for this new
  entry, we are done, otherwise we must split the
  leaf node.

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Insertion into a CF Tree (cont.)

• Modify the path to the leaf After inserting
  Ent into a leaf, we must update the CF
  information for each nonleaf entry on the path to
  the leaf. Without a split, this simply involves
  adding CF vectors to reflect the addition of
  Ent. A leaf split requires us to insert a new
  nonleaf entry into the parent node, to describe
  the newly created leaf. If the parent has space
  for this entry, at all higher levels, we only
  need to update the CF vectors to reflect the
  addition of Ent. In general, however, we may
  have to split the parent as well, and so up to
  the root. If the root I split, the tree height
  increases by one.

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The BIRCH Clustering Algorithm
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Phase 1 Revisited
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Performance of BIRCH

• Tested on three datasets used as base Work load
  with D as the weighted average diameter. Each
  dataset consists of K clusters of 2-d data
  points. A cluster is characterized by the number
  of data points in it (n), its radius (r), and its
  center (c).
• Running time is basically linear wrt N, and does
  not depend on the order (unlike CLARANS).

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Performance

• In conclusion, BIRCH uses much less memory, but
  is faster, more accurate, and less
  order-sensitive compared with CLARANS.

BIRCH Clusters of DS1
CLARA Clusters of DS1
Actual Clusters of DS1
CLARANS Clusters of DS1
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Application to Real Dataset

• BIRCH used to filter real images
• Two similar images of trees with a partly cloudy
  sky as he background, taken at two different
  wavelengths (512x1024 pixels each).
• NIR Near-infrared band
• VIS Visible wavelength band
• Soil scientists recieve hundreds of such image
  pairs and try to first filter the trees from the
  background, and then filter the trees into sunlit
  leaves, shadows and branches for statistical
  analysis.

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Application (cont.)

• Applied BIRCH to the (NIT,VIS) value pairs for
  all pixels in an image.
• Weighted equally obtain 5 clusters
• Very bright part of sky
• Ordinary part of sky
• Clouds
• Sunlit leaves
• Tree branches and shadows on the trees
• Pulled out the part of the data corresponding to
  (5) and used BIRCH again. This time NIR was
  weighted 10 times heavier than VIS.

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CURE Clustering Using REpresentatives

• A Hierarchical Clustering Algorithm that Uses
  Partitioning

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Partitional Clustering

• Find k clusters optimizing some criterion
• (for example, minimize the squared-error)

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Hierarchical Clustering

• Use nested partitions and tree structures
• Agglomerative Hierarchical Clustering
• Initially each point is a distinct cluster
• Repeated merge the closest clusters

D_avg
D_min
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CURE

• CURE proposed by Guha, Rastogi Shim, 1998
• A new hierarchical clustering algorithm that uses
  a fixed number of points as representatives
  (partition)
• Centroid based approach uses 1 pt to represent
  cluster gt too little information sensitive to
  data shapes
• All point based approach uses all points to
  cluster gt too much information sensitive to
  outliers
• A constant number c of well scattered points in a
  cluster are chosen, and then shrunk toward the
  center of the cluster by a specified fraction
  alpha
• The clusters with the closest pair of
  representative points are merged at each step
• Stops when there are only k clusters left, where
  k can be specified

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Six Steps in CURE Algorithm
Draw Random Sample
Partially Cluster Partitions
Partition Sample
Data
Cluster Partial Clusters
Label Data In Disk
Eliminate Outliers
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Example
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CUREs Advantages

• More accurate
• Adjusts well to geometry of non-spherical shapes.
  
• Scales to large datasets
• Less sensitive to outliers
• More efficient
• Space complexity O(n)
• Time complexity O(n2logn) (O(n2) if
  dimensionality of data points is small)

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Feature Random Sampling

• Key idea apply CURE to a random sample drawn
  from the data set rather than the entire data
  set.
• Advantages
• Smaller size
• Filtering outliers
• Concerns may miss out or incorrectly identify
  certain clusters!
• Experimental results show that, with moderate
  sized random samples, we were able to obtain very
  good clusters.

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Feature Partitioning for Speedup

• Partition the sample space into p partitions,
  each of size n/p.
• Partially cluster each partition until the final
  number of clusters in each partition reduces to
  n/(pq). (q gt 1)
• Collect all partitions and run a second
  clustering pass on the n/p partial clusters
• Tradeoff sample size vs. accuracy

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Feature Labeling Data on Disk

• Input is a randomly selected sample.
• Have to assign the appropriate cluster labels to
  the remaining data points
• Each data point is assigned to the cluster
  containing the representative point closest to it
• Advantage using multiple points enables CURE to
  correctly to distribute the data points when
  clusters are non-spherical or non-union

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Feature Outliers Handling

• Random sampling filters out a majority of the
  outliers.
• The remaining few outliers in the random sample
  are distributed all over the sample space and
  gets further isolated.
• The clusters which are growing very slowly are
  identified and eliminated as outliers.
• Use a second level pruning to eliminate
  merging-together outliers outliers form very
  small clusters.

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Experiments

1. Parameter Sensitivity
2. Comparison with BIRCH
3. Scale-up

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Dataset Setup

• Experiment with data sets
• Data set 1 contains one big and two small
  circles.
• Data set 2 consists of 100 clusters with centers
  arranged in a grid pattern and data points in
  each cluster following a normal distribution with
  mean at the cluster center.

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Sensitivity Experiment

• CURE is very sensitive to its user-specified
  parameters.
• Shrinking fraction alpha.
• Sample size s
• Number of representatives c

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Shrinking Factor Alpha

• Alpha lt 0.2, reduces to the centroid-based
  algorithm
• Alpha gt 0.7, becomes similar to the all-points
  approach.
• 0.2 0.7 is a good range for alpha

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Random Sample Size

• Tradeoff between random sample size and accuracy

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Number of Representatives

• For smaller values of c, the quality of
  clustering suffers.
• For values of c greater than 10, CURE always
  found the right clusters.

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CURE vs. BIRCH quality of clustering

• BIRCH cannot distinguish between the big and
  small clusters.
• MST (all-point approach) merges the two
  ellipsoids.
• CURE successfully discovers the clusters in Data
  set 1.

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CURE vs. BIRCH Execution Time

• Run both on dataset2
• CURE execution time is always lower than BIRCH
• Partitioning improves CUREs running time by gt
  50
• As sample size goes up, CUREs execution time
  only slightly increases due to fixed sample size

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CURE Scale-up Experiment
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Conclusion

• CURE and BIRCH are two hierarchical clustering
  algorithms
• CURE adjusts well to clusters having
  non-spherical shapes and wide variances in size.
• CURE can handle large databases efficiently.

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Acknowledgement

• Sudipto Guha, Rajeev Rastogi, Kyuseok Shim
  CURE An Efficient Clustering Algorithm for Large
  Databases. SIGMOD Conference 1998 73-84
• Jiawei Han and Micheline Kamber, Data Mining
  Concepts and Techniques, Morgan Kaufmann
  Publishers, 2000.
• Tian Zhang, Raghu Ramakrishnan, Miron Livny
  BIRCH An Efficient Data Clustering Method for
  Very Large Databases. SIGMOD Conf. 1996 103-114