Download A Powerpoint presentation on Clustering

Clustering Analysis
(of Spatial Data and using Peano Count Trees)
(Ptree technology is patented by NDSU)
Notes:
1. over 100 slides – not going to go through each in detail.
Clustering Methods
A Categorization of Major Clustering Methods
Partitioning methods
Hierarchical methods
Density-based methods
Grid-based methods
Model-based methods
Clustering Methods based on Partitioning
 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
– k-means (MacQueen’67): Each cluster is represented by the
center of the cluster
– k-medoids or PAM method (Partition Around Medoids)
(Kaufman & Rousseeuw’87): Each cluster is represented by 1
object in the cluster (~ the middle object or median-like object)
The K-Means Clustering Method
Given k, the k-means algorithm is implemented in 4 steps (assumes partitioning
criteria is: maximize intra-cluster similarity and minimize inter-cluster
similarity. Of course, a heuristic is used. Method isn’t really an optimization)
1.
Partition objects into k nonempty subsets (or pick k initial means).
2.
Compute the mean (center) or centroid of each cluster of the current
partition (if one started with k means, then this step is done).
centroid ~ point that minimizes the sum of dissimilarities from the mean or the
sum of the square errors from the mean.
Assign each object to the cluster with the most similar (closest) center.
3.
Go back to Step 2
4.
Stop when the new set of means doesn’t change (or some other stopping
condition?)
k-Means
Step 1
Step 2
Step 3
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The K-Means Clustering Method
 Strength
– Relatively efficient: O(tkn),
• n is # objects,
• k is # clusters
• t is # iterations.
Normally, k, t << n.
 Weakness
– Applicable only when mean is defined (e.g., a vector space)
– Need to specify k, the number of clusters, in advance.
– It is sensitive to noisy data and outliers since a small number of
such data can substantially influence the mean value.
The K-Medoids Clustering Method
 Find representative objects, called medoids, (must be an actual object
in the cluster, where as the mean seldom is).
 PAM (Partitioning Around Medoids, 1987)
– starts from an initial set of medoids
– iteratively replaces one of the medoids by a non-medoid
– if it improves the aggregate similarity measure, retain the swap.
Do this over all medoid-nonmedoid pairs
– PAM works for small data sets. Does not scale for large data sets
 CLARA (Clustering LARge Applications) (Kaufmann,Rousseeuw, 1990)
Sub-samples then apply PAM
 CLARANS (Clustering Large Applications based on RANdom
Search) (Ng & Han, 1994): Randomized the sampling
PAM (Partitioning Around Medoids) (1987)
 Use real object to represent the cluster
– Select k representative objects arbitrarily
– For each pair of non-selected object h and selected object i,
calculate the total swapping cost TCi,h
– For each pair of i and h,
• If TCi,h < 0, i is replaced by h
• Then assign each non-selected object to the most similar
representative object
– repeat steps 2-3 until there is no change
CLARA (Clustering Large Applications) (1990)
 CLARA (Kaufmann and Rousseeuw in 1990)
 It draws multiple samples of the data set, applies PAM on
each sample, and gives the best clustering as the output
 Strength: deals with larger data sets than PAM
 Weakness:
– Efficiency depends on the sample size
– A good clustering based on samples will not necessarily
represent a good clustering of the whole data set if the sample
is biased
CLARANS (“Randomized” CLARA) (1994)
 CLARANS (A Clustering Algorithm based on Randomized Search)
(Ng and Han’94)
 CLARANS draws sample of neighbors dynamically
 The clustering process can be presented as searching a graph
where every node is a potential solution, that is, a set of k medoids
 If the local optimum is found, CLARANS starts with new
randomly selected node in search for a new local optimum
(Genetic-Algorithm-like)
 Finally the best local optimum is chosen after some stopping
condition.
 It is more efficient and scalable than both PAM and CLARA
Distance-based partitioning has drawbacks

Simple and fast O(N)

The number of clusters, K, has to be arbitrarily chosen before it is known how
many clusters is correct.

Produces round shaped clusters, not arbitrary shapes (Chameleon data set below)

Sensitive to the selection of the initial partition and may converge to a local
minimum of the criterion function if the initial partition is not well chosen.
Correct result
K-means result
Distance-based partitioning (Cont.)
 If we start with A, B, and C as the
initial centriods around which the
three clusters are built, then we end
up with the partition {{A}, {B, C},
{D, E, F, G}} shown by ellipses.
 Whereas, the correct three-cluster
solution is obtained by choosing, for
example, A, D, and F as the initial
cluster means (rectangular clusters).
A Vertical Data Approach
 Partition the data set using rectangle P-trees (a gridding)
 These P-trees can be viewed as a grouping (partition) of data
 Pruning out outliers by disregard those sparse values
Input: total number of objects (N), percentage of outliers (t)
Output: Grid P-trees after prune
(1) Choose the Grid P-tree with smallest root count (Pgc)
(2) outliers:=outliers OR Pgc
(3) if (outliers/N<= t) then remove Pgc and repeat (1)(2)
 Finding clusters using PAM method; each grid P-tree is an object
– Note: when we have a P-tree mask for each cluster, the mean is just
the vector sum of the basic Ptrees ANDed with the cluster Ptree,
divided by the rootcount of the cluster Ptree
Distance Function
Data Matrix
n objects × p variables
X11 … X1f … X1p
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Xi1 … Xif … Xip
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Xn1 … Xnf … Xnp
Dissimilarity Matrix
n objects × n objects
0
d(2,1) 0
d(3,1) d(3,2) 0
.
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d(n,1) d(n,2) … d(n,n-1) 0
Euclidian Distance between object i and j :
d(i, j)  ( X i1  X j1 ) 2  ( X i 2  X j 2 ) 2      ( X ip  X jp ) 2
AGNES (Agglomerative Nesting)
 Introduced in Kaufmann and Rousseeuw (1990)
 Use the Single-Link (distance between two sets is the
minimum pairwise distance) method
 Merge nodes that are most similarity
 Eventually all nodes belong to the same cluster
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DIANA (Divisive Analysis)
 Introduced in Kaufmann and Rousseeuw (1990)
 Inverse order of AGNES (intitially all objects are in one cluster; then
it is split according to some criteria (e.g., maximize some aggregate
measure of pairwise dissimilarity again)
 Eventually each node forms a cluster on its own
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Contrasting Clustering Techniques
 Partitioning algorithms: Partition a
dataset to k clusters, e.g., k=3 
 Hierarchical alg: Create hierarchical
decomposition of ever-finer partitions.
e.g., top down (divisively).
bottom up (agglomerative)
Hierarchical Clustering
Step 0
a
Step 1
Step 2 Step 3 Step 4
ab
b
abcde
c
d
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cde
de
Agglomerative
Hierarchical Clustering (top down)
a
ab
b
abcde
c
cde
d
de
e
Step 4
Divisive
Step 3
Step 2 Step 1 Step 0
In either case, one gets a nice dendogram in which any maximal antichain (no 2 nodes linked) is a clustering (partition).
Hierarchical Clustering (Cont.)









Recall that any maximal anti-chain (maximal set of nodes in which
no 2 are chained) is a clustering (a dendogram offers many).

Hierarchical Clustering (Cont.)
But the “horizontal” anti-chains are the clusterings resulting from the
top down (or bottom up) method(s).
Hierarchical Clustering (Cont.)
 Most hierarchical clustering algorithms are variants of the singlelink, complete-link or average link.
 Of these, single-link and complete link are most popular.
– In the single-link method, the distance between two clusters is the minimum
of the distances between all pairs of patterns drawn one from each cluster.
– In the complete-link algorithm, the distance between two clusters is the
maximum of all pairwise distances between pairs of patterns drawn one
from each cluster.
– In the average-link algorithm, the distance between two clusters is the
average of all pairwise distances between pairs of patterns drawn one from
each cluster (which is the same as the distance between the means in the
vector space case – easier to calculate).
Distance Between Clusters
 Single Link: smallest distance between any pair of
points from two clusters
 Complete Link: largest distance between any pair
of points from two clusters
Distance between Clusters (Cont.)
Average Link: average distance between points
from two clusters
• Centroid: distance between centroids of
the two clusters
Single Link vs. Complete Link (Cont.)
Single link
works but not
complete link
Complete link
works but not
single link
Single Link vs. Complete Link (Cont.)
Complete link doesn’t
Single link works
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Single Link vs. Complete Link (Cont.)
Complete link does
Single link
doesn’t works
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1-cluster
noise
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Hierarchical vs. Partitional
 Hierarchical algorithms are more versatile than partitional
algorithms.
– For example, the single-link clustering algorithm works well on data
sets containing non-isotropic (non-roundish) clusters including wellseparated, chain-like, and concentric clusters, whereas a typical
partitional algorithm such as the k-means algorithm works well only
on data sets having isotropic clusters.
 On the other hand, the time and space complexities of the
partitional algorithms are typically lower than those of the
hierarchical algorithms.
More on Hierarchical Clustering Methods
 Major weakness of agglomerative clustering methods
– do not scale well: time complexity of at least O(n2), where n is the
number of total objects
– can never undo what was done previously (greedy algorithm)
 Integration of hierarchical with distance-based clustering
– BIRCH (1996): uses Clustering Feature tree (CF-tree) and
incrementally adjusts the quality of sub-clusters
– CURE (1998): selects well-scattered points from the cluster and
then shrinks them towards the center of the cluster by a specified
fraction
– CHAMELEON (1999): hierarchical clustering using dynamic
modeling
Density-Based Clustering Methods
 Clustering based on density (local cluster criterion), such as
density-connected points
 Major features:
–
–
–
–
Discover clusters of arbitrary shape
Handle noise
One scan
Need density parameters as termination condition
 Several interesting studies:
–
–
–
–
DBSCAN: Ester, et al. (KDD’96)
OPTICS: Ankerst, et al (SIGMOD’99).
DENCLUE: Hinneburg & D. Keim (KDD’98)
CLIQUE: Agrawal, et al. (SIGMOD’98)
Density-Based Clustering: Background
 Two parameters:
– : Maximum radius of the neighbourhood
– MinPts: Minimum number of points in an  -neighbourhood
of that point
 N(p):
{q belongs to D | dist(p,q)  }
 Directly (density) reachable: A point p is directly
density-reachable from a point q wrt. , MinPts if
– 1) p belongs to N(q)
– 2) q is a core point:
|N(q)|  MinPts
p
q
MinPts = 5
 = 1 cm
Density-Based Clustering: Background (II)
 Density-reachable:
– A point p is density-reachable from a point q (p)
wrt , MinPts if there is a chain of points p1, …, pn,
p1=q, pn=p such that pi+1 is directly densityreachable from pi
– q, q is density-reachable from q.
• Density reachability is reflexive and transitive, but
not symmetric, since only core objects can be density
reachable to each other.
p
q
p1
 Density-connected
– A point p is density-connected to a q wrt , MinPts
if there is a point o such that both, p and q are
density-reachable from o wrt , MinPts.
– Density reachability is not symmetric, Density
connectivity inherits the reflexivity and transitivity
and provides the symmetry. Thus, density
connectivity is an equivalence relation and therefore
gives a partition (clustering).
p
q
o
DBSCAN: Density Based Spatial Clustering of Applications with Noise
 Relies on a density-based notion of cluster: A cluster is defined as an equivalence
class of density-connected points.
– Which gives the transitive property for the density connectivity binary relation
and therefore it is an equivalence relation whose components form a partition
(clustering) according to the duality.
 Discovers clusters of arbitrary shape in spatial databases with noise
Outlier
Border
Core
 = 1cm
MinPts = 3
DBSCAN: The Algorithm
– Arbitrary select a point p
– Retrieve all points density-reachable from p wrt , MinPts.
– If p is a core point, a cluster is formed (note: it doesn’t matter which of the core
points within a cluster you start at since density reachability is symmetric on
core points.)
– If p is a border point or an outlier, no points are density-reachable from p and
DBSCAN visits the next point of the database. Keep track of such points. If
they don’t get “scooped up” by a later core point, then they are outliers.
– Continue the process until all of the points have been processed.
– What about a simpler version of DBSCAN:
• Define core points and core neighborhoods the same way.
• Define (undirected graph) edge between two points if they cohabitate a core nbrhd.
• The connectivity component partition is the clustering.
– Other related method? How does vertical technology help here? Gridding?
OPTICS
Ordering Points To Identify Clustering Structure
– Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99)
– http://portal.acm.org/citation.cfm?id=304187
– Addresses the shortcoming of DBSCAN, namely choosing
parameters.
– Develops a special order of the database wrt its density-based
clustering structure
– This cluster-ordering contains info equivalent to the density-based
clusterings corresponding to a broad range of parameter settings
– Good for both automatic and interactive cluster analysis, including
finding intrinsic clustering structure
OPTICS
Does this order
resemble the
Total Variation
order?
Reachability
-distance
undefined

‘

Cluster-order
of the objects
DENCLUE: using density functions
 DENsity-based CLUstEring by Hinneburg & Keim (KDD’98)
 Major features
– Solid mathematical foundation
– Good for data sets with large amounts of noise
– Allows a compact mathematical description of arbitrarily
shaped clusters in high-dimensional data sets
– Significant faster than existing algorithm (faster than
DBSCAN by a factor of up to 45 – claimed by authors ???)
– But needs a large number of parameters
Denclue: Technical Essence
 Uses grid cells but only keeps information about grid cells that do actually
contain data points and manages these cells in a tree-based access structure.
 Influence function: describes the impact of a data point within its
neighborhood.
– F(x,y) measures the influence that y has on x.
– A very good influence function is the Gaussian, F(x,y) = e
–d2(x,y)/2
– Others include functions similar to the squashing functions used in
neural networks.
– One can think of the influence function as a measure of the contribution
to the density at x made by y.
 Overall density of the data space can be calculated as the sum of the
influence function of all data points.
 Clusters can be determined mathematically by identifying density attractors.
 Density attractors are local maximal of the overall density function.
DENCLUE(D,σ,ξc,ξ)
1.
2.
3.
4.
Grid Data Set (use r = σ, the std. dev.)
Find (Highly) Populated Cells (use a
threshold=ξc) (shown in blue)
Identify populated cells (+nonempty cells)
Find Density Attractor pts, C*, using hill
climbing:
1.
2.
3.
4.
5.
5.
Randomly pick a point, pi.
Compute local density (use r=4σ)
Pick another point, pi+1, close to pi,
compute local density at pi+1
If LocDen(pi) < LocDen(pi+1), climb
Put all points within distance σ/2 of path,
pi, pi+1, …C* into a “density attractor
cluster called C*
Connect the density attractor clusters,
using a threshold, ξ, on the local densities
of the attractors.
A. Hinneburg and D. A. Keim. An Efficient Approach to Clustering in Multimedia Databases with Noise.
In Proc. 4th Int. Conf. on Knowledge Discovery and Data Mining. AAAI Press, 1998. & KDD 99
Workshop.
Comparison: DENCLUE Vs DBSCAN
BIRCH (1996)
 Birch: Balanced Iterative Reducing and Clustering using
Hierarchies, by Zhang, Ramakrishnan, Livny (SIGMOD’96
http://portal.acm.org/citation.cfm?id=235968.233324&dl=GUIDE&dl=ACM&idx=235968&part=periodical&WantType=per
iodical&title=ACM%20SIGMOD%20Record&CFID=16013608&CFTOKEN=14462336
 Incrementally construct a CF (Clustering Feature) tree, a
hierarchical data structure for multiphase clustering
– Phase 1: scan DB to build an initial in-memory CF tree (a multilevel compression of the data that tries to preserve the inherent
clustering structure of the data)
– Phase 2: use an arbitrary clustering algorithm to cluster the leaf
nodes of the CF-tree
 Scales linearly: finds a good clustering with a single scan
and improves quality with a few additional scans
 Weakness: handles only numeric data, and sensitive to the
order of the data record.
ABSTRACT
BIRCH
Finding useful patterns in large datasets has attracted considerable interest recently, and
one of the most widely studied problems in this area is the identification of clusters,
or densely populated regions, in a multi-dimensional dataset.
Prior work does not adequately address the problem of large datasets and minimization
of I/O costs.
This paper presents a data clustering method named BIRCH (Balanced Iterative
Reducing and Clustering using Hierarchies), and demonstrates that it is especially
suitable for very large databases.
BIRCH incrementally and dynamically clusters incoming multi-dimensional metric data
points to try to produce the best quality clustering with the available resources (i.e.,
available memory and time constraints).
BIRCH can typically find a good clustering with a single scan of the data, and improve
the quality further with a few additional scans.
BIRCH is also the first clustering algorithm proposed in the database area to handle
"noise" (data points that are not part of the underlying pattern) effectively.
We evaluate BIRCH's time/space efficiency, data input order sensitivity, and clustering
quality through several experiments.
Clustering Feature Vector
Clustering Feature: CF = (N, LS, SS)
N: Number of data points
LS: Ni=1=Xi
SS: Ni=1=Xi2
CF = (5, (16,30),(54,190))
Branching factor = max # children
Threshold = max diameter of leaf cluster
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(2,6)
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(4,7)
(3,8)
Iteratively put points into closest leaf until threshold is exceed, then split leaf.
Inodes summarize their subtrees and Inodes get split when threshold is exceeded.
Once in-memory CF tree is built, use another method to cluster leaves together.
Birch
Root
CF1
CF2 CF3
CF6
Branching factor, B=6
child1
child2 child3
child6
Threshold, L = 7
CF1
Non-leaf node
CF2 CF3
CF5
child1
child2 child3
child5
Leaf node
prev CF1 CF2
CF6 next
Leaf node
prev CF1 CF2
CF4 next
CURE (Clustering Using
REpresentatives )
 CURE: proposed by Guha, Rastogi & Shim, 1998
 http://portal.acm.org/citation.cfm?id=276312
– Stops the creation of a cluster hierarchy if a level consists of k clusters
– Uses multiple representative points to evaluate the distance between clusters
– adjusts well to arbitrary shaped clusters (not necessarily distance-based
– avoids single-link effect
Drawbacks of Distance-Based Method
 Drawbacks of square-error based clustering method
– Consider only one point as representative of a cluster
– Good only for convex shaped, similar size and density, and
if k can be reasonably estimated
Cure: The Algorithm
– Very much a hybrid method (involves pieces from many others).
– Draw random sample s.
– Partition sample to p partitions with size s/p
– Partially cluster partitions into s/pq clusters
– Eliminate outliers
• By random sampling
• If a cluster grows too slow, eliminate it.
– Cluster partial clusters.
– Label data in disk
ABSTRACT
Cure
Clustering, in data mining, is useful for discovering groups and identifying interesting
distributions in the underlying data. Traditional clustering algorithms either favor
clusters with spherical shapes and similar sizes, or are very fragile in the presence of
outliers.
We propose a new clustering algorithm called CURE that is more robust to outliers, and
identifies clusters having non-spherical shapes and wide variances in size.
CURE achieves this by representing each cluster by a certain fixed number of points
that are generated by selecting well scattered points from the cluster and then
shrinking them toward the center of the cluster by a specified fraction.
Having more than one representative point per cluster allows CURE to adjust well to
the geometry of non-spherical shapes and the shrinking helps to dampen the effects
of outliers.
To handle large databases, CURE employs a combination of random sampling and
partitioning. A random sample drawn from the data set is first partitioned and each
partition is partially clustered. The partial clusters are then clustered in a second pass
to yield the desired clusters.
Our experimental results confirm that the quality of clusters produced by CURE is
much better than those found by existing algorithms.
Furthermore, they demonstrate that random sampling and partitioning enable CURE to
not only outperform existing algorithms but also to scale well for large databases
without sacrificing clustering quality.
Data Partitioning and Clustering
– s = 50
– p=2
– s/p = 25

s/pq = 5
y
y
y
x
y
y
x
x
x
x
Cure: Shrinking Representative Points
y
y
x
 Shrink the multiple representative points towards the
gravity center by a fraction of .
 Multiple representatives capture the shape of the
cluster
x
Clustering Categorical Data: ROCK
http://portal.acm.org/citation.cfm?id=35174
5
ROCK: Robust Clustering using linKs,
by S. Guha, R. Rastogi, K. Shim (ICDE’99).
–
–
–
–
Agglomerative Hierarchical
Use links to measure similarity/proximity
Not distance based
O(n2  nmmma  n2 log n)
Computational complexity:
Basic ideas:
– Similarity function and neighbors:
Let T1 = {1,2,3}, T2={3,4,5}
Sim( T1, T 2) 
T1  T2
Sim(T1 , T2 ) 
T1  T2
{3}
1

 0.2
{1,2,3,4,5}
5
Abstract
ROCK
Clustering, in data mining, is useful to discover distribution patterns in the underlying
data.
Clustering algorithms usually employ a distance metric based (e.g., euclidean)
similarity measure in order to partition the database such that data points in the same
partition are more similar than points in different partitions.
In this paper, we study clustering algorithms for data with boolean and categorical
attributes.
We show that traditional clustering algorithms that use distances between points for
clustering are not appropriate for boolean and categorical attributes. Instead, we
propose a novel concept of links to measure the similarity/proximity between a pair
of data points.
We develop a robust hierarchical clustering algorithm ROCK that employs links and not
distances when merging clusters.
Our methods naturally extend to non-metric similarity measures that are relevant in
situations where a domain expert/similarity table is the only source of knowledge.
In addition to presenting detailed complexity results for ROCK, we also conduct an
experimental study with real-life as well as synthetic data sets to demonstrate the
effectiveness of our techniques.
For data with categorical attributes, our findings indicate that ROCK not only generates
better quality clusters than traditional algorithms, but it also exhibits good scalability
properties.
Rock: Algorithm
 Links: The number of common neighbors for the two pts
{1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,3,5}
{1,4,5}, {2,3,4}, {2,3,5}, {2,4,5}, {3,4,5}
3
{1,2,3}
{1,2,4}
 Algorithm
– Draw random sample
– Cluster with links
– Label data in disk
CHAMELEON
 CHAMELEON: hierarchical clustering using dynamic
modeling, by G. Karypis, E.H. Han and V. Kumar’99
 http://portal.acm.org/citation.cfm?id=621303
 Measures the similarity based on a dynamic model
– Two clusters are merged only if the interconnectivity and closeness
(proximity) between two clusters are high relative to the internal
interconnectivity of the clusters and closeness of items within the
clusters
 A two phase algorithm
– 1. Use a graph partitioning algorithm: cluster objects into a large
number of relatively small sub-clusters
– 2. Use an agglomerative hierarchical clustering algorithm: find the
genuine clusters by repeatedly combining these sub-clusters
ABSTRACT
CHAMELEON
Many advanced algorithms have difficulty dealing with highly variable clusters that do
not follow a preconceived model.
By basing its selections on both interconnectivity and closeness, the Chameleon
algorithm yields accurate results for these highly variable clusters.
Existing algorithms use a static model of the clusters and do not use information about
the nature of individual clusters as they are merged.
Furthermore, one set of schemes (the CURE algorithm and related schemes) ignores the
information about the aggregate interconnectivity of items in two clusters.
Another set of schemes (the Rock algorithm, group averaging method, and related
schemes) ignores information about the closeness of two clusters as defined by the
similarity of the closest items across two clusters.
By considering either interconnectivity or closeness only, these algorithms can select
and merge the wrong pair of clusters.
Chameleon's key feature is that it accounts for both interconnectivity and closeness in
identifying the most similar pair of clusters.
Chameleon finds the clusters in the data set by using a two-phase algorithm.
During the first phase, Chameleon uses a graph-partitioning algorithm to cluster the
data items into several relatively small subclusters.
During the second phase, it uses an algorithm to find the genuine clusters by repeatedly
combining these sub-clusters.
Overall Framework of CHAMELEON
Construct
Partition the Graph
Sparse Graph
Data Set
Merge Partition
Final Clusters
Grid-Based Clustering Method
 Using multi-resolution grid data structure
 Several interesting methods
– STING (a STatistical INformation Grid approach) by
Wang, Yang and Muntz (1997)
– WaveCluster by Sheikholeslami, Chatterjee, and Zhang
(VLDB’98)
• A multi-resolution clustering approach using
wavelet method
– CLIQUE: Agrawal, et al. (SIGMOD’98)
Vertical gridding
We can observe that almost all methods discussed so far suffer from the curse of cardinality (for very
large cardinality data sets, the algorithms are too slow to finish in the average life time!) and/or the curse
of dimensionality (points are all at ~ same distance).
The work-arounds employed to address the curses
sampling (throw out most of the points in a way that what remains is low enough cardinality for the
algorithm to finish and in such a way that the remaining “sample” contains all the information of the
original data set (Therein is the problem – that is impossible to do in general);
Gridding (agglomerate all points in a grid cell and treat them as one point (smooth the data set to this
gridding level). The problem with gridding, often, is that info is lost and the data structure that holds the
grid cell information is very complex. With vertical methods (e.g., P-trees), all the info can be retained and
griddings can be constructed very efficiently on demand. Horizontal data structures can’t do this.
Subspace restrictions (e.g., Principal Components, Subspace Clustering…)
Gradient based methods (e.g., the gradient tangent vector field of a response surface reduces the
calculations to the number of dimensions, not the number of combinations of dimensions.)
j-hi gridding: the j hi order bits identify a grid cells and the rest identify points in a particular cell.
Thus, j-hi cells are not necessarily cubical (unless all attribute bit-widths are the same).
j-lo gridding; the j lo order bits identify points in a particular cell and the rest identify a grid cell.
Thus, j-lo cells always have a nice uniform shape (cubical).
1-hi gridding of Vector Space, R(A1, A2, A3) in which all bit-widths are the same = 3
(so
each grid cell contains
= 64 potential points). Grid cells are identified by their Peano id (Pid) internally
the point’s cell coordinates are shown - called the grid cell id and
cell points
are id’ed by coordinates within their cell.
22 * 22 * 22
1
Pid = 001
0
gci=001
gci=001 gci=001 gci=001
gcp=
gcp=
gcp=
gcp= gci=001
gci=001 gci=001 gci=001
01,11,00 10,11,00 gcp=
11,11,00
00,11,00 gcp=
gcp=
gcp= gci=001
gci=001
gci=001 gci=001
11,11,01
00,11,01 01,11,01 gcp=
10,11,01 gcp=
gcp=
gcp= gci=001
gci=001
gci=001
gci=001
gci=001 gci=001
gci=001 gci=001
11,11,10
00,11,10 gcp=
10,11,10 gcp=
01,11,10 gcp=
gcp=
gcp=
gcp= gci=001
gcp= gci=001
gcp= gci=001
gci=001
00,10,00 01,10,00
11,10,00
11,11,11 gcp=
10,11,11 10,10,00
00,11,11 gcp=
01,11,11
gcp=
gci=001
gcp= gci=001
gci=001
00,10,01gci=001
11,10,01
01,10,01 10,10,01 gcp=
gcp=
gci=001
gcp= gci=001
gci=001
gcp= gci=001
gci=001 gci=001
gci=001
00,10,10
11,10,10
gcp=
10,10,10 gcp=
gcp=
01,10,10
gcp=
gcp=
gcp=
gcp=
gci=001
gci=001
gci=001
00,10,11 00,01,00
01,10,11 gcp=
11,10,11gci=001
01,01,00
10,10,11 gcp=
10,01,00
11,01.00
gcp=
gci=001
gcp=
gci=001
gci=001
gci=001
00,01,01 gcp=
01,01,01 10,01,01 11,01.01
gcp= gci=001
gcp= gci=001
gcp=
gci=001 gci=001
gci=001
gci=001
gci=001
01,01,10 10,01,10 11,01,10gci=001
00,01,10 gcp=
gcp=
gcp=
gcp=
gcp= gci=001
gcp=
gcp=
gcp=
gci=001
gci=001
01,01,11 01,00,00
11,01,11 gci=001
00,01,11 00,00,00
10,01,11
11,00,00
10,00,00 gcp=
gcp=
gcp= gci=001
gci=001
gcp=
gci=001
gci=001
01,00,01 10,00,01
00,00,01 gcp=
gcp=
11,00,01
gcp=
gcp= gci=001
gci=001
gci=001
00,00,10gci=001
01,00,10
11,00,10
10,00,10
gcp=
gcp=
gcp=
gcp=
00,00,11
A2
hi-bit
01,00,11
10,00,11
0
1
11,00,11
0
1
A1
hi-bit
A3
hi-bit
2-hi gridding of Vector Space, R(A1, A2, A3) in which all bitwidths are the same = 3
each grid cell contains 21 * 21 * 21 = 8 points).
(so
11
10
A2
Pid = 001.001
01
00
00
gci=
gci=
00,00,11 00,00,11
gci=
gci=
gcp=0,1,0 gcp=1,1,0
00,00,11 00,00,11
gci=gcp=1,1,1
gcp=0,1,1
gci=
00,00,11 00,00,11
gci=
gci=
gcp=0,0,0 gcp=1,0,0
00,00,11 00,00,11
gcp=0,0,1 gcp=1,0,1
00
01
10
11
01
A1
10
11
A3
1-hi gridding of R(A1, A2, A3), bitwidths of 3,2,3
Pid = 001
1
0
A2
hi-bit
gci=001
gci=001
gci=001 gci=001
gcp=
gcp=
gcp= gci=001
gcp= gci=001
gci=001
gci=001
11,1,00
10,1,00
00,1,00 gcp=
01,1,00 gcp=
gcp=
gcp=
gci=001
gci=001 gci=001
01,1,01 10,1,01 gci=001
00,1,01
11,1.01
gcp=
gcp= gci=001
gcp= gci=001
gcp=
gci=001 gci=001
gci=001 gci=001
gci=001
gci=001
00,1,10
01,1,10
10,1,10
11,1,10
gcp=
gcp=
gcp=
gcp=
gcp= gci=001
gcp=
gcp=
gcp=
gci=001
gci=001
gci=001
00,1,11 00,0,00
10,1,11
11,1,11
01,0,00
01,1,11 gcp=
11,0,00
10,0,00
gcp=
gci=001
gcp=
gci=001
gci=001 gci=001
gcp=
00,0,01
01,0,01
gcp=
11,0,01
gcp=
gcp= gci=001
10,0,01
gcp=
gci=001
gci=001
gci=001
00,0,10
11,0,10
01,0,10
10,0,10
gcp=
gcp=
gcp=
gcp=
01,0,11 10,0,11
11,0,11
00,0,11
0
0
1
1
A1
hi-bit
A3
hi-bit
2-hi gridding) of R(A1, A2, A3), bitwidths of 3,2,3 (each grid cell contains 21 * 20 * 21 = 4 potential pts).
11
00
10
gcp=
gcp=0,,0
01
gcp=
gcp=1,,0
0,,1
10
00
A2
2-hi-bit
01
1,,1
00
01
A1
2-hi-bit
10
11
11
Pid = 3.1.3
A3
2-hi-bit
HOBBit disks and rings:
(HOBBit = Hi Order Bifurcation Bit)
4-lo grid where A1,A2,A3 have bit-widths, b1+1, b2+1, b3+1,
HOBBit grid centers are points of the form
x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)
(exactly one per grid cell):
where xi,js range over all binary patterns
HOBBit disk about x, of radius 20 , H(x,20).
Note: we have switched the direction of A3
(x1,b1..x1,41010, x2,b2..x2,41011, x3,b3..x3,41011)
(x1,b1..x1,41010, x2,b2..x2,41011, x3,b3..x3,41010)
(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41011)
(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)
A2
A3
A1
 (x1,b1..x1,41011, x2,b2..x2,41011, x3,b3..x3,41011)
gcp=
1010,
gcp=1011,
1010,1011
1011,
1010gcp=
1010,
gcp=
1010,
1010,1011
1010,
1010
gcp=
1011,
gcp=1011,
1011,1011
1011,
1010gcp=
1011,
gcp=1010,
1011,1011
1010,
1010
(x1,b1..x1,41011, x2,b2..x2,41011, x3,b3..x3,41010)
(x1,b1..x1,41011, x2,b2..x2,41010, x3,b3..x3,41011)
(x1,b1..x1,41011, x2,b2..x2,41010, x3,b3..x3,41010)
H(x,21) HOBBit disk about a HOBBit grid center pt, x = (x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)
A2
(x1,b1..x1,41000, x2,b2..x2,41011, x3,b3..x3,41011)
, of radius 21
A3
(x1,b1..x1,41011, x2,b2..x2,41011, x3,b3..x3,41011)
(x1,b1..x1,41011, x2,b2..x2,41010, x3,b3..x3,41011)
(x1,b1..x1,41000, x2,b2..x2,41011, x3,b3..x3,41000)
A1
(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)
(x1,b1..x1,41011, x2,b2..x2,41000, x3,b3..x3,41011)
(x1,b1..x1,41011, x2,b2..x2,41000, x3,b3..x3,41010)
(x1,b1..x1,41011, x2,b2..x2,41000, x3,b3..x3,41001)
(x1,b1..x1,41000, x2,b2..x2,41000, x3,b3..x3,41000)
(x1,b1..x1,41011, x2,b2..x2,41000, x3,b3..x3,41000)
The Regions of H(x,21) are as follows:
A2
A3
A1
These REGIONS are labeled with dimensions in which length is increased
A2
(e.g., all three dimensions are increased below).
A3
A1
123-REGION
A2
A3
A1
13-REGION
A2
A3
A1
23-REG
A2
A3
A1
12-REGION
A2
A3
3-REG
A1
A2
A3
A1
2-REG
A2
A3
1-REGION
A1
H(x,20) =
123-REG
Of H(x,20)
A2
A3
A1
H(x,20
1-REGION
13-REGION
3-REG
A2 A
3
12-REGION
2-REG
A1
123-REGION
23-REG
Algorithm (for computing gradients):
1.
Select an “outlier threshold”, (pts without neighbors in their ot L-disk are outliers – That is,
there is no gradient at these outlier points (instantaneous rate of response change is zero).
2.
Create an j-lo grid with j=ot
3.
Pick a point, x in R. Build out alternating one-sided-rings centered at x until a neighbor is
found or radius ot is exceeded (in which case x is declared an outlier). If a neighbor is found
at a raduis, ri < ot 2j, f/ xk(x) is estimated as below:
1.
2.
3.
(see previous slides - where HOBBit disks are built out from HOBBit centers
x = ( x1,b1…x1,ot+11010 , … , xn,bn…xn,ot+11010 ), xi,js ranging over all binary patterns).
Note: one can use L-HOBBit or L ordinary distance.
Note: One-sided means that each successive build out increases aternatively only in the positive direction in all
dimensions then only in the negative direction in all dimensions.
Note: Building out HOBBit disks from a HOBBit center automatically gives “one-sided” rings (a built-out ring
is defined to be the built-out disk minus the previous built-out disk) as shown in the next few slides.
( RootcountD(x,ri) - RootcountD(x,ri)k ) / xk
where D(x,ri)k is D(x,ri-1) expanded
in all dimensions except k.
Alternatively in 3., actually calculate the mean (or median?) of the new points encountered in
D(x,ri) (we have a P-tree mask for the set so this it trivial) and measure the xk-distance.
NOTE: Might want to go 1 more ring out to see if one gets the same or a similar gradient
(this seems particularly important when j is odd (since the gradient then points the opposite way.)
NOTE: the calculation of xk can be done in various ways. Which is best?
Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1 =(2-1)/(-1) = -1
A2
H(x,21)
First new point
H(x,21)1
HOBBit center, x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)
A3
A1
( RootcountD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-1)/(-1) = -1
H(x,21)2
A2
H(x,21)
A3
A1
( RootcountD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-1)/(-1) = -1
H(x,21)3
A2
H(x,21)
A3
A1
Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1 =(2-1)/(-1) = -1
Est f/ xk(x)= ( RcD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-1)/(-1) = -1
Estimated gradient
Check other regions too?
Est f/ xk(x)=( RcD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-1)/(-1) = -1
A2
H(x,21)
A3
A1
First new point
HOBBit center, x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)
gradient
Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1 =(2-2)/(-1) = 0
A2
H(x,21)
First new point
H(x,21)1
HOBBit center, x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)
A3
A1
gradient
( RootcountD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-1)/(-1) = -1
H(x,21)2
A2
H(x,21)
A3
A1
gradient
( RootcountD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-1)/(-1) = -1
H(x,21)3
A2
H(x,21)
A3
A1
( RootcountD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-1)/(-1) = -1
Estimated gradient
Check other regions too?
gradient
( RootcountD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-1)/(-1) = -1
A2
H(x,21)
A3
A1
Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1 =(2-1)/(-1) = -1
A2
H(x,21)
First new point
H(x,21)1
HOBBit center, x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)
A3
A1
( RootcountD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-2)/(-1) = 0
H(x,21)2
A2
H(x,21)
A3
A1
( RootcountD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-2)/(-1) = 0
H(x,21)3
A2
H(x,21)
A3
A1
Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1 =(2-1)/(-1) = -1
Estimated gradient
A2
Check other regions too?
A3
A1
Intuitively, this Gradient estimation
method seems to work.
Next we consider a potential accuracy
improvement in which we take the
medoid of all new points as the
gradient
H(x,21)
(or, more accurately, as the point to
which we climb in any response
surface hill climbing technique)
Estimate the gradient arrowhead as being at the medoid of the new
point set (or, more correctly, estimate the next hill-climb step).
Note: If the original points are truly part of a strong cluster, the hill
climb will be excellent.
A2
H(x,21)
A3
A1
new points =
s
new points centroid =
Estimate the gradient arrowhead as being at the medoid of the new
point set (or, more correctly, estimate the next hill-climb step).
Note: If the original points are not truly part of a strong cluster, the
weak hill climb will indicate that.
A2
H(x,21)
A3
A1
new points =
s
new points centroid =
Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1 =(2-1)/(3) =
1/3
Est f/ xk(x)= ( RcD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-1)/(3) = 1/3
Est f/ xk(x)=( RcD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-1)/(3) = 1/3
H(x,22)1
A2
H(x,22)
A3
A1
Note that the gradient points in the right direction and is
very short (as it should be!)
First new
point
To evaluate how well the formula estimates the
gradient, it is important to consider all cases of the
new point appearing in one of these regions
(if  1 point appears, gradient components are
additive, so it suffices to consider 1?
H(x,2)1
H(x,2)1
H(x,2)1
H(x,2)1
H(x,2)1
H(x,2)
H(x,1)1
H(x,0)
H(x,2)1
H(x,1)3
H(x,1)13
H(x,1)2
H(x,1)12
H(x,1)123
H(x,1)23
H(x,2)1
To evaluate how well the formula estimates the gradient, it
is important to consider all cases of the new point appearing
in 1 of these regions (if  1 pt appears, gradient comps add)
H(x,2)23
H(x,1)1
H(x,0)
H(x,2)2
H(x,1)13H(x,1)3
H(x,2)123
H(x,2)12
H(x,2)13
H(x,1)H(x,2)
12 H(x,1)
2
H(x,1)123
H(x,1)
233
H(x,2)1
H(x,3)3
H(x,2)23
H(x,1)1
H(x,0)
H(x,2)2
H(x,1)13H(x,1)3
H(x,2)123
H(x,2)12
H(x,2)13
H(x,1)H(x,2)
12 H(x,1)
2
H(x,1)123
H(x,1)
233
H(x,3)1
H(x,3)13
H(x,3)123
H(x,3)13
H(x,2)1
H(x,3)13
H(x,3)13
Notice that the HOBBit
center moves more and
more toward the true
center as the grid size
increases.
H( x,23 )
Grid based Gradients and Hill Climbing
If we are using gridding to produce the gradient vector field of a response surface, might we
always vary xi in the positive direction only? “How can that be done most efficiently?”
1.
2.
3.
4.
j-lo gridding, building out HOBBit rings from HOBBit grid centers
(see previous slides where this approach was used.)
or
j-lo gridding. building out HOBBit rings from lo-value grid pts (ending in j 0-bits)
x = ( x1,b1…x1,j+10…0 , … , xn,bn…xn,j+10…0 )
Ordinary j-lo griddng, building out rings from lo-value ids (ending in j zero bits)
Ordinary j-lo gridding, uilding out Rings from true centers.
Other? (there are many other possibilities, but we will first explore 2.)
1.
2.
Using j-lo gridding with j=3 and lo-value cell identifiers, is shown on the next slide.
3.
Of course, we need not use HOBBit build out.
4.
With ordinary unit radius build out, the results are more exact, but are the calculations may
be more complex???
HOBBit j-lo rings using lo-value cell ids
x=(x1,b1…x1,j+10…0 ,…, xn,bn…xn,j+10…0)
H(x,2)23
1
H(x,2)12
H(x,2)123
1
H(x,2)12
1
H(x,2)13
1
H(x,2)13
H(x,1)23H(x,1)123
H(x,1)2 H(x,1)12
H(x,1)3
H(x,1)13
H(x,1)
1
H(x,0)
H(x,2)1
Ordinary j-lo rings using lo-value cell ids
x=(x1,b1…x1,j+10…0 ,…, xn,bn…xn,j+10…0)
= PDisk(x,3)^P’Disk(x,2)
Ring(x,2)
Ring(x,1)
= PDisk(x,2)^P’Disk(x,1)
wherePD(x,i) =
Disk(x,0)
Pxb^..^Pxj+1 ^P’j^..^P’i+1
k-Medoids Clustering Review:
 Find representative objects (medoids) (actual objects in the cluster - mean seldom is).
 PAM (Partitioning Around Medoids)
– Select k representative objects arbitrarily
– For each pair of non-selected object h and selected object i, calculate the total swapping cost TCi,h
– For each pair of i and h,
• If TCi,h < 0, i is replaced by h
• Then assign each non-selected object to the most similar representative object
– repeat steps 2-3 until there is no change
 CLARA (Clustering LARge Apps) draws multiple samples of the data set, applies PAM on each sample,
and gives the best clustering as the output. Strength: deals with larger data sets than PAM. Weakness:
Efficiency depends on the sample size. A good clustering based on samples will not necessarily represent
a good clustering of the whole data set if the sample is biased
 CLARANS (Clustering Large Apps based on RANdom Search) draws sample of neighbors dynamically.
The clustering process can be presented as searching a graph where every node is a potential solution,
that is, a set of k medoids. If the local optimum is found, CLARANS starts with new randomly selected
node in search for a new local optimum (Genetic-Algorithm-like). Finally the best local optimum is
chosen after some stopping condition. It is more efficient and scalable than both PAM and CLARA
A Vertical k-Medoids Clustering Algorithm
 Following PAM (to illustrate the main killer idea – but it can apply much more widely)
Select k component P-trees
1. The Goal here is to efficiently get one Ptree mask for each component
2. e.g., calculate the smallest j : the j-lo gridding has > k cells.
3. Agglomerate into precisely k components (by ORing Ptrees of cells with closest
means/mediods/corners(single_link…)
Where PAM uses: “For each pair of non-selected object h and selected object i, calculate total
swapping cost TCi,h. For each pair of i and h, if TCi,h < 0, i is replaced by h, then assign each
non-selected object to the most similar object.”, use:
1. Find Medoid of each component, Ci: Calculate TV(Ci,x) x  Ci. (create P-tree of all
points with min TV so far.  smaller TV, reset this P-tree to it. Ending up with a P-tree,
PtMi of “tieing-Medoids” for Ci. Calc TV(PtMi ,x) x  PtMi and pick its medoid (if there
are still multiple, repeat). This is Ci-medoid! Alternatively, just pick 1 pre-Medoid!
Note: this avoids expense of pairwise swappings, and avoids subsampling as in CLARA, CLARANS)
2. Put each point with its closest Medoid (building P-tree component masks as you do this).
3. Repeat 1 & 2 until (some stopping condition – such as: no change in Medoid set?)
4. Can we cluster at step 4 without a scan (create component P-trees??)
A Vertical k-Means Clustering Algorithm
As mentioned on previous slide, a Vertical k-Means algorithm goes similarily:
1. Select k component P-trees (The Goal here is to efficiently get one Ptree mask
for each component, e.g., calculate the smallest j : the j-lo gridding has > k cells
and agglomerate into precisely k components (by ORing Ptrees of cells with
closest means…)
2. Calculate the Means of each component, Ch, by
1. ANDing each basic Ptree with PCi
2. In dimension, Ak , calculate the k-component of the mean as
i=bk..0 2i * rc(PCh ^ Pk,i )
3. Put each pt with closest Mean (building P-tree component masks as you do this).
4. Repeat 2, 3 until (some stopping condition – such as: no change in Mean set?)
5. Can we cluster at step 4 without a scan (create component P-trees??)
Zillions of vertical hybrid clustering algs leap to mind (involving partitioning, hierarchical, density
methods)! Pick one!
Finding Density Attractors for Density Clustering Alg
Finding density attractors (and their attractor sets – which,
when agglomerated via a density threshold constitutes the
generic density clustering algorithm)
1. Pick a point.
2. Build out (HOBBit or Ordinary or?) rings one at a time until
the first k neighbors are found.
3. In that ring, computer the medoid points (as in step 1 of the
V-k-Medoids algorithm above)
4. if the Medoid increases the density, climb to it and goto 2
5. Else declare it a density attractor and goto 1
j-grids - P-tree relationship
ci=010
1-hi-gridding of R(A1, A2, A3).
Bitwidths: 3,2,3 (dim cardinalities: 8,4,8)
Using tree node-like identifiers
cell (tree) id (ci) of the form, c0c1…cd
point (coord) id (pi) of the form, p1,p2,p3
ci=110
ci=011
ci=111
ci=000
ci=100
ci=001
A2
00.1.00
00.1.01
00.1.10
A1
A3
01.1.01
01.1.01
01.1.10
11.1.00
10.1.00
11.1.01
10.1.01
10.1.10
ci=101
11.1.10
01.1.11
00.0.00
11.1.11
10.1.11 10.0.00
11.0.00
01.0.00
10.0.01
11.0.01
00.0.01
01.0.01
11.0.10
10.0.10
00.0.10
01.0.10
11.0.11
10.0.11
00.0.11
01.0.11
00.1.11
root
000
001
010
011
100
101
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111
11.1.11
11.1.10
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11.0.10
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10.1.10
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10.0.10
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11.1.00
11.0.01
11.0.00
10.1.01
10.1.00
10.0.01
10.0.00
01.1.11
01.1.10
01.0.11
01.0.10
00.1.11
00.1.10
00.0.11
00.0.10
01.1.01
01.1.00
01.0.01
01.0.00
00.1.01
00.1.00
00.0.01
00.0.00
A 1-hi-grid yields a P-tree with level-0 (cell level) fanout of 23 and level-1 (point level) fanout of 25. If leaves are segment labelled (not coords):
010.11
010.10
000.11
000.10
010.01
010.00
000.01
000.00
...
j-grids - P-tree relationship (Cont.)
One can view a standard P-tree as nested
1-hi-griddings, with compressed out
constant subtrees.
R(A1, A2, A3) with bitwidths: 3,2,3
010
110
011
111
000
A2
010
011
A1
A3
110
001
001
00
101
111
000
01
100
100
101
10
11
root
000
000
00
001
01
010
10
001
011
11
100
010
011
101
100
110
101
111
110
111
Gridding categorical data?
 The following bioinformatics (yeast genome) data
used was extracted mostly from the MIPS database
(Munich Information center for Protein Sequences)
 Left column shows features
–
–
treat these with hi-order bit (1 iff gene participates)
There may be more levels of hierarchy (e.g., function:
some genes actually cause the function when they
express in sufficient quantities, while others are
transcription factors for those primary genes. Primary
genes have hi bit on, tf’s have second bit on…)
 Right column shows # distinct feature values
–
Bitmap these
Feature
pathway
EC
complexes
function
localization
protein class
phenotype
interactions
Total Values
80
622
316
259
43
191
181
6347
Data Representation
gene-by-feature table.
 For a categorical feature, we consider each category as a
separate attribute or column by bit-mapping it.
 The resulting table has a total of
– 8039 distinct feature bit vectors (corresponding to “items” in
MBR) for
– 6374 yeast genes (corresponding to transactions in MBR)
STING: A Statistical Information Grid Approach
 Wang, Yang and Muntz (VLDB’97)
 The spatial area is divided into rectangular cells
 There are several levels of cells corresponding to different
levels of resolution
STING: A Statistical Information Grid Approach (2)
– Each cell at a high level is partitioned into a number of smaller
cells in the next lower level
– Statistical info of each cell is calculated and stored beforehand and
is used to answer queries
– Parameters of higher level cells can be easily calculated from
parameters of lower level cell
• count, mean, s, min, max
• type of distribution—normal, uniform, etc.
– Use a top-down approach to answer spatial data queries
– Start from a pre-selected layer—typically with a small number of
cells
– For each cell in the current level compute the confidence interval
STING: A Statistical Information Grid Approach (3)
– Remove the irrelevant cells from further consideration
– When finish examining the current layer, proceed to the next lower
level
– Repeat this process until the bottom layer is reached
– Advantages:
• Query-independent, easy to parallelize, incremental update
• O(K), where K is the number of grid cells at the lowest level
– Disadvantages:
• All the cluster boundaries are either horizontal or vertical, and
no diagonal boundary is detected
WaveCluster (1998)
 Sheikholeslami, Chatterjee, and Zhang (VLDB’98)
 A multi-resolution clustering approach which applies
wavelet transform to the feature space
– A wavelet transform is a signal processing technique that
decomposes a signal into different frequency sub-band.
 Both grid-based and density-based
 Input parameters:
– # of grid cells for each dimension
– the wavelet, and the # of applications of wavelet transform.
WaveCluster (1998)
How to apply wavelet transform to find clusters
– Summaries the data by imposing a multidimensional
grid structure onto data space
– These multidimensional spatial data objects are
represented in a n-dimensional feature space
– Apply wavelet transform on feature space to find the
dense regions in the feature space
– Apply wavelet transform multiple times which result in
clusters at different scales from fine to coarse
What Is Wavelet (2)?
Quantization
Transformation
WaveCluster (1998)
 Why is wavelet transformation useful for clustering
– Unsupervised clustering
It uses hat-shape filters to emphasize region where points cluster,
but simultaneously to suppress weaker information in their
boundary
– Effective removal of outliers
– Multi-resolution
– Cost efficiency
 Major features:
–
–
–
–
Complexity O(N)
Detect arbitrary shaped clusters at different scales
Not sensitive to noise, not sensitive to input order
Only applicable to low dimensional data
CLIQUE (Clustering In QUEst)
 Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98).
http://portal.acm.org/citation.cfm?id=276314
 Automatically identifying subspaces of a high dimensional data
space that allow better clustering than original space
 CLIQUE can be considered as both density-based and grid-based
– It partitions each dimension into the same number of equal length interval
– It partitions an m-dimensional data space into non-overlapping rectangular
units
– A unit is dense if the fraction of total data points contained in the unit
exceeds the input model parameter
– A cluster is a maximal set of connected dense units within a subspace
CLIQUE: The Major Steps
 Partition the data space and find the number of points
that lie inside each cell of the partition.
 Identify the subspaces that contain clusters using the
Apriori principle
 Identify clusters:
– Determine dense units in all subspaces of interests
– Determine connected dense units in all subspaces of interests.
 Generate minimal description for the clusters
– Determine maximal regions that cover a cluster of connected
dense units for each cluster
– Determination of minimal cover for each cluster
ABSTRACT
CLIQUE
Data mining applications place special requirements on clustering algorithms including:
the ability to find clusters embedded in subspaces of high dimensional data,
scalability, end-user comprehensibility of the results,
non-presumption of any canonical data distribution, and
insensitivity to the order of input records.
We present CLIQUE, a clustering algorithm that satisfies each of these requirements.
CLIQUE identifies dense clusters in subspaces of maximum dimensionality.
It generates cluster descriptions in the form of DNF expressions that are minimized for
ease of comprehension.
It produces identical results irrespective of the order in which input records are
presented and does not presume any specific mathematical form for data
distribution.
Through experiments, we show that CLIQUE efficiently finds accurate cluster in large
high dimensional datasets.
=3
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Vacation
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(week)
0 1 2 3 4 5 6 7
Vacation
(10,000)
0 1 2 3 4 5 6 7
age
60
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30
40
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age
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age
60
Strength and Weakness of CLIQUE
 Strength
– It automatically finds subspaces of the highest dimensionality such
that high density clusters exist in those subspaces
– It is insensitive to the order of records in input and does not
presume some canonical data distribution
– It scales linearly with the size of input and has good scalability as
the number of dimensions in the data increases
 Weakness
– The accuracy of the clustering result may be degraded at the
expense of simplicity of the method
Model-Based Clustering Methods
 Attempt to optimize the fit between the data and some
mathematical model
 Statistical and AI approach
– Conceptual clustering
• A form of clustering in machine learning
• Produces a classification scheme for a set of unlabeled objects
• Finds characteristic description for each concept (class)
– COBWEB (Fisher’87)
• A popular a simple method of incremental conceptual learning
• Creates a hierarchical clustering in the form of a classification tree
• Each node refers to a concept and contains a probabilistic description of that
concept
COBWEB Clustering Method
A classification tree
More on Statistical-Based Clustering
 Limitations of COBWEB
– The assumption that the attributes are independent of each other is
often too strong because correlation may exist
– Not suitable for clustering large database data – skewed tree and
expensive probability distributions
 CLASSIT
– an extension of COBWEB for incremental clustering of continuous
data
– suffers similar problems as COBWEB
 AutoClass (Cheeseman and Stutz, 1996)
– Uses Bayesian statistical analysis to estimate the number of
clusters
– Popular in industry
Other Model-Based Clustering Methods
 Neural network approaches
– Represent each cluster as an exemplar, acting as a
“prototype” of the cluster
– New objects are distributed to the cluster whose exemplar
is the most similar according to some dostance measure
 Competitive learning
– Involves a hierarchical architecture of several units
(neurons)
– Neurons compete in a “winner-takes-all” fashion for the
object currently being presented
Model-Based Clustering Methods
Self-organizing feature maps (SOMs)
 Clustering is also performed by having several units
competing for the current object
 The unit whose weight vector is closest to the current object
wins
 The winner and its neighbors learn by having their weights
adjusted
 SOMs are believed to resemble processing that can occur in
the brain
 Useful for visualizing high-dimensional data in 2- or 3-D
space
Hybrid Clustering
 Hybrid clustering combines partitioning clustering and
hierarchical clustering approach
 One of the common approaches
is to combine k-means method
and hierarchical clustering
 First partition the dataset into k
small clusters, and then merge
the clusters based on similarity
using hierarchical method.
K=
7
Problems of Existing Hybrid Clustering
 Predefine the number of preliminary
clusters, K.
 Unable to handle noisy data
500
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Similarity Measures
 Similarity is fundamental to the definition of a
cluster.
1. Minkowski Metric
2. Context Metrics:
1. Mutual neighbor distance
2. Conceptual similarity measure
Minkowski Metric
 Minkowski Metric = ||xi – xj||p
 It works well when a data set has “compact” or
“isolated” clusters
 The drawback is the tendency of the largest-scaled
feature to dominate the others.
 Solutions to this problem include normalization of the
continuous features (to a common range or variance) or
other weighting schemes.
Mutual neighbor distance (MND)
 MND comes from real life observations and is based on a nonsymmetric similarity (e.g., friendship level).
 Two persons A and B group together as close friends if they
mutually feel that the other is his closest friend.
 If A feels that B is not such a close friend to him, then even though
B may feel that A is his closest friend, the bond of friendship
between them is comparatively weak.
 The strength of the bond of friendship between two persons is a
function of mutual feeling rather than one-way feeling.
Mutual neighbor distance (MND)
MND(xi, xj) = NN(xi, xj) + NN(xj, xi)
where NN(xi, xj) is the neighbor-ness value (or similarity) of xj to xi
 The MND is not a metric (it does not satisfy the triangle inequality). It
satisfies the first two conditions of a metric:
1. MND(xi, xj) >= 0, and MND(xi, xj) = 0  xi = xj positive definite
2. MND(xi, xj) = MND(xj, xi)
symmetric
–
Note the symmetry was manufactured by defining MND as the sum of the two
non-symmetric NN measures.
 In spite of this, MND has been successfully applied in several clustering
applications.
 This observation supports the viewpoint that the dissimilarity does not
need to be a metric.
MND Distance (Cont.)
 Figure 4:
– NN(A, B) = 1, NN(B, A) = 1, MND(A,B) = 2
– NN(B, C) = 1, NN(C, B) = 2, MND(B, C) = 3
 Figure 5:
– NN(A, B) = 1, NN(B, A) = 4, MND(A,B) = 5
– NN(B, C) = 1, NN(C, B) = 2, MND(B, C) = 3
Conceptual Similarity Measure
 In the case of conceptual clustering, the similarity
between xi and xj is defined as
s( xi, xj ) = f( xi, xj, σ, ε )
 where  is the context (the set of surrounding points, and
 is a set of pre-defined concepts.
 The conceptual similarity measure is the most general
similarity measure.
Conceptual Similarity Measure (Cont.)
 The Euclidean distance
between points A and B is less
than that between B and C.
 However, B and C can be
viewed as “more similar” than
A and B because B and C
belong to the same concept
(ellipse) and A belongs to a
different concept (rectangle).
Problems and Challenges
Some progress has been made in scalable clustering methods
– Partitioning: k-means, k-medoids, CLARANS
– Hierarchical: BIRCH, CURE
– Density-based: DBSCAN, CLIQUE, OPTICS
– Grid-based: STING, WaveCluster
– Model-based: Autoclass, Denclue, Cobweb
 Current clustering techniques do not address all the
requirements adequately
 Constraint-based clustering analysis: Constraints exist in
data space (bridges and highways) or in user queries
 Still the curse of cardinality stands as
THE difficult problem!
What Is Outlier Discovery?
 What are outliers?
– The set of objects are considerably dissimilar from the
remainder of the data
– Example: Sports: Michael Jordon, Wayne Gretzky, ...
 Problem
– Find top n outlier points
 Applications:
–
–
–
–
Credit card fraud detection
Telecom fraud detection
Customer segmentation
Medical analysis
Outlier Discovery:
Statistical Approaches
Assume a model underlying distribution that generates
data set (e.g. normal distribution)
 Use discordancy tests depending on
– data distribution
– distribution parameter (e.g., mean, variance)
– number of expected outliers
 Drawbacks
– most tests are for single attribute
– In many cases, data distribution may not be known
Outlier Discovery: Distance-Based Approach
 Introduced to counter the main limitations imposed by statistical methods
– We need multi-dimensional analysis without knowing data distribution.
 Distance-based outlier: A DB(p, D)-outlier is an object O in a dataset T such that at
least fraction p (usually ~99/100 or so) of the objects in T lie at a distance greater
than D from O (basically: Too many of the points lie far from O)
 A dual formulation is, basically: Too few of the points lie close to O, so O is a
DB(p, D)-outlier if at least fraction p (usually ~1/100 or so) of the objects in T lie at
a distance less than D from O
»
 Algorithms for mining distance-based outliers
– Index-based algorithm
– Nested-loop algorithm
– Cell-based algorithm
Outlier Discovery: Deviation-Based Approach
 Identifies outliers by examining the main characteristics of
objects in a group
 Objects that “deviate” from this description are considered
outliers
 sequential exception technique
– simulates the way in which humans can distinguish unusual objects
from among a series of supposedly like objects
 OLAP data cube technique
– uses data cubes to identify regions of anomalies in large
multidimensional data
Outlier Detection
Huge amount of Data
Unexpected Knowledge/Anomaly Interest
Outlier Detection
 Mining rare events, deviant objects and exceptions
 Find anomaly interests in many domains:
•
•
•
•
Criminal activities in electronic commerce
Intrusion in network
Pest infestation in agriculture
Detecting bugs in software, etc
 An example in business
• British Telecom broke a multimillion dollar fraud
Outlier Analysis (Cont.)
Outlier Analysis
Statistic-based
Distance-based
Distribution
Depth
(Barnett 1994) (Preparata 1988)
Knorr and Ng
(VLDB 1998)
Density-based
Breunig’s LOF
(SIGMOD 2000)
Ramaswamy
(MOD 2000)
Angiulli
(TKDE2005)
Cluster-based
Papadimitriou’s
LOCI
(ICDE 2002)
Jiang’s MST
(PRL)
He’s CBLOF
(PRL)
Distance-based Outlier Detection
 Distance-based outlier definition, DB (p, D), by Knorr and Ng (1997)
– p: a number percentage; D: distance threshold
– Unify distribution-based outlier definitions
– Most well known definition
 Outlier detection methods
–
–
–
–
–
Nested loop method (Knorr and Ng, 1998)
Cell structure-based method (Knorr and Ng, 1998)
Partition-based method (Ramaswamy, 2000)
Work well for small datasets
Not efficient for large datasets
 Our proposed work followed aims to improve the efficiency of the DB
outlier detection process
Outlier Definition
 Knorr and Ng’s Definition
– x is an outlier if {y X |d (y-x) ≥ D & |y| ≥ p*|X|}
– d(y,x): distance between y and x
|y| ≥ p*|X|
x
|y| < (1-p)*|X|
complement
·D
Figure 1. Knorr’s definition
x
·D
Figure 2. Equivalent definition
 Equivalent Description
– x is an outlier if {y X |d (y-x) < D & |y|<p*|X|}
Our Definition
 Definition 1: Disk neighborhood
– Define the Disk-neighborhood of x with radius r as a set
satisfying
DiskNbr (x, r) = {y X |d(y-x) r}
where, y is any point in DiskNbr(x,r) and d is the distance between y and x.
– Define points in DiskNbr (x, r) as r-neighbors of x.
 Definition 2: DiskNbr-based outlier
– A point x is considered as a DB (p, D) outlier iff
|DiskNbr(x,D)|≤ (1-p)*|X|.
where |DiskNbr (x,D)| is the number of D-neighbors of x,
|X| is the total size of dataset X
– (1-p)*|X| is a number threshold given user defined p.
Proposition 1
 If |DiskNbr (x, D/2)| ≥ (1-p)*|X|, then all the D/2-neighbors of
x are not outliers.
Proof: For  q  DiskNbr(x,D/2)
– DiskNbr (q, D)  DiskNbr (x, D/2)
– |DiskNbr(q, D)| ≥ |DiskNbr (x, D/2)| ≥ (1-p)*|X|
q is not an outlier.
 Mark D/2-neighbors as non-outliers
 A pruning rule
DiskNbr (x, D/2)
D
q
x
D/2
D
Figure 3. D/2-neighbors of x marked as non-outliers
Proposition 2
 If |DiskNbr (x, 2D)| < (1-p)* |X|, then all D-neighbors of
x are outliers.
Proof: For q  DiskNbr (x, D)

– DiskNbr (q, D)
DiskNbr (x, 2D)
– |DiskNbr (q, D)| ≤ |DiskNbr(x, 2D)| < (1-p)* |X|
q is an outlier
 All points in DiskNbr(x,D) are outliers
 Identify outliers by-neighborhood, faster than by point
DiskNbr(x,D)
D
q
x
D
2D
Figure 4. All D-Neighbors of x are outliers
The DBODLP Algorithm
Step 1: Select a point x arbitrarily from X;
Step 2: Search D-neighbors of x and calculate
|DiskNbr(x,D)|:
– |DiskNbr(x,D)| < (1-p)*|X|
• r <- 2D
• Compute |DiskNbr (x, 2*D)|
• IF |DiskNbr(x,2*D)| < (1-p)*|X|
insert D-neighbors into the outlier set.
ELSE only x is an outlier.
– |DiskNbr(x,D)| ≥ (1-p)*|X|
·2D
·D x
x·D/2
·D
Figure 5. An example
• r <- D/2;
• Calculate |DiskNbr(x,D/2)|.
• IF |DiskNbr(x,D/2)| ≥ (1-p) * |X|
mark all D/2-neighbors as non-outliers.
ELSE only x is non outlier.
Step 3: The process is conducted iteratively until all points
in dataset X are examined.
The DBODLP Algorithm
The algorithm is implemented based on a vertical
storage and index model
Vertical Data Structures
Vertical DBODLP
 DBODLP is implemented based on P-Tree
structure
 Predicate Range Tree is used to calculate the
number of disk nearest neighbors
– PDiskNbr(x,D)= Py>x-D AND Py≤x+D
– |DiskNbr(x,D)|=COUNT (PDiskNbr(x,D))
 The experiments are done in DataMIME
Architecture for the DataMIME™ System
(DataMIMEtm = P-tree based Data base and data mining system)
YOUR DATA
YOUR DATA
MINING
Query and Data
mining
Meta
Data File
P-Tree
Feeders
Data Integration
Language
DIL
DII (Data Integration Interface)
Ptree (Predicates) Query
Language
DMI (Data Mining Interface)
Data Repository
lossless, compressed, distributed, verticallystructured P-tree database
Preliminary experimental analysis
scalability of NL, PODM and PODMP
Comparison of NL,PODM, PODMP
1800
2000
1600
1400
Run Time
1000
(in second)
1200
500
0
run time
1500
n
256
1024
4096
16384
65536
NL
0.06
0.8
20.03
336.5
1657.9
PODM
0.12
1.15
7.32
65.4
803.2
PODMP
0.11
1.05
5.3
30.2
267.52
1000
NL
800
PODM
600
PODMP
400
200
0
-200256
1024
Data Size
Figure 6. Run Time Comparison
4096
16384
65536
data size
Figure 7. Scalability Comparisons
1. Our method has almost an order of magnitude of
improvement over the nested loop method in terms of run
time
2. All methods produce same outlier set
3. Our method scales better than others
Summary
 Cluster analysis groups objects based on their similarity
and has wide applications
 Measure of similarity can be computed for various types of
data
 Clustering algorithms can be categorized into partitioning
methods, hierarchical methods, density-based methods,
grid-based methods, and model-based methods
 Outlier detection and analysis are very useful for fraud
detection, etc. and can be performed by statistical, distancebased or deviation-based approaches
 There are still lots of research issues on cluster analysis,
such as constraint-based clustering
References (1)
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high dimensional data for data mining applications. SIGMOD'98
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clustering structure, SIGMOD’99.
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clusters in large spatial databases. KDD'96.
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SIGMOD'98.
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References (2)
 L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster
Analysis. John Wiley & Sons, 1990.
 E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets.
VLDB’98.
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Clustering. John Wiley and Sons, 1988.
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approach for very large spatial databases. VLDB’98.
 W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial Data
Mining, VLDB’97.
 T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering method for
very large databases. SIGMOD'96.