Download Document

A Summary to Current
Clustering Methods and
OPTICS: Ordering Points To
Identify the Clustering Structure
Presented by
Ho Wai Shing
Overview




Introduction
Current Clustering Techniques
OPTICS
Discussions
Introduction

What is clustering?


Given: a dataset with N points in a d dimensional space
Task: find a natural partitioning of the
points into a number (k ) of closely related
groups (clusters) and noise
Introduction

Example Application

To find similar electronic parts from their
design blue-prints:



use Fourier transform to transform contours of
parts into coefficients
do clustering on the coefficients
discuss this later in the talk
Introduction

What are the main concerns?




Efficiency
Effectiveness
Scalability
Interactivity
Current Techniques

can be classified into groups

hierarchical vs partitioning


bottom-up merging vs flat partitioning
centroid-based vs density-based

1 or more representative points for a cluster
Several Clustering Algorithms





k -mean
BIRCH
DBSCAN
CURE / C2P
OPTICS
k-mean



partitions the space into k clusters
each cluster is represented by the mean
of the points belong to this cluster
iteratively refines the k representative
points until reaching a local minimal on
total distances within clusters
k-mean



the clusters must be convex, and should
have similar size (may not be the case
in real data)
need to scan the database many times
(slow)
easily disturbed by outliers (every point
counts in calculating the mean)
an example dataset
BIRCH

use CF-Tree to summarize the data
points so that everything are in memory



points will be merged to a leaf entry of the
CF-Tree if they are similar
points will be stored in an “extension” if no
similar leaves can be found
build clusters over those leaves instead
of the original data points
an example dataset
entries in CF-Tree Leaves (contains N, LS, SS)
an example dataset
…
…
entries in CF-Tree Leaves (contains N, LS, SS)
BIRCH






basically hierarchical
one of the fastest algorithm available
scan the data only once
can remove some outliers
can be used as a pre-clustering step
the result depends on inserting order
DBSCAN



the first density-based algorithm
without grid
clusters = collections of densityconnected points
definitions:



directly density-reachable
density-reachable
density-connected
r is directly density-reachable from q
DBSCAN




start with an arbitrary point,
perform k-NN (k-nearest-neighbor)
search for that point
if it is dense then we grow that point
into a cluster
find another point until we exhaust all
the points
DBSCAN

good:




can find arbitrary shaped clusters
intuitive definition of clusters
reasonable complexity if index is available for kNN search
bad:

difficult to determine input parameters Eps and
MinPts

suffers from “Chaining Effect”
The Chaining Problem
chain
CURE


instead of using all points in calculating
the distance between a point and a
cluster like DBSCAN, CURE uses a set of
representative points within a cluster
this could reduce the chaining effect
CURE diagram: no longer
chains
actual points
points are close,
but
representatives
ain’t
C2P


much better than CURE in terms of efficiency
(O(n2lgn) to O(nlgn + m2lgm))
accomplished by a O(nlgn) pre-clustering
phase



add links between all the points and their nearest
neighbours
condense each connected graph into a single point
repeat until m points remains
OPTICS




Ordering Points To Identify the
Clustering Structure
in SIGMOD 99
by Ankerst et al.
A generalization of DBSCAN + a
visualization technique
OPTICS

Motivation:



input parameters (e.g., Eps) are difficult to
be determined
one global parameter setting may not fit all
the clusters
it’s good to allow users to have flexibility in
selecting clusters
OPTICS

definitions

core-distance of a point p


the distance between the point p and its
MinPts’th neighbour
reachability-distance of a point p w.r.t.
another point o

the distance between o and p, with a lower
bound of core-dist(o)
OPTICS



start from an arbitrary point, sorts the points
according to the reachability-distance
this sorting can be used to produce densitybased clusters with 0 < Eps < Epsinput
Reachability plot can be used to provide a
good visualization tool for analyzing clusters
OPTICS reachability plot
OPTICS 16-d reachability plot
Discussions


All the methods described above fail at
high-dimensional cases
“The curse of dimensionality”



distances between all the points are nearly
the same
grids are usually not dense (O(2d) grids vs
O(n) points)), clusters tend to be divided
by grids
no efficient indices for k-NN search
Discussions

key observation:



not all dimensions are meaningful in
clustering
some clusters may exist under a subset of
dimensions while the others exist under
another subset of dimensions
leads to:


feature selection
subspace clustering / projected clustering
References







M Ankerst, M M Breunig and H-P Kriegel, J Sander, OPTICS: Ordering Points To
Identify the Clustering Structure, SIGMOD’99
T Zhang, R Ramakrishnan and M Livny, BIRCH: An Efficient Data Clustering
Method for Very Large Databases, SIGMOD’96
S Guha, R Rastogi and K Shim, CURE: An Efficent Clustering Algorithm for Large
Databases SIGMOD’98
C C Aggarwal and P S Yu, Finding Generalized Projected Clusters in High
Dimensional Spaces, SIGMOD’00
R Agrawal, J Gehrke, D Gunopulos and P Raghavan, Automatic Subspace
Clustering of High Dimensional Data for Data Mining Applications, SIGMOD’98
M Ester, H-P Kriegel, J Sander and X Xu, A Density-Based Algorithm for
Discovering Clusters in Large Spatial Database with Noise, KDD’96
A Nanopoulos, Y Theodoridis and Y Manolopoulos, C2P: Clustering based on
Closest Pairs, VLDB’01
Questions?