Optics ordering points to identify the clustering structure

(Paper Presentation)
OPTICS-Ordering Points To Identify The
Clustering Structure
Presenter
Anu Singha
Asiya Naz
Rajesh Piryani
South Asian University

OUTLINE
 Introduction
 Definition (Directly Density Reachable, Density Reachable, Density
Connected,
 OPTICS Algorithm
 Example
 Graphical Results
April 30,2012 2

CLUSTERING
 Goal
 Group objects into meaningful subclasses as part of an exploratory process
to insight into data or as a preprocessing step for other algorithms.
 Clustering Strategies
 Hierarchical
 Partitioning
 k-means
 Density Based
April 30,2012 3

DENSITY BASED CLUSTERING
 Density-based Clustering locates regions of high density that are separated from
one another by regions of low density.
 Density = number of points within a specified radius (Eps)
April 30,2012 4

DENSITY BASED CLUSTERING
Flat Clustering
one level of clusters
Hierarchical Clustering
nested clusters
e.g. density-based clustering algorithm
DBSCAN [KDD 96]
e.g. density-based clustering algorithm
OPTICS [SIGMOD 99]
April 30,2012 5

INTRODUCTION
 DBSCAN can cluster objects given input parameters such as
 (Eps) (the maximum radius of a neighborhood) and
 MinPts (the minimum number of points required in the neighborhood of a
core object),
 it encumbers users with the responsibility of selecting parameter values that will
lead to the discovery of acceptable clusters.
 Such parameter settings are usually empirically set and difficult to determine.
 Moreover, real-world, high-dimensional data sets often have very skewed
distributions such that their intrinsic clustering structure may not be well
characterized by a single set of global density parameters.
April 30,2012 6

INTRODUCTION
 density-based clusters are monotonic with respect to the neighborhood
threshold.
 In DBSCAN, for a fixed MinPts value and two neighborhood thresholds,
 (Eps) 1 < (Eps) 2, a cluster C with respect to (Eps)1 and
 MinPts must be a subset of a cluster C’ with respect to (Eps) 2 and MinPts.
 This means that if two objects are in a density-based cluster, they must also be
in a cluster with a lower density requirement.
 Different clusters may have very different densities
 Clusters may be in hierarchies
April 30,2012 7
To overcome the difficulty in using one set of global parameters in clustering analysis, a
cluster analysis method called OPTICS was proposed.

OPTICS
 in figure 3, where
 C1 and C2 are density-based clusters
with respect to e2 < e1
 and C is a density based cluster with
respect to e1 completely containing the
sets C1 and C2.
 for a constant MinPts-value, density-
based clusters with respect to a higher
density (i.e. a lower value for e) are
completely contained in density-
connected sets with respect to a lower
density (i.e. a higher value for e).
April 30,2012 8

OPTICS
 To produce a consistent result
 obey a specific order in which objects are processed when expanding a cluster.
 select an object which is density-reachable with respect to the lowest ε value
 to guarantee that clusters w.r.t higher density (i.e. smaller e values) are finished first.
 OPTICS works in principle like such an extended DBSCAN algorithm for an
infinite number of distance parameters εi which are smaller than a “generating
distance” ε (i.e. 0 ≤ εi ≤ ε).
 The only difference is that we do not assign cluster memberships.
 Instead, we store the order in which the objects are processed and the
information which would be used by an extended DBSCAN algorithm to assign
cluster memberships if this were at all possible for an infinite number of
parameters).
April 30,2012 9

OPTICS
 OPTICS does not explicitly produce a data set clustering.
 It outputs a cluster ordering.
 It is linear list of all objects under analysis and
 represents the density-based clustering structure of the data.
 Objects in a denser cluster are listed closer to each other in the cluster ordering.
 Ordering is equivalent to density-based clustering obtained from a wide range of
parameter settings.
 Thus, OPTICS does not require the user to provide a specific density threshold.
 The cluster ordering can be used to extract basic clustering information (e.g.,
cluster centers, or arbitrary-shaped clusters), derive the intrinsic clustering
structure, as well as provide a visualization of the clustering
April 30,2012 10

OPTICS (CONTINUED..)
 To construct the different clusterings simultaneously, the objects are processed
in a specific order.
 This order selects an object that is density-reachable with respect to the lowest
(Eps) value so that clusters with higher density (lower (Eps)) will be finished
first.
 Based on this idea, OPTICS needs two important pieces of information per
object:
 Core Distance
 Reachability Distance
April 30,2012 11
It was presented by Mihael Ankerst, Markus M. Breunig,Hans-Peter
Kriegel and Jörg Sander.

TERMINOLOGIES
 ε-Neighborhood
 Objects within a radius of ε from an object. (epsilon-neighborhood)
 Core objects
 ε-Neighborhood of an object contains at least MinPts of objects
April 30,2012 12
q p
εε
ε-Neighborhood of p
ε-Neighborhood of q
p is a core object (MinPts = 4)
q is not a core object

TERMINOLOGIES
 Directly Density Reachable
 An object q is directly density-reachable from object p if q is within the ε-
Neighborhood of p and p is a core object
April 30,2012 13
q p
εε
 q is directly density-reachable from p
 p is not directly density- reachable from q?

TERMINOLOGIES
 Density Reachable
 An object p is density-reachable from q w.r.t ε and MinPts if there is a
chain of objects p1,…,pn, with p1=q, pn=p such that pi+1is directly density-
reachable from pi w.r.t ε and MinPts for all 1 <= i <= n
April 30,2012 14
p
 q is density-reachable from p
 p is not density- reachable from q>
 Transitive closure of direct density-Reachability,
asymmetric
q

TERMINOLOGIES
 Definition: core-distance
 Definition: reachability-distance






otherwise)(dist
|),(rangeQuery|if
)(distcore ,
oMinPts
MinPtso
oMinPts


reach dist ( , ) max(core dist ( ),dist( , )), ,   MinPts MinPtsp o o p o
core-distance(o)
o
reachability-distance(p,o)
p
p
reachability-distance(p,o)

MinPts = 5
April 30,2012 15

ABOUT OPTICS COMPUTATION
 It computes an ordering of all objects in a given database. And
 It stores the core-distance and a suitable reachability-distance for each object
in the database.
 OPTICS maintains a list called OrderSeeds to generate the output ordering.
 Objects in OrderSeeds
 are sorted by the reachability-distance from their respective closest core
objects,
 that is, by the smallest reachability-distance of each object.
April 30,2012 16

ABOUT OPTICS ALGORITHM
 Begin with an arbitrary object from the input database as the current object, p.
 It retrieves the ε-neighborhood of p, determines the core-distance, and sets
the reachability-distance to undefined.
 The current object, p, is then written to output.
 If p is not a core object,
 OPTICS simply moves on to the next object in the OrderSeeds list (or the
input database if OrderSeeds is empty).
April 30,2012 17

ABOUT OPTICS ALGORITHM
 If p is a core object,
 then for each object, q, in the ε-neighborhood of p,
 OPTICS updates its reachability-distance from p
 and inserts q into OrderSeeds if q has not yet been processed.
 The iteration continues until the input is fully consumed and OrderSeeds is
empty.
April 30,2012 18

ALGORITHM
 OPTICS (SetOfObjects, e, MinPts, OrderedFile)
 OrderedFile.open();
 FOR i FROM 1 TO SetOfObjects.size DO
 Object := SetOfObjects.get(i);
 IF NOT Object.Processed THEN
 ExpandClusterOrder(SetOfObjects, Object, e, MinPts,
OrderedFile)
 OrderedFile.close();
 END; // OPTICS
April 30,2012 19

PROCEDURE FOR
ExpandClusterOrder
 ExpandClusterOrder(SetOfObjects, Object, ε, MinPts, OrderedFile);
 neighbors := SetOfObjects.neighbors(Object, ε);
 Object.Processed := TRUE;
 Object.reachability_distance := UNDEFINED;
 Object.setCoreDistance(neighbors, ε, MinPts);
 OrderedFile.write(Object);
 IF Object.core_distance <> UNDEFINED THEN
 OrderSeeds.update(neighbors, Object);
 WHILE NOT OrderSeeds.empty() DO
 currentObject := OrderSeeds.next();
 neighbors:=SetOfObjects.neighbors(currentObject, ε);
 currentObject.Processed := TRUE;
 currentObject.setCoreDistance(neighbors, ε, MinPts);
 OrderedFile.write(currentObject);
 IF currentObject.core_distance<>UNDEFINED THEN
 OrderSeeds.update(neighbors, currentObject);
 END; // ExpandClusterOrder
April 30,2012 20
object is simply written to the file OrderedFile with its coredistance and its
current reachability-distance.

OrderSeeds::update()
 OrderSeeds::update(neighbors, CenterObject);
 c_dist := CenterObject.core_distance;
 FORALL Object FROM neighbors DO
 IF NOT Object.Processed THEN
 new_r_dist:=max(c_dist,CenterObject.dist(Object));
 IF Object.reachability_distance=UNDEFINED THEN
 Object.reachability_distance := new_r_dist;
 insert(Object, new_r_dist);
 ELSE // Object already in OrderSeeds
 IF new_r_dist<Object.reachability_distance THEN
 Object.reachability_distance := new_r_dist;
 decrease(Object, new_r_dist);
 END; // OrderSeeds::update
April 30,2012 21

 Having generated the augmented cluster-ordering of a database with respect to e
and MinPts,
 extract any density-based clustering from this order with respect to MinPts and a
clustering- distance ε ’ ≤ε
 by simply “scanning” the cluster-ordering
 and assigning cluster-memberships depending on the reachability- distance and the core-
distance of the objects.
 That an extraction is possible only demonstrates that the cluster-ordering of a
data set actually contains the information about the intrinsic clustering structure
of that data set (up to the generating distance ε) .
April 30,2012 22

ExtractDBSCAN-Clustering
(ClusterOrderedObjs, ε’, MinPts)
 ExtractDBSCAN-Clustering (ClusterOrderedObjs, ε’, MinPts)
 // Precondition: ε ' ≤ generating dist ε for ClusterOrderedObjs
 ClusterId := NOISE;
 FOR i FROM 1 TO ClusterOrderedObjs.size DO
 Object := ClusterOrderedObjs.get(i);
 IF Object.reachability_distance > ε’ THEN
 // UNDEFINED > ε
 IF Object.core_distance ≤ ε’ THEN
 ClusterId := nextId(ClusterId);
 Object.clusterId := ClusterId;
 ELSE
 Object.clusterId := NOISE;
 ELSE // Object.reachability_distance ≤ ε’
 Object.clusterId := ClusterId;
 END; // ExtractDBSCAN-Clustering
April 30,2012 23

OPTICS ALGORITHM EXAMPLE
A I
B
J
K
L
R
M
P
N
C
F
D
E
G H
44

reach
seedlist:
• Example Database (2-dimensional, 16 points)
• ε= 44, MinPts = 3
April 30,2012 24

A
I
B
J
K
L
R
M
P
N
C
F
D
E
G
H
44

reach
seedlist:
A
I
B
J
K
L
R
M
P
N
C
F
D
E
G
H
A
44


core-
distance
(B,40) (I, 40)
• ε= 44, MinPts = 3
April 30,2012 25

44

reach
A
44

B
A
I
B
J
K
L
R
M
P
N
C
F
D
E
G
H
seedlist: (I, 40) (C, 40)
• ε= 44, MinPts = 3
April 30,2012 26

44

reach
A
44

B
A
I
B
J
K
L
R
M
P
N
C
F
D
E
G
H
I
seedlist: (J, 20) (K, 20) (L, 31) (C, 40) (M, 40) (R, 43)
• ε= 44, MinPts = 3
April 30,2012 27

44

reach
A
44

B I
A
I
B
J
K
L
R
M
P
N
C
F
D
E
G
H
J
seedlist: (L, 19) (K, 20) (R, 21) (M, 30) (P, 31) (C, 40)
• ε= 44, MinPts = 3
April 30,2012 28

44

reach
A
44

B I J
A
I
B
J
K
L
R
M
P
N
C
F
D
E
G
H
L
…
seedlist: (M, 18) (K, 18) (R, 20) (P, 21) (N, 35) (C, 40)
• ε= 44, MinPts = 3
April 30,2012 29

A
I
B
J
K
L
R
M
P
N
C
F
D
E G
H
seedlist: -
A B I J L M K N R P C D F G E H
44
reach

• ε= 44, MinPts = 3
April 30,2012 30

A
I
B
J
K
L
R
M
P
N
C
F
D
E
G
H
seedlist: -
A B I J L M K N R P C D F G E H
44
reach

• ε= 44, MinPts = 3
April 30,2012 31

GRAPHICAL REPRESENTATION
 A data set’s cluster ordering can be represented graphically.
 It helps to visualize and understand the clustering structure in a data set.
April 30,2012 32

GRAPHICAL REPRESENTATION
 In Figure
 reachability plot for a simple 2-D data set, which presents a general
overview of how the data are structured and clustered.
 The data objects are plotted in the clustering order (horizontal axis) together
with their respective reachability-distances (vertical axis).
 The three Gaussian “bumps” in the plot reflect three clusters in the data set.
April 30,2012 33

ALGORITHM PERFROMANCE
 performed an extensive performance test using different data sets and different
parameter settings.
 simply turned out that the run-time of OPTICS was almost constantly 1.6 times
the run-time of DBSCAN.
 not surprising since the run-time for OPTICS as well as for DBSCAN is heavily
dominated by
 the run-time of the ε -neighborhood queries
 which must be performed for each object in the database, i.e. the run-time
for both algorithms is O(n * run-time of an e-neighborhood query).
April 30,2012 34

 To retrieve the e-neighborhood of an object o, a region query with the center o
and the radius e is used.
 Without any index support, to answer such a region query, a scan through the
whole database has to be performed.
 In this case, the run-time of OPTICS would be O(n2).
 If a tree-based spatial index can be used, the run-time is reduced to O (n log n)
April 30,2012 35

 The height of such a tree-based index is O(log n) for a database of n objects in
the worst case and, at least in low-dimensional spaces, a query with a “small”
query region has to traverse only a limited number of paths.
 Furthermore, if we have a direct access to the e-neighborhood, e.g. if the objects
are organized in a grid, the run-time is further reduced to O(n) because in a grid
the complexity of a single neighborhood query is O(1).
April 30,2012 36

CONCLUSION
 OPTICS computes an augmented cluster- ordering of the database objects.
 The main advantage of approach, when compared to the clustering algorithms
proposed in the literature, is that, do not limit to one global parameter setting.
 Instead, the augmented cluster-ordering contains information which is
equivalent to the density based clusterings corresponding to a broad range of
parameter settings and thus is a versatile basis for both automatic and interactive
cluster analysis.
April 30,2012 37

REFERENCES
 [1] Mihael Ankerst, Markus M. Breunig, Hans-Peter Kriegel, Jörg Sander,
“OPTICS: Ordering Points To Identify the Clustering Structure” , Proc. ACM
SIGMOD’99 Int. Conf. on Management of Data, Philadelphia PA, 1999.
 [2] Data Mining Concepts and Techniques by Han Kamber Pei , Third Edition
 [3] Stefan Brecheisen, Hans-Peter Kriegel, Martin Pfeifle, “Efficient Density-
Based Clustering of Complex Objects“
 [4] Class Lecture Slides about Density Clustering -DBSCAN
April 30,2012 38

THANK YOU
FOR YOUR CO-OPERATION
April 30,2012 39

Optics ordering points to identify the clustering structure

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