han/eick: clustering ii 1 clustering part2 continued 1. birch skipped 2. density-based clustering...

Han/Eick: Clustering II 1

Clustering Part2 continued

1. BIRCH skipped

2. Density-based Clustering --- DBSCAN and DENCLUE

3. GRID-based Approaches --- STING and CLIQUE

4. SOM skipped

5. Cluster Validation other set of transparencies

6. Outlier/Anomaly Detection other set of transparencies

Slides in red will be used for Part1.


BIRCH (1996)

Birch: Balanced Iterative Reducing and Clustering using Hierarchies, by Zhang, Ramakrishnan, Livny (SIGMOD’96)

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 multi-level 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 the quality with a few additional scans

Weakness: handles only numeric data, and sensitive to the order of the data record.


Clustering Feature Vector

Clustering Feature: CF = (N, LS, SS)

N: Number of data points

LS: Ni=1=Xi

SS: Ni=1=Xi

2

1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

CF = (5, (16,30),(54,190))

(3,4)(2,6)(4,5)(4,7)(3,8)


CF TreeCF1

child1

CF3

child3

CF2

child2

CF6

child6

CF1

child1

CF3

child3

CF2

child2

CF5

child5

CF1 CF2 CF6prev next CF1 CF2 CF4

prev next

B = 7

L = 6

Root

Non-leaf node

Leaf node Leaf node


Chapter 8. Cluster Analysis

What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary


Density-Based Clustering Methods

Clustering based on density (local cluster criterion), such as density-connected points or based on an explicitly constructed density function

Major features: Discover clusters of arbitrary shape Handle noise Usually one scan Need density estimation parameters

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)


DBSCAN Two parameters:

Eps: Maximum radius of the neighbourhood MinPts: Minimum number of points in an Eps-

neighbourhood of that point NEps(p): {q belongs to D | dist(p,q) <= Eps} Directly density-reachable: A point p is directly

density-reachable from a point q wrt. Eps, MinPts if 1) p belongs to NEps(q) 2) core point condition:

|NEps (q)| >= MinPts

pq

MinPts = 5

Eps = 1 cm


Density-Based Clustering: Background (II)

Density-reachable: A point p is density-reachable

from a point q wrt. Eps, MinPts if there is a chain of points p1, …, pn, p1 = q, pn = p such that pi+1 is directly density-reachable from pi

Density-connected A point p is density-connected

to a point q wrt. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o wrt. Eps and MinPts.

p

qp1

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 a maximal set of density-connected points

Discovers clusters of arbitrary shape in spatial databases with noise

Core

Border

Outlier

Eps = 1cm

MinPts = 5

Density reachablefrom core point

Not density reachablefrom core point


DBSCAN: The Algorithm

Arbitrary select a point p

Retrieve all points density-reachable from p wrt Eps and MinPts.

If p is a core point, a cluster is formed.

If p ia not a core point, no points are density-reachable from p and DBSCAN visits the next point of the database.

Continue the process until all of the points have been processed.


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)

But needs a large number of parameters


Influence function: describes the impact of a data point within its neighborhood.

Overall density of the data space can be calculated as the sum of the influences of all data points.

Clusters can be determined mathematically by identifying density attractors; object that are associated with the same density attractor belong to the same cluster

Density attractors are local maximal of the overall density function.

Uses grid-cells to speed up computations; only data points in neighboring grid-cells are used to determine the density for a point.

Denclue: Technical Essence


DENCLUE Influence Function and its Gradient

Example

N

i

xxdD

Gaussian

i

exf1

2

),(2

2

)(

N

i

xxd

iiD

Gaussian

i

exxxxf1

2

),(2

2

)(),(

f x y eGaussian

d x y

( , )( , )

2

22


Example: Density Computation

D={x1,x2,x3,x4}

fDGaussian(x)= influence(x1) + influence(x2) + influence(x3) +

influence(x4)=0.04+0.06+0.08+0.6=0.78

x1

x2

x3

x4x 0.6

0.08

0.06

0.04

y

Remark: the density value of y would be larger than the one for x

2

2

2

),(

),(inf yxd

eyxluence

Han/Eick: Clustering II

Example Non-Parametric DE in R Demo Rcode: setwd("C:\\Users\\C. Eick\\Desktop")a<-read.csv("c8.csv")require("spatstat")require("ppp")d<-data.frame(a=a[,1],b=a[,2],c=a[,3])plot(d$a,d$b)w <- owin(poly=list (list(x=c(0,530,701,640,0),y=c(0,42,20,400,420)),list(x=c(320,430,310), y=c(215,200,190)),list(x=c(10,70,170,20), y=c(200,220,170, 175))) )z<-ppp(d[,1],d[,2],window=w, marks=factor(d[,3]))plot(z)summary(z)q<-quadratcount(z, nx=12,ny=10)plot(q)den<-density(z, sigma=80)plot(den)den<-density(z, sigma=30)plot(den)den<-density(z, sigma=15)plot(den)den<-density(z, sigma=12)plot(den)den<-density(z, sigma=10)plot(den)den<-density(z, sigma=4)plot(den)

documentation for function ‘density’: http://127.0.0.1:28030/library/spatstat/html/density.ppp.html

15

http://127.0.0.1:28030/library/spatstat/html/density.ppp.html


Density Attractor


Examples of DENCLUE Clusters


Basic Steps DENCLUE Algorithms

1. Determine density attractors2. Associate data objects with density

attractors using hill climbing ( initial clustering)

3. Merge the initial clusters further relying on a hierarchical clustering approach (optional)


Density-based Clustering: Pros and Cons

+: can discover clusters of arbitrary shape+: not sensitive to outliers and supports outlier detection+: can handle noise +-: medium algorithm complexities-: finding good density estimation parameters is frequently difficult; more difficult to use than K-means. -: usually, do not do well in clustering high-dimensional datasets. -: cluster models are not well understood (yet)


Chapter 8. Cluster Analysis

What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary


Steps of Grid-based Clustering Algorithms

Basic Grid-based Algorithm1. Define a set of grid-cells2. Assign objects to the appropriate grid

cell and compute the density of each cell.

3. Eliminate cells, whose density is below a certain threshold t.

4. Form clusters from contiguous (adjacent) groups of dense cells (usually minimizing a given objective function).


Advantages of Grid-based Clustering Algorithms

fast: No distance computations Clustering is performed on summaries

and not individual objects; complexity is usually O(#-populated-grid-cells) and not O(#objects)

Easy to determine which clusters are neighboring

Shapes are limited to union of rectangular grid-cells


Grid-Based Clustering Methods

Several interesting methods (in addition to the basic grid-based algorithm) STING (a STatistical INformation Grid approach)

by Wang, Yang and Muntz (1997) CLIQUE: Agrawal, et al. (SIGMOD’98)


STING: A Statistical Information Grid Approach

Wang, Yang and Muntz (VLDB’97) The spatial area 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)

Main contribution of STING is the proposal of a data structure that can be used for many purposes (e.g. SCMRG, BIRCH kind of uses it)

The data structure is used to form clusters based on queries

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 Clusters are formed by merging cells that match a given

query description ( next slide)


STING: Query Processing(3)

Used a top-down approach to answer spatial data queries1. Start from a pre-selected layer—typically with a small

number of cells2. From the pre-selected layer until you reach the bottom

layer do the following: For each cell in the current level compute the confidence

interval indicating a cell’s relevance to a given query; If it is relevant, include the cell in a cluster If it irrelevant, remove cell from further consideration otherwise, look for relevant cells at the next lower layer

3. Combine relevant cells into relevant regions (based on grid-neighborhood) and return the so obtained clusters as your answers.


STING: A Statistical Information Grid Approach (3)

Advantages: Query-independent, easy to parallelize,

incremental update O(K), where K is the number of grid cells at

the lowest level Can be used in conjunction with a grid-based

clustering algorithm Disadvantages:

All the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected


Subspace Clustering

Clustering in very high-dimensional spaces is very difficult

High dimensional attribute spaces tend to be sparseit is hard to find any clusters

It is very difficult to create summaries from clusters in very difficult

This creates the motivation for subspace clustering: Find interesting subspaces (areas that are dense

with respect to the attributes belonging to the subspace)

Find clusters for each interesting Remark: multiple, overlapping clusters might be

obtained; basically one clustering for each subspace.


CLIQUE (Clustering In QUEst) Agrawal, Gehrke, Gunopulos, Raghavan

(SIGMOD’98). 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.


Sala

ry

(10,

000)

20 30 40 50 60age

54

31

26

70

20 30 40 50 60age

54

31

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Vac

atio

n(w

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age

Vac

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Salary 30 50

= 3


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

Quite expensive


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


References (1)

R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. SIGMOD'98

M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973. M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to

identify the clustering structure, SIGMOD’99. P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World

Scietific, 1996 M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for

discovering clusters in large spatial databases. KDD'96. M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial

databases: Focusing techniques for efficient class identification. SSD'95. D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine

Learning, 2:139-172, 1987. D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An

approach based on dynamic systems. In Proc. VLDB’98. S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for

large databases. SIGMOD'98. A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.


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.

G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to Clustering. John Wiley and Sons, 1988.

P. Michaud. Clustering techniques. Future Generation Computer systems, 13, 1997.

R. Ng and J. Han. Efficient and effective clustering method for spatial data mining. VLDB'94.

E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large data sets. Proc. 1996 Int. Conf. on Pattern Recognition, 101-105.

G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution clustering 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.

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