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Ch. 15: Data Clustering

Ch.15.1: Motivation

- Organisms with similar genomes

A B C ~ evolutionary chains

|A,B| < |A,C|

A

B D

C E

F

Human genome ~ 1GB probability for F is extremely small

Assumption: C stems from closest neighbor

Prof. Bayer, DWH-SS2000, Clustering

More Examples

- Stocks with similar behavior in market
- Buying basket discovery
- Documents on similar topics

nearest neighbors?

similarity or distance measures?

Prof. Bayer, DWH-SS2000, Clustering

Supervised vs. Unsupervised Classification

Supervised: set of classes (clusters) is given, assign new pattern (point) to proper cluster, label it with label of its cluster

Examples: classify bacteria to select proper antibiotics, assign signature to book and place in proper shelf

Unsupervised: for given set of patterns, discover a set of clusters (training set) and assign addtional patterns to proper cluster

Examples: buying behavior, stock groups, closed groups of researchers citing each other, more?

Prof. Bayer, DWH-SS2000, Clustering

Components of a Clustering Task

- Pattern representation (feature extraction), e.g. key words, image features, genes in DNS sequences
- Pattern proximity: similarity, distance
- Clustering (grouping) algorithm
- Data abstraction: representation of a cluster, label for class, prototype, properties of a class
- Assessment of quality of output

Prof. Bayer, DWH-SS2000, Clustering

Examples

- Clustering of documents or research groups by citation index, evolution of papers. Problem: find minimal set of papers describing essential ideas
- Clustering of items from excavations according to cultural epoques
- Clustering of tooth fragments in anthropology
- Carl von Linne: Systema Naturae, 1735, botanics, later zoology
- etc ...

Prof. Bayer, DWH-SS2000, Clustering

Ch. 15.2: Formal Definitions

Pattern: (feature vector, measurements, observations, data points) X = ( x1, ... , xd) d dimensions, measurements, often d not fixed, e.g. for key words

Attribute, Feature: xi

Dimensionality: d

Pattern Set: H = { X1, X2, ... , Xn}

H often represented as nd pattern matrix

Class: set of similar patterns, pattern generating process in nature, e.g. growth of plants

Prof. Bayer, DWH-SS2000, Clustering

More Definitions

Hard Clustering: classes are disjoint, every pattern gets a unique label from

L = { l1, l2, ... , ln} with li{ 1, ... , k }see Fig. 1

Fuzzy Clustering: pattern Xigets a fractional degree of membership fijfor each output cluster j

Distance Measure: metric or proximity function or similarity function in feature space to quantify similarity of patterns

Prof. Bayer, DWH-SS2000, Clustering

Ch. 15.3: Pattern Representation and Feature Selection

Human creativity: Select few, but most relevant features !!!

Cartesian or polar Coordinates? See Fig. 3

Document retrieval: key words (which) or citations? What is a good similarity function? Use of thesaurus?

Zoology: Skeleton, lungs instead of body shape or living habits: dolphins, penguins, ostrich!!

Prof. Bayer, DWH-SS2000, Clustering

Types of Features

- Quantitative: continuous, discrete, intervals, fuzzy
- Qualitative: enumeration types (colors), ordinals (military ranks), general features like (hot,cold), (quiet, loud) (humid, dry)
- Structure Features: oo hierarchies likevehicle carBenzS400 vehicle boatsubmarine

Prof. Bayer, DWH-SS2000, Clustering

Ch. 15.4: Similarity Measures

Similar ~ small distance

Similarity function: not necessarily a metric, triangle inequality dist (A,B) + dist (B,C) dist (A,C) may be missing, quasi metric.

Euclidean Distance

dist2 (Xi,Xj) = (k=1d |xi,k - xj,k |²)1/2 = ||Xi - Xj ||2

Special case of Minkowski metric

distp (Xi,Xj) = (k=1d |xi,k - xj,k |p)1/p = ||Xi - Xj ||p

Prof. Bayer, DWH-SS2000, Clustering

Proximity Matrix: for n patterns with symmetric similarity:

n * (n-1)/2 similarity values

Representation problem:

mixture of continuous and discontinuous attributes, e.g.

dist((white, -17), (green, 25))

use wavelength as value for colors and then Euclidean distance???

Prof. Bayer, DWH-SS2000, Clustering

Other Similarity Functions

Context Similarity:

s(Xi , Xk ) = f(Xi , Xk, E) for environment E

e.g. 2 cars on a country road

2 climbers on 2 different towers of 3 Zinnen mountain

neighborhood distance of Xkw.r. to Xi = nearest neighbor number = NN(Xi , Xk )

mutual neighborhood distance

MND(Xi , Xk ) = NN(Xi , Xk ) + NN(Xk , Xi )

Prof. Bayer, DWH-SS2000, Clustering

Lemma: MND(Xi , Xk ) = MND(Xk , Xi )

MND(Xi , Xi ) = 0

Note: MND is not a metric, triangle inequality is missing!

see Fig. 4:

NN(A,B) = NN(B,A) = 1 MND(A,B) = 2

NN(B,C) = 2; NN(C,B) = 1 MND(B,C) = 3

see Fig. 5:

NN(A,B) = 4; NN(B,A) = 1 MND(A,B) = 5

NN(B,C) = 2; NN(C,B) = 1 MND(B,C) = 3

Prof. Bayer, DWH-SS2000, Clustering

(Semantic) Concept Similarity

s (Xi , Xk ) = f (Xi , Xk , C, E)

for concept C, environment E

Examples for C: ellipse, rectangle, house, car, tree ... See Fig. 6

Prof. Bayer, DWH-SS2000, Clustering

Structural Similarity of Patterns

(5 cyl, Diesel, 4000 ccm)

(6 cyl, gasoline, 2800 ccm)

dist ( ... ) ???

dist (car, boat) ???

how to cluster car engines?

Dynamic, user defined clusters via query boxes?

Prof. Bayer, DWH-SS2000, Clustering

Ch. 15.5: Clustering Methods

- agglomerativemerge clusters versus divisivesplit clusters
- monotheticall features at once versus polythetic one feature at a time
- hard clusters pattern insingle class versus fuzzy clusterspattern in several classes
- incremental add pattern at a time versusnon-increm. add patterns at once (important for large data sets!!)

Prof. Bayer, DWH-SS2000, Clustering

Ch. 15.5.1: Hierarchical Clusteringby Agglomeration

Single Link:

C2

C1

dist (C1, C2) =

min { dist (X1, X2 ) : X1C1, X2C2 }

Prof. Bayer, DWH-SS2000, Clustering

C2

C1

dist (C1, C2) =

max { dist (X1, X2 ) : X1C1, X2C2 }

Merge clusters with smallest distance in both cases!

Examples: Figures 9 to 13

Prof. Bayer, DWH-SS2000, Clustering

Hierarchical algglomerative clustering algorithmfor single link and complete link clustering

1. Compute proximity matrix between pairs of patterns, initialize each pattern as a cluster

2. Find closest pair of clusters, i.e. sort n2 distances with O(n2*log n), merge clusters, update proximity matrix (how, complexity?)

3. if all patterns are in one cluster then stopelse goto step 2

Note: proximity matrix requires O(n2) space and for distance computation at least O(n2) time for n patterns (even without clustering process), not feasible for large datasets!

Prof. Bayer, DWH-SS2000, Clustering

Ch. 15.5.2 Partitioning Algorithms

Remember: agglomerative algorithms compute a sequence of partitions from finest (1 pattern per cluster) to coarsest (all patterns in a single cluster). The number of desired clusters is chosen at the end by cutting the dendogram at a certain level.

Partitioning algorithms fix the number k of desired clusters first, choose k starting points (e.g. randomly or by sampling) and assign the patterns to the closest cluster.

Prof. Bayer, DWH-SS2000, Clustering

Def.: A k-partition has k (disjoint) clusters.

There are nk different k-partitions for a set of n patterns.

What is a good k-partition?

Squared Error Criterion: Assume pattern set H is divided into k clusters labeled by L and li {1, ... ,k}

Let cj be the centroid of cluster j, then the squared error is:

e2 (H,L) = j=1ki=1nj ||Xij - cj ||2

Xij = jth pattern of cluster j

Prof. Bayer, DWH-SS2000, Clustering

Sqared Error Clustering Algorithm (k-means)

- 1. Choose k cluster centers somehow
- 2. Assign each pattern to ist closest cluster center O(n*k)
- 3. Recompute new cluster centers cj’ and e2
- 4. If convergence criterion not satisfied then goto step 2 else exit
- Convergence Criteria:
- few reassignments
- little decrease of e2

Prof. Bayer, DWH-SS2000, Clustering

- convergence, speed of convergence?
- local minimum of e2 instead of global minimum? This is just a hill climbing algorithm.
- therefore several tries with different sets of starting centroids
- Complexities
- time: O(n*k*l) l is number of iterations
- space: O (n) disk and O(k) main storage space

Prof. Bayer, DWH-SS2000, Clustering

Note: simple k-means algorithm is very sensitive to initial choice of clusters

-> several runs with different initial choices

-> split and merge clusters, e.g. merge 2 clusters with closest centers and split cluster with largest e2

Modified algorithm is ISODATA algorithm:

Prof. Bayer, DWH-SS2000, Clustering

1. choose k cluster centers somehow, heuristics

2. assign each pattern to its closest cluster center O(n*k)

3. recompute new cluster centers cj’ and e2

4. if convergence criterion not satisfied then goto step 2 else exit

5. merge and split clusters according to some heuristics

Yields more stable results than k-means in practical cases!

Prof. Bayer, DWH-SS2000, Clustering

Minimal Spanning Tree clustering

1. Compute minimal spanning tree in O (m*log m) where m is the number of edges in graph, i.e. m = n2

2. Break tree into k clusters by removing the k-1 most expensive edges from tree

Ex: see Fig. 15

Prof. Bayer, DWH-SS2000, Clustering

hard clusters ~ equivalence classes

1. Take one point as representative

2. Set of members

3. Centroid (Fig. 17)

4. Some boundary points

5. Bounding polygon

6. Convex hull (fuzzy)

7. Decision tree or predicate (Fig. 18)

Prof. Bayer, DWH-SS2000, Clustering

Genetic Algorithms with k-means Clustering

Pattern setH = { X1, X2, ... , Xn}

labeling L = { l1, l2, ... , ln} with li{ 1, ... , k }

generate one or several labelings Li to start

solution ~ genome or chromosomeB= b1 b2 ... bn

B is binary encoding of L with

bi = fixed length binary representation of li

Note: there are 2 n ld k points in the search space for solutions, gigantic! Interesting cases n >> 100. Removal of symmetries and redundancies does not help much.

Prof. Bayer, DWH-SS2000, Clustering

Fitnessfunction: inverse of squared error function. What is the optimum?

Genetic operations:

crossover of two chromosomes b1 b2 ...| bi ... bn c1 c2 ...|ci ... cn

results in two new solutions, decide by fitness function

b1 b2 ...| ci ... cn

c1 c2 ...| bi ... bn

Prof. Bayer, DWH-SS2000, Clustering

mutation: invert 1 bit, this guarantees completeness of search procedure. Distance of 2 solutions = number of different bits

selection: probabilistic choice from a set of solutions, e.g. seeds that grow into plants and replicate (natural selection). Probabilistic choice of centroids for clustering.

exchange of genes

replication of genes at another place

Integrity constraints for survival (mongolism) and fitness functions for quality. Not all genetic modifications of nature are used in genetic algorithms!

Prof. Bayer, DWH-SS2000, Clustering

S1 = 01000 S2 = 11111

S1 = 01|000 S2 = 11|111

crossover yields

S3 = 01111 S4 = 11000

for global search see Fig. 21

Prof. Bayer, DWH-SS2000, Clustering

Difficulties

- how to choose k?
- how to choose starting centroids?
- after clustering, recomputation of centroids
- centroids move and Voronoi partitioning changes, e.i. reassignment of patterns to clusters
- etc.

Prof. Bayer, DWH-SS2000, Clustering

Example 1: Relay stations for mobile phones

- Optimal placement of relay stations ~ optimal k-clustering!
- Complications:
- points correspond to phones
- positions are not fixed
- number of patterns is not fixed
- how to choose k ?
- distance function complicated: 3D geographic model with mountains and buildings, shadowing, ...

Prof. Bayer, DWH-SS2000, Clustering

Example 2: Placement of Warehouses for Goods

- points correspond to customer locations
- centroids correspond to locations of warehouses
- distance function is delivery time from warehouse multiplied by number of trips, i.e. related to volume of delivered goods
- multilevel clustering, e.g. for post office, train companies, airlines (which airports to choose as hubs), etc.

Prof. Bayer, DWH-SS2000, Clustering

Ch. 15.6: Clustering of Large Datasets

Experiments reported in Literature

clustering of 60 patterns into 5 clusters and comparison of various algorithms: using the encoding of genetic algorithms:

length of one chromosome: 60 * ceiling(ld(5)) = 180 bits

each chromosome ~ 1 solution, i.e.

260*ceiling(ld(5)) = 2180 ~ 100018 ~ 1054

points in the search space for optimal solution

Isolated experiments with 200 points to cluster

Prof. Bayer, DWH-SS2000, Clustering

clustering pixels of 500500 image = 250.000 points

documents in Elektra: > 1 Mio documents

see Table 1

above problems prohibitive for most algorithms, only candidate so far is k-means

Prof. Bayer, DWH-SS2000, Clustering

Properties of k-means Algorithm

- Time O (n*k*l)
- Space O (k + n ) to represent H, L, dist (Xi, centroid(li ))
- solution is independent of order in which centroids are chosen, order in which points are processed
- high potential for parallelism

Prof. Bayer, DWH-SS2000, Clustering

use p processors Proci, each Proci knows centroids

C1, C2, ...Ck

points are partitioned (round robin or hashing) into p groups G1, G2, ... Gp

processor Prociprocesses group Gi

parallel k-means has time complexity 1/p O(n*k*l)

Prof. Bayer, DWH-SS2000, Clustering

Use GA to generate p different initial clusterings

C1i, C2i, ... , Cki for i = 1, 2, ... , p

Proci computes solution for the seed C1i, C2i, ... , Cki

and fitness function for its own solution, which determines the winning clustering

Prof. Bayer, DWH-SS2000, Clustering

Large Experiments (main memory)

- classification of < 40.000 points, basic ideas:
- use random sampling to find good initial centroids for clustering
- keep summary information in balanced tree structures

Prof. Bayer, DWH-SS2000, Clustering

Algorithms for Large Datasets (on disk)

- Divide and conquer: cluster subsets of data separately, then combine clusters
- Store points on disk, use compact cluster representations in main memory, read points from disk, assign to cluster, write back with label
- Parallel implementations

Prof. Bayer, DWH-SS2000, Clustering

Ch. 15.7: Nearest Neighbor Clustering

Idea: every point belongs to the same cluster as its nearest neighbor

Advantage: instead of computing n2 distances in O(n2 ) compute nearest neighbor in O (n*log(n))

Note: this works in 2-dimensional space with sweep line paradigm and Voronoi diagrams, unknown complexity in multidimensional space

Prof. Bayer, DWH-SS2000, Clustering

Nearest Neighbor Clustering Algorithm

1. Initialize: point = its own cluster

2. Compute nearest neighbor Y of X, represent as (X, Y, dist) O(n*log(n)) for 2dimexpected O(n*log(n)) for d dim??? I/O complexity O(n)

3. Sort (X, Y, dist) by dist O (n*log(n))

4. Assign point to cluster of nearest neighbor ( i.g. merge cluster with nearest cluster and compute: new centroid, diameter, cardinality of cluster, count number of clusters)

5. if not done then goto 2

Prof. Bayer, DWH-SS2000, Clustering

Termination Criteria

- distance between clusters
- size of clusters
- diameter of cluster
- squared error function

Analysis: with every iteration the number of clusters is decreased to at most 1/2 of previous number, i.e. at most O( log(n)) iterations, total complexity: O(n*log2(n)) resp. O(n*log (n))

we could compute complete dendrogram for nearest neighbor clustering!

Prof. Bayer, DWH-SS2000, Clustering

Efficient Computation of Nearest Neighbor

2-dimensional: use sweep line paradigm and Voronoi diagrams

d-dimensional: so far just an idea, try as DA? Use generalization of sweep line to sweep zone in combination with UB-tree and caching similar to Tetris algorithm

Prof. Bayer, DWH-SS2000, Clustering

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