This presentation is the property of its rightful owner.
1 / 156

Clustering Methods PowerPoint PPT Presentation

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

Clustering Methods

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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)

• Partition objects into k nonempty subsets (or pick k initial means).

• 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.

• Go back to Step 2

• Stop when the new set of means doesn’t change (or some other stopping condition?)

10

9

8

7

6

5

4

3

2

1

0

0

1

2

3

4

5

6

7

8

9

10

Step 1

Step 2

Step 3

10

9

8

7

6

5

4

3

2

1

0

0

1

2

3

4

5

6

7

8

9

10

Step 4

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 roundshaped 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

X11 … X1f … X1p

. . .

. . .

. . .

Xi1 … Xif … Xip

. . .

. . .

. . .

Xn1 … Xnf … Xnp

0

d(2,1) 0

d(3,1) d(3,2) 0

. . .

. . .

. . .

. . .

. . .

d(n,1) d(n,2) … d(n,n-1) 0

• Data Matrix

n objects × p variables

• Dissimilarity Matrix

n objects × n objects

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

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

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)

Agglomerative

a

Step 1

Step 2

Step 3

Step 4

Step 0

a b

b

a b c d e

c

c d e

d

d e

e

Hierarchical Clustering (top down)

a

a b

b

a b c d e

c

c d e

d

d e

e

Divisive

Step 3

Step 2

Step 1

Step 0

Step 4

In either case, one gets a nice dendogram in which any maximal anti-chain (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.)

• 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

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

1

1

1

1

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

2

2

1

1

2

2

2

2

1-cluster noise 2-cluster

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 well-separated, 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

p

MinPts = 5

 = 1 cm

q

• 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

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 density-reachable 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.

• Density-connected

• A point p is density-connected to a q wrt , MinPtsif 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

p1

q

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

 = 1cm

MinPts = 3

Core

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)

• 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:

• 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.

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=periodical&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 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 quality with a few additional scans

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

BIRCH

ABSTRACT

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:CF = (N, LS, SS)

N: Number of data points

LS: Ni=1=Xi

SS: Ni=1=Xi2

Branching factor = max # children

Threshold = max diameter of leaf cluster

Clustering Feature Vector

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

(3,4)

(2,6)

(4,5)

(4,7)

(3,8)

Birch

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.

Root

CF1

CF2

CF3

CF6

child1

child2

child3

child6

Non-leaf node

CF1

CF2

CF3

CF5

child1

child2

child3

child5

Leaf node

Leaf node

prev

CF1

CF2

CF6

next

prev

CF1

CF2

CF4

next

Branching factor, B=6

Threshold, L = 7

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

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

Cure

ABSTRACT

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.

y

y

y

x

y

y

x

x

• s = 50

• p = 2

• s/p = 25

• s/pq = 5

x

x

Cure: Shrinking Representative Points

y

y

x

x

• Shrink the multiple representative points towards the gravity center by a fraction of .

• Multiple representatives capture the shape of the cluster

Clustering Categorical Data: ROCKhttp://portal.acm.org/citation.cfm?id=351745

• 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

• Computational complexity:

• Basic ideas:

• Similarity function and neighbors:

Let T1 = {1,2,3}, T2={3,4,5}

ROCK

Abstract

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

• Algorithm

• Draw random sample

• Label data in disk

{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}

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 interconnectivityand 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

CHAMELEON

ABSTRACT

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

Sparse Graph

Partition the 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 celland 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 22 * 22 * 22 = 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.

1

gci=001

gcp=

01,11,00

gci=001

gcp=

11,11,00

gci=001

gcp=

00,11,00

gci=001

gcp=

10,11,00

gci=001

gcp=

11,11,01

gci=001

gcp=

01,11,01

gci=001

gcp=

00,11,01

gci=001

gcp=

10,11,01

Pid = 001

gci=001

gcp=

11,11,10

gci=001

gcp=

00,11,10

gci=001

gcp=

10,11,10

gci=001

gcp=

01,11,10

gci=001

gcp=

00,10,00

gci=001

gcp=

11,10,00

gci=001

gcp=

01,10,00

gci=001

gcp=

11,11,11

gci=001

gcp=

10,11,11

gci=001

gcp=

00,11,11

gci=001

gcp=

10,10,00

gci=001

gcp=

01,11,11

gci=001

gcp=

00,10,01

gci=001

gcp=

11,10,01

gci=001

gcp=

01,10,01

gci=001

gcp=

10,10,01

gci=001

gcp=

00,10,10

gci=001

gcp=

11,10,10

gci=001

gcp=

10,10,10

gci=001

gcp=

01,10,10

gci=001

gcp=

00,10,11

gci=001

gcp=

00,01,00

gci=001

gcp=

01,10,11

gci=001

gcp=

11,10,11

gci=001

gcp=

01,01,00

gci=001

gcp=

10,10,11

gci=001

gcp=

10,01,00

gci=001

gcp=

11,01.00

gci=001

gcp=

00,01,01

gci=001

gcp=

01,01,01

gci=001

gcp=

10,01,01

gci=001

gcp=

11,01.01

gci=001

gcp=

01,01,10

gci=001

gcp=

00,01,10

A3

hi-bit

gci=001

gcp=

10,01,10

gci=001

gcp=

11,01,10

0

gci=001

gcp=

01,01,11

gci=001

gcp=

11,01,11

gci=001

gcp=

00,01,11

gci=001

gcp=

10,01,11

gci=001

gcp=

00,00,00

gci=001

gcp=

11,00,00

gci=001

gcp=

01,00,00

gci=001

gcp=

10,00,00

gci=001

gcp=

01,00,01

gci=001

gcp=

00,00,01

gci=001

gcp=

10,00,01

gci=001

gcp=

11,00,01

0

gci=001

gcp=

00,00,10

gci=001

gcp=

01,00,10

gci=001

gcp=

11,00,10

gci=001

gcp=

10,00,10

gci=001

gcp=

00,00,11

gci=001

gcp=

11,00,11

gci=001

gcp=

01,00,11

gci=001

gcp=

10,00,11

1

A2

hi-bit

0

1

A1

hi-bit

2-hi gridding of Vector Space, R(A1, A2, A3) in which all bitwidths are the same = 3 (so each grid cell contains 21 * 21 * 21 = 8 points).

A1

11

10

A2

Pid = 001.001

01

00

A3

gci=

00,00,11

gcp=0,1,0

gci=

00,00,11

gcp=1,1,0

gci=

00,00,11

gcp=1,1,1

gci=

00,00,11

gcp=0,1,1

01

gci=

00,00,11

gcp=0,0,0

gci=

00,00,11

gcp=1,0,0

00

10

gci=

00,00,11

gcp=0,0,1

gci=

00,00,11

gcp=1,0,1

11

00

01

10

11

1-hi gridding of R(A1, A2, A3), bitwidths of 3,2,3

Pid = 001

gci=001

gcp=

11,1,00

gci=001

gcp=

10,1,00

gci=001

gcp=

00,1,00

gci=001

gcp=

01,1,00

1

gci=001

gcp=

01,1,01

gci=001

gcp=

00,1,01

gci=001

gcp=

10,1,01

gci=001

gcp=

11,1.01

gci=001

gcp=

00,1,10

gci=001

gcp=

01,1,10

A3

hi-bit

gci=001

gcp=

10,1,10

gci=001

gcp=

11,1,10

0

gci=001

gcp=

00,1,11

gci=001

gcp=

10,1,11

gci=001

gcp=

11,1,11

gci=001

gcp=

01,0,00

gci=001

gcp=

01,1,11

gci=001

gcp=

00,0,00

gci=001

gcp=

11,0,00

gci=001

gcp=

10,0,00

gci=001

gcp=

00,0,01

gci=001

gcp=

01,0,01

gci=001

gcp=

11,0,01

gci=001

gcp=

10,0,01

0

gci=001

gcp=

00,0,10

gci=001

gcp=

11,0,10

gci=001

gcp=

01,0,10

gci=001

gcp=

10,0,10

gci=001

gcp=

01,0,11

gci=001

gcp=

11,0,11

gci=001

gcp=

10,0,11

gci=001

gcp=

00,0,11

1

A2

hi-bit

0

1

A1

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=

1,,0

gcp=

0,,0

A3

2-hi-bit

gcp=

0,,1

gcp=

1,,1

01

01

10

00

11

A2

2-hi-bit

00

01

10

11

A1

2-hi-bit

Pid = 3.1.3

A2

A1

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 (exactly one per grid cell):

x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)where xi,js range over all binary patterns

Note: we have switched the direction of A3

(x1,b1..x1,41010, x2,b2..x2,41011, x3,b3..x3,41011)

 (x1,b1..x1,41011, x2,b2..x2,41011, x3,b3..x3,41011)

gcp=

1011,

1011,

1011

gcp=

1010,

1011,

1011

gcp=

1010,

1011,

1010

gcp=

1011,

1011,

1010

(x1,b1..x1,41011, x2,b2..x2,41011, x3,b3..x3,41010)

(x1,b1..x1,41010, x2,b2..x2,41011, x3,b3..x3,41010)

gcp=

1010,

1010,

1011

gcp=

1011,

1010,

1011

gcp=

1010,

1010,

1010

gcp=

1011,

1010,

1010

(x1,b1..x1,41011, x2,b2..x2,41010, x3,b3..x3,41011)

(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41011)

(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)

(x1,b1..x1,41011, x2,b2..x2,41010, x3,b3..x3,41010)

A3

H(x,21)HOBBit disk about a HOBBit grid center pt, x = , of radius 21

(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)

A2

A3

(x1,b1..x1,41000, x2,b2..x2,41011, x3,b3..x3,41011)

(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)

A2

A3

A1

The Regions of H(x,21) are as follows:

These REGIONS are labeled with dimensions in which length is increased

(e.g., all three dimensions are increased below).

A2

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

A3

2-REG

12-REGION

A1

123-REGION

23-REG

• Select an “outlier threshold”, (pts without neighbors in their ot L-disk are outliers – That is, there is no gradientat these outlier points (instantaneous rate of response change is zero).

• Create an j-lo grid with j=ot(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).

• 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:

• 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 ) / xkwhere 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

A3

A1

H(x,21)

H(x,21)1

First new point

HOBBit center, x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)

A2

A3

A1

( RootcountD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-1)/(-1) = -1

H(x,21)2

H(x,21)

A2

A3

A1

( RootcountD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-1)/(-1) = -1

H(x,21)3

H(x,21)

Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1=(2-1)/(-1) = -1

Estimated gradient Check other regions too?

Est f/ xk(x)= ( RcD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-1)/(-1) = -1

Est f/ xk(x)=( RcD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-1)/(-1) = -1

A2

A3

A1

H(x,21)

First new point

HOBBit center, x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)

Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1=(2-2)/(-1) = 0

A2

A3

A1

First new point

H(x,21)

H(x,21)1

HOBBit center, x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)

( RootcountD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-1)/(-1) = -1

A2

A3

A1

H(x,21)2

H(x,21)

( RootcountD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-1)/(-1) = -1

A2

A3

A1

H(x,21)3

H(x,21)

Estimated gradient Check other regions too?

( RootcountD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-1)/(-1) = -1

( RootcountD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-1)/(-1) = -1

A2

A3

A1

H(x,21)

Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1=(2-1)/(-1) = -1

A2

A3

A1

H(x,21)

H(x,21)1

First new point

HOBBit center, x=(x1,b1..x1,41010, x2,b2..x2,41010, x3,b3..x3,41010)

A2

A3

A1

( RootcountD(x,ri) - RootcountD(x,ri)2 ) / x2 = (2-2)/(-1) = 0

H(x,21)2

H(x,21)

A2

A3

A1

( RootcountD(x,ri) - RootcountD(x,ri)3 ) / x3 = (2-2)/(-1) = 0

H(x,21)3

H(x,21)

Est f/ xk(x) = (RcD(x,ri) - RootcountD(x,ri)1 ) / x1=(2-1)/(-1) = -1

Estimated gradient Check other regions too?

A2

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

(or, more accurately, as the point to which we climb in any response surface hill climbing technique)

H(x,21)

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

A3

A1

H(x,21)

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

A3

A1

H(x,21)

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

A2

A3

A1

Note that the gradient points in the right direction and is very short (as it should be!)

First new point

H(x,22)

H(x,22)1

H(x,1)1

H(x,1)2

H(x,1)3

H(x,1)13

H(x,2)1

H(x,1)123

H(x,1)23

H(x,2)1

H(x,2)1

H(x,2)1

H(x,2)1

H(x,2)1

H(x,0)

H(x,1)12

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 one of these regions

(if  1 point appears, gradient components are

additive, so it suffices to consider 1?

H(x,2)

H(x,1)1

H(x,1)2

H(x,1)3

H(x,1)13

H(x,2)2

H(x,1)123

H(x,1)23

H(x,2)23

H(x,2)12

H(x,2)3

H(x,2)123

H(x,2)13

H(x,0)

H(x,1)12

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,1)1

H(x,1)2

H(x,1)3

H(x,1)13

H(x,2)2

H(x,1)123

H(x,1)23

H(x,3)3

H(x,2)23

H(x,3)13

H(x,2)12

H(x,3)1

H(x,2)3

H(x,3)13

H(x,3)13

H(x,3)13

H(x,2)123

H(x,2)13

H(x,3)123

H(x,0)

H(x,1)12

H(x,2)1

H( x,23 )

Notice that the HOBBit center moves more and more toward the true center as the grid size increases.

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?”

• 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.)

• Using j-lo gridding with j=3 and lo-value cell identifiers, is shown on the next slide.

• Of course, we need not use HOBBit build out.

• 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,1)1

H(x,1)1

H(x,2)23

H(x,2)1

H(x,2)123

H(x,2)1

H(x,1)2

H(x,1)2

H(x,2)1

H(x,2)3

H(x,1)3

H(x,1)3

H(x,1)13

H(x,1)13

H(x,2)12

H(x,2)1

H(x,1)123

H(x,1)123

H(x,1)23

H(x,1)23

H(x,2)1

H(x,2)13

H(x,2)1

H(x,2)2

H(x,0)

H(x,0)

H(x,1)12

H(x,1)12

H(x,2)1

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)

Ring(x,2)

Ring(x,1)

= PDisk(x,3)^P’Disk(x,2)

= PDisk(x,2)^P’Disk(x,1)

wherePD(x,i) =

Pxb^..^Pxj+1 ^P’j^..^P’i+1

Disk(x,0)

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

• 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.

• 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:

• 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)

• Put each point with its closest Medoid (building P-tree component masks as you do this).

• Repeat 1 & 2 until (some stopping condition – such as: no change in Medoid set?)

• 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:

• 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…)

• Calculate the Means of each component, Ch, by

• ANDing each basic Ptree with PCi

• In dimension, Ak , calculate the k-component of the mean as i=bk..0 2i * rc(PCh ^ Pk,i )

• Put each pt with closest Mean (building P-tree component masks as you do this).

• Repeat 2, 3 until (some stopping condition – such as: no change in Mean set?)

• 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)

• Pick a point.

• Build out (HOBBit or Ordinary or?) rings one at a time until the first k neighbors are found.

• In that ring, computer the medoid points (as in step 1 of the V-k-Medoids algorithm above)

• if the Medoid increases the density, climb to it and goto 2

• Else declare it a density attractor and goto 1

j-grids - P-tree relationship

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

A2

ci=111

ci=110

ci=101

ci=100

ci=011

ci=010

ci=001

ci=000

root

A1

000

001

010

011

100

101

110

111

10.0.10

00.0.00

000.00

00.0.01

000.01

00.1.00

010.00

00.1.01

010.01

01.0.00

000.10

01.0.01

000.11

01.1.00

010.10

01.1.01

010.11

00.0.10

00.0.11

00.1.10

00.1.11

01.0.10

01.0.11

01.1.10

01.1.11

10.0.00

10.0.01

10.1.00

10.1.01

11.0.00

11.0.01

11.1.00

11.1.01

10.0.11

10.1.10

10.1.11

11.0.10

11.0.11

11.1.10

11.1.11

. . .

A3

00.1.00

11.1.00

01.1.01

10.1.00

00.1.01

01.1.01

11.1.01

10.1.01

01.1.10

00.1.10

10.1.10

11.1.10

01.1.11

11.1.11

00.1.11

00.0.00

10.1.11

01.0.00

11.0.00

10.0.00

00.0.01

01.0.01

10.0.01

11.0.01

00.0.10

01.0.10

11.0.10

10.0.10

00.0.11

11.0.11

01.0.11

10.0.11

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):

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

A2

010

001

011

000

100

101

111

110

011

100

000

001

111

110

101

010

root

A1

000

001

010

011

100

101

110

111

000

001

010

011

100

101

110

111

00

01

10

11

A3

00

10

01

11

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

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

• Query-independent, easy to parallelize, incremental update

• O(K), where K is the number of grid cells at the lowest level

• 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

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

CLIQUE

ABSTRACT

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.

Vacation(week)

7

6

5

4

3

2

1

age

0

20

30

40

50

60

Vacation

30

50

Salary

age

Salary (10,000)

7

6

5

4

3

2

1

age

0

20

30

40

50

60

 = 3

Strength and Weakness of CLIQUE

• Strength

• It automatically finds subspaces of thehighest 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

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

• 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.

• Hybrid clustering combines partitioning clustering and hierarchical clustering approach

K = 7

Problems of Existing Hybrid Clustering

• Predefine the number of preliminary clusters, K.

• Unable to handle noisy data

Similarity Measures

• Similarity is fundamental to the definition of a cluster.

• Minkowski Metric

• Context Metrics:

• Mutual neighbor distance

• 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 non-symmetric 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:

• MND(xi, xj) >= 0, and MND(xi, xj) = 0  xi = xjpositive definite

• 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 xiand xjis defined as

• 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.

s( xi, xj ) = f( xi, xj, σ, ε )

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

• 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

• British Telecom broke a multimillion dollar fraud

• Outlier Analysis (Cont.)

Outlier Analysis

Cluster-based

Statistic-based

Density-based

Distance-based

Distribution

(Barnett 1994)

LOCI

(ICDE 2002)

Depth

(Preparata 1988)

Breunig’s LOF

(SIGMOD 2000)

Knorr and Ng

(VLDB 1998)

Ramaswamy

(MOD 2000)

Angiulli

(TKDE2005)

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| < (1-p)*|X|

|y| ≥ p*|X|

complement

x

x

·D

·D

Figure 2. Equivalent definition

Figure 1. Knorr’s 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

DiskNbr (x, D/2)

D

q

D/2

x

D

• 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

• Figure 3. D/2-neighbors of x marked as non-outliers

Proposition 2

DiskNbr(x,D)

D

q

D

x

2D

• 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

• Figure 4. All D-Neighbors of x are outliers

The DBODLP Algorithm

·2D

x

·D

·D/2

x

·D

Figure 5. An example

Step 1: Select a point x arbitrarily from X;

Step 2: Search D-neighbors of x andcalculate |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|

• 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 ona 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)

Query and Data mining

Data Integration Language

DIL

Ptree (Predicates) Query Language

DII (Data Integration Interface)

DMI (Data Mining Interface)

Data Repository

lossless, compressed, distributed, vertically-structured P-tree database

Meta Data File

P-Tree Feeders

Preliminary experimental analysis

Figure 6. Run Time Comparison

Figure 7. Scalability Comparisons

• Our method has almost an order of magnitude of improvement over the nested loop method in terms of run time

• 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, distance-based or deviation-based approaches

• There are still lots of research issues on cluster analysis, such as constraint-based clustering

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.