Discovering Interesting Regions in Spatial Data Sets

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Discovering Interesting Regions inSpatial Data
Sets

• Christoph F. Eick for Data Mining Class
• Motivation Examples of Region Discovery
• Region Discovery Framework
• A Fitness For Hotspot Discovery
• Other Fitness Functions
• A Family of Clustering Algorithms for Region
  Discovery
• Summary

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Discovering Interesting Regions inSpatial Data
Sets

• Christoph F. Eick for Data Mining Class
• Motivation Examples of Region Discovery
• Region Discovery Framework
• A Fitness For Hotspot Discovery
• Other Fitness Functions
• A Family of Clustering Algorithms for Region
  Discovery
• Summary

3
Next 2-3 Classes

1. Region Discovery Framework
2. DBSCAN
3. Hierarchical Clustering
4. Clustering Algorithms for Region Discovery
   Clever,
5. Critical Issues with Respect to Clustering
6. Programming Project-specific Discussion
7. Similarity Assessment

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1. Motivation Examples of Region Discovery
Application 1 Hot-spot Discovery this
presentation, EVJW07 Application 2 Regional
Association Rule Mining and Scoping DEWY06,
DEYWN07 Application 3 Find Interesting Regions
with respect to a Continuous Variable Application
4 Regional Co-location Mining EPWSN07 Applicati
on 5 Find representative regions (Sampling)
b1.01
RD-Algorithm
b1.04
Wells in Texas Green safe well with respect to
arsenic Red unsafe well
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2. Region Discovery Framework

• We assume we have spatial or spatio-temporal
  datasets that have the following structure
• (x,y,z,tltnon-spatial attributesgt)
• e.g. (longitude, lattitude, class_variable)
  or (longitude, lattitude, continous_variable)
• Clustering occurs in the (x,y,z,t)-space
  regions are found in this space.
• The non-spatial attributes are used by the
  fitness function but neither in distance
  computations nor by the clustering algorithm
  itself.
• For the remainder of the talk, we view region
  discovery as a clustering task and assume that
  regions and clusters are the same

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Region Discovery Framework Continued

• The algorithms we currently investigate solve the
  following problem
• Given
• A dataset O with a schema R
• A distance function d defined on instances of R
• A fitness function q(X) that evaluates clustering
  Xc1,,ck as follows
• q(X) ?c?X reward(c)?c?X interestingness(c)size(
  c)? with bgt1
• Objective
• Find c1,,ck ? O such that
• ci?cj? if i?j
• Xc1,,ck maximizes q(X)
• All cluster ci?X are contiguous (each pair of
  objects belonging to ci has to be
  delaunay-connected with respect to ci and to d)
• c1?,,?ck ? O
• c1,,ck are usually ranked based on the reward
  each cluster receives, and low reward clusters
  are frequently not reported

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Challenges for Region Discovery

1. Recall and precision with respect to the
   discovered regions should be high
2. Definition of measures of interestingness and of
   corresponding parameterized reward-based fitness
   functions that capture what domain experts find
   interesting in spatial datasets
3. Detection of regions at different levels of
   granularities (from very local to almost global
   patterns)
4. Detection of regions of arbitrary shapes
5. Necessity to cope with very large datasets
6. Regions should be properly ranked by relevance
   (reward) in many application only the top-k
   regions are of interest
7. Design and implementation of clustering
   algorithms that are suitable to address
   challenges 1, 3, 4, 5 and 6.

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3. Fitness Function for Hot Spot Discovery

• Class of Interest Unsafe_Well
• Prior Probability 20
• ?1 0.5, ?2 1.5
• R 1, R- 1
• ß 1.1, ?1.

10
30
Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5
c 50 200 200 350 200
P(c, Unsafe) 20/50 40 40/200 20 10/200 5 30/350 8.6 100/20050
Reward
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4. Fitness Functions for Other Region Discovery
Tasks
4.1 Creating Contour Maps for Water Temperature
(Temp)
Fig. 1 Sea Surface Temperature on July 7 2002
Var2.2 Reward 48,5 Rank 3

Mean11.2
A single region and its summary

• Examples in the data set WT have the form
  (x,y,temp) var(c,temp) denotes the variance of
  variable temp in region c
• interestingness(c)
• IF
  var(c,temp)gtvar(WT,temp)
• THEN 0
• ELSE
  min(1, log20(var(WT,temp)/var(c,temp)))?
• with ? being a parameter (with default
  1)
• Basically, regions receive rewards if their
  variance is lower than the variance of the
  variable temperature for the whole data set, and
  regions whose variance is at least 20 times less
  receive the maximum reward of 1.

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4.2 Finding Regions with High Water Temperature
Differences

• Examples in the data set WT have the form
  (x,y,Temp)
• Fitness function Let c be a cluster to be
  evaluated
• interestingness(c)
• IF var(c,temp)ltvar(WT,temp)
• THEN 0
• ELSE min(1, log20(var(c,temp)/var(WT,tem
  p)))? )
• with ? being a parameter (with default
  1)

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4.3 Programming Project Fitness Functions Purity
r1
r2
(6, 2, 2)
r3
(0, 0, 5)
(2,2,1)
We assume we have 3 classes in r1 we have 6
objects of class1, 3 objects of class 2, and 2
objects of class1
We assume th0.5 and ?2 i(r1)
(0.6-0.5)20.01 i(r2)(1-0.5)20.25 i(r3)0 q
(X)q(r1,r2,r3) 0.0110b 0.255b
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Programming Project Fitness Functions Variance
r3 Var(r3)1100
r1 var(r1)80
O Var(O)100
r2 Var(r2)200
r4 Var(r4)20
We assume ?1 and b10 i(r1) 0 i(r2)log10(2)0.
3010 i(r3)1 i(r4)0
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Programming Project Function MSE
r1
r2
(2,2) (4,4)
(-1,-1) (-7,-7) (-4,-4)
MSE(r1)(1212121212)/22 MSE(r2)(3
23232321200)/312
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4.4 Regional Co-location Mining
R1
R2
Regional Co-location
R3
R4
Task Find Co-location patterns for the following
data-set.
Global Co-location and are
co-located in the whole dataset
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A Reward Function for Binary Co-location

• Task Find regions in which the density of 2 or
  more classes is elevated. In general, multipliers
  lC are computed for every region r, indicating
  how much the density of instances of class C is
  elevated in region r compared to Cs density in
  the whole space, and the interestness of a region
  with respect to two classes C1 and C2 is assessed
  proportional to the product lC1lC2
• Example Binary Co-Location Reward Framework
• lC(r)p(C,r)/prior(C)
• ?C1,C2 1/((prior(C1)prior(C2)) maximum
  multiplier
• kC1,C2(r) IF lC1(r)lt1 or lC2(r )lt1 THEN 0
• ELSE sqrt((lC1(r)1)(lC2(r)1))/(
  ?C1,C2 1)
• interestingness(r) maxC1,C2C1?C2 (kC1,C2(c))

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The Ultimate Vision of the Presented Research
DomainExpert
Spatial Databases
Measure ofInterestingness Acquisition Tool
Database Integration Tool
Fitness Function
Data Set
Family of Clustering Algorithms
Region DiscoveryDisplay
Ranked Set of Interesting Regions and their
Properties
Visualization Tools
Architecture Region Discovery Engine
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How to Apply the Suggested Methodology

1. With the assistance of domain experts determine
   structure of dataset to be used.
2. Acquire measure of interestingness for the
   problem of hand (this was purity, variance, MSE,
   probability elevation of two or more classes in
   the examples discussed before)
3. Convert measure of interestingness into a
   reward-based fitness function. The designed
   fitness function should assign a reward of 0 to
   boring regions. It is also a good idea to
   normalize rewards by limiting the maximum reward
   to 1.
4. After the region discovery algorithm has been
   run, rank and visualize the top k regions with
   respect to rewards obtained (interestingness(c)si
   ze(c)?), and their properties which are usually
   task specific.

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5. A Family of Clustering Algorithms for Region
Discovery

• Supervised Partitioning Around Medoids (SPAM).
• Representative-based Clustering Using Randomized
  Hill Climbing (CLEVER)
• Supervised Clustering using Evolutionary
  Computing (SCEC)
• Agglomerative Hierarchical Supervised Clustering
  (SCAH)
• Hierarchical Grid-based Supervised Clustering
  (SCHG)
• Supervised Clustering using Multi-Resolution
  Grids (SCMRG)
• Representative-based Clustering with Gabriel
  Graph Based Post-processing (MOSAIC)
• Supervised Clustering using Density Estimation
  Techniques (SCDE)

Remark For a more details about SCEC, SPAM,
SRIDHCR see EZZ04, ZEZ06 the PKDD06 paper
briefly discusses SCAH, SCHG, SCMRG
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SCAH (Agglomerative Hierarchical)
Inputs A dataset Oo1,...,on A distance Matrix
D d(oi,oj) oi,oj ? O , Output Clustering
Xc1,,ck  Algorithm 1) Initialize
Create single object clusters ci oi, 1 i
n Compute merge candidates based on nearest
clusters 2) DO FOREVER a) Find the pair
(ci, cj) of merge candidates that improves q(X)
the most b) If no such pair exist terminate,
returning Xc1,,ck c) Delete the two
clusters ci and cj from X and add the cluster ci
? cj to X d) Update inter-cluster
distances incrementally e) Update merge
candidates based on inter-cluster distances
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SCHG (Hierarchical Grid-based)
Remark Same as SCAH, but uses grid cells as
initial clusters Inputs A dataset
Oo1,...,on A grid structure G Output Clusterin
g Xc1,,ck   Algorithm 1) Initialize
Create clusters making each single non-empty grid
cell a cluster Compute merge candidates (all
pairs of neighboring grid cells) 2) DO FOREVER
a) Find the pair (ci, cj) of merge candidates
that improves q(X) the most b) If no such
pair exist terminate, returning Xc1,,ck
c) Delete the two clusters ci and cj from X and
add the cluster cci ? cj to X d)
Update merge candidates ?c?X (MC(c,c) ? MC(c,
ci) ? MC(c, cj ))
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Ideas SCMRG (Divisive, Multi-Resolution Grids)
Cell Processing Strategy 1. If a cell receives
a reward that is larger than the sum of its
rewards its ancestors return that cell.
2. If a cell and its ancestor do not receive
any reward prune 3. Otherwise, process the
children of the cell (drill down)
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Code SCMRG
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Problems with SCAH
Too restrictive definition of merge candidates
XXX OOO OOO XXX
No look ahead
Non-contiguous clusters
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6. Summary

1. A framework for region discovery that relies on
   additive, reward-based fitness functions and
   views region discovery as a clustering problem
   has been introduced.
2. Evidence concerning the usefulness of the
   framework for hot spot discovery problems has
   been presented.
3. As a by-product some known and not so well known
   flaws of hierarchical clustering algorithms have
   been identified.
4. The ultimate vision of this research is the
   development of region discovery engines that
   assist earth scientists in finding interesting
   regions in spatial datasets.

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Why should people use Region Discovery Engines
(RDE)?

• RDE finds sub-regions with special
  characteristics in large spatial datasets and
  presents findings in an understandable form. This
  is important for
• Focused summarization
• Find interesting subsets in spatial datasets for
  further studies
• Identify regions with unexpected patterns
  because they are unexpected they deviate from
  global patterns therefore, their regional
  characteristics are frequently important for
  domain experts
• Without powerful region discovery algorithms,
  finding regional patters tends to be haphazard,
  and only leads to discoveries if ad-hoc region
  boundaries have enough resemblance with the true
  decision boundary
• Exploratory data analysis for a mostly unknown
  dataset
• Co-location statistics frequently blurred when
  arbitrary region definitions are used, hiding the
  true relationship of two co-occurring phenomena
  that become invisible by taking averages over
  regions in which a strong relationship is watered
  down, by including objects that do not contribute
  to the relationship (example High crime-rates
  along the major rivers in Texas)
• Data set reduction focused sampling