Mining spatiotemporal cascade patterns

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Mining spatio-temporal cascade patterns
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Outline

• Motivation
• Modeling Spatio-Temporal Cascade Pattern
• Spatio Temporal Cascade Pattern Mining Problem
• Challenges
• Related Work
• Contributions
• Key Concepts
• Proposed Approach
• Validation Methodology
• Validation of Interest Measures
• Analytical Validation of Algorithms
• Conclusion and Further Work

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Motivation

• Application Domains Crime (Crime Linkage),
  Military (Insurgent Attack Patterns), ecology
  (preservation of endangered species), power
  systems (cascading blackouts)
• Example Military
• Understanding insurgent Attack Patterns
• Understanding global and local trends in attacks.
• Predicting future locations of attacks.
• Example Crime
• Crime linkage analysis.
• Relationships between different types of crime
  (Drunk Driving, Hit and Run, Homicide, Shop
  Breaking)
• Example Power Systems
• Transmission Data Analysis
• Identifying sequences of faults for a potential
  cascading black out.
• Preventing blackouts.

Source http//www.ferc.gov/EventCalendar/Files/20
040414101846-blackout.pps
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Motivation Tactical Crime Analysis

• Geographic Profiling Step of Scenario
  SelectionRossmo2006
• Scenario selection Optimal subset of Crime
  Sites to be profiled.
• Performs Crime series Linkage Analysis
• Requires Human Intervention and manually
  building separate scenarios.
• No Existing measure for the validity of crime
  linkage
• Outliers Complicates the problem.

• Tactical Crime Analyst Questions
• Can we prune unnecessary data in an un-biased
  way ?
• Are a particular set of Crimes part of a single
  series?
• Are these serial crimes committed by one
  individual or a group of individuals operating
  together ?
• How can we identify and eliminate outliers ?

Source http//pt.wikipedia.org/wiki/Ataques_a_Tir
os_em_Beltway
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Modeling Spatio Temporal Cascade Patterns Iraq
Insurgency Example

• Cascade Patterns
• Series of Attacks by US Troops. (A)
• Series of attacks on US Troops.(B)
• Series of attacks on civilian facilities.(C)
• Series of suicide attacks. (D)
• Series of Communal Clashes.(E)
• Attacks on Iraqi troops.(F)

Source www.tribuneindia.com
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Modeling Spatio Temporal Cascade Patterns
Lincoln Crime sample dataset
0100 AM
0130 AM
0200 AM
0230 AM
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Sample Data Records
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Modeling Spatio Temporal Cascade Patterns An
Example Cascade Pattern

• Output Cascade Pattern
  

Bar Closing(B)
AutoCrime(A)
Assault (C),Vandalism (D)
OtherCrimes(E)
D
A
B
C
D
A
Continuous Sub-cascades
D
B
E
C
A
E
C
Cascade Pattern
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Spatio Temporal Cascade Pattern Mining Problem

• Given
• A spatio-temporal event database.
• A set of M spatio-temporal event types.
• A spatial neighborhood relation S?, A temporal
  Neighborhood relation T ? .
• A Time window ( interval) T w (gt T ? )
• A spatio-temporal co-occurrence prevalence
  threshold
• A spatio-temporal link prevalence threshold
• A Cascade Prevalence Threshold CP?
• Find
• All Spatio-temporal Cascade (ST- Cascade)
  patterns with cascade prevalence gt CP?,
• where, all its component spatio-temporal
  co-occurrence prevalence gt and all
  its inter component links have a spatio-temporal
  link prevalence gt
• Objective
• Statistical Significance of interest measures.
• Minimize Computational Cost
• Constraints
• Correctness
• Completeness
• Monotonic Composite Multi-dimensional Interest
  Measure

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Challenges

• Conceptual Challenges
• Spatial footprint of events over time changes
  with different problem settings.
• A cascade pattern is non-linear . (Example.,
  Cascading Power Failure)
• Concurrent processes can together form a
  cascade. (Example A group of serial criminals
  working together and committing different crimes)
• Requires different timing considerations to
  capture concurrency , prolonged influence and
  lack of influence.
• Statistical challenges
• Cascading behavior is multidimensional (involves
  both space and time) Interest measures need to
  capture this aspect.
• Timing constraints and neighborhood definitions
  vary across problem domains Interest measures
  need to be flexible and correct.
• Identifying the right statistical measure from
  spatial statistics to compare proposed interest
  measures.
• Existence of non-linearity and concurrent
  processes across space and time Requires
  Composite Multi-dimensional Interest measures
• Risk of generating spurious patterns.
• Desirable computational properties such as
  monotonicity.
• Computational challenges
• Composite multi-dimensional interest measures are
  computationally complex.
• Patterns are exponential in the number of event
  types.
• Patterns are exponential in the nature of the
  timing constraints and maximum time span length
  of the dataset.

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Classification of Related Work and Proposed
Pattern
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Modeling ST-Cascade Patterns Related ST-Patterns
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Related Work Topological Patterns Wang et al.
2005

• Topological Pattern Co-occurrence of m feature
  types over a spatio-temporal neighborhood.

Example Bar Closing Drunk Driving, Hit and
Run, Accident

• Properties
• All events occur in the same spatio-temporal
  neighborhood.
• No concept of separate time intervals.
• Concept of Space and time neighborhood.
• Interest Measure Prevalence (Participation
  Index over a Space Time Neighborhood)

Topological Patterns Wang et al. 2005
ST Cascade

• Limitation
• Does not recognize sequences of different
  spatio-temporal co-occurrences
• (no ordering between different
  co-occurrences)

• Example
• Can catch crime patterns co-occurring in a
  spatio-temporal neighborhood.
• Cannot identify an ordering over different
  spatio-temporal co-occurrences.

Contains
Spatio-temporal Co-occurrence
A Topological Pattern
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Related Work Sequential Patterns from
Spatio-temporal Event Datasets, Huang et al. 2008

• Spatio temporal Sequential Pattern Sequence of
  spatio-temporal events based on a temporal
  ordering.

Example

• Properties
• Defines Follow Predicate for time ordering
  using a Particular neighborhood N(e)
• Defines During Predicate for intervals
  (possibly subset) must have to define another
  neighbor hood N1(e) to find subsets.
• Interest measure weak anti monotonic.

• Limitation
• Sequence Index and Density Ratio can be only a
  single type of neighborhood function at a time
  for a particular pattern.
• Requires a-priori specification of the
  respective neighbor hood definitions between
  different event types to obtain sequences of
  subsets.
• Due to these reasons Sequence Index cannot be
  used as the significance measure for capturing
  cascades.
• No notion of a continuous sub-sequence where
  continuity referes to the presence of common
  subsets of event types.
• Rules out presence of both the Follow and
  During relationship between a pair of event
  types.

• Example
• Can identify ordered sets of spatio-temporal
  event types that is a link between different
  types.
• Can identify links, but cannot identify
  co-occurrences of crime types and hence cannot
  identify ordering between different co-occurring
  sets.

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Sequential Patterns from Spatio-temporal Event
Datasets
Spatio-temporal Cascade Patterns
Finds
Finds
Cannot find both of these from the same dataset
D and D in the above pattern correspond to
different event instance sets in this case, in
Cascades it is not, in crime also it is not!!
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A distinguishing Example

• Three Criterion
• Neighborhood Relationship
• Interest Measure
• Pattern Semantics

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A distinguishing Example
During
Follow
Sequence Index different values for both.
Hence cannot capture patterns where both exist.
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Related Work Collocation Episodes Cao et al.
2006

• Collocation Episodes finding inter-movement
  regularities of different object types.

Example (Bar Closing, DrunkDriving)?(DrunkDriving
,HitRun)

• Properties
• A reference feature is used for defining the
  sequence of episodes.
• Defined for moving object types.

• Limitation
• A reference feature type is required for finding
  such patterns
• Trajectories known apriori.

• Example
• Can catch patterns which have common event type.
• Cannot identify patterns that do not have common
  event type.
• Assumes that a sequence of episodes are
  connected by a common reference type.

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Collocation Episodes
Spatio-temporal Cascade Patterns

• Inputs
• Trajectories of Moving objects

• Inputs
• Spatio-temporal Event Database

Finds
Finds
Constraints

• Reference Feature Type D

Cannot Find
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Related Work Spatio-Temporal Cross Correlation
Function, Ma et al. 2006 (ST Cross K Function)

• Temporal Extension to the Cross K Function -
  statistical measure for quantifying cause
  effect relationships of different event types

• Properties
• Extends Ripleys K function (Ripley et al. 1976)
  by adding time.
• One Tail case events of other type occurring
  only after a current event
• Two Tail case- events before and after the
  current event.

• Limitations
• Defined only for pairs of event types.
• High Computational cost for computing the K
  function.

• Example
• Can identify only pairs of spatio-temporally
  correlated event types.

ST Cross K Function
ST Cascades
A
D
D
A
A
B
C
B
C
Two-Tail Effect
A
A
One-Tail Effect
B
E
B
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Contributions

• Modeled Spatio-Temporal Cascade Patterns
• Definition of monotonic composite interest
  measure Cascade Prevalence
• Prove interest measures preserve anti-monotone
  property of Cascades and statistical
  significance.
• Development of a novel ST Cascade Miner Algorithm.

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Key Concepts -2 Spatio-Temporal Co-occurrence

• Spatio-Temporal Participation Ratio

• Spatio-Temporal Prevalence (Spatio-temporal
  Participation Index)

• A Spatio-temporal Co-occurrence is prevalent if

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Key Concepts -3

• Spatio-Temporal Link Relationship is an
  ordering between spatio-temporal co-occurrences
  or spatio-temporal event types.

• B and A are event types which
• Satisfy a spatial neighborhood relation S?
• Are within a time window Tw
• Do not satisfy a time neighborhood relationT?

• Definition 1 Spatio-temporal Link Prevalence of
  a pattern

min( of instances of D satisfying a spatial
neighborhood relationship with B and occurring
after B)/ (total of instances of D), (of
instances of B in spatial neighborhood of D and
occurring before D)/(total of instances of B)
min4/4,3/5 3/5

• Definition 2 prevalent Spatio-temporal Cascade

3/5
3/5
Prevalent ST Cascades
ST Cascade
ST Cascade
3/5
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ST Cascade
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Key Concepts -4

• Definition 3 Cascade Prevalence of a ST Cascade

Intersecting at the number of common instances of
B
Cascade Prevalence (instances of B in

) / (total
instances of B)
If, prevalent( ) and prevalent(
)


Intersecting at the number of common instances of
E
Cascade Prevalence (instances of E in

) / (total
instances of E)
If, prevalent( ) and prevalent(
)
Cascade Prevalence min (instances of E in

) / (total instances of E) ,
(instances of B in
) / (total
instances of B)
If, prevalent( ) and
prevalent( )
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In all of the above cases Tw gt T?, if Tw T? ?
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Key Concepts -4 (contd , Examples)
if Tw T? then we have (or patterns are just
connected by common type)
A
D
D


B
C
If, prevalent( ) and
prevalent( )
Cascade Prevalence (instances of D in

) / (total
instances of D)



E
B
If, prevalent( ) and
prevalent( )
Cascade Prevalence (instances of B in

) / (total instances of B)
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Key Concepts -4 (contd, Examples)
A
D
D


B
C
Cascade Prevalence min(( instances of D in

)/ (total instances of D), (
instances of B in
)/ (total
instances of B) )


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Evaluation of Interest Measures
Properties

• Monotonicity
• Spatio-temporal prevalence (PI)
• Link prevalence.
• Cascade Prevalence (CP)
• Statistical Significance to ST Cross K Function
  (Ma et al. 2006)
• Spatio-temporal prevalence (PI)
• Link prevalence is defined only between 2 types.
• Cascade Prevalence (CP)

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Evaluation of Interest Measures
Properties

• Lemma 1
• Spatio-temporal Prevalence is Monotonic.

2/5
2/5
1
Spatio-temporal Link Prevalence is similar to
Spatio-temporal Prevalence with a different Time
window hence, has same properties.
2/5

• Lemma 2
• Cascade Prevalence preserves is Monotonic

3/5
3/5
2/5
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Spatiotemporal prevalence is an upper bound to ST
K-Function - Illustration
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Spatio-Temporal Link Prevalence is an upper bound
to ST K-Function - Illustration
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Cascade Prevalence is an upper bound to ST
K-Function - Illustration
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Proposed Approach Naïve Algorithm

• Step 1 Preprocessing Create instance level
  graph
• Step 2 Candidate Generation
• Candidate Co-occurrences
• Candidate Spatio-temporal Links
• Candidate continuous single link cascades
• Step 3 Repeatedly merge all continuous k-1
  linked cascades to form continuous k linked
  cascades.
• Step4 Pruning Step

• Limitations
• Step 2 No pruning
• Step 3 merge all k-1 link cascades
• Step 4 looks at all candidates.

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Proposed Approach Naïve Algorithm

• Inputs
• All of the Inputs in problem definition
• Output
• All Prevalent Cascade Patterns
• Pseudo code
• preprocessing step
• generate_all_instance
  _co_occurrences_ pplying S? and T ? .
• generate_all_instance_
  links_applying S? and T w .
• mining step
• candidate generation
• generate all event type
  co-occurrences
• generate all event type
  links
• generate all continuous sub
  cascades with one link add it to set S.
• merging step
• intialize k 2
• repeat until S ! null set
• C all k-1 linked
  continuous sub cascades
• repeat until C!null
  set
• merge all
  continuous k-1 linked continuous sub cascades
  with one another

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Conclusions and Further Work

• Optimized algorithm for Mining ST Cascades
• Proof of Correctness and Completeness.
• Experimental Evaluation.