Spatio-Temporal Predicates

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Spatio-Temporal Predicates
Martin Erwig and Markus Schneider IEEE
TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
Presented by Mamadou Hassimiou Diallo
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Overview

• Challenges
• Dealing with large collections of relatively
  simple geometric objects
• Modeling, storing, querying spatio-temporal
  objects
• ExampleDatabase information about the flights
  of airplanes and about weather conditionsQuery
  whether an airplane crossed a certain storm
• Focus
• Nature and formal definition of spatio-temporal
  relationships gt Spatio-temporal predicates
• Spatio-temporal predicates describe developments
  of spatial topological relationships
• Approach
• Integrates the two-dimensional topological
  relationships and the change of spatial
  information over time
• Framework Spatio-temporal predicates obtained by
  temporal aggregation of elementary spatial
  predicates and sequential composition

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Foundations Basic and Spatial Data Types

• Formal definition
• Based on point set
• Used for storing, retrieving, manipulating, and
  querying spatial objects
• Data types
• Undefined
• Boolean
• Points elements of the Euclidean plane
• Lines two-dimensional curves
• Regions point sets with a two-dimensional extent
  and are bounded by lines

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Foundations Basic and Spatial Data Types

• Problem
• Modeling regions as arbitrary point sets
• Can result in undesired geometric anomalies
• Solution
• Point set topology regularization process
• Point set topology point set different parts
• Point set A
• Parts
• Boundary interior closure Not(A)
• interior eliminate dangling points, dangling
  lines, and boundary parts
• closure eliminate cuts and punctures
• Regularity a point set is regular closed if
• Regularization function
• Regions type

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Foundations Topological Predicates

• Spatial data modeling and reasoning
• Topological predicates between spatial objects in
  the two-dimensional space
• 9 possible intersections of boundary interior,
  and exterior
• Matrix for evaluating two dimensional spaces A
  and B
• 29 512 different configurations
• 8 meaningful configurations

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Foundations Topological Predicates
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Foundations Topological Predicates
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Foundations Spatio-Temporal Data Types

• Definition
• continuous model of time time R
• temporal function t(a) time ---gt a (all total
  functions from time a)
• Algebraic model
• Moving points t(point), evolving lines t(lines),
  evolving regions t(region)
• Focus moving points and evolving regions
• Temporal lifting
• Flat functions -----gt temporal functions
• Flat function f a1 xx an ------gt ß
• Lifted function ?f t(a1) x x t(an) ------gt
  t(ß)
• Example
• Distance ?distance
• distance (point) x (region) -----gt real
• Distance t(point) x t(region) -----gt t(real)

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Spatio-Temporal Predicates Nature

• Definition
• Predicates can be used to express facts (true or
  false)
• spatial predicate function spatial objects
  -----gt boolean
• temporally lifted spatial predicate function
  spatio-temporal objects -----gt temporal Booleans
• A spatio-temporal predicate is a function of type
  t(a) x t(ß) ---gt ? for a, ß in point, region
• Examples
• inside point x region ---gt bool (Yields true,
  undefined, false)
• ? inside Point x Region ---gt Bool (Yields true,
  undefined, false)
• Always-inside true iff î inside true for all
  times
• this definition is a bit problematic

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Spatio-Temporal Predicates Temporal Aggregation

• Universal and existential aggregation
• Operator spatial predicate -----gt
  spatio-temporal predicate
• (a x ß ---gt bool) -----gt (t(a) x t(ß) ---gt B)
• Existential quantification semantic is true iff
  p is true for the values of S1 and S2 at some
  time
• lambda-notation
• Universal quantification semanticdepends on
  time, t1 U t2, t1, t2, t1 ? t2
• Creation Operators
• spatio-temporal predicates from spatial
  predicates
• arrowhead indicates which object's lifetime

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Spatio-Temporal Predicates Basic Spatio-Temporal
Predicates

• Can be defined by temporal lifting and
  aggregation
• Default expected aggregation behavior (universal
  quantifier)
• Relaxing symmetric definitions for Meet and
  Overlap
• Predicate that yields true for two arbitrary
  spatio-temporal objects

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Spatio-Temporal Predicates Developments

• Developments
• Sequences of spatio-temporal predicates
• Example A moving point P is located at time t1
  outside of an evolving region R and changes
  (continuously) its location
• If P, at time t3, is inside of R ----gt P enters R
• If P, at some time t5, is again outside of R
  ----gt P crosses R
• P is located on the border of R at some time t2
  and at some time t4
• Table Development of P
• Observations

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Spatio-Temporal Predicates Development

• Observation
• developments of objects a need to restrict the
  validity of spatio-temporal predicates to
  intervals
• Definition Predicate Constriction
• Let P be a spatio-temporal predicate, and let I
  be a (half-) open or closed interval. Then,
• Example Inside (P, R) is false, Inside (P, R) is
  false
• Two classes of predicates

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Spatio-Temporal Predicates Development

• Combination operations
• from defines a spatio-temporal predicate that
  for some time t0 checks p and then enforces P for
  all t gt t0
• until P must hold until p is true at some time
  t0
• then is true if there is some time point t0 when
  p is true so that P holds before and Q holds
  after t0
• Definition Temporal Composition
• Let p be a spatial predicate, and let P and Q be
  spatio-temporal predicates. Then,
• Example

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Questions?
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Spatio-Temporal Data Modeling

• Spatio-Temporal Objects
• Examples
• flight of an airplane, the migration of whales,
  the raging of a storm, or the spreading of a fire
  region
• Characteristic features spatial entities
  changing over time continuously
• Changes motion, shrinking, growing, shape
  transformation, splitting, merging, disappearing,
  or reappearing of spatio-temporal objects
• Temporal changes modifications of mutual
  topological relationships over time
• disjoint, intersect