Post on 05-Jan-2016
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Spatio-Temporal PredicatesMartin Erwig and Markus Schneider
IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
Presented byMamadou Hassimiou Diallo
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Overview Challenges
Dealing with large collections of relatively simple geometric objects Modeling, storing, querying spatio-temporal objects Example:
Database: 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 ===>
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
2^9 = 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: τ(α) = time ---> α (all total functions from time α)
Algebraic model Moving points τ(point), evolving lines τ(lines), evolving regions τ(region) Focus: moving points and evolving regions
Temporal lifting Flat functions -----> temporal functions Flat function f: α1 x…x αn ------> β Lifted function f: τ(α1) x … x τ(αn) ------> τ(β) Example
Distance = distance distance : (point) x (region) -----> real Distance : τ(point) x τ(region) -----> τ(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 -----> boolean temporally lifted spatial predicate = function: spatio-temporal objects ----->
temporal Booleans A spatio-temporal predicate is a function of type τ(α) x τ(β) ---> Β for α, β in
{point, region}
Examples inside: point x region ---> bool (Yields: true, undefined, false) inside: Point x Region ---> 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 -----> spatio-temporal predicate (α x β ---> bool) -----> (τ(α) x τ(β) ---> B) Existential quantification semantic:
is true iff p is true for the values of S1 and S2 at some time
lambda-notation: Universal quantification semantic:
depends on: time, t1 U t2, t1, t2, t1 Π 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 ----> P enters R If P, at some time t5, is again outside of R ----> 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 > 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