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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