Spatio-Temporal Predicates

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

2. 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

3. 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

4. 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:

5. 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

6. Foundations: Topological Predicates

7. Foundations: Topological Predicates

8. 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)

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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:

15. Questions?

16. 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