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

2. 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 conditions Query: 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. 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. 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. 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. 6 Foundations: Topological Predicates

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8. 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. 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. 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. 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. 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. 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. 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. 15 Questions?

16. 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