Lagrangian Xgraphs: A logical data-model for Spatio-temporal Network Data Acknowledgement: Venkata Gunturi, Shashi Shekhar University of Minnesota, Minneapolis

Outline of the Talk What is Spatio-temporal Network (STN) data? What is Spatio-temporal Network (STN) data? Value addition potential of STN data Value addition potential of STN data Problem Definition Problem Definition Challenges Challenges Limitations of Related Work Limitations of Related Work Proposed Lagrangian Xgraphs Proposed Lagrangian Xgraphs Concluding Remarks Concluding Remarks

What is Spatial-temporal Network (STN) Data? STN data is result of interactions (across time) of entity(s) with a network embedded in space. Large number of urban sensors produce a variety of datasets. E.g., GPS navigation devices, Loop detector data, Social media etc. Some are mobile, some are stationary, All of them capture diverse characteristics of a network in a urban scenario Sample STN datasets over Transportation Network Temporally detailed roadmaps. Traffic signal and coordination data. GPS tracks annotated with engine measurement data. Motivation: Collective wisdom from these datasets could support valuable use-cases, e.g., eco-routing

From Traditional Roadmaps Source: Google Maps Dinky town Roadmap Corresponding Digital Representation Intersection between 5 th Ave SE and 4 th St Intersection between 5 th Ave SE and 5 th St 5 th Ave SE edge Attributes of 5 th Ave SE road segment between N4 and N7 N7 N4

To Temporally Detailed (TD) Roadmaps Contains typical travel-time under traffic equilibrium conditions Per minute speed/travel time values 100 million road segments in US NAVTEQ’s highly compressed weekly speed profile data Source: ESRI and NAVTEQ

GPS traces Sources: Mobile devices Smart phones, in car/truck GPS devices, GPS collars  Coupled with engine measurements VGI  Commuter preferred routes under non-equilibrium conditions Estimate traffic signal delays? Ramp meters Coordinated signals Left turn delays Waiting at signals

Outline of the Talk What is Spatio-temporal Network (STN) data? What is Spatio-temporal Network (STN) data? Value addition potential of STN data Value addition potential of STN data Problem Definition Problem Definition Challenges Challenges Limitations of Related Work Limitations of Related Work Proposed Lagrangian Xgraphs Proposed Lagrangian Xgraphs Concluding Remarks Concluding Remarks

McKinsey Conjecture and Preliminary Evidence U.P.S. Embraces High-Tech Delivery Methods (July 12, 2007) By “The research at U.P.S. is paying off. ……..— saving roughly three million gallons of fuel in good part by mapping routes that minimize left turns.”

Outline of the Talk What is Spatio-temporal Network (STN) data? What is Spatio-temporal Network (STN) data? Value addition potential of STN data Value addition potential of STN data Problem Definition Problem Definition Challenges Challenges Limitations of Related Work Limitations of Related Work Proposed Lagrangian Xgraphs Proposed Lagrangian Xgraphs Concluding Remarks Concluding Remarks

Input Input –A collection of Spatio-temporal Network datasets –Use case queries (e.g. compare candidate routes) Output Output –A unified logical model across these datasets Objective Objective –Travel related concepts are expressed upfront –Suitable for common routing algorithms e.g. Dijsktra’s, A* P ROBLEM D EFINITION

P ROBLEM I LLUSTRATION : A T C ONCEPTUAL L EVEL Logical Model for STN datasets over Transportation Network Usually entities like Roads, Signals, Streets are modeled using lines strings and polygons. Queried through OGIS operators Not suitable for comparing candidate routes. Modeling as Spatial/Spatio-temporal networks? GPS DATA Delay Data TD roadmaps

Outline of the Talk What is Spatio-temporal Network (STN) data? What is Spatio-temporal Network (STN) data? Value addition potential of STN data Value addition potential of STN data Problem Definition Problem Definition Challenges Challenges Limitations of Related Work Limitations of Related Work Proposed Lagrangian Xgraphs Proposed Lagrangian Xgraphs Concluding Remarks Concluding Remarks

C HALLENGES OF “S EQUENCE OF ” R ELATION Logical Model for STN datasets over Transportation Network Current spatial/spatio-temporal models work for M=2 What if M>2? e.g. GPS traces and Traffic signal coordination What if M >2?

Outline of the Talk What is Spatio-temporal Network (STN) data? What is Spatio-temporal Network (STN) data? Value addition potential of STN data Value addition potential of STN data Problem Definition Problem Definition Challenges Challenges Limitations of Related Work Limitations of Related Work Proposed Lagrangian Xgraphs Proposed Lagrangian Xgraphs Concluding Remarks Concluding Remarks

After waiting at SG1, SG2 and SG3 become wait-free!  Non-local interactions (SG1 not a neighbor of SG2)  Typical delay measured over S-B-C-E-D will have wait only at SG1  Not true for journeys starting after intersection B or intersection C Limitations of Related Work: Non-decomposable Properties of N-ary relations Holistic Property: Properties measured over a larger instance loose their semantic meaning when broken down into properties of small instances Sample N-ary relation: Typical delay experienced in series of coordination signals Sample N-ary relation: Typical delay experienced in series of coordination signals

Typical Representational model used by current network databases, e.g., Oracle spatial, ArcGIS etc. Query: What is the typical travel-time experienced on Hiawatha Ave (between S and D)? Result: Between 21mins – 25mins 30secs Current related work not suitable for representing holistic properties which cannot be decomposed Cannot represent signal coordination upfront! Limitations of Related Work: Non-decomposable Properties of N-ary Relations

Outline of the Talk What is Spatio-temporal Network (STN) data? What is Spatio-temporal Network (STN) data? Value addition potential of STN data Value addition potential of STN data Problem Definition Problem Definition Challenges Challenges Limitations of Related Work Limitations of Related Work Proposed Lagrangian Xgraphs Proposed Lagrangian Xgraphs Concluding Remarks Concluding Remarks

Proposed Approach: Lagrangian Xgraphs Summary of proposed approach Holistic properties are modeled as series of overlapping “sub-journeys” Each “sub-journey” is contains one non- local interaction Suitable for non-decomposable properties of N-ary relations. 3mins 8mins 5mins

Travel Related Concepts: Lagrangian vs Eulerian frame of reference Eulerian Frame: Perspective of a fixed observe, e.g., traffic observatory Eulerian Frame: Perspective of a fixed observe, e.g., traffic observatory What is cost of following routes at 5:00pm I-35W Hiawatha Route Legend: A-I-D: UMN-I35W-Airport A-H-D: UMN-Hiawatha-Airport Digital Road Map Path Cost from Traveler Pers. Cost at 5:00pm Fixed Obs. A-I-D 27 mins20 mins A-H-D 25 mins

Legend: A-I-D: UMN-I35W-Airport A-H-D: UMN-Hiawatha-Airport Digital Road Map Travel Related Concepts: Lagrangian vs Eulerian frame of reference Lagrangian Frame: Perspective of a traveler travelling through the network Lagrangian Frame: Perspective of a traveler travelling through the network What is cost of following routes at 5:00pm I-35W Hiawatha Route Path Cost from Traveler Pers. Cost at 5:00pm Fixed Obs. A-I-D mins A-H-D ??25 mins

Legend: A-I-D: UMN-I35W-Airport A-H-D: UMN-Hiawatha-Airport Digital Road Map Lagrangian Frame: Perspective of a traveler travelling through the network Lagrangian Frame: Perspective of a traveler travelling through the network Travel Related Concepts: Lagrangian & Eulerian frame of reference What is cost of following routes at 5:00pm I-35W Hiawatha Route Path Cost from Traveler Pers. Cost at 5:00pm Fixed Obs. A-I-D =2720 mins A-H-D ??25 mins

Legend: A-I-D: UMN-I35W-Airport A-H-D: UMN-Hiawatha-Airport Digital Road Map Path Cost from Traveler Pers. 5:00PM Snapshot A-I-D 27 mins20 mins A-H-D 25 mins What is cost of following routes at 5:00pm I-35W Hiawatha Route Travel Related Concepts: Lagrangian & Eulerian frame of reference

Distance inferred from a GPS track can be decomposed into distances along individual road segments Travel Related Concepts: Decomposable vs Holistic Properties Decomposable: Property measured over a larger instance can be broken down into properties of small instances Decomposable: Property measured over a larger instance can be broken down into properties of small instances

Travel Related Concepts: Decomposable vs Holistic Properties What about travel-time inferred from a GPS track? from a GPS track?  Time spent on a segment depends on the initial velocity attained before entering the segment!  Holistic Property: Properties measured over a larger instance loose their semantic meaning when broken down into properties of small instances

Taxonomy of Travel Related Concepts Captured in STN Datasets All STN datasets capture data along two dimensions. TD roadmaps Signal Delay Data GPS DATA

Traveler’s Frame of Reference For Comparing Candidate Routes Candidate routes are evaluated from the perspective of a person moving through the transportation network. Candidate routes are evaluated from the perspective of a person moving through the transportation network. A-C-D is shorter for t=1 : Lagrangian Frame needs to be upfront

Langrangian Xgraph: Formal Definition Lagrangian Xgraph: {Xnodes, Xedges} Lagrangian Xgraph: {Xnodes, Xedges} Xnodes: Underlying entities at specific space-time coordinates. Xnodes: Underlying entities at specific space-time coordinates. – –Xv1, Xv2, Xv3… Xedges: Express a Lagrangian relation (i.e.,’as-traveled’ or ‘typical- experience-in-travel’) relationship among a group a Xnodes Xedges: Express a Lagrangian relation (i.e.,’as-traveled’ or ‘typical- experience-in-travel’) relationship among a group a Xnodes –Xei = {Xvs, Xv1, Xv2, Xv3…, Xvk, Xvd1, Xvd2,..,Xvdj} First and Last set of Xnodes in an Xedge are marked separately Xedges are classified based on these TD roadmaps  Shoot Xedges TD roadmaps  Shoot Xedges GPS Traces  Shoot and Stem Xedges GPS Traces  Shoot and Stem Xedges Trafffic Signal Delays  Bush and Flower Xedges Trafffic Signal Delays  Bush and Flower Xedges Get, Set and Join operators (only Xedges) For Xndoes and Xedges

Sample Langrangian Xgraph for Signal Coordination (1/2) Xnodes: Underlying road segments between two road intersections at specific departure-times. Xnodes: Underlying road segments between two road intersections at specific departure-times. Xedges: Express a ‘as-traveled’ or ‘typical-experience-in-travel’ relationship among a group a Xnodes Xedges: Express a ‘as-traveled’ or ‘typical-experience-in-travel’ relationship among a group a Xnodes 3mins 8mins 5mins Xnode ED6: Road segment ED for departure-time 7:03am at E

Xedge SB0 and (ED32, ED33, ED34, ED 35) as first and last Xnodes: Xedge SB0 and (ED32, ED33, ED34, ED 35) as first and last Xnodes: –An Xedge representing: “If one leaves at S at 7:00am  he/she can start traversing segment E-D at times 7:16, 7:16:30, 7:17, or 7:17:30” Sample Langrangian Xgraph for Signal Coordination (2/2)

Outline of the Talk What is Spatio-temporal Network (STN) data? What is Spatio-temporal Network (STN) data? Value addition potential of STN data Value addition potential of STN data Problem Definition Problem Definition Challenges Challenges Limitations of Related Work Limitations of Related Work Proposed Lagrangian Xgraphs Proposed Lagrangian Xgraphs Concluding Remarks Concluding Remarks

Conclusion Increased proliferation of sensors  Increased proliferation of sensors  –Spatio-temporal datasets capturing diverse phenomena on a transportation network Collectively they can add significant value to societal use-cases. Collectively they can add significant value to societal use-cases. However, they pose modeling challenges due to holistic nature of properties captured in these datasets. However, they pose modeling challenges due to holistic nature of properties captured in these datasets. Proposed Lagrangian Xgraphs Proposed Lagrangian Xgraphs –can model both decomposable and holistic properties