Continual Neighborhood Tracking for Moving Objects Yoshiharu Ishikawa Hiroyuki Kitagawa Tooru Kawashima University of Tsukuba, Japan Using Adaptive Distances

Organization Background and Overview Our Approach Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work

Background Progress of Digital Cartography Development of GPS Technologies Wide Use of PDA and Hand-held Devices New Types of Information Services: Providing neighborhood information to moving objects (people with PDAs, cars with navigation systems) considering their locations and trajectories

Motivating Example (1) Neighborhood Query: A user at point x wants to find nearby gas stations Typical Approach: retrieve gas stations with their distances less than 200 meters from x x A spatial query based on the Euclidean distance

Motivating Example (2) A What’s Wrong? If we know user’s past and future trajectories, we can provide more appropriate information past trajectory future trajectory

Our Idea (1) A Use of an ellipsoid region to represent a neighborhood query An ellipsoid region is computed based on the past/future trajectories A neighborhood query is specified as a spatial query with an ellipsoid distance

Our Idea (2) Neighborhood Info Retrieval System start point destination start pointdestination initial query parameters : Data objects : sampled estimated positions of the moving object Sample positions are taken by unit-time basis At each sample position, a spatial query is generated The system perform queries continuously

Problems and Solutions How can we generate appropriate spatial queries? Introduction of influence model of trajectory points Proposal of query derivation models How about efficiency? Use of spatial indexes for efficient query processing Low-cost query update procedure for continuous queries

Organization Background and Overview Our Approach Influence model of trajectory points Query derivation model Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work

Representation of Location Information (1) Object locations are represented by d-D vectors : no. of dimensions

current point start point destination Representation of Location Information (2) Locations of a moving object: Assumption: past/future trajectory points are given in unit-time basis : current time : estimated arrival time : departure time

Influence Model of Trajectory Points (1) current position We usually set high importance on current neighborhood points

Influence Model of Trajectory Points (2) current position A user may be interested in near future neighborhood where he or she will arrive soon

Influence Model of Trajectory Points (3) The influence model sets the highest weight “1” on location information at time t =  +  (  unit times after the current time  ) The influence values decay exponentially towards past and future with parameters  and, respectively time Influence Value τ+σ τ+σ+1 τ+σ+2 τ+σ － 1 τ+σ － 2

current point start point destination Influence value for each point when  = 1 Influence Model of Trajectory Points (4)     ’-1      ’-2 highest weight point since  = 1

Organization Background and Overview Our Approach Influence model of trajectory points Query derivation model Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work

Query Derivation Model Neighborhood queries for moving objects are issued to a spatial database A spatial query is fixed specifying query center q two models (cur, avg) distance function D three models (EU, OV, HB) query task range query and k-nn query q D

Derivation of Query Centers (1) Model cur: set the point with the highest importance to the query center current position query center q

Derivation of Query Centers (2) Model avg: weighted average based on influence values current position query center q Setting of parameters  and changes the query center

Query Derivation Model Neighborhood queries for moving objects are issued to a spatial database A spatial query is fixed specifying query center q two models (cur, avg) distance function D three models (EU, OV, HB) query task range query and k-nn query q D

Distance Function Derivation Models (1) Model EU: Euclid distance-based model Pros - simple and intuitive - easy to compute Cons - do not consider past/future - trajectory information

Ellipsoid Distance Appropriate setting of the distance matrix A allows flexible tuning of distances We derive an appropriate matrix A using past/future trajectory information

Distance Function Derivation Models (2) Model OV: ellipsoid distance-based model derive a distance matrix M that reflects the sample point distribution nearby the query point [19] C is the weighted covariance matrix

Distance Function Derivation Models (3) Model OV: ellipsoid distance-based model pros allows retrieval along the trajectory since the derived distance is an extended version of the Mahalanobis distance [8, 20] cons: not robust compared to the Euclidean distance When an object is moving along a straight line or staying in some place, the matrix C becomes an ill- conditioned matrix: therefore, we cannot derive the distance matrix M!

Model HB: hybrid model integrates the benefits of EU and OV models Distance Function Derivation Models (4) : unit matrix becomes an regular matrix regularization

Query Derivation Model Neighborhood queries for moving objects are issued to a spatial database A spatial query is fixed specifying query center q two models (cur, avg) distance function D three models (EU, OV, HB) query task range query and k-nn query q D

Query Task (1) Range Query: At each point, retrieve objects within distance 

Query Task (2) k-Nearest Neighbor Query: At each point retrieve nearest k objects when k = 3

Organization Background and Overview Our Approach Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work

Experiment 1: Observation of Behaviors Query generation example for the trajectory (blue line) Target points are shown in green points Queries are generated based on the hybrid model

x Experiment 1 (1) Comparison of Euclidean distance and ellipsoid distance

Experiment 1 (2) Set the “near future” point as query center initial parameters σ= 0, μ=0.5 ν=0.5, λ= 1.0 modified parameters σ= 5, μ=0.4 ν=0.4, λ= 1.0 x y

Experiment 1 (3) Set high weights on future trajectory initial parameters σ= 0, μ=0.4 ν=0.4, λ= 1.0 refined parameters σ= 0, μ=0.4 ν=0.9, λ= 1.0 x

Experiment 1 (4) x Use of the regularization parameter initial parameters σ= 0, μ=0.4 ν=0.4, λ= 1.0 refined parameters σ= 0, μ=0.4 ν=0.4, λ= 0.7

Experiment 2: Simulation Based on Trace Data (1) Car driving trace data is used to compute queries

Experiment 2: Simulation Based on Trace Data (2) Each isosurface represents the query generated at the point

Organization Background and Overview Our Approach Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work

Query Processing Based on Spatial Indexes Most of spatial indexes do not support ellipsoid distance-based queries We extend the approach of Seidl & Kriegel [30] to support ellipsoid distance-based queries with conventional spatial indexes Assumptions: only three generic retrieval functions are supported by the underlying spatial indexes

Generic Retrieval Functions (1) rect_search(r): retrieve objects within the specified rectangle region r r

Generic Retrieval Functions (2) dist_search(q,  ): retrieve objects within distance  from q using the Euclidean distance 

Generic Retrieval Functions (3) knn_search(q, k): retrieve nearest k objects from the query center q using the Euclidean distance

Minimal Bounding Box (MBB) for Ellipsoid Isosurface [30] MBB that tightly encloses the ellipsoid ellip(M, q,  ) j-th dimension i-th dimension : (i, i) element of the inverse of M ellip(M, q,  )

Minimal Bounding Sphere (MBS) for Ellipsoid Isosuraface [30] MBS that tightly encloses the ellipsoid ellip(M, q,  ) : the smallest eigenvalue of M ellip(M, q,  )

Query Processing (1) Range query processing with MBB approximation 

Query Processing (1)  Range query processing with MBB approximation

Query Processing (2) k-NN query (k = 3)

Query Processing (2) k-NN query (k = 3) 1.Perform k-NN query based on the Euclidean distance 2. Derive an ellipsoid that tightly encloses k-NN objects 3. Perform a range query with MBS (or MBB) that tightly encloses the ellipsoid region 4. Select nearest k objects from the retrieved objects using the ellipsoid distance

Experiment: Retrieval I/O Cost with Spatial Indexes (1) I/O cost evaluation using R-tree (GiST) Target dataset (green points): 39,226 crossroad points of Maryland County in U.S. Query: 62 blue points along the road I/O costs are compared for sequential scan ellipsoid distance query with the support of spatial indexes k-NN (k = 1, 10, …, 150) results are shown

Experiment: Retrieval I/O Cost with Spatial Indexes (2) Average page I/O cost per query

Organization Background and Overview Our Approach Examples Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work

Query Update In each query point, a slightly different query is generated The query center and the distance function will change Naïve update strategy Derive the query center and the distance function from scratch The generation cost is quite large It requires calculation from past/future trajectory information Can we update queries incrementally? Answer: Yes, but periodic reorganization is required

Incremental Query Update (1) Basic Idea Decompose statistics used to generate a query into past part and future part At each update, make “one step shift” from the future part to the past part Exponential decay factors allow a simple and efficient procedure

Incremental Query Update (2) Example: Incremental update of query center for model avg Decompose x|  as

Incremental Query Update (3) Then update using the following formulas We can make an incremental update for covariance matrix (C) in a similar manner

Incremental Query Update (4) Incremental query update procedure allows constant update cost for fixed dimensionality d Bad news: two problems A moving object may reach early or late to the next point. Moreover, it may change the estimated route. A number of incremental updates will result in incorrect query generation since the proposed incremental update procedure amplifies the noise. Practical update procedure Use incremental update procedure for short period and recalculate statistics periodically

Organization Background and Overview Our Approach Examples Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work

Conclusions Generation of Neighborhood Tracking Queries Based on Adaptive Distances (Ellipsoid Distances) Introduction of Influence Decay Model of Trajectory Points Proposal of Spatial Query Generation Models Efficient Query Evaluation with Spatial Indexes Query Update Method for Continual Query Processing

Future Work Development of parameter set-up method that considers query workloads and query tasks Use of previous query results (cached results) for efficient continual query processing Development of Prototype System

Prototype System Under development on top of ArcView GIS Support of dynamic location feeding from GPS