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Spatial Range Querying for Gaussian-Based Imprecise Query Objects

Spatial Range Querying for Gaussian-Based Imprecise Query Objects. Yoshiharu Ishikawa , Yuichi Iijima Nagoya University Jeffrey Xu Yu The Chinese University of Hong Kong. Outline. Background and Problem Formulation Related Work Query Processing Strategies Experimental Results Conclusions.

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Spatial Range Querying for Gaussian-Based Imprecise Query Objects

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  1. Spatial Range Querying forGaussian-Based Imprecise Query Objects Yoshiharu Ishikawa, Yuichi IijimaNagoya University Jeffrey Xu Yu The Chinese University of Hong Kong

  2. Outline • Background and Problem Formulation • Related Work • Query Processing Strategies • Experimental Results • Conclusions

  3. ImpreciseLocation Information • Sensor Environments • Frequent updates may not be possible • GPS-based positioning consumes batteries • Robotics • Localization using sensing andmovement histories • Probabilistic approach hasvagueness • Privacy • Location Anonymity

  4. Location-based Range Queries • Location-based Range Queries • Example: Find hotels located within 2km from Yuyuan Garden • Traditional problem in spatial databases • Efficient query processing using spatial indices • Extensible to multi-dimensional cases (e.g., image retrieval) • What happen if the location of query object is uncertain? q

  5. Probabilistic Range Query (PRQ) (1) • Assumptions • Location of query object qis specified as a Gaussian distribution • Target data: static points • Gaussian Distribution • Σ: Covariance matrix

  6. Probabilistic Range Query (PRQ) (2) • Probabilistic Range Query (PRQ) • Find objects such that the probabilities thattheir distances from qare less than δ are greater than θ q

  7. Probabilistic Range Query (PRQ) (3) • Is distance between q and p within d? pdf of q (Gaussian distribution) p Numerical integraiton is required

  8. Naïve Approach for Query Processing d d d d d • Exchanging roles • Pr[p is within d from q]= Pr[q is within d fromp] • Naïve approach • For each object p,integrate pdf forsphere region R • R : sphere withcenter p andradius δ • If the result q,it is qualified • Quite costly! R q p

  9. Outline • Background and Problem Formulation • Related Work • Query Processing Strategies • Experimental Results • Conclusions

  10. Related Work • Query processing methods for uncertain (location) data • Cheng, Prabhakar, et al. (SIGMOD’03, VLDB’04, …) • Tao et al. (VLDB’05, TODS’07) • Parker, Subrahmanian, et al. (TKDE’07, ‘09) • Consider arbitrary PDFs or uniform PDFs • Target objects may be uncertain • Research related to Gaussian distribution • Gauss-tree [Böhm et al., ICDE’06] • Target objects are based on Gaussian distributions

  11. Outline • Background and Problem Formulation • Related Work • Query Processing Strategies • Experimental Results • Conclusions

  12. Outline of Query Processing • Generic query processing strategy consists of three phases • Index-Based Search: Retrieve all candidate objects using spatial index (R-tree) • Filtering: Using several conditions, some candidates are pruned • Probability Computation: Perform numerical integration (Monte Carlo method) to evaluate exact probability • Phase 3 dominates processing cost • Filtering (phase 2) is important for efficiency

  13. Query Processing Strategies • Three strategies • Rectilinear-Region-Based Approach (RR) • Oblique-Region-Based Approach (OR) • Bounding-Function-Based Approach (BF) • Combination of strategies is also possible

  14. Rectilinear-Region-Based (RR) (1) • Use the concept of -region • Similar concepts are used in query processing for uncertain spatial databases •  -region: Ellipsoidal region for which the result of the integration becomes 1 – 2q: • The ellipsoidal regionis the q -region

  15. Rectilinear-Region-Based (RR) (2) • Query processing • Given a query, -region is computed: it is suffice if we have rθ-tablefor “normal” Gaussian pdf • “Normal” Gaussian:S = I, q = 0 • Given , it returns appropriate rθ • Derive MBR for -region and perform Minkowski Sum • Retrieve candidates then perform numerical integration d c a q q d b  -region

  16. Rectilinear-Region-Based (RR) (3) • Geometry of bounding boxwhere (Σ)iiis the (i, i) entry of Σ xj xi wj q wj wi wi

  17. Oblique-Region-Based (OR) (1) • Use of oblique rectangle • Query processing based on axis transformation • Not effective for phase 1 (index-based search): Only used for filtering (phase 2) c d c d d a b d q b a q

  18. Oblique-Region-Based (OR) (2) • Step 1: Rotate candidate objects • Based on the result of eigenvalue decomposition of Σ-1 • Step 2: Check whether each object is inside of the rectangle • λi: Eigenvalue of Σ-1 for i-th dimension q

  19. Bounding-Function-Based (BF) (1) d d d • Basic idea • Covariance matrix S = I (“normal” Gaussian pdf) • Isosurface of pdf has a spherical shape • Approach • Let a be the radius for which the integration result is q • If dist(q, p) ≤athen psatisfies the condition • Construct a table that gives (d, q ) abeforehand Pr< q Pr= q a q Pr> q

  20. Bounding-Function-Based (BF) (2) • General case • isosurface has an ellipsoidal shape • Approach • Use of upper- and lower-bounding functions for pdf • They have spherericalisosurfaces • Derived from covariance matrix q

  21. Bounding Functions • Original Gaussian pdf • Upper- and lower-bounding functions T T T Isosurfacehas aspherical shape T Note: λT = min{λi} T holds T λ = max{λi}

  22. Bounding-Function-Based (BF) (3) d d T • aT (a ): Radius with which the integration result of upper- (lower-) bounding function isq Originalpdf Prob. (integrationresult) = q T a Upper-bounding function T a q Lower-bounding function 2d 2d

  23. Bounding-Function-Based (BF) (4) • Theoretical result • Let ST be a spherical region with radius and its center relative to the origin is βT, and assume that ST satisfies the following equation: • Using table that gives (d, q ) a, we can get βT: • Then we can get

  24. Bounding-Function-Based (BF) (5) • Step 1: Use of R-tree • {b, c, d} are retrieved ascandidates • Step 2: Filtering using aT • b is deleted • Step 2’: Filtering using a • We can determine d as an answer withoutnumerical integration • Step 3: Numerical integration • Performed on {c} b a T T a q T d c a

  25. Outline • Background and Problem Formulation • Related Work • Query Processing Strategies • Experimental Results • and Conclusions

  26. Experiments on 2D Data (1) • Map of Long Beach, CA • Normalized into [0, 1000]  [0, 1000] • 50,747 entries • Indexed by R-tree • Covariance matrix • γ : Scaling parameter • Default: γ = 10

  27. Example Query • Find objects within distance δ = 50 with probability threshold θ = 1%

  28. Experiments on 2D Data (2) • Numerical integration dominates the total cost • R-tree-based search is negligible • ALL is the most effective strategy δ = 25 θ = 0.01

  29. Experiments on 2D Data (3) • Filtering regions (δ = 25, θ = 0.01, γ = 10) BF (upper) OR y RR x BF (lower) Integration region for ALL I

  30. Experiments on 2D Data (4) • Filtering regions for different uncertainty setting (δ = 25, θ = 0.01) γ = 1 : Nearly exact γ = 10 : Medium uncertainty γ = 100 : Uncertain

  31. Experiments on 9D Data (1) • Motivating Scenario: Example-Based Image Retrieval • User specifies sample images • Image retrieval system estimates his interest as a Gaussian distribution

  32. Experiments on 9D Data (2) • Data set: Corel Image Features data set • From UCI KDD Archive • Color Moments data • 68,040 9D vectors • Euclidean-distance based similarity • Experimental Scenario: Pseudo-Feedback • Select a random query object, then retrieve k-NN query (k = 20) as sample images • Derive the covariance matrix from samples : Sample covariance matrix κ : Normalization parameter

  33. Experiments on 9D Data (3) • Parameters • δ = 0.7: For exact case, it retrieves 15.3 objects • θ = 40% • Number of candidates (ANS: answer objs) Too many candidates to retrieve only 3.9 objects!

  34. Experiments on 9D Data (4) • Reason: Curse of dimensionality • Plot shows existence probability for pnorm for different radii and dimensions Location of query object is too vague: In medium dimension, it is quite apart from its distribution center on average Example: For 9D case, the probability that query object is within distance two is only 9%

  35. Outline • Background and Problem Formulation • Related Work • Query Processing Strategies • Experimental Results • Conclusions

  36. Conclusions • Spatial range query processing methods for imprecise query objects • Location of query object is represented by Gaussian distribution • Three strategies and their combinations • Reduction of numerical integration is important • Problem is difficult for medium- and high-dimensional data • Our related work • Probabilistic Nearest Neighbor Queries (MDM’09)

  37. Spatial Range Querying forGaussian-Based Imprecise Query Objects Yoshiharu Ishikawa, Yuichi IijimaNagoya University Jeffrey Xu Yu The Chinese University of Hong Kong

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