Handling Uncertain Spatial Data: Comparisons between Indexing Structures. Bir Bhanu, Rui Li, Chinya Ravishankar and Jinfeng Ni

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1 Handlng Uncertan Spatal Data: Comparsons between Indexng Structures Br Bhanu, Ru L, Chnya Ravshankar and Jnfeng N Abstract Managng and manpulatng uncertanty n spatal databases are mportant problems for varous practcal applcatons. Unlke the tradtonal fuzzy approaches n relatonal databases, n ths paper we propose a probablty-based method to model and ndex uncertan spatal data where every object s represented by a probablty densty functon (PDF). To ndex PDFs, we construct an optmzed Gaussan mxture herarchy () and two varants of uncertan R-tree. We provde a comprehensve comparson among these three ndces and plan R-tree on TIGER/Lne Southern Calforna landmark pont dataset. We fnd that uncertan R-tree s the best for fxed query and s sutable for both certan and uncertan queres. Moreover, s sutable not only for spatal databases, but also for mult-dmensonal ndexng applcatons lke content based mage retreval, where R-tree s neffcent n hgh dmensons. Index Terms ndex structures, R-tree, spatal database, uncertanty. I. INTRODUCTION Geographc nformaton system (GIS) s a system of computer software, hardware and data, and personnel to help manpulate, analyze and present nformaton that s ted to a spatal locaton. Spatal database s the system whch organzes spatal nformaton n GIS []. In GIS applcatons, t s generally agreed that there are several types of error (uncertanty) whch determne the overall accuracy of fnal products. A bennal Spatal Accuracy Symposum s held specfcally on ths topc []. However, n the fled of nformaton technology scant attenton has been pad for handlng uncertanty n spatal databases [3]. Even though there s more awareness and some understandng of uncertanty sources n spatal data, at present there s not a complete system whch can store and operate on uncertan Ths work was supported by NSF Informaton Technology Research Grant The contents of the nformaton do not reflect the poston or polcy of the U.S. Government. B. Bhanu s wth the Center for Research n Intellgent System, Unversty of Calforna, Rversde, CA 5 (e-mal: bhanu@crs.ucr.edu). R. L s wth the Department of Electrcal Engneerng, Unversty of Calforna, Rversde, CA 95 (e-mal: rl@vslab.ucr.edu). C. Ravshankar s wth the Department of Computer Scence & Engneerng, Unversty of Calforna, Rversde, CA 95 (e-mal: rav@cs.ucr.edu). J. N s wth the Department of Computer Scence & Engneerng, Unversty of Calforna, Rversde, CA 95 (e-mal: jn@cs.ucr.edu). spatal data. ArcInfo and Oracle Extensons are only suted for certan spatal data []. In ths paper, we present our approaches on ndexng uncertan spatal data. Most of the exstng approaches for management of probablstc data are based on the relatonal model and use fuzzy set theory [4, 5]. They are useful for representng uncertanty at the symbolc level. However, n addton to symbolc uncertanty, sensor-processng tasks nvolve uncertantes at both numerc and exstence levels. Supportng these types of uncertanty n the current relatonal model usng fuzzy logc s fundamentally dffcult. So n our approach, we use probablty densty functons (PDFs) to represent uncertan data, whch means every object s a random varable. In spatal databases, R-tree s the most often used ndexng structure, whch s a depth-balanced tree whose nodes are represented by Mnmum Boundng Rectangles (MBR). Guttman [6] frst proposed R-tree and Greene and Beckmann [7, 8] optmzed t by reducng margn and MBR overlap. But R-tree famly ndexes fxed data only. In order to handle the uncertan nformaton related to each object, we have to use R- tree varant or desgn new ndex. In ths paper, we buld two varants of uncertan R-tree and an optmzed Gaussan mxture herarchy () based on Gaussan mxture model (GMM). The rest of the paper s organzed as follows. The uncertanty representaton and ndex constructon are explaned n Secton II. Secton III gves the expermental results on TIGER/Lne Southern Calforna landmark pont dataset. Secton IV concludes the paper. A. PDF representaton II. TECHNICAL APPROACH In conventonal spatal databases, each object s represented by a feature vector n n dmensonal feature space. But when the data are uncertan, a dfferent representaton s requred. In out approach, we use a PDF to represent each uncertan object. If an object needs n features to descrbe, then t s an n dmensonal (feature vector) random varable, as gven n (). where f =,, N [ f f,, f ] T, =, N s the number of objects For smplcty, we assume that the features are ndependent of each other and each feature s PDF ( f j, j =,..., n ) s known. The process of gettng the PDFs s called uncertanty n ()

2 modelng and t s a part of our on gong related work. Our system needs to handle both certan and uncertan data. So we do not gve a specfc name to uncertan data. Feature vector s the name for both certan and uncertan data n ths paper. For feature vectors, metrcs lke Eucldean dstance, Manhattan dstance etc. are used to measure smlarty. Snce our uncertan objects are random varables represented by PDFs, we defne the smlarty of feature vectors as the probablty that two random varables are the same, as shown n (). In ths equaton, D and Q are two objects. D s the object n the database and Q s the query. s a threshold vector descrbng the maxmal error the system can tolerate to stll regard D as smlar to Q. Paper [9] presents the smlarty measure n detal. smlarty ( D, Q) = Pr( D Q < ) () D, Q are two feature vectors B. System dagram In our method, we assume that the uncertanty s small [0], therefore the dsturbed objects roughly keep the orgnal dstrbuton. Thus, we can use the feature vector means to construct the ndex, and attach uncertan nformaton to the data entres. At ths step, an ndex whch handles uncertanty s constructed, as shown by the dash-lne box n Fg.. Index whch handles uncertanty Uncertanty s attached to each data entry Feature vectors Feature vector means are extracted Indexng constructon Get nearest node(s) Gather all data entres as the canddate set Refne the query result wthn canddate set based on the smlarty measure NN search results Fg.. Index structure and NN search procedure. Query When a query comes n, the nearest node(s) n the ndex s found. All the data entres belongng to these nodes (wth uncertanty) are gathered as the canddate set. Ths process s called flter. In the refne step, the smlarty between the query and each data entry n the canddate set s calculated and sorted. The entres correspondng to the largest smlarty are the query result. Ths s called nearest neghbors (NN) search. Here, we are only nterested n NN because t s the bass for other comprehensve queres. The flter step dffers n dfferent ndex constructons. We construct two ndex systems: (a) Optmzed Gaussan mxture herarchy () based on Gaussan mxture model, and (b) Two R-tree varants called uncertan R-tree () and. These are explaned n part C and D n ths secton. C. Optmzed Gaussan Mxture Herarchy () We assume all the objects n the n dmensonal feature space follow some dstrbuton. From the probablty theory, any dstrbuton can be approxmated by a weghted sum of several Gaussan dstrbutons [], as shown n (3), where dstrbuton (x) f x, p s approxmated by C Gaussan dstrbutons: ( ) =,,C. α s the weght for each Gaussan dstrbuton represented by mean vector u and covarance matrx Σ : C C p ( x) = α f ( x) = α N( u, Σ ) (3) Fgueredo and Jan [] proposed a varant of Expectaton Maxmzaton (EM) algorthm to automatcally fnd the number of clusters and to perform clusterng. We use ths algorthm to get the Gaussan components. It nvolves the followng steps: In the frst step, the entre dataset s clustered nto several groups. In the subsequent steps, a bottom-up bnary tree s bult based on these Gaussan components. Snce every leaf node s represented by a Gaussan dstrbuton, we represent every nner node by the Gaussan mxture of all ts offsprng leaves. When a query comes n, we start from the root node, calculate the query s smlarty wth the left chld and the rght chld, and then select the branch wth hgher smlarty. Ths process s repeated untl a leaf node s reached. Then all the data entres belongng to ths leaf are gathered as the canddate set. If there are not enough data entres n the canddate set, all data belongng to the sblng of ths leaf node are added. Refnement s performed wthn the canddate set to get the NN search result. Detal explanaton of tree constructon and NN search algorthm are gven n [9]. D. Uncertan R-tree The uncertan R-tree constructon s based on all the feature vector means and uncertan nformaton s attached to each data entry. To support NN searches, two flter strateges: and are developed. ) : Nearest leaves are returned, even f they do not belong to the same parents. The number of leaves s decded by the requred canddate set sze. All data entres of these leaf nodes are gathered together as the canddate set for our uncertan query. ) : The nearest leaf s found, then ts ancestors are backtracked untl the one that has enough data entry

3 offsprng for refnement s met. All the data entres of these offsprng construct the canddate set. The canddate set n both and s refned to get the NN for the query. and have the same canddate set extracton strategy, so only ther comparson s far, whereas wll theoretcally acheve hgher precson. The comparson n the next secton wll testfy ths result. III. EXPERIMENTAL RESULTS A. Dataset and uncertanty assgnment We take the TIGER/Lne southern Calforna landmark pont dataset n the experment, as shown n Fg. and Fg. 3. It contans 8703 D coordnates (longtude and lattude n degrees) and ts precson (uncertanty) s 67 feet [0], whch s Uncertanty s added as a D Gaussan nose to each pont, as shown n (4): Nose ~ N [ 0,0] T δ x, 0 0 δ y (4) for dfferent uncertanty cases. There s a parameter defnng the mnmum canddate set sze, called MCS_sze. ~5 nearest neghbor(s) are returned as the result to the query. B. Comparson of and uncertan R-tree The NN search s made on,, and plan R-tree. Ther performance measures are precson, I/O cost and CPU cost. The experments are performed under two dfferent uncertantes (0.0005, ) and two MCS_szes (40, 60). The program s wrtten n C++ and the system confguraton s as below: System: Sun Mcrosystems sun4u OS: Solars.8 Memory: 048MB ) Precson: # of correct results actually returned Precson = # of results should be returned Precson 5 Fg.. Countes n Southern Calforna. Fg. 4. σ= , MCS_sze = 40. Precson Fg. 3. Landmarks n Southern Calforna. In all the experments, the test data are fxed (the orgnal data) and the tranng data are nosy, whereδ x, δ for each y pont are randomly selected from [0, ] or [0, ] 5 Fg. 5. σ= , MCS sze = 60. 3

4 σ = σ = Precson Precson Fg. 6. σ= 0.005, MCS_sze = 40. Fg. 8. precson on dfferent σ. 6 4 σ = σ = Precson 5 Fg. 7. σ= 0.005, MCS_sze = 60. Precson Fg. 9. precson on dfferent σ. From Fg. 4 - Fg. 7, we can make the followng observatons: (a) always gves the best performance, followed by and. As mentoned n Secton II.D (Uncertan R- tree), and have the same flter strategy. (b) has hgher precson performance than, especally when MCS_sze s larger. So Gaussan Mxture model s more approprate than MBRs n PDF ndexng. (c) Plan R-tree gves the worst precson and t s not acceptable, so n the followng comparsons, plan R-tree s removed. From Fg. 8 and Fg. 9 we can see that when the uncertanty ncreases from to (MCS_sze = 60), the precson performance of degrades much more than that of (5% vs. 0.5%). So s more stable than uncertan R-tree. ) I/O cost -- the average page read/wrte for -NN query sgma = 0.005, MCS_sze = 40 sgma = 0.005, MCS_sze = 60 sgma = , MCS_sze = 40 sgma = , MCS_sze = 60 Fg. 0. I/O cost comparson among all ndces. As shown n Fg. 0, and need more I/O cost than. The tree has more levels than R-tree (7 vs. 4). The more I/O cost of comes from the back trackng, but t s stll comparable wth. 4

5 ) CPU cost -- average tme (second) for -NN query. sgma = 0.005, MCS_sze= 40 sgma = 0.005, MCS_sze = 60 sgma = , MCS_sze = 40 sgma = , MCS_sze = 60 Fg.. CPU cost comparson among all ndces. Fg. ndcates that and are comparable on tme complexty and they are more effcent than, ths s because the back trackng of s tme consumng. From the three comparsons above, we see that when the query s fxed, s the best n general, followed by and. Plan R-tree s not acceptable wth respect to precson performance. But as mentoned n Secton I, a complete system should be able to handle both fxed and uncertan queres. Thus, the choce of ndex structure depends on the applcaton. If no uncertan query s asked, s the best choce, otherwse only s sutable. IV. CONCLUSIONS Uncertanty n spatal databases s gettng more and more attenton, but most of the exstng approaches are based on relatonal models usng fuzzy set theory. Ths method s only sutable for handlng uncertanty n symbolc level. In order to support uncertantes at numerc and exstence levels, we proposed a new uncertanty model and new ndexng schemes. In ths paper, we represented uncertan objects wth PDFs and constructed an optmzed Gaussan mxture herarchy based on Gaussan mxture model and an uncertan R-tree. After a comprehensve comparson based on NN search precson, I/O cost and CPU cost, uncertan R-tree () s the best for fxed queres. But s sutable for both certan and uncertan queres. Moreover, the s not sutable for only spatal databases, but also for other multdmensonal ndex applcatons lke content based mage and vdeo retreval. REFERENCES [] Rgaus, P., M. Scholl, and A. Vosard, Spatal Databases: wth Applcaton to GIS. San Francsco, Calforna: Morgan aufmann, 00. [] Internatonal Symposum on Spatal Accuracy Assessment n Natural Resources and Envronmental Scences. ault.html [3] Zanolo, C., et al., Introducton to Advanced Database Systems: Morgan-aufmann, 997. [4] Schneder, M. Uncertanty Management for Spatal Data n Databases: Fuzzy Spatal Data Types. n 6th Int. Symposum on Advances n Spatal Databases (SSD). 999: Sprnger Verlag. [5] Robnson, V.B., A Perspectve on Managng Uncertanty n Geographc Informaton System wth Fuzzy Sets. IEEE Transactons n Geographc Informaton System, (): p. -5. [6] Guttman, A. R-trees: A Dynamc Index Structure for Spatal Searchng. n SIGMOD Boston, MA,. [7] Greene, D. An Implementaton and Performance Analyss of Spatal Data Access Methods. n Proceedngs of the Ffth Internatonal Conference on Data Engneerng. 989: IEEE Computer Socety Washngton, DC, USA. [8] Beckmann, N., et al. The R*-tree: an effcent and robust access method for ponts and rectangles. n ACM SIGMOD Internatonal Conference on Management of Data Atlantc Cty, NJ. [9] Bhanu, B., R. L, C. Ravshankar, M. urth, and J. N. Indexng Structure for Handlng Uncertan Spatal Database. n 6 th Internatonal Symposum on Spatal Accuracy Assessment n Natural Resources and Envronmental Scences Portland, Mane, USA. [0] Brown, R.H. and E. Ehrlch, TIGER/Lne(TM) Fles, : Washngton, D. C. (03/30/04) [] Duda, R.O., Hart, P.E., and D.G. Stork, Pattern Classfcaton. New York: A Wley-Interscence Publcaton, 000. [] Fgueredo, M.A. and A. Jan, Unsupervsed Learnng of Fnte Mxture Models. IEEE Transactons on Pattern Analyss and Machne Intellgence, 00. 4(3): p