隨著近年來無限通訊以及全球衛星定位系統的進步，配備這些裝備的移動物體（例如汽車）可以傳達本身資訊給予中央系統。藉由這些配備，查詢移動物體的位置成為目前最為重要的課題。目前已經有不少學者針對這個問題提出相當多的方法，然而，這些方法中的模型卻是假設物體只會依照單一路徑移動。正常情況下，一個物體的位置是不得而知的，直到這個物體將資訊傳至中央系統中。對中央系統而言，一旦資訊的傳遞結束後，這個物體的位置會隨著時間的增加而漸漸變的模糊；直到下一次的資訊傳遞，中央系統才又重新完全掌握此物體的確切位置。在這樣的情況下，查詢這些物體的位置將會產生模糊或容易錯誤的結果；但是，若是我們使用機率模型表示這些移動物體，在查詢這些物體時，我們可以使用機率保證查詢的正確性。在這篇論文中，我們針對一般移動物體的特性，選擇了最適宜的機率模型以及利用方程式表示物體的位置。然而在一般應用下，物體的數量是非常龐大的，因為查詢所導致的計算量將是難以接受。為了解決這各問題，我們將這些物體將來會到達的位置建成索引，加速找尋這些物體的速度。另外，我們也提出近似法，能夠快速的檢驗這些物體是否符合查詢所要的結果。根據實驗的結果，這篇論文所提出的方法可以大量減少檢索物體的數目，以及降低檢索物體時所耗費的資源。

With the advances in wireless communication and global positioning systems, today’s moving objects such as moving cars have the ability to update their location and velocity information to a central server. Range queries for querying the current and future positions of the moving objects are becoming increasingly necessary. Existing methods have been developed to support range queries, but they unreasonably assume that an object only moves according to its predicted single path. In general, the certain location of an object is unknown until the object updates its location information to the server. After the update, the uncertainty of the object’s location starts increasing until its next update. Although querying these uncertain data results in imprecise answers, these answers can be possibly estimated with probability guarantees by uncertainty models. In this paper, we study an uncertainty model, which is a function that expresses the possible movements with corresponding probabilities of the moving objects. Unfortunately, due to the complexity of the probability evaluation and the large number of objects to examine, the process of querying with probabilities is very costly. To overcome this problem, we map the uncertain movements of all objects to another space for an easy indexing. Our proposed method first eliminates infeasible answers by querying on the index. Then, for evaluating the remaining objects, an approximate examination with an error bound is employed to lower the overhead of the probability evaluation. The experimental study shows that our technique reduces the number of object examinations and the cost of the probability evaluation.

[1] R. Cheng, D. V. Kalashnikov, and S. Prabhakar. Evaluating Probabilistic Queries over Imprecise Data. In Proceedings of the ACM SIGMOD International Conference on Management of Data, pp 551-562, 2003.
[2] R. Cheng, D. V. Kalashnikov, and S. Prabhakar. Querying Imprecise Data in Moving Object Environments. In IEEE Transactions on Knowledge and Data Engineering, pp 1112-1127, 2004.
[3] R. Cheng, Y. Xia, S. Prabhakar, R. Shah, and J. S. Vitter. Efficient Indexing Methods for Probabilistic Threshold Queries over Uncertain Data. In Proceedings of the 30th International Conference on Very Large Data Bases, pp 876-887, 2004.
[4] W. Feller. An Introduction to Probability Theory and Its Applications 2nd edn. (Wiley), pp 340-391, 1957.
[5] A. Guttman. R-Trees: A Dynamic Index Structure for Spatial Searching. In Proceedings of the ACM SIGMOD International Conference on Management of Data, pp 47-57, 1984.
[6] V. Gaede, O. Günther. Multidimensional Access Methods. In ACM Computing Surveys, pp 170-231, 1998.
[7] J. Goldstein, R. Ramakrishnan, U. Shaft, J. B. Yu. Processing Queries by Linear Constraints. In Proceedings of the 16th ACM PODS Symposium on Principles of Database Systems, pp 257-267, 1997.
[8] P. V. C. Hough. Method and Means for Recognizing Complex Patterns, U. S. Patent No. 306964, 1962.
[9] H. V. Jagadish. On Indexing Line Segments. In Proceedings of 16th International Conference on Very Large Data Bases, pp 614-625, 1990.
[10] S. Karlin and H.M. Taylor. A First Course in Stochastic Processes 2nd edn. (Academic Press), pp. 340-391, 1975.
[11] G. Kollios, D. Papadopoulos, D. Gunopulos, and J. Tsotras. Indexing Mobile Objects using Dual Transformations. In The International Journal on Very Large Data Bases, pp 238-256, 2005.
[12] Z. Lei, C. U. Saraydar, and N. B. Mandayam. Paging Area Optimization based on Interval Estimation in Wireless Personal Communication Networks. In Mobile Networks and Applications, pp 85-99, 2000.
[13] A. Papoulis. Probability, Random Variables and Stochastic Processes 3rd edn. (McGraw-Hill), 1991.
[14] H. Packard. Normal and Inverse Normal Distribution for the HP-67. “http://www.hpmuseum.org/software/67pacs/67ndist.htm”.
[15] C. Rose. Minimizing the Average Cost of Paging and Registration: A Timer-based Method. In Wireless Networks, pp 109-116, 1996.
[16] S. Saltenis, C. S. Jensen, S. T. Leutenegger, and M. A. Lopez. Indexing the Positions of Continuously Moving Objects. In Proceedings of the ACM SIGMOD International Conference on Management of Data, pp 331-342, 2000.
[17] A. P. Sistla, O. Wolfson, S. Chamberlain, and S. Dao. Modeling and Querying Moving Objects. In Proceedings of the 13th International Conference on Data Engineering, pp 422-432, 1997.
[18] Y. Tao, C. Faloutsos, D. Papadias, and B. Liu. Prediction and Indexing of Moving Objects with Unknown Motion Patterns. In Proceedings of the ACM SIGMOD International Conference on Management of Data, pp 611-622, 2004.
[19] Y. Tao, D. Papadias, and J. Sun. The TPR*-Tree: An Optimized Spatio-Temporal Access Method for Predictive Queries. In Proceedings of 29th International Conference on Very Large Data Bases, pp 790-801, 2003.
[20] Y. Tao, R. Cheng, X. Xiao, W. K. Ngai, B. Kao, and S. Prabhakar. Indexing Multi-Dimensional Uncertain Data with Arbitrary Probability Density. In Proceedings of the 31st international conference on Very large data bases, pp 922-933, 2005.