Spatio-temporal Range Queries Research Articles

MSTGI: a multi-scale spatio-temporal grid index model for remote-sensing big data retrieval

ABSTRACT To promote the transformation of remote sensing (RS) data into geoscience knowledge, it is necessary to provide better data discovery capabilities, especially when large amounts of RS data have been accumulated. Spatio-temporal range query is one of them to enable data discovery on RS images by the spatio-temporal range. Although existing RS image indexing methods are suitable for data discovery based on spatio-temporal range queries, they do not fully consider the problem of low data retrieval efficiency when the temporal and spatial range scales are unbalanced. To address this problem, we propose a multi-scale spatio-temporal grid index model (MSTGI). First, we divide the time dimension into three levels according to the granularity of year, month and day, and build an independent grid index structure for the object at different levels. Second, we design an adaptive hierarchical indexing strategy to perform parallel retrieval in an appropriate combination of partitions. MSTGI linearizes the grid obtained after global geospatial subdivision using Hilbert curves. Our experimental results, obtained on the Landsat series satellite dataset, reveal that the proposed method improves query efficiency levels by approximately 12.582% and 87.754% compared with GeoSOT-ST and AGMC at various spatial scales, respectively, and reaches 100% recall.

HGST: A Hilbert-GeoSOT Spatio-Temporal Meshing and Coding Method for Efficient Spatio-Temporal Range Query on Massive Trajectory Data

In recent years, with the widespread use of location-aware handheld devices and the development of wireless networks, trajectory data have shown a trend of rapid growth in data volume and coverage, which has led to the prosperous development of location-based services (LBS). Spatio-temporal range query, as the basis of many services, remains a challenge in supporting efficient analysis and calculation of data, especially when large volumes of trajectory data have been accumulated. We propose a Hilbert-GeoSOT spatio-temporal meshing and coding method called HGST to improve the efficiency of spatio-temporal range queries on massive trajectory data. First, the method uses Hilbert to encode the grids obtained based on the GeoSOT space division model, and then constructs a unified time division standard to generate the space–time location identification of trajectory data. Second, this paper builds a novel spatio-temporal index to organize trajectory data, and designs an adaptive spatio-temporal scaling and coding method based on HGST to improve the query performance on indexed records. Finally, we implement a prototype system based on HBase and Spark, and develop a Spark-based algorithm to accelerate the spatio-temporal range query for huge trajectory data. Extensive experiments on a real taxi trajectory dataset demonstrate that HGST improves query efficiency levels by approximately 14.77% and 34.93% compared with GeoSOT-ST and GeoMesa at various spatial scales, respectively, and has better scalability under different data volumes.

HBase-based spatial-temporal index model for trajectory data

The development of global positioning technology and the popularization of smart mobile terminals has led to a trend of rapid growth in the data volume and coverage of trajectory data. This type of data has the characteristics of fast update speed, high dimensional characteristics, and a large amount of information that can be mined. Many technology companies will use trajectory data to provide location-based services, such as vehicle scheduling and road condition estimation. However, the storage and query efficiency of massive trajectory data have increasingly become bottlenecks in these applications, especially for large-scale spatiotemporal query scenarios. This paper solves this problem by designing a trajectory data index model based on GeoSOT-ST spatiotemporal coding. Based on this model, the HBase-based trajectory data storage scheme and spatiotemporal range query technology are studied, and MapReduce is used as the calculation engine to complete the query attribute condition filtering in parallel. Comparative experiments prove that the index model proposed in this paper can achieve efficient trajectory data query management.

GeoMapViz: a framework for distributed management and geospatial data visualization based on massive spatiotemporal data streams

Spatiotemporal big data have multisource, heterogeneous, high-dimensional and spatiotemporal associations. Due to the limited computing and network resources, while the spatiotemporal data to be rendered are large and dynamic, efficient visual analysis has always been a popular topic and has had difficulty in the research of spatiotemporal big data. As one of the important means of big data visualization, thermal maps play an important role in expressing data flow, information flow, and trajectory flow. At the same time, the development of a distributed computing framework also provides technical support for the online calculation and visualization of spatiotemporal data streams. In response to the above problems, this paper designs and implements GeoMapViz, a distributed management based on massive spatiotemporal data streams and a multiscale geographic spatial visualization framework, which is oriented by the expression of thermal maps of massive point datasets. First, based on the concept of the tile pyramid model and spatiotemporal cube, we propose a thermal map sequential tile pyramid (TS_Tile) model, which realizes scalable storage and efficient retrieval of data flow. GeoMapViz adopts a high-performance Flink stream computing cluster to implement the large-scale parallel construction of hierarchical tile pyramids, implements distributed storage and index construction of data based on HBase and Geomesa, and uses Geoserver to manage the map service to provide a spatiotemporal range query interface. Finally, through using an open dataset as a system simulation test, the results show that the TS_Tile model can effectively organize large-scale, time-space and multidimensional thermal map data, and the query and visualization of the heatmap can reach a subsecond response. Furthermore, GeoMapViz supports the integration of the thermal map and original flow and provides a feasible solution for the visual analysis of large-scale spatiotemporal data.

Query Optimization for Distributed Spatio-Temporal Sensing Data Processing

The unprecedented development of Internet of Things (IoT) technology produces humongous amounts of spatio-temporal sensing data with various geometry types. However, processing such datasets is often challenging due to high-dimensional sensor data geometry characteristics, complex anomalistic spatial regions, unique query patterns, and so on. Timely and efficient spatio-temporal querying significantly improves the accuracy and intelligence of processing sensing data. Most existing query algorithms show their lack of supporting spatio-temporal queries and irregular spatial areas. In this paper, we propose two spatio-temporal query optimization algorithms based on SpatialHadoop to improve the efficiency of query spatio-temporal sensing data: (1) spatio-temporal polygon range query (STPRQ), which aims to find all records from a polygonal location in a time interval; (2) spatio-temporal k nearest neighbors query (STkNNQ), which directly searches the query point’s k closest neighbors. To optimize the STkNNQ algorithm, we further propose an adaptive iterative range optimization algorithm (AIRO), which can optimize the iterative range of the algorithm according to the query time range and avoid querying irrelevant data partitions. Finally, extensive experiments based on trajectory datasets demonstrate that our proposed query algorithms can significantly improve query performance over baseline algorithms and shorten response time by 81% and 35.6%, respectively.

Road-aware Indexing for Trajectory Range Queries

Answering spatio-temporal range queries (RQs) on trajectory databases, i.e., finding all trajectories that intersect given ranges, is crucial in many real-world applications. Various kinds of indexes have been proposed to accelerate RQs. However, existing indexes typically use Euclidean distance to prune irrelevant regions without concerning the underlying road network information. Nevertheless, as vehicle trajectories are generated on road network edges, the road network could be seen as meta knowledge of trajectories and be used to index and query trajectories. To this end, we propose RP-Tree, a <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</u> oad network-aware <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</u> artition <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tree</u> to support efficient RQs. The basic idea is partitioning a road network graph into hierarchical subgraphs and generate a balanced tree structure, where each tree node maintains its associated trajectories. We compactly index the spatio-temporal information of trajectories on the corresponding road network edges. Then, we design efficient search algorithms to support RQs by pruning irrelevant trajectories through subgraph range borders associated with RP-Tree nodes. Last but not least, we scale RP-Tree to very large datasets by devising approximate algorithms with bounded confidence at an interactive speed. Experimental results on three real-world datasets from Porto, Chengdu, and Beijing show that our method outperform baselines by 1 to 2 orders of magnitude.

Trajectory-aware spatio-temporal range query processing for unmanned aerial vehicle networks

Unmanned aerial vehicle (UAV) network is widely used in environmental monitoring, target searching and rescuing, logistics, and other fields due to its characteristics of large-scale coverage, rapid deployment and strong resistance to destruction. When users are interested in sensory data in certain areas covered by UAV networks, they can send a spatio-temporal range query with time and geography constraints through the ground station. For example, obtaining the temperature information around fire points in the forest within an hour before the fire bursts out. Then, UAVs that carry the query results will return the data to the ground station through multi-hop routing. However, most of the existing spatio-temporal range query algorithms are designed for static networks. How to conduct spatio-temporal range queries in the UAV networks is still an open problem. In this paper, we propose a Trajectory-Aware Spatio-Temporal range query processing algorithm (TAST) for UAV networks. The ground station takes advantage of the pre-planned trajectory information of UAVs to build the topology change model of the UAV network, which can reflect the changes of UAVs’ communication links and neighbors. Furthermore, the static topology change graph (TCG) is proposed for optimizing the routing of query results in the spatio-temporal query processing. Next, we propose a Trajectory-Aware Spatio-Temporal range query processing algorithm based on packet Splitting (TASTS), which is used to split large query results into multiple small packets called unit packets, and each unit packet is transmitted back to the ground station independently and efficiently. Finally, we evaluated our algorithms through simulation experiments. The simulation results show that TAST and TASTS perform well in terms of query success rate, query efficiency and overhead ratio.

Hilbert Curve and Cassandra Based Indexing and Storing Approach for Large-Scale Spatiotemporal Data

Objectives Because of the fast growing acquisition of real-time spatiotemporal data for various applications such as smart city or real-time air-quality monitoring, the traditional database technologies can-not satisfy the higher standards for large-scale data indexing, querying, and storing operations. As the via-ble alternative, NoSQL databases that are scalable and possess fast input/output capabilities offer potential solutions to accommodate the needs. Methods We propose a Hilbert curve and Cassandra technologies based approach for efficient indexing and storing of large-scale spatiotemporal datasets aiming to provide an effective framework for processing, querying, and analyzing large amount of data with spatial and temporal features. For example, the dataset of vehicle trajectories contains valuable spatial and temporal features those are being employed in the real world. The collected spatiotemporal datasets are preprocessed in order to fit the proposed structures for different applications. Specifically, two types of query applications com -monly used in the real world are the spatiotemporal range query and query upon vehicle IDs respectively. Two corresponding indexing structures are designed and implemented in order to accommodate the requests. S2 Geometry Library open sourced by Google is utilized to divide the earth surface into grids, and data points fall in grids are assigned with the specific IDs as the keys. The keys and columns are so designed by applying the Hilbert curve and Cassandra techniques that the resultant structures will physically store the spatially neighboring data points close to each other, and they are more suitable for large-scale spatiotempo-ral data querying and analyzing applications. Results The datasets acquired from the real applications are used to conduct the computational experiments to validate the efficiency of the proposed approach. The que-ry efficiency and the time consumed to store large amount of spatiotemporal data are investigated and bench-marked against some existing database technologies. Conclusions The computational experiments reveal the superiority of the proposed approach comparing to the existing methodologies, the required time to store (insert) data in the database is reduced by 6 times while the time needed to query data is decreased by at least 10 times. The efficiency of the proposed methodology is validate further by applying it to query the vehicle trajectories gathering the real-time air quality data.

Reference-Based Framework for Spatio-Temporal Trajectory Compression and Query Processing

The pervasiveness of GPS-enabled devices and wireless communication technologies results in massive trajectory data, incurring expensive cost for storage, transmission, and query processing. To relieve this problem, in this paper we propose a novel framework for compressing trajectory data, REST ( Re ference-based S patio- t emporal trajectory compression), by which a raw trajectory is represented by concatenation of a series of historical (sub-)trajectories (called reference trajectories) that form the compressed trajectory within a given spatio-temporal deviation threshold. In order to construct a reference trajectory set that can most benefit the subsequent compression, we propose three kinds of techniques to select reference trajectories wisely from a large dataset such that the resulting reference set is more compact yet covering most footprints of trajectories in the area of interest. To address the computational issue caused by the large number of combinations of reference trajectories that may exist for resembling a given trajectory, we propose efficient greedy algorithms that run in the blink of an eye and dynamic programming algorithms that can achieve the optimal compression ratio. Compared to existing work on trajectory compression, our framework has few assumptions about data such as moving within a road network or moving with constant direction and speed, and better compression performance with fairly small spatio-temporal loss. In addition, by indexing the reference trajectories directly with an in-memory R-tree and building connections to the raw trajectories with inverted index, we develop an extremely efficient algorithm that can answer spatio-temporal range queries over trajectories in their compressed form. Extensive experiments on a real taxi trajectory dataset demonstrate the superiority of our framework over existing representative approaches in terms of both compression ratio and efficiency.

GeoSOT-Based Spatiotemporal Index of Massive Trajectory Data

With the rapid development of global positioning technologies and the pervasiveness of intelligent mobile terminals, trajectory data have shown a sharp growth trend both in terms of data volume and coverage. In recent years, increasing numbers of LBS (location based service) applications have provided us with trajectory data services such as traffic flow statistics and user behavior pattern analyses. However, the storage and query efficiency of massive trajectory data are increasingly creating a bottleneck for these applications, especially for large-scale spatiotemporal query scenarios. To solve this problem, we propose a new spatiotemporal indexing method to improve the query efficiency of massive trajectory data. First, the method extends the GeoSOT spatial partitioning scheme to the time dimension and forms a global space–time subdivision scheme. Second, a novel multilevel spatiotemporal grid index, called the GeoSOT ST-index, was constructed to organize trajectory data hierarchically. Finally, a spatiotemporal range query processing method is proposed based on the index. We implement and evaluate the index in MongoDB. By comparing the range query efficiency and scalability of our index with those of the other two space–time composite indexes, we found that our approach improves query efficiency levels by approximately 40% and has better scalability under different data volumes.

Packed-Memory Quadtree: A cache-oblivious data structure for visual exploration of streaming spatiotemporal big data

The visual analysis of large multidimensional spatiotemporal datasets poses challenging questions regarding storage requirements and query performance. Several data structures have recently been proposed to address these problems that rely on indexes that pre-compute different aggregations from a known-a-priori dataset. Consider now the problem of handling streaming datasets, in which data arrive as one or more continuous data streams. Such datasets introduce challenges to the data structure, which now has to support dynamic updates (insertions/deletions) and rebalancing operations to perform self-reorganizations. In this work, we present the Packed-Memory Quadtree (PMQ), a novel data structure designed to support visual exploration of streaming spatiotemporal datasets. PMQ is cache-oblivious to perform well under different cache configurations. We store streaming data in an internal index that keeps a spatiotemporal ordering over the data following a quadtree representation, with support for real-time insertions and deletions. We validate our data structure under different dynamic scenarios and compare to competing strategies. We demonstrate how PMQ could be used to answer different types of visual spatiotemporal range queries of streaming datasets.

Towards fusing uncertain location data from heterogeneous sources

Properly incorporating location-uncertainties --- which is, fully considering their impact when processing queries of interest --- is a paramount in any application dealing with spatio-temporal data. Typically, the location-uncertainty is a consequence of the fact that objects cannot be tracked continuously and the inherent imprecision of localization devices. Although there is a large body of works tackling various aspects of efficient management of uncertainty in spatio-temporal data --- the settings consider homogeneous localization devices, e.g., either a Global Positioning System (GPS), or different sensors (roadside, indoor, etc.).In this work, we take a first step towards combining the uncertain location data --- i.e., fusing the uncertainty of moving objects location --- obtained from both GPS devices and roadside sensors. We develop a formal model for capturing the whereabouts in time in this setting and propose the Fused Bead (FB) model, extending the bead model based solely on GPS locations. We also present algorithms for answering traditional spatio-temporal range queries, as well as a special variant pertaining to objects locations with respect to lanes on road segments --- augmenting the conventional graph based road network with the width attribute. In addition, pruning techniques are proposed in order to expedite the query processing. We evaluated the benefits of the proposed approach on both real (Beijing taxi) and synthetic (generated from a customized trajectory generator) data. Our experiments demonstrate that the proposed method of fusing the uncertainties may eliminate up to 26 % of the false positives in the Beijing taxi data, and up to 40 % of the false positives in the larger synthetic dataset, when compared to using the traditional bead uncertainty models.

Uncertainty for Privacy and 2-Dimensional Range Query Distortion

In this work, we study the problem of privacy-preservation data publishing in moving objects databases. In particular, the trajectory of a mobile user in a plane is no longer a polyline in a two-dimensional space, instead it is a two-dimensional surface of fixed width <TEX>$2A_{min}$</TEX>, where <TEX>$A_{min}$</TEX> defines the semi-diameter of the minimum spatial circular extent that must replace the real location of the mobile user on the XY-plane, in the anonymized (kNN) request. The desired anonymity is not achieved and the entire system becomes vulnerable to attackers, since a malicious attacker can observe that during the time, many of the neighbors' ids change, except for a small number of users. Thus, we reinforce the privacy model by clustering the mobile users according to their motion patterns in (u, <TEX>${\theta}$</TEX>) plane, where u and <TEX>${\theta}$</TEX> define the velocity measure and the motion direction (angle) respectively. In this case, the anonymized (kNN) request looks up neighbors, who belong to the same cluster with the mobile requester in (u, <TEX>${\theta}$</TEX>) space: Thus, we know that the trajectory of the k-anonymous mobile user is within this surface, but we do not know exactly where. We transform the surface's boundary poly-lines to dual points and we focus on the information distortion introduced by this space translation. We develop a set of efficient spatiotemporal access methods and we experimentally measure the impact of information distortion by comparing the performance results of the same spatiotemporal range queries executed on the original database and on the anonymized one.

Indexing in-network trajectory flows

Indexing moving objects (MO) is a hot topic in the field of moving objects databases since many years. An impressive number of access methods have been proposed to optimize the processing of MO-related queries. Several methods have focused on spatio-temporal range queries, which represent the foundation of MO trajectory queries. Surprisingly, only a few of them consider that the objects movements are constrained. This is an important aspect for several reasons ranging from better capturing the relationship between the trajectory and the network space to more accurate trajectory representation with lower storage requirements. In this paper, we propose T-PARINET, an access method to efficiently retrieve the trajectories of objects moving in networks. T-PARINET is designed for continuous indexing of trajectory data flows. The cornerstone of T-PARINET is PARINET, an efficient index for historical trajectory data. The structure of PARINET is based on a combination of graph partitioning and a set of composite B+-tree local indexes. Because the network can be modeled using graphs, the partitioning of the trajectory data makes use of graph partitioning theory and can be tuned for a given query load and a given data distribution in the network space. The tuning process is built on a good quality cost model that is supplied with PARINET. The advantage of having a cost model is twofold; it allows a better integration of the index into the query optimizer of any DBMS, and it permits tuning the index structure for better performance. The tuning process can be performed before the index creation in the case of historical data or online in the case of indexing data flows. In fact, massive online updates can degrade the index quality, which can be measured by the cost model. We propose a specific maintenance process that results into T-PARINET. We study different types of queries and provide an optimized configuration for several scenarios. T-PARINET can easily be integrated into any RDBMS, which is an essential asset particularly for industrial or commercial applications. The experimental evaluation under an off-the-shelf DBMS shows that our method is robust. It also significantly outperforms the reference R-tree-based access methods for in-network trajectory databases.

Anonymization of moving objects databases by clustering and perturbation

Preserving individual privacy when publishing data is a problem that is receiving increasing attention. Thanks to its simplicity the concept of k-anonymity, introduced by Samarati and Sweeney [1], established itself as one fundamental principle for privacy preserving data publishing. According to the k-anonymity principle, each release of data must be such that each individual is indistinguishable from at least k−1 other individuals. In this article we tackle the problem of anonymization of moving objects databases. We propose a novel concept of k-anonymity based on co-localization, that exploits the inherent uncertainty of the moving object's whereabouts. Due to sampling and imprecision of the positioning systems (e.g., GPS), the trajectory of a moving object is no longer a polyline in a three-dimensional space, instead it is a cylindrical volume, where its radius δ represents the possible location imprecision: we know that the trajectory of the moving object is within this cylinder, but we do not know exactly where. If another object moves within the same cylinder they are indistinguishable from each other. This leads to the definition of ( k , δ ) -anonymity for moving objects databases. We first characterize the ( k , δ ) -anonymity problem, then we recall NWA ( N ever W alk A lone ), a method that we introduced in [2] based on clustering and spatial perturbation. Starting from a discussion on the limits of NWA we develop a novel clustering method that, being based on EDR distance [3], has the important feature of being time-tolerant. As a consequence it perturbs trajectories both in space and time. The novel method, named W 4 M ( W ait for M e ), is empirically shown to produce higher quality anonymization than NWA , at the price of higher computational requirements. Therefore, in order to make W 4 M scalable to large datasets, we introduce two variants based on a novel (and computationally cheaper) time-tolerant distance function, and on chunking. All the variants of W 4 M 1 1 Software freely available at: www-kdd.isti.cnr.it/W4M/. are empirically evaluated in terms of data quality and efficiency, and thoroughly compared to their predecessor NWA . 2 2 Software freely available at: www-kdd.isti.cnr.it/NWA/. Data quality is assessed both by means of objective measures of information distortion, and by more usability oriented measure, i.e., by comparing the results of (i) spatio-temporal range queries and (ii) frequent pattern mining, executed on the original database and on the ( k , δ ) -anonymized one. Experimental results over both real-world and synthetic mobility data confirm that, for a wide range of values of δ and k, the relative distortion introduced by our anonymization methods is kept low. Moreover, the techniques introduced to make W 4 M scalable to large datasets, achieve their goal without giving up data quality in the anonymization process.

A meta-index for querying distributed moving object database servers

Distributed moving object database servers offer a feasible solution to the scalability problems of centralized database systems. In these potentially large-scale systems, querying about the time-varying location of specific moving objects can be particularly expensive in terms of running time. This work proposes a meta-index based strategy that can significantly speed up the processing of these queries. The meta-index acts as an entry point for spatio-temporal queries and quickly drives the search process to the database servers that contain solutions. It also enables very fast approximated solutions to queries such as top- k and spatio-temporal range queries.

이동 객체 궤적의 색인을 위한 개선된 분할 알고리즘

최근 GPS, 이동 전화, 무선 네트워크 등의 발달로 인해 넓은 공간상에서 시간의 흐름에 따라 변화하는 이동 객체에 대한 위치 정보를 수집하여 실생활에 활용하는 다양한 위치 기반 서비스의 사용이 늘어나고 있다. 그와 함께 대용량의 이동 객체를 빠르게 검색하기 위한 효율적인 색인 방법의 필요성이 대두 됨에 따라 관련된 많은 연구가 현재 진행 중이다. 본 논문에서는 이동 객체의 궤적에 대한 색인 과정에서 필요한 개선된 궤적 분할 방법을 제안한다. 궤적의 적절한 분할 위치를 찾아 근사치 영역을 나타내는 최소 경계 사각형(MBR)을 만드는 과정에서 평균적인 질의의 크기를 고려하여 형성되는 확장된 최소 경계 사각형(EMBR)의 영역을 이용한다. 이에 따라 EMBR의 총면적이 최소에 가까운 분할을 만들어내어 색인 구성 후 질의 수행 과정 동안에 불필요한 탐색 공간을 감소시키는 이점을 보이게 된다. 본 논문에서 제안하는 궤적 분할방법의 우수성을 입증하기 위해 최적의 궤적 분할 방법과 기존의 궤적 분할 방법을 구현하여 각각의 EMBR 면적을 비교 분석한다. 비교 결과 제안하는 궤적 분할 방법이 기존의 방법보다 최적의 분할에 더 가까운 EMBR의 총면적을 나타내는 것을 알 수 있었다. Recently, use of various position base servicesthat collect position information for moving object and utilize in real life is increasing by the development of wireless network technology. Accordingly, new index structures are required to efficiently retrieve the consecutive positions of moving objects. This paper addresses an improved trajectory split algorithm for the purpose of efficiently supporting spatio-temporal range queries using index structures that use Minimum Bounding Rectangles(MBR) as trajectory approximations. We consider volume of Extended Minimum Bounding Rectangles (EMBR) to be determined by average size of range queries. Also, Use a priority queue to speed up our process. This algorithm gives in general sub-optimal solutions with respect to search space. Our improved trajectory split algorithm is going to derive minimizing volume of EMBRs better than previously proposed split algorithm.

Indexing of Spatio-Temporal Telemetric Data Based on Adaptive Multi-Dimensional Bucket Index

This paper describes the idea of a multi-dimensional bucket index designed for efficient indexing of telemetric readings. This data structure answers spatio-temporal range queries concerning utility usage within user selected region and time. In addition, it has a capability to adjust to incoming data and therefore is suitable to process data of highly dynamic nature. The paper also presents a stochastic prediction method to estimate utility usage in the near future.

Indexing Spatio-Temporal Trajectories with Efficient Polynomial Approximations

Complex queries on trajectory data are increasingly common in applications involving moving objects. MBR or grid-cell approximations on trajectories perform suboptimally since they do not capture the smoothness and lack of internal area of trajectories. We describe a parametric space indexing method for historical trajectory data, approximating a sequence of movement functions with single continuous polynomial. Our approach works well, yielding much finer approximation quality than MBRs. We present the PA-tree, a parametric index that uses this method, and show through extensive experiments that PA-trees have excellent performance for offline and online spatio-temporal range queries. Compared to MVR-trees, PA-trees are an order of magnitude faster to construct and incur I/O cost for spatio-temporal range queries lower by a factor of 2-4. SETI is faster than our method for index construction and timestamp queries, but incurs twice the I/O cost for time interval queries, which are much more expensive and are the bottleneck in online processing. Therefore, the PA-tree is an excellent choice for both offline and online processing of historical trajectories

Visual spatio-temporal function-based querying

Visual interfaces are very important for human interactions in cyberworlds. Visual spatio-temporal querying should be one of the basic tools for data mining and retrieval in cyberworlds. In this paper, we propose a novel function-based query model for arbitrary shape spatio-temporal querying. The queries are defined as geometric shapes changing over time. In our model, data are interpreted geometrically as multidimensional points with time dimension or as moving points. The queries are formulated with geometric objects and operations over them to form the query solid changing over time. The proposed query model allows us to pose arbitrary shape spatio-temporal range queries. With the uniform geometric model we integrate visual mining and querying of time-dependent data employing 3D visualization tools. It allows for creating an intuitive visual interface using 2D projections of 3D query shapes. Our approach combines visualization of spatio-temporal data with visualization of the range query formulation employing very compact function-based query model. The implemented visual query system and its visual interface are proposed and described. An example of application of the system in analysis of simulation results in molecular dynamics is considered.