The explosion in the availability of GPS-enabled devices has resulted in an abundance of trajectory data. In reality, however, majority of these trajectories are collected at a low sampling rate and only provide partial observations on their actually traversed routes. Consequently, they are mired with uncertainty. In this paper, we develop a technique called Infer Tra to infer uncertain trajectories from network-constrained partial observations. Rather than predicting the most likely route, the inferred uncertain trajectory takes the form of an edge-weighted graph and summarizes all probable routes in a holistic manner. For trajectory inference, Infer Tra employs Gibbs sampling by learning a Network Mobility Model (NMM) from a database of historical trajectories. Extensive experiments on real trajectory databases show that the graph-based approach of Infer Tra is up to 50 more accurate, 20 times faster, and immensely more versatile than state-of-the-art techniques.
More recently, a technique @cite_6 was designed to study the trajectory inference problem in a setting where trajectories are not constrained by a network. Due to the focus on network-free trajectories, @cite_6 is not applicable to our problem.
@cite_6
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In order to track moving objects in long range against occlusion, interruption, and background clutter, this paper proposes a unified approach for global trajectory analysis. Instead of the traditional frame-by-frame tracking, our method recovers target trajectories based on a short sequence of video frames, e.g., 15 frames. We initially calculate a foreground map at each frame obtained from a state-of-the-art background model. An attribute graph is then extracted from the foreground map, where the graph vertices are image primitives represented by the composite features. With this graph representation, we pose trajectory analysis as a joint task of spatial graph partitioning and temporal graph matching. The task can be formulated by maximizing a posteriori under the Bayesian framework, in which we integrate the spatio-temporal contexts and the appearance models. The probabilistic inference is achieved by a data-driven Markov chain Monte Carlo algorithm. Given a period of observed frames, the algorithm simulates an ergodic and aperiodic Markov chain, and it visits a sequence of solution states in the joint space of spatial graph partitioning and temporal graph matching. In the experiments, our method is tested on several challenging videos from the public datasets of visual surveillance, and it outperforms the state-of-the-art methods. This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations. We present an interpretable framework for path prediction that learns scene-specific causations behind agents' behaviors. We exploit two sources of information: the past motion trajectory of the agent of interest and a wide top-down view of the scene. We propose a Clairvoyant Attentive Recurrent Network (CAR-Net) that learns "where to look" in the large image when solving the path prediction task. While previous works on trajectory prediction are constrained to either use semantic information or hand-crafted regions centered around the agent, our method has the capacity to select any region within the image, e.g., a far-away curve when predicting the change of speed of vehicles. To study our goal towards learning observable causality behind agents' behaviors, we have built a new dataset made of top view images of hundreds of scenes (e.g., F1 racing circuits) where the vehicles are governed by known specific regions within the images (e.g., upcoming curves). Our algorithm successfully selects these regions, learns navigation patterns that generalize to unseen maps, outperforms previous works in terms of prediction accuracy on publicly available datasets, and provides human-interpretable static scene-specific dependencies. The advances in location-acquisition technologies have led to a myriad of spatial trajectories. These trajectories are usually generated at a low or an irregular frequency due to applications' characteristics or energy saving, leaving the routes between two consecutive points of a single trajectory uncertain (called an uncertain trajectory). In this paper, we present a Route Inference framework based on Collective Knowledge (abbreviated as RICK) to construct the popular routes from uncertain trajectories. Explicitly, given a location sequence and a time span, the RICK is able to construct the top-k routes which sequentially pass through the locations within the specified time span, by aggregating such uncertain trajectories in a mutual reinforcement way (i.e., uncertain + uncertain → certain). Our work can benefit trip planning, traffic management, and animal movement studies. The RICK comprises two components: routable graph construction and route inference. First, we explore the spatial and temporal characteristics of uncertain trajectories and construct a routable graph by collaborative learning among the uncertain trajectories. Second, in light of the routable graph, we propose a routing algorithm to construct the top-k routes according to a user-specified query. We have conducted extensive experiments on two real datasets, consisting of Foursquare check-in datasets and taxi trajectories. The results show that RICK is both effective and efficient. We present an interpretable framework for path prediction that leverages dependencies between agents’ behaviors and their spatial navigation environment. We exploit two sources of information: the past motion trajectory of the agent of interest and a wide top-view image of the navigation scene. We propose a Clairvoyant Attentive Recurrent Network (CAR-Net) that learns where to look in a large image of the scene when solving the path prediction task. Our method can attend to any area, or combination of areas, within the raw image (e.g., road intersections) when predicting the trajectory of the agent. This allows us to visualize fine-grained semantic elements of navigation scenes that influence the prediction of trajectories. To study the impact of space on agents’ trajectories, we build a new dataset made of top-view images of hundreds of scenes (Formula One racing tracks) where agents’ behaviors are heavily influenced by known areas in the images (e.g., upcoming turns). CAR-Net successfully attends to these salient regions. Additionally, CAR-Net reaches state-of-the-art accuracy on the standard trajectory forecasting benchmark, Stanford Drone Dataset (SDD). Finally, we show CAR-Net’s ability to generalize to unseen scenes. The main objective of this paper is to develop an efficient method for learning and reproduction of complex trajectories for robot programming by demonstration. Encoding of the demonstrated trajectories is performed with hidden Markov model, and generation of a generalized trajectory is achieved by using the concept of key points. Identification of the key points is based on significant changes in position and velocity in the demonstrated trajectories. The resulting sequences of trajectory key points are temporally aligned using the multidimensional dynamic time warping algorithm, and a generalized trajectory is obtained by smoothing spline interpolation of the clustered key points. The principal advantage of our proposed approach is utilization of the trajectory key points from all demonstrations for generation of a generalized trajectory. In addition, variability of the key points' clusters across the demonstrated set is employed for assigning weighting coefficients, resulting in a generalization procedure which accounts for the relevance of reproduction of different parts of the trajectories. The approach is verified experimentally for trajectories with two different levels of complexity. We propose a method for global multi-target tracking that can incorporate higher-order track smoothness constraints such as constant velocity. Our problem formulation readily lends itself to path estimation in a trellis graph, but unlike previous methods, each node in our network represents a candidate pair of matching observations between consecutive frames. Extra constraints on binary flow variables in the graph result in a problem that can no longer be solved by min-cost network flow. We therefore propose an iterative solution method that relaxes these extra constraints using Lagrangian relaxation, resulting in a series of problems that ARE solvable by min-cost flow, and that progressively improve towards a high-quality solution to our original optimization problem. We present experimental results showing that our method outperforms the standard network-flow formulation as well as other recent algorithms that attempt to incorporate higher-order smoothness constraints. A new method is proposed for exploiting causal independencies in exact Bayesian network inference. A Bayesian network can be viewed as representing a factorization of a joint probability into the multiplication of a set of conditional probabilities. We present a notion of causal independence that enables one to further factorize the conditional probabilities into a combination of even smaller factors and consequently obtain a finer-grain factorization of the joint probability. The new formulation of causal independence lets us specify the conditional probability of a variable given its parents in terms of an associative and commutative operator, such as "or", "sum" or "max", on the contribution of each parent. We start with a simple algorithm VE for Bayesian network inference that, given evidence and a query variable, uses the factorization to find the posterior distribution of the query. We show how this algorithm can be extended to exploit causal independence. Empirical studies, based on the CPCS networks for medical diagnosis, show that this method is more efficient than previous methods and allows for inference in larger networks than previous algorithms. A decision philosophy that seeks the avoidance of error by trading off belief of truth and value of information is applied to the problem of recognizing tracks from multiple targets (MTT). A successful MTT methodology should be robust in that its performance degrades gracefully as the conditions of the collection become less favorable to optimal operation. By stressing the avoidance, rather than the explicit minimization, of error, the authors obtain a decision rule for trajectory-data association that does not require the resolution of all conflicting hypotheses when the database does not contain sufficient information to do so reliably. This rule, coupled with a set-valued Kalman filter for trajectory estimation, results in a methodology that does not attempt to extract more information from the database than it contains. > It is difficult to extract meaningful patterns from massive trajectory data. One of the main challenges is to characterise, compare and generalise trajectories to find overall patterns and trends. The major limitation of existing methods is that they do not consider topological relations among trajectories. This research proposes a graph-based approach that converts trajectory data to a graph-based representation and treats them as a complex network. Within the context of vehicle movements, the research develops a sequence of steps to extract representative points to reduce data redundancy, interpolate trajectories to accurately establish topological relationships among trajectories and locations, construct a graph (or matrix) representation of trajectories, apply a spatially constrained graph partitioning method to discover natural regions defined by trajectories and use the discovered regions to search and visualise trajectory clusters. Applications with a real data set shows that our new approach can effectively facilitate the understanding of spatial and spatiotemporal patterns in trajectories and discover novel patterns that existing methods cannot find. Spatio-temporal and geo-referenced datasets are growing rapidly, with the rapid development of some technology, such as GPS, satellite systems. At present, many scholars are very interested in the clustering of the trajectory. Existing trajectory clustering algorithms group similar trajectories as a whole and can't distinguish the direction of trajectory. Our key finding is that clustering trajectories as a whole could miss common sub-trajectories and trajectory has direction information. In many applications, discovering common sub-trajectories is very useful. In this paper, we present a trajectory clustering algorithm CTHD (clustering of trajectory based on hausdorff distance). In the CTHD, the trajectory is firstly described by a sequence of flow vectors and partitioned into a set of sub-trajectory. Next the similarity between trajectories is measured by their respective Hausdorff distances. Finally, the trajectories are clustered by the DBSCAN clustering algorithm. The proposed algorithm is different from other schemes using Hausdorff distance that the flow vectors include the position and direction. So it can distinguish the trajectories in different directions. The experimental result shows the phenomenon. In traffic research, management, and planning a number of path-based analyses are heavily used, e.g., for computing turn-times, evaluating green waves, or studying traffic flow. These analyses require retrieving the trajectories that follow the full path being analyzed. Existing path queries cannot sufficiently support such path-based analyses because they retrieve all trajectories that touch any edge in the path. In this paper, we define and formalize the strict path query. This is a novel query type tailored to support path-based analysis, where trajectories must follow all edges in the path. To efficiently support strict path queries, we present a novel NET work-constrained TRAjectory index (NETTRA). This index enables very efficient retrieval of trajectories that follow a specific path, i.e., strict path queries. NETTRA uses a new path encoding scheme that can determine if a trajectory follows a specific path by only retrieving data from the first and last edge in the path. To correctly answer strict path queries existing network-constrained trajectory indexes must retrieve data from all edges in the path. An extensive performance study of NETTRA using a very large real-world trajectory data set, consisting of 1.7 million trajectories (941 million GPS records) and a road network with 1.3 million edges, shows a speed-up of two orders of magnitude compared to state-of-the-art trajectory indexes. The MAP (maximum a posteriori hypothesis) problem in Bayesian networks is to find the most likely states of a set of variabls given partial evidence on the complement of that set. Standard structure-based inference methods for finding exact solutions to MAP, such as variable elimination and join-tree algorithms, have complexities that are exponential in the constrained treewidth of the network. A more recent algorithm, proposed by Park and Darwiche, is exponential only in the treewidth and has been shown to handle networks whose constrained treewidth is quite high. In this paper we present a new algorithm for exact MAP that is not necessarily limited in scalability even by the treewidth. This is achieved by leveraging recent advances in compilation of Bayesian networks into arithmetic circuits, which can circumvent treewidth-imposed limits by exploiting the local structure present in the network. Specifically, we implement a branch-and-bound search where the bounds are computed using linear-time operations on the compiled arithmetic circuit. On networks with local structure, we observe orders-of-magnitude improvements over the algorithm of Park and Darwiche. In particular, we are able to efficiently solve many problems where the latter algorithm runs out of memory because of high treewidth. We introduce a new problem of continuous, coverage-aware trajectory optimization under localization and sensing uncertainty. In this problem, the goal is to plan a path from a start state to a goal state that maximizes the coverage of a user-specified region while minimizing the control costs of the robot and the probability of collision with the environment. We present a principled method for quantifying the coverage sensing uncertainty of the robot. We use this sensing uncertainty along with the uncertainty in robot localization to develop C-OPT, a coverage-optimization algorithm which optimizes trajectories over belief-space to find locally optimal coverage paths. We highlight the applicability of our approach in multiple simulated scenarios inspired by surveillance, UAV crop analysis, and search-and-rescue tasks. We also present a case study on a physical, differential-drive robot. We also provide quantitative and qualitative analysis of the paths generated by our approach. Motivated by the increasing availability of vehicle trajectory data, we propose learn-to-route, a comprehensive trajectory-based routing solution. Specifically, we first construct a graph-like structure from trajectories as the routing infrastructure. Second, we enable trajectory-based routing given an arbitrary (source, destination) pair. In the first step, given a road network and a collection of trajectories, we propose a trajectory-based clustering method that identifies regions in a road network. If a pair of regions are connected by trajectories, we maintain the paths used by these trajectories and learn a routing preference for travel between the regions. As trajectories are skewed and sparse, and although the introduction of regions serves to consolidate the sparse data, many region pairs are not connected by trajectories. We thus transfer routing preferences from region pairs with sufficient trajectories to such region pairs and then use the transferred preferences to identify paths between the regions. In the second step, we exploit the above graph-like structure to achieve a comprehensive trajectory-based routing solution. Empirical studies with two substantial trajectory data sets offer insight into the proposed solution, indicating that it is practical. A comparison with a leading routing service offers evidence that the paper's proposal is able to enhance routing quality. We consider the problem of constructing street networks from geo-referenced trajectory data: Given a set of trajectories in the plane, compute a street network that represents all trajectories in the set. We present a simple and practical incremental algorithm that is based on partial matching of the trajectories to the graph. We use minimumlink paths to reduce the complexity of the reconstructed graph. We provide quality guarantees and experimental results based on both real and synthetic data. For the partial matching we introduce a new variant of partial Frechet distance. The increasing availability of GPS-embedded mobile devices has given rise to a new spectrum of location-based services, which have accumulated a huge collection of location trajectories. In practice, a large portion of these trajectories are of low-sampling-rate. For instance, the time interval between consecutive GPS points of some trajectories can be several minutes or even hours. With such a low sampling rate, most details of their movement are lost, which makes them difficult to process effectively. In this work, we investigate how to reduce the uncertainty in such kind of trajectories. Specifically, given a low-sampling-rate trajectory, we aim to infer its possible routes. The methodology adopted in our work is to take full advantage of the rich information extracted from the historical trajectories. We propose a systematic solution, History based Route Inference System (HRIS), which covers a series of novel algorithms that can derive the travel pattern from historical data and incorporate it into the route inference process. To validate the effectiveness of the system, we apply our solution to the map-matching problem which is an important application scenario of this work, and conduct extensive experiments on a real taxi trajectory dataset. The experiment results demonstrate that HRIS can achieve higher accuracy than the existing map-matching algorithms for low-sampling-rate trajectories. Trajectory planning and optimization is a fundamental problem in articulated robotics. Algorithms used typically for this problem compute optimal trajectories from scratch in a new situation. In effect, extensive data is accumulated containing situations together with the respective optimized trajectories--but this data is in practice hardly exploited. This article describes a novel method to learn from such data and speed up motion generation, a method we denote tajectory pediction. The main idea is to use demonstrated optimal motions to quickly predict appropriate trajectories for novel situations. These can be used to initialize and thereby drastically speed-up subsequent optimization of robotic movements. Our approach has two essential ingredients. First, to generalize from previous situations to new ones we need a situation descriptor--we construct features for such descriptors and use a sparse regularized feature selection approach to improve generalization. Second, the transfer of previously optimized trajectories to a new situation should not be made in joint angle space--we propose a more efficient task space transfer. We present extensive results in simulation to illustrate the benefits of the new method, and demonstrate it also with real robot hardware. Our experiments in diverse tasks show that we can predict good motion trajectories in new situations for which the refinement is much faster than an optimization from scratch. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. The increasing pervasiveness of location-acquisition technologies (GPS, GSM networks, etc.) is leading to the collection of large spatio-temporal datasets and to the opportunity of discovering usable knowledge about movement behaviour, which fosters novel applications and services. In this paper, we move towards this direction and develop an extension of the sequential pattern mining paradigm that analyzes the trajectories of moving objects. We introduce trajectory patterns as concise descriptions of frequent behaviours, in terms of both space (i.e., the regions of space visited during movements) and time (i.e., the duration of movements). In this setting, we provide a general formal statement of the novel mining problem and then study several different instantiations of different complexity. The various approaches are then empirically evaluated over real data and synthetic benchmarks, comparing their strengths and weaknesses.