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A novel anomaly detection procedure based on the Ornstein–Uhlenbeck (OU) mean-reverting stochastic process is presented. The considered anomaly is a vessel that deviates from a planned route, changing its nominal velocity $\boldsymbol{v}_0$ . In order to hide this behavior, the vessel switches off its automatic identification system (AIS) device for a time $T$ and then tries to revert to the previous nominal velocity $\boldsymbol{v}_0$ . The decision that has to be made is declaring that a deviation either happened or not, relying only upon two consecutive AIS contacts. Furthermore, the extension to the scenario in which multiple contacts (e.g., radar) are available during the time period $T$ is also considered. A proper statistical hypothesis testing procedure that builds on the changes in the OU process long-term velocity parameter $\boldsymbol{v}_0$ of the vessel is the core of the proposed approach and enables the solution of the anomaly detection problem. Closed analytical forms are provided for the detection and false alarm probabilities of the hypothesis test.

If two radar sensors observe the same target their measurements can be combined to produce a fused target-state estimate that is of higher quality than that from one radar alone. If there are multiple targets whose information is shared, a necessary first step to fusion is to “assign” each measurement from the first sensor to that at the other in such a way that both refer to the same underlying object, a task generally accomplished by minimizing a global cost involving distance. An assignment error occurs when the measurement originated by target i at the first radar is wrongly associated to a measurement originated by target $j$ (not i) at the second radar. Naturally, when such an error occurs the result is fusion of information describing disparate objects, resulting in degraded estimation performance and poor self-assessment in terms of posterior uncertainty. Here we address the issue, and derive approximate assignment error probability. Remarkably, performance depends only upon the parameters combined to a single scalar constant.

In this paper, a sequential Bayesian framework is proposed to address the task of joint anomaly detection and tracking for surveillance applications in the presence of clutter. This is achieved by modeling the anomaly as a switching unknown control input which goes into action by modifying the expected dynamics of a target and ceases its activity (becomes nonexistent) under nominal behavior. Random Finite Sets (RFS) make it possible to represent the switching nature of the object anomalous behavior and derive a hybrid Bernoulli filter (HBF) that sequentially updates the joint posterior density of a Bernoulli RFS for the unknown velocity input and the object kinematic state. In addition, the proposed HBF has been customized for maritime anomaly detection by using a piecewise Ornstein-Uhlenbeck (OU) stochastic process as dynamic model of vessels. We illustrate the effectiveness of the proposed filter, implemented in Gaussian-mixture form, and compare its performance in a maritime surveillance example with the Interacting Multiple Model Probabilistic Data Association Filter (IMM - PDAF) for different levels of clutter.

In this letter, we study the problem of estimating the long-run mean of the Ornstein–Uhlenbeck (OU) stochastic process and its effect on the long-term prediction of future vessel states, which is a crucial problem for Maritime Situational Awareness (MSA). We employ a sample mean estimator (SME) to estimate the key OU parameter from the observations, computing the closed-form SME covariance error in both the random and constant sampling time regimes, providing a fundamental building block of the overall long-term state prediction covariance. We show also that the SME is: $\sqrt{n}$ -consistent when the sampling time is random; asymptotically efficient when the sampling time is constant; and very close to the Cramer–Rao lower bound in the cases of practical interest for MSA.

This paper presents an unsupervised approach to extract maritime Patterns of Life (PoL) from historical Automatic Identification System (AIS) data based on a low-dimensional synthetic representation of ship routes. Recent advances in long-term vessel motion modeling through Ornstein-Uhlenbeck mean-reverting stochastic processes allow to encode knowledge about maritime traffic via a compact graph-based model where waypoints are graph vertices and the connections between them, i.e., the navigational legs, are graph edges. The resulting directed graph ultimately leads to the detection and statistical characterization of recurrent maritime traffic patterns. To demonstrate its effectiveness and applicability to real-world case studies, the proposed methodology has been tested on two extensive AIS datasets, collected in the areas of two operational trials of EU-H2020’s MARISA (Maritime Integrated Surveillance Awareness) project.

We present a novel method for predicting long-term target states based on mean-reverting stochastic processes. We use the Ornstein-Uhlenbeck (OU) process, leading to a revised target state equation and to a time scaling law for the related uncertainty that in the long term is shown to be orders of magnitude lower than under the nearly constant velocity (NCV) assumption. In support of the proposed model, an analysis of a significant portion of real-world maritime traffic is provided.

We present a hypothesis test to associate ship track measurements to an edge of a given graph that statistically models common traffic routes in a given area of interest. The association algorithm is based on the hypothesis that ship velocities are modeled by mean-reverting stochastic processes. Prior knowledge about the traffic is provided by the graph in form of probability density functions of the mean-reverting kinematic parameters for each node and edge of the graph, that are exploited in the formalization of the association algorithm. Tests on real Automatic Identification System (AIS) data show a qualitatively good association performance. Future developments of this work include the development of specific quantitative metrics to assess the association performance.

Covariance matrix estimation is a crucial task in adaptive signal processing applied to several surveillance systems, including radar and sonar. In this paper we propose a dynamic environment learning strategy to track both the covariance matrix and its class; the class represents a set of structured covariance matrices. We assume that the posterior distribution of the covariance given the class, is basically a mixture of inverse Wishart, while the class posterior distribution evolves according to a Markov chain. The proposed multi-class inverse Wishart mixture filter is shown to outperform the class-clairvoyant maximum likelihood estimator in terms of covariance estimate accuracy, as well as the Bayesian information criterion rule in terms of classification performance.

Inspired by the fair regularity of the motion of ships, we present a method to derive a representation of the commercial maritime traffic in the form of a graph, whose nodes represent way-point areas, or regions of likely direction changes, and whose edges represent navigational legs with constant cruise velocity. The proposed method is based on the representation of a ship's velocity with an Ornstein-Uhlenbeck process and on the detection of changes of its long-run mean to identify navigational way-points. In order to assess the graph representativeness of the traffic, two performance metrics are introduced, leading to distinct graph construction criteria. Finally, the proposed method is validated against real-world Automatic Identification System data collected in a large area.

This paper presents a solution to the problem of sequential joint anomaly detection and tracking of a target subject to switching unknown path deviations. Based on a dynamic model described by Ornstein-Uhlenbeck (OU) stochastic processes, the anomaly is represented by a target (e.g., a marine vessel) that deviates from a preset route by changing its nominal mean velocity. The Random Finite Set (RFS) framework is used to represent the switching nature of target's anomalous behavior in the presence of spurious measurements and detection uncertainty. Combining these two ingredients, the problem of jointly detecting target's path deviations and estimating its kinematic state can be formulated within the Bayesian framework, and analytically solved by means of a hybrid Bernoulli filter that sequentially updates the joint posterior density of the unknown OU velocity input (a Bernoulli RFS) and of the target's state random vector. We illustrate the effectiveness of the proposed filter, implemented in Gaussian-mixture form, in a simulated scenario of vessel tracking for maritime traffic monitoring.

In this work, we consider the problem of algorithmically predicting rendezvous among vessels based on their trajectory forecasts in a maritime environment. The problem is treated as hypothesis testing on the expected value of the distance between trajectories. We relate this quantity to the first and second degree Wasserstein distances between trajectory forecast distributions. These distributions are obtained using integrated Ornstein-Uhlenbeck process models with the trajectory measurements collected so far. Building upon these results, we propose an algorithm which traverses the trajectories observed so far for detecting rendezvous over a rolling time horizon. We demonstrate the efficacy of the proposed algorlthm using simulations.

We propose an unsupervised procedure to automatically extract a graph-based model of commercial maritime traffic routes from historical Automatic Identification System (AIS) data. In the proposed representation, the main elements of maritime traffic patterns, such as maneuvering regions and sea-lanes, are represented, respectively, with graph vertices and edges. Vessel motion dynamics are defined by multiple Ornstein- Uhlenbeck (OU) processes with different long-run mean parameters, which in our approach can be estimated with a change detection procedure based on Page's test, aimed to reveal the spatial points representative of velocity changes. A density-based clustering algorithm (DBSCAN) is then applied to aggregate the detected changes into groups of similar elements and reject outliers. To validate the proposed graph-based representation of the maritime traffic, two performance criteria are tested against a real-world trajectory data set collected off the Iberian Coast and the English Channel. Results show the effectiveness of the proposed approach, which is suitable to be integrated at any level of a JDL system.