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Trajectory outlier detection is one of the fundamental data mining techniques used to analyze the trajectory data of the Global Positioning System. A comprehensive literature review of trajectory outlier detectors published between 2000 and 2022 led to a conclusion that conventional trajectory outlier detectors suffered from drawbacks, either due to the detectors themselves or the pre-processing methods for the variable-length trajectory inputs utilized by detectors. To address these issues, we proposed a feature extraction method called middle polar coordinates (MiPo). MiPo extracted tabular features from trajectory data prior to the application of conventional outlier detectors to detect trajectory outliers. By representing variable-length trajectory data as fixed-length tabular data, MiPo granted tabular outlier detectors the ability to detect trajectory outliers, which was previously impossible. Experiments with real-world datasets showed that MiPo outperformed all baseline methods with 0.99 AUC on average; however, it only required approximately 10% of the computing time of the existing industrial best. MiPo exhibited linear time and space complexity. The features extracted by MiPo may aid other trajectory data mining tasks. We believe that MiPo has the potential to revolutionize the field of trajectory outlier detection.

A polar coordinate transformation is considered, which transforms the complex geometries into a unit disc. Some basic properties of the polar coordinate transformation are given. As applications, we consider the elliptic equation in two-dimensional complex geometries. The existence and uniqueness of the weak solution are proved, the Fourier–Legendre spectral-Galerkin scheme is constructed and the optimal convergence of numerical solutions under H 1 -norm is analyzed. The proposed method is very effective and easy to implement for problems in 2D complex geometries. Numerical results are presented to demonstrate the high accuracy of our spectral-Galerkin method.

Due to the influence of nonlinear radiation distortion and geometric deformation, achieving multimodal image matching remains a challenging task. To address these issues, this paper proposes a method called radiation invariant phase correlation (RIPC) to simultaneously estimate the rotation, scale, and displacement changes of multimodal image pairs. Firstly, based on the local structure characteristics of the image itself, we harness the nonlinear invariance of kernel canonical correlation analysis to devise the multimodal local self-correlation (MLSC) descriptor. This descriptor is resilient to nonlinear radiative differences, as well as local rotation and scale variations. Subsequently, we incorporate the log-polar coordinate transformation to capture the overall rotation and scale changes in the image, enabling independent representation of these factors on the Cartesian coordinate system. Finally, drawing upon the continuity of displacement estimation, as well as rotation and scale estimation, we construct a five-dimensional descriptor tailored for phase correlation. Extensive experiments conducted on five open-source datasets demonstrate that our proposed method surpasses state-of-the-art (SOTA) techniques in matching performance. Furthermore, our RIPC method achieves matching accuracy within 2-pixel threshold, which underscores its effectiveness in multimodal remote sensing image matching.

Physics‐informed neural networks (PINNs) are increasingly being used in various scientific disciplines. However, dealing with non‐stationary physical processes remains a significant challenge in such models, whereas fluid motions are typically non‐stationary. In this study, a PINN‐based method was designed and optimized to solve non‐stationary fluid dynamics with shallow water equations in a polar coordinate system (PINN‐SWEP). It was developed and validated with a classic circular basin case that is well‐documented in scientific literature. In the validation case, the wind‐induced water surface fluctuations are less than 1 cm, posing challenges in modeling. However, our PINN‐SWEP model can accurately simulate such tiny water surface fluctuations and resolve complex fluid motions based on limited and sparse data. A boundary discontinuity problem associated with the use of a polar coordinate system is further discussed and improved, thereby enhancing the applicability of PINN in water research. The methodology can provide an alternative solution for numerical or analytical solutions with high accuracy. Plain Language Summary Winds generate flows and waves over open water, with dimensions ranging from small ripples to large ocean waves. The wind‐induced fluid motions can be described using governing equations. These equations are usually difficult to solve mathematically. Therefore, we propose a machine learning (ML) model that combines the governing equations and sparse data to reproduce the wind‐induced fluid motions. This data‐physics hybrid model is validated by simulating classic wind‐induced fluid motions in a circular basin. Our method shows good performance, indicating a promising new approach that can coexist with conventional models. Additionally, machine learning models are prone to encountering a boundary discontinuity issue when solving problems defined in polar coordinate systems. This issue has been solved ingeniously, expanding the applicability of ML methods in geophysical and water research. Key Points A data‐physics hybrid machine learning model is developed to solve shallow water equations in a polar coordinate system The validation using a classic circular basin case demonstrates the capability of our model to solve non‐stationary hydrodynamics A boundary discontinuity problem due to the polar coordinate system is discussed and improved

Density peaks clustering (DPC) algorithm provides an efficient method to quickly find cluster centers with decision graphs. In recent years, due to its unique parameters, no iteration, and good robustness, it has been widely studied and applied. However, it also has some shortcomings, such as no adaptability, inadaptability to high-dimensional data and accuracy is easily affected. For reducing the higher time complexity of DPC, we introduce the polar coordinates to DPC (PC-DPC). Firstly, obtain the distance from every point to third-party point and the cosine value of the angle formed with the third-party vector, and reorder the points by distances and cosine values. Then, select other points in the adjacent sequence number of each point to calculate distances, and build a sparse distance matrix. Finally, the sparse distance matrix is used as the input of DPC to obtain clustering results. Theoretical analysis and experiments show that, compared with DPC and other algorithms, PC-DPC greatly reduces running time of DPC while maintaining clustering precision.

The article discusses the improvement of methods for discrete local optimization of highway tracing in the context of planning urban transport networks. Tracing and building the optimal configuration of the highway network is one of the key tasks in city planning. The main goal is to determine the shortest route for moving vehicles. Delivery of people to their destination and goods to consumers in a short time. The discretely local optimization of the network for three given points is considered. Network tracing for these points is achieved by building a polar Steiner network. Along with the orthogonal and Euclidean distance, as the research results have shown, the polar distance is important in practice. To introduce the polar distance, let us consider a certain plane with a fixed polar coordinate system. For some practical reasons, on a plane with a polar coordinate system, we leave only two directions of movement free. Movements are allowed along concentric circles drawn from the center coinciding with the pole, and along rays emanating from the pole. A bundle of straight lines with a support at the pole and the set of all concentric circles drawn from the center form an orthogonal polar grid. To solve this problem, various variants of geometric network models with a polar metric for three points are systematized and generalized, taking into account the weight of the specified points. The synthesis of an optimal highway route connecting specified points is a necessary component of optimizing the city's transport networks. To solve the problem, a network configuration with a polar metric consisting of radial segments and circular arcs is considered. The total length of the segment of arcs and circles should be minimal. The optimal network configuration is achieved by adding an additional Steiner point. The network constructed in the polar coordinate system will be called the \"Steiner Polar Network\". Geometric models of local optimization are an effective and visual means of developing various network tracing options within functional zones. From several network tracing options, a network is selected that meets the pre-defined planning requirements. It allows you to analyze and make the right decision in determining the promising directions for the development of the city's transport network.

An interesting deformation of Jackiw–Teitelboim (JT) gravity has been proposed by Witten by adding a potential term U(ϕ) as a self-coupling of the scalar dilaton field. During calculating the path integral over fields, a constraint comes from integration over ϕ as R(x)+2=2αδ(x→-x→′). The resulting Euclidean metric suffered from a conical singularity at x→=x→′. A possible geometry is modeled locally in polar coordinates (r,φ) by ds2=dr2+r2dφ2,φ≅φ+2π-α. In this letter we show that there exists another family of ”exact” geometries for arbitrary values of the α. A pair of exact solutions are found for the case of α=0. One represents the static patch of the AdS and the other one is the non-static patch of the AdS metric. These solutions were used to construct the Green function for the inhomogeneous model with α≠0. We address a type of phase transition between different patches of the AdS in theory because of the discontinuity in the first derivative of the metric at x=x′. We extended the study to the exact space of metrics satisfying the constraint R(x)+2=2∑i=1kαiδ(2)(x-xi′) as a modulus diffeomorphisms for an arbitrary set of deficit parameters (α1,α2,…,αk). The space is the moduli space of Riemann surfaces of genus g with k conical singularities located at xk′, denoted by Mg,k.

This paper reports machine learning models for predicting metal fatigue life under uncertainty by extracting stress–strain data from hysteresis loops. First, the hysteresis loops of Q235B under strain-controlled constant amplitude loading are analyzed. The values of stress and strain in six key points are extracted from each hysteresis loop at the earliest stages of the fatigue process, and transformed into polar coordinates. Second, the uncertainty is quantified by extending the applied strain amplitude and the selected stress–strain values to intervals. A great deal of data are generated randomly in each interval for coping with the challenge of a small fatigue test dataset. Third, three machine learning models are constructed, where the parameters of the back-propagation neural network model are optimized by using the leave-one-out cross-validation technique, and the models of support vector regression and random forest are selected carefully. The point and interval predictions of the low-cycle-fatigue life of Q235B are reported to reveal the feasibility and advantage of the proposed models. The results help to identify how to understand the fatigue behavior of materials by combining machine learning models and stress–strain hysteresis loops.

In this paper, a curve approximation approach using bio-inspired polar coordinate bald eagle search algorithm (PBES) is proposed. PBES algorithm is inspired by the spiral mechanism of bald eagle during predation. By introducing polar coordinate, the spiral predation process of the bald eagle will become more intuitive, which is more conducive for the algorithm to polar coordinate optimization problems. The initialization stage of PBES algorithm is modified to make the distribution of initialized individuals more uniform and some parameters are introduced to strengthen the exploration and exploitation capabilities of algorithm. The performance of the PBES algorithm is tested in three aspects: polar coordinate transcendental equation, curve approximation and robotic manipulator. The experimental results show that the PBES algorithm is superior to the well-known metaheuristic algorithms as it is effectively applicable for curve approximation problem.

This paper proposes a modified optimization algorithm called polar coordinate salp swarm algorithm (PSSA). The main inspiration of PSSA is the aggregation chain and foraging trajectory of salp is spiral. Some curves are extremely complex when represented in Cartesian coordinate system, but if they are expressed in polar coordinates, it becomes very simple and easy to handle, and polar coordinates are widely used in scientific computing and engineering issues. It will be more intuitive and convenient if use polar coordinates to define the foraging and aggregation process of salps. At the same time, different from other algorithms proposed in the past, the PSSA directly initialize individuals in polar space instead of using mapping functions to convert to polar coordinates, change the position of particles by updating polar angles and polar diameters. This algorithm is tested on two complex polar coordinate equations, several curve approximation problems and engineering design problems using PSSA. The experimental results illustrated that the proposed PSSA algorithm is superior to the state-of-the-art metaheuristic algorithms in terms of the performance measures.