Noisy Trajectories Research Articles

Gesture2Text: A Generalizable Decoder for Word-Gesture Keyboards in XR Through Trajectory Coarse Discretization and Pre-Training.

Text entry with word-gesture keyboards (WGK) is emerging as a popular method and becoming a key interaction for Extended Reality (XR). However, the diversity of interaction modes, keyboard sizes, and visual feedback in these environments introduces divergent word-gesture trajectory data patterns, thus leading to complexity in decoding trajectories into text. Template-matching decoding methods, such as SHARK2 [32], are commonly used for these WGK systems because they are easy to implement and configure. However, these methods are susceptible to decoding inaccuracies for noisy trajectories. While conventional neural-network-based decoders (neural decoders) trained on word-gesture trajectory data have been proposed to improve accuracy, they have their own limitations: they require extensive data for training and deep-learning expertise for implementation. To address these challenges, we propose a novel solution that combines ease of implementation with high decoding accuracy: a generalizable neural decoder enabled by pre-training on large-scale coarsely discretized word-gesture trajectories. This approach produces a ready-to-use WGK decoder that is generalizable across mid-air and on-surface WGK systems in augmented reality (AR) and virtual reality (VR), which is evident by a robust average Top-4 accuracy of 90.4% on four diverse datasets. It significantly outperforms SHARK2 with a 37.2% enhancement and surpasses the conventional neural decoder by 7.4%. Moreover, the Pre-trained Neural Decoder's size is only 4 MB after quantization, without sacrificing accuracy, and it can operate in real-time, executing in just 97 milliseconds on Quest 3.

A New Trajectory Clustering Method for Mining Multiple Periodic Patterns from Complex Oceanic Trajectories

Oceanic trajectories frequently exhibit multiple periodic patterns across various time intervals, e.g., tidal variations, mesoscale eddies, and El Niño events correspond to diurnal, seasonal, and interannual fluctuations in environmental factors. To explore hidden spatiotemporal multiple periodic behaviors in noisy ocean data, we propose a novel trajectory clustering method, namely DTID-STFC. It first identifies dense time intervals (DTIs) in which trajectories occur frequently. Subsequently, within each DTI, it utilizes spectral embedding to project trajectories onto a latent subspace and proposes three-way fuzzy clustering to obtain results. We evaluate the proposed method on simulated datasets and compare it with traditional and state-of-the-art trajectory clustering approaches. Experimental results indicate that it outperforms other methods across all five metrics. Moreover, when applying the DTID-STFC method to the analysis of mesoscale cyclonic eddies in the South China Sea and vessel data, it demonstrates more discernible results than traditional methods, and it aligns well with physical oceanographic processes. This proposed method offers valuable insights into identifying periodic behaviors from complex and noisy spatiotemporal oceanic trajectory data.

Predictive deep learning models for analyzing discrete fractional dynamics from noisy and incomplete data

We study the accuracy of machine learning methods for inferring the parameters of noisy fractional Wu–Baleanu trajectories with some missing initial terms. Our model is based on a combination of convolutional and recurrent neural networks (LSTM), which permits the extraction of characteristics from trajectories while preserving time dependency. We show that these approach exhibit good accuracy results despite the poor quality of the data.

Social System Inference From Noisy Observations

This article studies social system inference from a single noisy trajectory of public evolving opinions, wherein observation noise leads to the statistical dependence of samples on time and coordinates. We first propose a cyber-social system that comprises individuals in a social network and a set of information sources in a cyber layer, whose opinion dynamics explicitly takes the asymmetric cognitive bias including confirmation bias and negativity bias and the process noise into account. Based on the proposed cyber-social model, we then study the sample complexity of least-square auto-regressive model estimation, which governs the length of a single observed trajectory that is sufficient for the identified model to achieve the prescribed levels of accuracy and confidence (PAC). Building on the identified social model, we then investigate social inference, with a particular focus on the weighted network topology and the model parameters of asymmetric cognitive bias. Finally, the theoretical results and the effectiveness of the proposed inference framework are validated by the U.S. Senate Member Ideology data.

Transformation of traditional art education model in multimedia environment

Abstract Along with the continuous development of multimedia technology and the deepening of teaching information, virtual reality design factors are crucial to enhancing the impact of the teaching experience. The purpose of this paper is to address traditional art education’s transformational needs and develop an art education teaching system that utilizes virtual reality technology. It also uses the eye-tracking instrument to collect student eye movement data, and after preprocessing the data, such as missing data and noisy trajectory points, it counts the results of each eye movement index in the teaching process. The study showed that some of the students’ eye movement indicators during teaching had a strong correlation with the level of cognitive engagement in learning (P<0.05), and there was no covariance between the indicators. Following the implementation of the teaching mode, the students teaching art under multimedia exhibited a significantly higher level of cognitive engagement in learning (9.74 points) compared to those in the traditional teaching mode (7.14 points). Meanwhile, the eye movement experiment found that the difference in students’ attention concentration in the artwork and work introduction sections was not significant (P > 0.05) when the design elements of the teaching interface were paired with the top of the bottom of the picture and the middle size and the top of the picture and the bottom of the picture and the large size. This paper provides references and their significance for related research, as well as the design of a virtual simulation platform for art teaching.

Extract latent features of single-particle trajectories with historical experience learning

Single-particle tracking has enabled real-time, in situ quantitative studies of complex systems. However, inferring dynamic state changes from noisy and undersampling trajectories encounters challenges. Here, we introduce a data-driven method for extracting features of subtrajectories with historical experience learning (Deep-SEES), where a single-particle tracking analysis pipeline based on a self-supervised architecture automatically searches for the latent space, allowing effective segmentation of the underlying states from noisy trajectories without prior knowledge on the particle dynamics. We validated our method on a variety of noisy simulated and experimental data. Our results showed that the method can faithfully capture both stable states and their dynamic switch. In highly random systems, our method outperformed commonly used unsupervised methods in inferring motion states, which is important for understanding nanoparticles interacting with living cell membranes, active enzymes, and liquid-liquid phase separation. Self-generating latent features of trajectories could potentially improve the understanding, estimation, and prediction of many complex systems.

Towards Trajectory Forecasting From Detection.

Trajectory forecasting for traffic participants (e.g., vehicles) is critical for autonomous platforms to make safe plans. Currently, most trajectory forecasting methods assume that object trajectories have been extracted and directly develop trajectory predictors based on the ground truth trajectories. However, this assumption does not hold in practical situations. Trajectories obtained from object detection and tracking are inevitably noisy, which could cause serious forecasting errors to predictors built on ground truth trajectories. In this paper, we propose to predict trajectories directly based on detection results without relying on explicitly formed trajectories. Different from traditional methods which encode the motion cues of an agent based on its clearly defined trajectory, we extract the motion information only based on the affinity cues among detection results, in which an affinity-aware state update mechanism is designed to manage the state information. In addition, considering that there could be multiple plausible matching candidates, we aggregate the states of them. These designs take the uncertainty of association into account which relax the undesirable effect of noisy trajectory obtained from data association and improve the robustness of the predictor. Extensive experiments validate the effectiveness of our method and its generalization ability to different detectors or forecasting schemes.

Robust Learning with Noisy Ship Trajectories by Adaptive Noise Rate Estimation.

Ship trajectory classification is of great significance for shipping analysis and marine security governance. However, in order to cover up their illegal fishing or espionage activities, some illicit ships will forge the ship type information in the Automatic Identification System (AIS), and this label noise will significantly impact the algorithm's classification accuracy. Sample selection is a common and effective approach in the field of learning from noisy labels. However, most of the existing methods based on sample selection need to determine the noise rate of the data through prior means. To address these issues, we propose a noise rate adaptive learning mechanism that operates without prior conditions. This mechanism is integrated with the robust training paradigm JoCoR (joint training with co-regularization), giving rise to a noise rate adaptive learning robust training paradigm called A-JoCoR. Experimental results on real-world trajectories provided by the Danish Maritime Authority verified the effectiveness of A-JoCoR. It not only realizes the adaptive learning of the data noise rate during the training process, but also significantly improves the classification performance compared with the original method.

A dynamical systems approach for most probable escape paths over periodic boundaries

Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler–Lagrange system, which we call the River. Using the Maslov index we identify a subset of the River which is comprised of local minimizers. The Onsager–Machlup (OM) functional, which is treated as a perturbation of the Friedlin–Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of the paper is focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.

MaxCal can infer models from coupled stochastic trajectories of gene expression and cell division

Gene expression is inherently noisy due to small numbers of proteins and nucleic acids inside a cell. Likewise, cell division is stochastic, particularly when tracking at the level of a single cell. The two can be coupled when gene expression affects the rate of cell division. Single-cell time-lapse experiments can measure both fluctuations by simultaneously recording protein levels inside a cell and its stochastic division. These information-rich noisy trajectory data sets can be harnessed to learn about the underlying molecular and cellular details that are often not known a priori. A critical question is: How can we infer a model given data where fluctuations at two levels—gene expression and cell division—are intricately convoluted? We show the principle of maximum caliber (MaxCal)—integrated within a Bayesian framework—can be used to infer several cellular and molecular details (division rates, protein production, and degradation rates) from these coupled stochastic trajectories (CSTs). We demonstrate this proof of concept using synthetic data generated from a known model. An additional challenge in data analysis is that trajectories are often not in protein numbers, but in noisy fluorescence that depends on protein number in a probabilistic manner. We again show that MaxCal can infer important molecular and cellular rates even when data are in fluorescence, another example of CST with three confounding factors—gene expression noise, cell division noise, and fluorescence distortion—all coupled. Our approach will provide guidance to build models in synthetic biology experiments as well as general biological systems where examples of CSTs are abundant.

A TDV attention-based BiGRU network for AIS-based vessel trajectory prediction

Automatic identification system (AIS) is a vessel-based system for the automatic broadcast and reception of vessel information, and it also supports data for trajectory prediction. Since the vessel's sailing route is flexible and changeable and the AIS broadcast is unconfirmed, the trajectory varies greatly and the original AIS data contains some noisy trajectory, which leads to low prediction accuracy and stability. Therefore, to solve the above problem, this paper proposes a trajectory prediction method based on bidirectional gate recurrent unit (BiGRU) and trajectory direction vector (TDV) with attention mechanism. This paper firstly proposes a TDV to associate latitude and longitude with the course and speed. Then the paper proposes an attention mechanism to self-adaptively update weight to the TDV in different stages to eliminate unreasonable predicted trajectory points. Finally, this paper combines the TDV attention mechanism and the BiGRU network to train a vessel trajectory prediction model.

Computing high-dimensional invariant distributions from noisy data

The invariant distribution is an important object in the study of randomly perturbed dynamical systems. The existing methods, including traditional finite difference or finite element methods as well as recently developed machine learning-based methods, all require the knowledge of the dynamical equations or adequate equilibrium data for estimating the invariant distribution. In this work, we propose a data-driven method for inferring the dynamics and simultaneously learning the invariant distribution from noisy trajectory data of the dynamical system. The data is not necessarily at equilibrium and may be collected from the transient period of the dynamics. The proposed method combines the idea of maximum likelihood estimation and a decomposition of the force field as suggested by the Fokker-Planck equation. The drift term (or force field), which is in the form of a decomposition with the constraint specified by the Fokker-Planck equation, and the diffusion term are learned from the data using an alternate updating algorithm. The generalized potential, which is the negative logarithm of the invariant distribution multiplied by the noise, is then obtained from the potential part of the decomposition. The proposed method is able to deal with high-dimensional dynamical systems and the small noise regime, without prior knowledge of the dynamical equations. The proposed method is demonstrated by four numerical examples, including a practical 10-dimensional biological network model.

1D-DGAN-PHM: A 1-D denoising GAN for Prognostics and Health Management with an application to turbofan

The performance of prognostics is closely related to the quality of condition monitoring signals (e.g., temperature, pressure, or vibration signals), which reveal the degradation of the system of interest. However, typical condition monitoring signals include noise and outliers. Disentangling noise from these signals is essential to obtain the actual degradation trajectories. Different denoising methods have been proposed in prognostics. Conventional denoising methods have low complexity but usually do not preserve edge information and do not involve physical considerations. A promising deep learning approach is denoising generative models. This approach is popular in Computer Vision, which has been shown to outperform other classical techniques but has seldom been used in prognostics on 1-D signals. In this paper, we propose the 1-D Denoising Generative Adversarial Network for Prognostics and Health Management (1D-DGAN-PHM). The 1D-DGAN-PHM is trained on synthetic data generated by a custom data generator that infuses physics-of-failure knowledge in paired samples of noisy and noise-free trajectories. The network consists of two components, a denoising generator and a discriminator. The denoising generator aims to learn to denoise a 1-D input signal. The discriminator guides the learning by comparing noise-free signals with signals from the denoising generator. Advantages of the 1D-DGAN-PHM include the physics-of-failure information in the synthetic data generator and the model sophistication. In this work, we apply the 1D-DGAN-PHM to denoise the raw signals derived from NASA’s C-MAPSS simulator of an aircraft turbofan engine. Baseline methods are Moving Average, Median filter, Savitzky–Golay filter, and a denoising autoencoder. The 1D-DGAN-PHM produces smooth trajectories and preserves the initial linear degradation of the signals. The 1D-DGAN-PHM has the most significant improvement in prognosability (on average, 0.73 to 0.81). Data from the 1D-DGAN-PHM resulted in the best MAE (29 to 25 cycles) and RMSE (score of 39 to 36) for a Random Forest.The code is publicly available at 1D-DGAN-PHM.

Human Mobility Prediction with Calibration for Noisy Trajectories

Human mobility prediction is a key task in smart cities to help improve urban management effectiveness. However, it remains challenging due to widespread intractable noises in large-scale mobility data. Based on previous research and our statistical analysis of real large-scale data, we observe that there is heterogeneity in the quality of users’ trajectories, that is, the regularity and periodicity of one user’s trajectories can be quite different from another. Inspired by this, we propose a trajectory quality calibration framework for quantifying the quality of each trajectory and promoting high-quality training instances to calibrate the final prediction process. The main module of our approach is a calibration network that evaluates the quality of each user’s trajectories by learning their similarity between them. It is designed to be model-independent and can be trained in an unsupervised manner. Finally, the mobility prediction model is trained with the instance-weighting strategy, which integrates quantified quality scores into the parameter updating process of the model. Experiments conducted on two citywide mobility datasets demonstrate the effectiveness of our approach when dealing with massive noisy trajectories in the real world.

CSR-PERT: Joint framework for MRI and HARDI data reconstruction using perturbed radial trajectory estimated from compressively sensed measurements

Radial sampling pattern is an important signal acquisition strategy in magnetic resonance imaging (MRI) owing to better immunity to motion-induced artifacts and less pronounced aliasing due to undersampling compared to the Cartesian sampling. These advantages of radial sampling can be exploited in acquisition of multidimensional signals such as High Angular Resolution Diffusion Imaging (HARDI), with tremendous scope of acceleration. Despite such benefits, gradient delays lead to samples being acquired from unknown miscentered radial trajectories, severely degrading the image reconstruction quality. In the present work, we propose Csr-Pert that is a joint framework, wherein these perturbed radial trajectories are estimated and utilized for image reconstruction from the compressively sensed measurements of (i) MRI data and (ii) HARDI data. The proposed Csr-Pert method is tested on one real MRI dataset with trajectory deviations and is observed to perform better than the existing state-of-the-art method at high acceleration factors up to 8. To the best of our knowledge, this is the first work to address the problem of estimating perturbed trajectories using the compressively sensed MRI and HARDI data. The method is also tested for varying combinations of trajectory deviations and sampling proportions. It is observed to yield very good quality HARDI reconstruction for a wide variety of scenarios. We have also demonstrated the robustness of the proposed method on real datasets in clinical settings assuming perturbed as well as noisy trajectories.

Detecting Lagrangian coherent structures from sparse and noisy trajectory data

Many complex flows such as those arising from the collective motion of ocean plastics in geophysics or motile cells in biology are characterized by sparse and noisy trajectory datasets. We introduce techniques for identifying Lagrangian coherent structures (LCSs) of hyperbolic and elliptic nature in such datasets. Hyperbolic LCSs, which represent surfaces with maximal attraction or repulsion over a finite amount of time, are computed through a regularized least-squares approximation of the flow map gradient. Elliptic LCSs, which identify regions of coherent motion such as vortices and jets, are extracted using DBSCAN – a popular data clustering algorithm – combined with a systematic parameter selection strategy. We deploy these methods on various benchmark analytical flows and real-life experimental datasets ranging from oceanography to biology and show that they yield accurate results, despite sparse and noisy data. We also provide a lightweight computational implementation of these techniques as a user-friendly and straightforward Python code.

Asymptotic Theory of \(\boldsymbol \ell _1\) -Regularized PDE Identification from a Single Noisy Trajectory

Asymptotic Theory of \(\boldsymbol \ell _1\) -Regularized PDE Identification from a Single Noisy Trajectory

Deep learning for location prediction on noisy trajectories

Deep learning for location prediction on noisy trajectories

OBD-Data-Assisted Cost-Based Map-Matching Algorithm for Low-Sampled Telematics Data in Urban Environments

A myriad of connected vehicles collects large-scale telematics data throughout cities, enabling data-based infrastructure planning. To truly benefit from this emerging technology, it is important to integrate pervasive telematics data with map data to produce more tractable and readable information for traffic flows and safety. Map-matching algorithms enable the projection of noisy trajectory data onto map data as a means of integrating telematics data. However, map-matching poses challenges due to higher levels of positioning errors and complex road networks. The authors propose a novel map-matching algorithm that can fuse in-vehicle data with trajectory data to improve the efficiency and accuracy of the algorithm. The proposed algorithm combines the probabilistic and weight-based map-matching frameworks. The novelty of the proposed algorithm includes (i) an adaptive segment candidate search mechanism based on in-vehicle speed information, (ii) adaptive matching parameters to reflect the variations in the Global Positioning System (GPS) noise levels, (iii) a novel transition probability that uses in-vehicle speed data, and (iv) a backend data query system for the shortest routes. Map-matching results were validated based on ground-truth data collected using an in-vehicle sensing device developed by the authors, as well as comparing with a commonly-used off-the-shelf map-matching platform. The proposed algorithm is proven to be robust, with an accuracy of 97.45%, particularly where map data are denser and GPS noise is high.

Inferring potential landscapes from noisy trajectories of particles within an optical feedback trap

SummaryWhile particle trajectories encode information on their governing potentials, potentials can be challenging to robustly extract from trajectories. Measurement errors may corrupt a particle’s position, and sparse sampling of the potential limits data in higher energy regions such as barriers. We develop a Bayesian method to infer potentials from trajectories corrupted by Markovian measurement noise without assuming prior functional form on the potentials. As an alternative to Gaussian process priors over potentials, we introduce structured kernel interpolation to the Natural Sciences which allows us to extend our analysis to large datasets. Structured-Kernel-Interpolation Priors for Potential Energy Reconstruction (SKIPPER) is validated on 1D and 2D experimental trajectories for particles in a feedback trap.