Noisy Trajectories Research Articles

CT-Mapper: Mapping sparse multimodal cellular trajectories using a multilayer transportation network

Mobile phone data have recently become an attractive source of information about mobility behavior. Since cell phone data can be captured in a passive way for a large user population, they can be harnessed to collect well-sampled mobility information. In this paper, we propose CT-Mapper, an unsupervised algorithm that enables the mapping of mobile phone traces over a multimodal transport network. One of the main strengths of CT-Mapper is its capability to map noisy sparse cellular multimodal trajectories over a multilayer transportation network where the layers have different physical properties and not only to map trajectories associated with a single layer. Such a network is modeled by a large multilayer graph in which the nodes correspond to metro/train stations or road intersections and edges correspond to connections between them. The mapping problem is modeled by an unsupervised HMM where the observations correspond to sparse user mobile trajectories and the hidden states to the multilayer graph nodes. The HMM is unsupervised as the transition and emission probabilities are inferred using respectively the physical transportation properties and the information on the spatial coverage of antenna base stations. To evaluate CT-Mapper we collected cellular traces with their corresponding GPS trajectories for a group of volunteer users in Paris and vicinity (France). We show that CT-Mapper is able to accurately retrieve the real cell phone user paths despite the sparsity of the observed trace trajectories. Furthermore our transition probability model is up to 20% more accurate than other naive models.

Extracting Stops from Noisy Trajectories: A Sequence Oriented Clustering Approach

Trajectories, representing the movements of objects in the real world, carry significant stop/move semantics. The detection of trajectory stops poses a critical problem in the study of moving objects and becomes even more challenging due to the inevitable noise recorded along with true data. To extract stops with a variety of shapes and sizes from single trajectories with noise, this paper presents a sequence oriented clustering approach, in which noise points within the sequence of a stop can be identified and classified as a part of the stop. In our method, two key concepts are first introduced: (1) a core sequence that defines sequence density based not only on proximity in space but also continuity in time as well as the duration over time; and (2) an Eps-reachability sequence that aggregates core sequences that overlap or meet over time. Then, three criteria are presented to merge Eps-reachability sequences interrupted by noise. Further, an algorithm, called SOC (Sequence Oriented Clustering), is developed to automatically extract stops from a single trajectory. In addition, a reachability graph is designed that visually illustrates the spatio-temporal clustering structure and levels of a trajectory. Finally, the proposed algorithm is evaluated against two baseline methods through extensive experiments based on real world trajectories, some with serious noise, and the results show that our approach is fairly effective in recognizing trajectory stops.

Multi-Color Single Particle Tracking for Determining Protein Interaction Lifetimes

The lifetime of interactions between membrane proteins can be measured using two-color or multi-color single particle tracking. The spectral separation of the labels allows closely spaced proteins to be localized with the same precision of isolated fluorophores, typically ∼ 20 nm. Proteins that move together can then be identified as dimers and the lifetime of their interaction measured. I will discuss the important practical aspects of collecting and analyzing data from multi-color tracking experiments including channel registration, tracking, and the statistical analysis of the noisy trajectories to infer interaction lifetimes.

Noisy Lagrangian Tracers for Filtering Random Rotating Compressible Flows

The recovery of a random turbulent velocity field using Lagrangian tracers that move with the fluid flow is a practically important problem. This paper studies the filtering skill of $$L$$ -noisy Lagrangian tracers in recovering random rotating compressible flows that are a linear combination of random incompressible geostrophically balanced (GB) flow and random rotating compressible gravity waves. The idealized random fields are defined through forced damped random amplitudes of Fourier eigenmodes of the rotating shallow-water equations with the rotation rate measured by the Rossby number $$\varepsilon $$ . In many realistic geophysical flows, there is fast rotation so $$\varepsilon $$ satisfies $$\varepsilon \ll 1$$ and the random rotating shallow-water equations become a slow–fast system where often the primary practical objective is the recovery of the GB component from the Lagrangian tracer observations. Unfortunately, the $$L$$ -noisy Lagrangian tracer observations are highly nonlinear and mix the slow GB modes and the fast gravity modes. Despite this inherent nonlinearity, it is shown here that there are closed analytical formulas for the optimal filter for recovering these random rotating compressible flows for any $$\varepsilon $$ involving Ricatti equations with random coefficients. The performance of the optimal filter is compared and contrasted through mathematical theorems and concise numerical experiments with the performance of the optimal filter for the incompressible GB random flow with $$L$$ -noisy Lagrangian tracers involving only the GB part of the flow. In addition, a sub-optimal filter is defined for recovering the GB flow alone through observing the $$L$$ -noisy random compressible Lagrangian trajectories, so the effect of the gravity wave dynamics is unresolved but effects the tracer observations. Rigorous theorems proved below through suitable stochastic fast-wave averaging techniques and explicit formulas rigorously demonstrate that all these filters have comparable skill in recovering the slow GB flow in the limit $$\varepsilon \rightarrow 0$$ for any bounded time interval. Concise numerical experiments confirm the mathematical theory and elucidate various new features of filter performance as the Rossby number $$\varepsilon $$ , the number of tracers $$L$$ and the tracer noise variance change.

Extraction of rotational correlation times from noisy single molecule fluorescence trajectories.

Monitoring single molecule probe rotations is an increasingly common approach to studying dynamics of complex systems, including supercooled liquids. Even with advances in fluorophore design and detector sensitivity, such measurements typically exhibit low signal to noise and signal to background ratios. Here, we simulated and analyzed orthogonally decomposed fluorescence signals of single molecules undergoing rotational diffusion in a manner that mimics experimentally collected data of probes in small molecule supercooled liquids. The effects of noise, background, and trajectory length were explicitly considered, as were the effects of data processing approaches that may limit the impact of noise and background on assessment of environmental dynamics. In many cases, data treatment that attempts to remove noise and background were found to be deleterious. However, for short trajectories below a critical signal to background threshold, a thresholding approach that successfully removed data points associated with noise and spared those associated with signal allowed for assessment of environmental dynamics that was as accurate and precise as would be achieved in the absence of noise.

Robust Estimation Fusion in Wireless Senor Networks with Outliers and Correlated Noises

This paper addresses the problem of estimation fusion in a distributed wireless sensor network (WSN) under the following conditions: (i) sensor noises are contaminated by outliers or gross errors; (ii) process noise and sensor noises are correlated; (iii) cross-correlation among local estimates is unknown. First, to attack the correlation and outliers, a correlated robust Kalman filtering (coR2KF) scheme with weighted matrices on innovation sequences is introduced as local estimator. It is shown that the proposed coR2KF takes both conventional Kalman filter and robust Kalman filter as a special case. Then, a novel version of our internal ellipsoid approximation fusion (IEAF) is used in the fusion center to handle the unknown cross-correlation of local estimates. The explicit solution to both fusion estimate and corresponding covariance is given. Finally, to demonstrate robustness of the proposed coR2KF and the effectiveness of IEAF strategy, a simulation example of tracking a target moving on noisy circular trajectories is included.

MATHEMATICAL TREATMENTS THAT SOLVE SINGLE MOLECULES

In this article, we talk about the ways that scientists can solve single molecule trajectories. Solving single molecules, that is, finding the model from the data, is complicated at least as much as measuring single molecules. We must filter the noise and take care of every step in the analysis when constructing the most accurate model from the data. Here, we present valuable solutions. Ways that solve clean discrete data are first presented. We review here our reduced dimensions forms (RDFs): unique models that are canonical forms of discrete data, and the statistical and numerical toolbox that builds a RDF from finite, clean, two-state data. We then review our most recent filter that "tackles" the noise when measuring two state noisy photon trajectories. The filter is a numerical algorithm with various special statistical treatments that is based on a general likelihood function that we have developed recently. We show the strengths of the filter (also over other approaches) and talk about its various new variants. This filter (with minor adjustments) can solve the noise in any discrete state trajectories, yet, extensions are needed in "tackling" the noise from other data, e.g. continuous data. Only the combined procedures enable creating the most accurate model from noisy discrete trajectories from single molecules. These concepts and methods (with adjustments) are valuable also when solving continuous trajectories and fluorescence resonance energy transfer trajectories. We also present a set of simple methods that can help any scientist with treating the trajectory perhaps encouraging applying the involved methods. The involved methods will appear in software that we are developing now, helping therefore the experimentalists utilizing these methods on real data. Comparisons with other known methods in this field are made. [Formula: see text]Special Issue Comment: This article about mathematical treatments when solving single molecules is related to the reviews in this Special Issue about measuring enzymes67 and about FRET experiments2 and about the software QUB.6

Trajectory Data Analyses for Pedestrian Space-time Activity Study

It is well recognized that human movement in the spatial and temporal dimensions has direct influence on disease transmission(1-3). An infectious disease typically spreads via contact between infected and susceptible individuals in their overlapped activity spaces. Therefore, daily mobility-activity information can be used as an indicator to measure exposures to risk factors of infection. However, a major difficulty and thus the reason for paucity of studies of infectious disease transmission at the micro scale arise from the lack of detailed individual mobility data. Previously in transportation and tourism research detailed space-time activity data often relied on the time-space diary technique, which requires subjects to actively record their activities in time and space. This is highly demanding for the participants and collaboration from the participants greatly affects the quality of data(4). Modern technologies such as GPS and mobile communications have made possible the automatic collection of trajectory data. The data collected, however, is not ideal for modeling human space-time activities, limited by the accuracies of existing devices. There is also no readily available tool for efficient processing of the data for human behavior study. We present here a suite of methods and an integrated ArcGIS desktop-based visual interface for the pre-processing and spatiotemporal analyses of trajectory data. We provide examples of how such processing may be used to model human space-time activities, especially with error-rich pedestrian trajectory data, that could be useful in public health studies such as infectious disease transmission modeling. The procedure presented includes pre-processing, trajectory segmentation, activity space characterization, density estimation and visualization, and a few other exploratory analysis methods. Pre-processing is the cleaning of noisy raw trajectory data. We introduce an interactive visual pre-processing interface as well as an automatic module. Trajectory segmentation(5) involves the identification of indoor and outdoor parts from pre-processed space-time tracks. Again, both interactive visual segmentation and automatic segmentation are supported. Segmented space-time tracks are then analyzed to derive characteristics of one's activity space such as activity radius etc. Density estimation and visualization are used to examine large amount of trajectory data to model hot spots and interactions. We demonstrate both density surface mapping(6) and density volume rendering(7). We also include a couple of other exploratory data analyses (EDA) and visualizations tools, such as Google Earth animation support and connection analysis. The suite of analytical as well as visual methods presented in this paper may be applied to any trajectory data for space-time activity studies.

A Behavioural Approach in EIV Identification: the SISO Case

AbstractErrors-in-Variables (EIV) models consider the presence of additive errors on the measures of all measurable attributes of a process. Traditional identification procedures for these processes rely on the properties of the covariance matrix of the observations and on its decomposition into the sum of a matrix associated with the (unknown) noiseless sequences and of the observation noise covariance matrix. This paper makes reference, in a behavioural framework, to the observed noisy trajectories and performs a decomposition of these trajectories into a regular part, defininig the associated behaviour of the process, and a noise part. The Monte Carlo simulations that have been performed show that the proposed approach leads to accurate estimates of both the system parameters and the noise variances.

Target tracking in wireless sensor networks using adaptive measurement quantization

Quantization/compression is usually adopted in wireless sensor networks (WSNs) since each sensor node typically has very limited power supply and communication bandwidth. We consider the problem of target tracking in a WSN with quantized measurements in this paper. Attention is focused on the design of measurement quantizer with adaptive thresholds. Based on the probability density function (PDF) of the signal amplitude measured at a random location and by maximizing the entropy, an adaptive design method for quantization thresholds is proposed. Due to the nonlinear measuring and quantization models, particle filtering (PF) is adopted in the fusion center (FC) to estimate the target state. Posterior Cramer-Rao lower bounds (CRLBs) for tracking accuracy using quantized measurements are also derived. Finally, a simulation example on tracking single target with noisy circular trajectories is provided to illustrate the effectiveness of the proposed approach.

The regularizing effects of resetting in a particle system for the Burgers equation

We study the dissipation mechanism of a stochastic particle system for the Burgers equation. The velocity field of the viscous Burgers and Navier-Stokes equations can be expressed as an expected value of a stochastic process based on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3 (2008) 330-345]. In this paper we study a particle system for the viscous Burgers equations using a Monte-Carlo version of the above; we consider N copies of the above stochastic flow, each driven by independent Wiener processes, and replace the expected value with $\frac{1}{N}$ times the sum over these copies. A similar construction for the Navier-Stokes equations was studied by Mattingly and the first author of this paper [Iyer and Mattingly Nonlinearity 21 (2008) 2537-2553]. Surprisingly, for any finite N, the particle system for the Burgers equations shocks almost surely in finite time. In contrast to the full expected value, the empirical mean $\frac{1}{N}\sum_1^N$ does not regularize the system enough to ensure a time global solution. To avoid these shocks, we consider a resetting procedure, which at first sight should have no regularizing effect at all. However, we prove that this procedure prevents the formation of shocks for any $N\geq2$, and consequently as $N\to\infty$ we get convergence to the solution of the viscous Burgers equation on long time intervals.

Fractal snapshot components in chaos induced by strong noise

In systems exhibiting transient chaos in coexistence with periodic attractors, the inclusion of weak noise might give rise to noise-induced chaotic attractors. When the noise amplitude exceeds a critical value, an extended attractor appears along the fractal unstable manifold of the underlying nonattracting chaotic set. A further increase of noise leads to a fuzzy nonfractal pattern. By means of the concept of snapshot attractors and random maps, we point out that the fuzzy pattern can be decomposed into well-defined fractal components, the snapshot attractors belonging to a given realization of the noise and generated by following an ensemble of noisy trajectories. The pattern of the snapshot attractor and its characteristic numbers, such as the finite time Lyapunov exponents and numerically evaluated fractal dimensions, change continuously in time. We find that this temporal fluctuation is a robust property of the system which hardly changes with increasing ensemble size. The validity of the Kaplan-Yorke formula is also investigated. A superposition of about 100 snapshot attractors provides a good approximant to the fuzzy noise-induced attractor at the same noise strength.

Trajectory clustering by means of Earth Mover's Distance

AbstractWe propose a method for trajectory classification based on a general cluster-based methodology, that can be used both off-line in an unsupervised fashion, both on-line, classifying new trajectories or part of them. We use the Earth Mover's Distance (EMD) and we adapt it in order to employ it as a tool for trajectory clustering. We propose a novel effective method to identify the clusters’ representatives by means of the p—median location problem. This methodology is able to manage different length and noisy trajectories and takes velocity profiles and stops into account. We discuss the experimental results and we compare our approach with other trajectory clustering methods.

Posterior CramÉr–Rao Lower Bounds for Target Tracking in Sensor Networks With Quantized Range-Only Measurements

We consider the problem of target tracking in a wireless sensor network (WSN) that consists of randomly distributed range-only sensors. Quantized measurements are usually adopted in such a network to attack the problem of limited power supply and communication bandwidth. Assuming that local sensor noises are mutually independent, we derive the posterior Cramer-Rao lower bound (CRLB) on the mean squared error (MSE) of target tracking in WSNs with quantized range-only measurements. Recursion of posterior CRLB on tracking based on both constant velocity (CV) and constant acceleration (CA) model for target dynamics and a general range-only measuring model for local sensors are obtained. Due to the analytical difficulties, particle filter is applied to approximate the theoretical bounds. To illustrate the posterior CRLB, an example on tracking a target with noisy circular trajectories is given.

Robustness of the geometric phase under parametric noise

We study the robustness of the geometric phase in the presence of parametric noise. For this purpose we consider a simple case study, namely a semiclassical particle that moves adiabatically along a closed loop in a static magnetic field acquiring the Dirac phase. Parametric noise comes from the interaction with a classical environment, which adds a Brownian component to the path followed by the particle. After defining a gauge-invariant Dirac phase, we discuss the first and second moments of the distribution of the Dirac phase angle coming from the noisy trajectory.

Video synchronization and its application to object transfer

In video post-production applications, camera motion analysis and alignment are important in order to ensure the geometric correctness and temporal consistency. In this paper, we trade some generality in estimating and aligning camera motion for reduced computational complexity and increased image-based nature. The main contribution is to use fundamental ratios to synchronize video sequences of distinct scenes captured by cameras undergoing similar motions. We also present a simple method to align 3D camera trajectories when the fundamental ratios are not able to match the noisy trajectories. Experimental results show that our method can accurately synchronize sequences even when the scenes are totally different and have dense depths. An application on 3D object transfer is also demonstrated.

Functional Variance Processes

We introduce the notion of a functional variance process to quantify variation in functional data. The functional data are modeled as samples of smooth random trajectories observed under additive noise. The noise is assumed to be composed of white noise and a smooth random process—the functional variance process—which gives rise to smooth random trajectories of variance. The functional variance process is a tool for analyzing stochastic time trends in noise variance. As a smooth random process, it can be characterized by the eigenfunctions and eigenvalues of its autocovariance operator. We develop methods to estimate these characteristics from the data, applying concepts from functional data analysis to the residuals obtained after an initial smoothing step. Asymptotic justifications for the proposed estimates are provided. The proposed functional variance process extends the concept of a variance function, an established tool in nonparametric and semiparametric regression analysis, to the case of functional data. We demonstrate that functional variance processes offer a novel data analysis technique that leads to relevant findings in applications, ranging from a seismic discrimination problem to the analysis of noisy reproductive trajectories in evolutionary biology.

The Exit Problem in a Nonlinear System Driven by 1/f Noise: The Delay Locked Loop

The frequency generated by high frequency oscillators contains a small but significant noise component known as phase noise, also known as oscillator noise or phase jitter. The phase noise belongs to the family of stochastic processes with spectra $1/f^\alpha$, which exhibits scaleinvariance (or self-similarity) and a long-term correlation structure that decays polynomially in time. Both the phase and thermal noises cause errors in receivers that contain the oscillators. In particular, they cause losses of lock in phase tracking systems such as the phase locked loop in coherent systems, which include cellular phones, global positioning systems (GPS), and radar (e.g., synthetic aperture radar (SAR)), and in the delay locked loop (DLL), which is an important component of code division multiple access receivers and interface to modern memory modules, such as double data rate synchronous dynamic random access memory. The mean time to lose lock (MTLL) is well known to be an important design objective for various tracking loops. The evaluation of the MTLL is known in the mathematical literature as the exit problem for a dynamical system driven by noise, which is the problem of calculating the mean time for the noisy trajectories to reach the boundary of the domain of attraction of a stable point of the noiseless dynamics. In this paper we develop an analytic approach to the evaluation of the leading order term for MTLL of a second order DLL, due to both the non-Markovian $1/f^\alpha$ noise and to thermal white noise. The method is applicable to more general systems driven by a wide class of phase noises. The keys to the solution of this exit problem are the construction of a series of higher order Markovian processes that converge to the non-Markovian $1/f^\alpha$ noise and the asymptotic solution to a multidimensional elliptic boundary value problem that the mean first passage time (MFPT) satisfies.

Fast high-quality numerical shadowing of chaotic maps using synchronization

For a chaotic, dynamical system, a typical noisy trajectory diverges exponentially from the true trajectory with the same initial conditions. Nonetheless, the noisy trajectory, as representative of the dynamical system, may have credibility if there exists a true trajectory, corresponding to a slightly different initial condition, that stays close to the noisy trajectory for long periods of time. For finding such shadowing trajectories, a synchronization based method is presented. We call it the synchronize-and-pullback algorithm. Several numerical examples are shown illustrating the method. Finally, an application of the proposed shadowing technique for noise filtration in the context of communication is presented.

Nonstationary Kalman Filter for Estimation of Accurate and Consistent Car-Following Data

Difficulty in obtaining accurate car-following data has traditionally been regarded as a considerable drawback in understanding real phenomena and has affected the development and validation of traffic microsimulation models. Recent advancements in digital technology have opened up new horizons in the conduct of research in this field. Despite the high degrees of precision of these techniques, estimation of time series data of speeds and accelerations from positions with the required accuracy is still a demanding task. The core of the problem is filtering the noisy trajectory data for each vehicle without altering platoon data consistency; i.e., the speeds and accelerations of following vehicles must be estimated so that the resulting intervehicle spacings are equal to the real one. Otherwise, negative spacings can also easily occur. The task was achieved in this study by considering vehicles of a platoon as a sole dynamic system and reducing several estimation problems to a single consistent one. This process was accomplished by means of a nonstationary Kalman filter that used measurements and time-varying error information from differential Global Positioning System devices. The Kalman filter was fruitfully applied here to estimation of the speed of the whole platoon by including intervehicle spacings as additional measurements (assumed to be reference measurements). The closed solution of an optimization problem that ensures strict observation of the true intervehicle spacings concludes the estimation process. The stationary counterpart of the devised filter is suitable for application to position data, regardless of the data collection technique used, e.g., video cameras.