Gauss-Markov Process Research Articles

Harmonic oscillators, waves and Gaussian processes

Classical harmonic oscillators are ubiquitous mechanical systems, appearing in many setups, in particular, as Gauss–Markov processes. The connection between wave motion and harmonic oscillators leads us to a connection between waves and a special class of infinitely dimensional Gauss–Markov processes.It is the aim of this work to explore this connection in the simplest case, namely, that of a linear wave on a string with fixed end points. We shall see that the initial conditions are unknown and distributed according to a Gaussian measure, the string is an infinite dimensional Gaussian Markov process.A byproduct of the setup, is that the logarithm of the characteristic function of a Gaussian wave is a concave function ( it is the negative) of the Fenchel–Lagrange dual of the energy, that could perhaps be thought of as the “entropy” of the solution to the wave equation.

Multiple Markov Gaussian processes

Multiple Markov Gaussian processes

On the Entropy of Fractionally Integrated Gauss–Markov Processes

This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1).

Effect of gusty inflow on the jet-switching characteristics of a plunging foil

The effect of stochastic inflow fluctuations on the jet-switching characteristics of a harmonically plunging elliptic foil at a low Reynolds number regime has been analyzed in the present study. The inflow fluctuations are generated by simulating an Ornstein–Uhlenbeck process—a stationary Gauss–Markov process—with a chosen correlation function. In the absence of fluctuations, quasi-periodic movement of the wake vortices plays a key role in bringing out jet-switching at κh ≥ 1.5. However, fluctuating inflow alters the organized arrangement of the vortex street even at a lower κh (κh = 1.0), giving way to an advanced jet-switching onset. More frequent switching with a larger deflection angle is also observed at κh = 1.5 as compared to the no fluctuation case. Effects of inflow timescales on the deflection angle and switching frequency of the wake are investigated by varying the input correlation-lengths. The underlying flow physics are investigated through a qualitative study of the near-field interactions as well as various quantitative measures derived from the unsteady flow-field.

BROWNIAN MOTION MINUS THE INDEPENDENT INCREMENTS: REPRESENTATION AND QUEUING APPLICATION

This paper relaxes assumptions defining multivariate Brownian motion (BM) to construct processes with dependent increments as tractable models for problems in engineering and management science. We show that any Gaussian Markov process starting at zero and possessing stationary increments and a symmetric smooth kernel has a parametric kernel of a particular form, and we derive the unique unbiased, jointly sufficient, maximum-likelihood estimators of those parameters. As an application, we model a single-server queue driven by such a process and derive its transient distribution conditional on its history.

Optimal linear recursive estimators for stochastic uncertain systems with time-correlated additive noises and packet dropout compensations

This paper is concerned with the state estimation problem over a packet-dropping network for stochastic uncertain systems with time-correlated additive noises. Additive process and measurement noises are both depicted by first-order Gauss-Markov processes. Stochastic parameter uncertainties are described by correlated white multiplicative noises. When a sensor measurement transmitted over networks is lost, its predictor is used for compensation. The optimal linear recursive full-order state filter, predictor and smoother are proposed under the linear minimum variance (LMV) criterion by an innovation analysis approach. They are calculated in terms of the filter of the product of the multiplicative noise and state, estimators of the process and measurement noises, and cross-covariance matrices for the state and/or noises. The steady-state estimators are also studied. A sufficient condition for convergence of optimal linear estimators is given. The simulation results verify the effectiveness of the proposed algorithms.

Mobility Modeling and Stochastic Property Analysis of Airborne Network

In literature, a few attempts have been made to model the mobility of the airborne networks. However, existing models are mostly application specific and oversimplify the system dynamics. This paper proposes a comprehensive mobility model for the airborne networks where the mobility of the airborne node is presented as a Gauss–Markov process comprising of three key parameters—the speed, angular velocity, and pitch angle. The consideration of the Gauss–Markov process and angular velocity enables the model to address the mobility more realistically and, thus, makes it capable of generating apparently all possible types of trajectories. Additionally, this model designs more realistic boundary effects to be practical. This paper also presents the stochastic analysis of the proposed model. Finally, by analyzing the extensive simulation results, it has been affirmed that the proposed mobility model can be used as a fundamental template to model and analyze a wide range of practical airborne networks regardless of the application including both homogeneous and heterogeneous networks.

Mitigation Techniques for Impulsive Noise With Memory Modeled by a Two State Markov-Gaussian Process

Impulsive noise, a common impediment preventing the system from achieving error-free transmission, is significant in many wireless and power line communication environments. Although the performance of several mitigation techniques for orthogonal frequency division multiplexing (OFDM)-based multicarrier communication systems impaired by memoryless impulsive noise are widely acknowledged, we note that OFDM is outperformed by its single-carrier counterpart when the impulses are very strong and/or they occur frequently, which is likely to exist in contemporary communication systems including smart grid communications. On the other hand, many communication technologies used in the smart grid do not employ OFDM and likewise, the assumption of memoryless noise is not valid for such communication scenarios. Memoryless noise models cannot take into account one of the main features of the actual noise, i.e., the time-correlation among the noise samples. The aim of this article is to compare and analyze several mitigation techniques such as clipping, blanking, and combined clipping-blanking to mitigate the noxious effects of bursty impulsive noise for low-density parity-check coded single-carrier communication systems. Moreover, we propose a log-likelihood ratio (LLR)-based impulsive noise mitigation for the considered scenario. In this context, provided simulation results highlight the superiority of the LLR-based mitigation scheme over the clipping/blanking schemes.

Causal Structure Learning in Continuous Systems.

Real causal systems are complicated. Despite this, causal learning research has traditionally emphasized how causal relations can be induced on the basis of idealized events, i.e., those that have been mapped to binary variables and abstracted from time. For example, participants may be asked to assess the efficacy of a headache-relief pill on the basis of multiple patients who take the pill (or not) and find their headache relieved (or not). In contrast, the current study examines learning via interactions with continuous dynamic systems, systems that include continuous variables that interact over time (and that can be continuously observed in real time by the learner). To explore such systems, we develop a new framework that represents a causal system as a network of stationary Gauss–Markov (“Ornstein–Uhlenbeck”) processes and show how such OU networks can express complex dynamic phenomena, such as feedback loops and oscillations. To assess adult's abilities to learn such systems, we conducted an experiment in which participants were asked to identify the causal relationships of a number of OU networks, potentially carrying out multiple, temporally-extended interventions. We compared their judgments to a normative model for learning OU networks as well as a range of alternative and heuristic learning models from the literature. We found that, although participants exhibited substantial learning of such systems, they committed certain systematic errors. These successes and failures were best accounted for by a model that describes people as focusing on pairs of variables, rather than evaluating the evidence with respect to the full space of possible structural models. We argue that our approach provides both a principled framework for exploring the space of dynamic learning environments as well as new algorithmic insights into how people interact successfully with a continuous causal world.

A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the first passage time density through specific boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the first passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.

Modeling of Real Time Kinematics localization error for use in 5G networks

In 5G networks information about localization of a user equipment (UE) can be used not only for emergency calls or location-based services, but also for the network optimization applications, e.g., network management or dynamic spectrum access by using Radio Environment Maps (REM). However, some of these applications require much better localization accuracy than currently available in 4G systems. One promising localization method is Global Navigation Satellite System (GNSS)-based Real-Time Kinematics (RTK). While the signal received from satellites is the same as in traditional GNSS, a new reception method utilizing real-time data from a nearby reference station (e.g., 5G base station) results in cm-level positioning accuracy. The aim of this paper is to obtain a model of the RTK localization error for smartphone-grade GNSS antenna under open-sky conditions, that can be used in 5G network simulators. First, a tutorial-style overview of RTK positioning, and satellite orbits prediction is provided. Next, an RTK localization simulator is implemented utilizing GNSS satellites constellations. Results are investigated statistically to provide a simple, yet accurate RTK localization error framework, which is based on two Gauss-Markov process generators parametrized by visible satellites geometry, UE motion, and UE-satellite distance error variance.

Graph Learning for Spatiotemporal Signals With Long- and Short-Term Characterization

Mining natural associations from high-dimensional spatiotemporal signals plays an important role in various fields including biology, climatology, and financial analysis. However, most existing works have mainly studied time-independent signals without considering the correlations of spatiotemporal signals that achieve high learning accuracy. This paper aims to learn graphs that better reflect underlying data relations by leveraging the long- and short-term characteristics of spatiotemporal signals. First, a spatiotemporal signal model is presented that considers both spatial and temporal relations. In particular, we integrate a low-rank representation and a Gaussian Markov process to describe the temporal correlations. Then, the graph learning problem is formulated as a joint low-rank component estimation and graph Laplacian inference. Accordingly, we propose a low rank and spatiotemporal smoothness-based graph learning method (GL-LRSS), which introduces a spatiotemporal smoothness prior into time-vertex signal analysis. By jointly exploiting the low rank of long-time observations and the smoothness of short-time observations, the overall learning performance can be effectively improved. Experiments on both synthetic and real-world datasets demonstrate substantial improvements in the learning accuracy of the proposed method over the state-of-the-art low-rank component estimation and graph learning methods.

Air-to-ground 3D channel modeling for UAV based on Gauss-Markov mobile model

Air-to-ground (A2G) communication has been attracted great attention due to the gradual popularization of unmanned aerial vehicle (UAV). In this paper, a 3D geometry-based stochastic model (GBSM) for A2G channels is proposed. Meanwhile, the Gauss-Markov mobile model is adopted to generate dynamic trajectories. According to different scattering environments, we establish a reference model and a statistical simulation model for A2G channels, and derive their respective time correlation functions. In particular, the purpose of establishing the statistical simulation model is to reduce complexity, and we verify the consistency of the two models. Then, we analyze the dynamic motion scenarios generated by Gauss-Markov process, and their effects on A2G channel correlation. Finally, the good conformities of time correlation curves with the existing measurement data and sphere model verify the reliability of our proposed models.

Adaptive Beam Tracking With the Unscented Kalman Filter for Millimeter Wave Communication

Millimeter wave (mmWave) communication links for 5G cellular technology require high beamforming gain to overcome channel impairments and achieve high throughput. While much work has focused on estimating mmWave channels and designing beamforming schemes, the time dynamic nature of mmWave channels quickly renders estimates stale and increases sounding overhead. We model the underlying time dynamic state space of mmWave channels and design sounding beamformers suitable for tracking in a Kalman filtering framework. Given an initial channel estimate, filtering efficiently leads to refined estimates and allows forward prediction for higher sustained beamforming gain during data transmission. From tracked prior channel estimates, adaptively chosen optimal and constrained suboptimal beams reduce sounding overhead while minimizing estimation error.

A Semi-Markov Leaky Integrate-and-Fire Model

In this paper, a Leaky Integrate-and-Fire (LIF) model for the membrane potential of a neuron is considered, in case the potential process is a semi-Markov process. Semi-Markov property is obtained here by means of the time-change of a Gauss-Markov process. This model has some merits, including heavy-tailed distribution of the waiting times between spikes. This and other properties of the process, such as the mean, variance and autocovariance, are discussed.

A Novel Adaptive Kalman Filtering Approach to Human Motion Tracking With Magnetic-Inertial Sensors

This article presents an adaptive Kalman filtering approach to estimate the orientation of human body segments by using magnetic-inertial measurement units (MIMUs) with three-axis gyroscope, accelerometer, and magnetometer. In order to mitigate the negative impact of the external accelerations and ferromagnetic disturbances, the disturbances are modeled by first-order Gauss–Markov processes, and two parallel adaptive Kalman filters containing disturbance models are implemented to eliminate the disturbances. An adaptive factor is introduced to adjust the process noise covariance matrix to compensate for modeling errors induced by unmodeled gyroscope biases and the erroneous process noise covariance matrices. Furthermore, the outliers are checked and discarded by hypothesis tests on the innovation, and the measurement is rejected when the innovation exceeds a given confidence interval. Then, the estimated gravity acceleration and geomagnetic field are used to calculate the orientation quaternion using triaxial attitude determination algorithm. Finally, experiments under different motion scenarios are carried out to evaluate the effectiveness of the proposed method, and the performance of the proposed method is discussed in comparison with existing ones. A forward kinematics three-dimensional (3-D) human motion reconstruction method is proposed to drive the human skeleton model to reproduce human motion with a visual interface based on ROS platform.

Pinned Diffusions and Markov Bridges

In this article, we consider a family of real-valued diffusion processes on the time interval [0, 1] indexed by their prescribed initial value $$x \in \mathbb {R}$$ and another point in space, $$y \in \mathbb {R}$$. We first present an easy-to-check condition on their drift and diffusion coefficients ensuring that the diffusion is pinned in y at time $$t=1$$. Our main result then concerns the following question: can this family of pinned diffusions be obtained as the bridges either of a Gaussian Markov process or of an Ito diffusion? We eventually illustrate our precise answer with several examples.

On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes

We investigate the main statistical parameters of the integral over time of the fractional Brownian motion and of a kind of pseudo-fractional Gaussian process, obtained as a classical Gauss–Markov process from Doob representation by replacing Brownian motion with fractional Brownian motion. Possible applications in the context of neuronal models are highlighted. A fractional Ornstein–Uhlenbeck process is considered and relations with the integral of the pseudo-fractional Gaussian process are provided.

Propagation of grid-scale density model uncertainty to orbital uncertainties

Accurate knowledge of the orbital state-vector uncertainty is required for the computation of reliable collision probabilities, the association of tracks, as well as the optimal planning of sensor resources. In low Earth Orbits (LEO) neutral atmospheric density uncertainty is the main contributor to orbital state vector uncertainty. Grid-scale model uncertainty on the other hand is, in most cases, the dominating component of atmospheric density uncertainty.Prior research in the field of density uncertainty has been broad. Nowadays multiple authors seem to agree that a first-order stochastic Gauss-Markov process, also known as Ornstein-Uhlenbeck process (OUP), is an appropriate stochastic representation for density model uncertainties. While many have studied orbital variations due to stochastic grid-scale density uncertainty, it is believed that the work at hand provides for the first time explicit analytic equations for the estimation of orbital uncertainties due to an underlying Ornstein-Uhlenbeck process that has been optimized for the representation of atmospheric density. It is shown that the long-term in-track position error due to this modified OUP grows with ~t3.The presented derivations and their validation build upon prior work by Emmert et al. and Sagnières and Sharf. Numerical Monte-Carlo simulations are used to validate the findings with two different semi-empirical density models. We also demonstrate that the resulting orbital uncertainty due to the modified OUP in atmospheric density can be estimated using a classical OUP in the relative density error.

A stochastic model for interacting neurons in the olfactory bulb

We focus on interacting neurons organized in a block-layered network devoted to the information processing from the sensory system to the brain. Specifically, we consider the firing activity of olfactory sensory neurons, periglomerular, granule and mitral cells in the context of the neuronal activity of the olfactory bulb. We propose and investigate a stochastic model of a layered and modular network to describe the dynamic behavior of each prototypical neuron, taking into account both its role (excitatory/inhibitory) and its location within the network. We adopt specific Gauss-Markov processes suitable to provide reliable estimates of the firing activity of the different neurons, given their linkages. Furthermore, we study the impact of selective excitation/inhibition on the information transmission by means of simulations and numerical estimates obtained through a Volterra integral approach.