Gauss-Markov Process Research Articles

On the Construction of a Special Class of Time-Inhomogeneous Diffusion Processes

A special class of time-inhomogeneous diffusion processes, generated starting from Gauss–Markov processes conditioned on the same initial state, is considered. This class includes many interesting diffusion processes with time-dependent infinitesimal drift and variance, for which the transition probability density function is explicitly determined. Moreover, closed form results for the first-passage-time density through suitable time-varying boundaries are obtained. Special cases, generated starting from Wiener and Ornstein–Uhlenbeck processes, are considered and widely discussed.

Frequency domain detection of sinusoidal signal under coloured Gaussian noise

Signal detection in coloured Gaussian noise usually requires preprocessing the received signal firstly to remove its correlation, resulting in a complicated calculation process. This study transforms the received signal into the frequency domain for detection and systematically studies both the real part (imaginary part) test and the approximate amplitude test (AAT). The study shows that the detection performance of the former is greatly affected by the initial phase of the signal, while that of the latter is basically independent of the initial phase. Frequency domain detection and the best detection are compared under a Gauss–Markov process, and the simulation results show that the detection performance of the AAT is similar to the best detection. In addition, the calculation of the AAT is obviously improved, so the AAT is easier to be applied in engineering compared to the best detection.

Tracking and Control of Gauss–Markov Processes over Packet-Drop Channels with Acknowledgments

We consider the problem of tracking the state of Gauss–Markov processes over rate-limited erasure-prone links. We concentrate first on the scenario in which several independent processes are seen by a single observer. The observer maps the processes into finite-rate packets that are sent over the erasure-prone links to a state estimator, and are acknowledged upon packet arrivals. The aim of the state estimator is to track the processes with zero delay and with minimum mean square error (MMSE). We show that, in the limit of many processes, greedy quantization with respect to the squared error distortion is optimal. That is, there is no tension between optimizing the MMSE of the process in the current time instant and that of future times. For the case of packet erasures with delayed acknowledgments, we connect the problem to that of compression with side information that is known at the observer and may be known at the state estimator—where the most recent packets serve as side information that may have been erased, and demonstrate that the loss due to a delay by one time unit is rather small. For the scenario where only one process is tracked by the observer–state estimator system, we further show that variable-length coding techniques are within a small gap of the many-process outer bound. We demonstrate the usefulness of the proposed approach for the simple setting of discrete-time scalar linear quadratic Gaussian control with a limited data-rate feedback that is susceptible to packet erasures.

Research on Time-Correlated Errors Using Allan Variance in a Kalman Filter Applicable to Vector-Tracking-Based GNSS Software-Defined Receiver for Autonomous Ground Vehicle Navigation

The global navigation satellite system (GNSS) has been applied to many areas, e.g.,the autonomous ground vehicle, unmanned aerial vehicle (UAV), precision agriculture, smart city,and the GNSS-reflectometry (GNSS-R), being of considerable significance over the past few decades.Unfortunately, the GNSS signal performance has the high risk of being reduced by the environmentalinterference. The vector tracking (VT) technique is promising to enhance the robustness in highdynamics as well as improve the sensitivity against the weak environment of the GNSS receiver.However, the time-correlated error coupled in the receiver clock estimations in terms of the VT loopcan decrease the accuracy of the navigation solution. There are few works present dealing with thisissue. In this work, the Allan variance is accordingly exploited to specify a model which is expectedto account for this type of error based on the 1st-order Gauss-Markov (GM) process. Then, it is usedfor proposing an enhanced Kalman filter (KF) by which this error can be suppressed. Furthermore,the proposed system model makes use of the innovation sequence so that the process covariancematrix can be adaptively adjusted and updated. The field tests demonstrate the performance of theproposed adaptive vector-tracking time-correlated error suppressed Kalman filter (A-VTTCES-KF).When compared with the results produced by the ordinary adaptive KF algorithm in terms of the VTloop, the real-time kinematic (RTK) positioning and code-based differential global positioning system(DGPS) positioning accuracies have been improved by 14.17% and 9.73%, respectively. On the otherhand, the RTK positioning performance has been increased by maximum 21.40% when comparedwith the results obtained from the commercial low-cost U-Blox receiver.

Comment on “Bayesian MMSE Estimation of a Gaussian Source in the Presence of Bursty Impulsive Noise”

In the above-titled letter by Alam et al. (see ibid., vol.22, no.9, p.1846–49, September 2018), the authors claimed to derive the minimum-mean-squared-error (MMSE) estimator for a scalar Gaussian source in the presence of correlated noise, modeled by a two-state Gauss-Markov process. In the proposed approach, first, the state of the noise process at time epoch k is determined according to the maximum a posteriori (MAP) decision rule based on the whole sequence of observations. Then, given the hard decision on the noise state, the corresponding MMSE estimator (i.e., conditional mean) is employed to estimate the source signal at time epoch k . However, performing estimation based on the hard decision of the MAP detector is not optimal in terms of minimizing the mean-squared-error (MSE). In this comment, we show that the optimal MMSE estimator for the considered problem is obtained as a weighted average of the MMSE estimators corresponding to distinct states of the noise probable at time epoch k and the weights are equal to the a posteriori probabilities of the noise state computed based on the whole sequence of observations.

On the limiting spectral density of random matrices filled with stochastic processes

Abstract We discuss the limiting spectral density of real symmetric random matrices. In contrast to standard random matrix theory, the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic process. Under assumptions on this process which are satisfied, e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures on finite state spaces, or by Gaussian Markov processes, we show that the limiting spectral distribution depends on the way the matrix is filled with the stochastic process. If the filling is in a certain way compatible with the symmetry condition on the matrix, the limiting law of the empirical eigenvalue distribution is the well-known semi-circle law. For other fillings we show that the semi-circle law cannot be the limiting spectral density.

Variance swaps with double exponential Ornstein-Uhlenbeck stochastic volatility

Recent empirical studies find that with a high sampling frequency, stochastic volatility models can be differentiated from each other and the log-normal stochastic volatility model can capture the volatility of real market better than the Heston model. Unless certain restriction such as fast mean reversion or zero mean reversion is assumed, however, the log-normal stochastic volatility model does not allow an analytic solution for the characteristic function. When we consider derivative products such as variance swaps, the restriction may cause a significant gap between theoretical price and actual price. On the other hand, there are a good amount of studies showing that a multi-factor stochastic volatility model can capture the market more accurately than a single-factor stochastic volatility model. So, in this paper, we propose a two-factor log normal stochastic volatility model in which log volatility is given by an Ornstein-Uhlenbeck process which is a mean-reverting, temporally homogeneous, Markov Gaussian process. We derive an exact semi-analytic solution as well as an approximate analytic solution for the fair strike price of discretely sampled variance swap by using the first and second moments of the moment generating function. We check the validity of our solution through Monte Carlo simulation and show how to estimate the model parameters by implementing calibration to real market data and compare our two-factor model with the one-factor version and also the Heston and double Heston models. Furthermore, the sample discretization risk is examined for the four different models.

Sum-Product Unmixing for Hyperspectral Analysis With Endmember Variability

Models of endmember variability capture the notion that multiple spectra may represent a single class or material, and while these models are physically realistic, they often give rise to excessive computational complexity during the spectral unmixing process. In this letter, we present a computationally tractable supervised unmixing method that uses Gauss–Markov processes to model endmember variability and interband correlation properties within endmembers. We use a probabilistic graphical model over multiple Gauss–Markov processes to capture the mixing effects of a spectral sensor and employ sum-product message passing to develop an accelerated statistical unmixing algorithm. The computational complexity of the proposed unmixing algorithm is only linear in the number of bands, making it suitable for both hyperspectral (hundreds of bands) and ultraspectral (thousands of bands) applications. Unmixing examples with measured reflectance spectra show sizable performance improvements when accounting for interband correlation using the proposed method, and empirical results quantify orders of magnitude reduction in complexity compared to alternative methods.

Probability Characteristics of the Absolute Maximum of the Discontinuous Homogeneous Gaussian Random Field

A new method is introduced in order to determine the limiting characteristics of the homogeneous Gaussian random field with the non-differentiable correlation function at zero. This method is based on the representation of a field in the small neighborhood of the maximum point by the superposition of the statistically independent locally Markov Gaussian random processes. Then, using the expressions for the distribution function of the Markov Gaussian process, it is possible to obtain the asymptotic approximations for both the probability of the threshold crossing by the absolute maximum of the homogeneous Gaussian random field and the distribution function of the absolute maximum of the field. It is shown that the proposed approximations are monotonic and valid in a much wider range of both threshold values and the area of the definitional domain of the random field, unlike the commonly known approximations. The applicability borders of the introduced theoretical formulas are established by means of the statistical computer simulation.

가우스-마르코프 프로세스 기반 랜덤 도로 조도 측정

This paper presents the characterization of a random road roughness based on the direct measurements of the road roughness using a laser-type displacement sensor. A road profile is typically classified by its wavelength into micro texture, macro texture, mega texture, and road roughness. Since the road roughness acts as the disturbance input to the suspension system, it is essential to characterize the properties of the road roughness for analyzing the ride quality and handling the characteristics of the vehicle. To model the interaction between the tire and road excitation, the road roughness is assumed as the Gauss–Markov process in this study. The measurement results of the road roughness show how the road surface can act as a disturbance input to the vehicle suspension systems, and a good agreement with the Gauss–Markov process. The road roughness model based on the Gauss–Markov process can be utilized for Monte Carlo simulation and the process noise modeling of discrete Kalman filter in the suspension system.

Accuracy enhancement of high-rate GNSS positions using a complete ensemble empirical mode decomposition-based multiscale multiway PCA

In order to improve the possibility of correct interpretation of some geophysical phenomena such seismology, it is important to reduce systematic errors such as common mode error (CME) and multipath errors, and to improve the accuracy of high-rate (1–5 Hz) displacements. This paper focuses on accuracy enhancement and noise assessment of high-rate Global Navigation Satellite System (GNSS) position time series of regional network of GNSS stations. Complete ensemble empirical mode decomposition-based multiscale multiway principal component analysis (C-MSMPCA) denoising method is introduced to denoise the high-rate (1–5 Hz) GNSS position time series, whose performance was compared with those of modified sidereal filtering (MSF), stacking filtering, PCA and the recently developed multiscale multiway PCA (MSMPCA). It turns out that C-MSMPCA can significantly eliminate the high-frequency random noise and remove the low-frequency CME, and other systematic errors such as multipath errors. Furthermore, for denoising high-rate GNSS time series, C-MSMPCA is more precise than the other denoising methods. Irrespective of sampling rates and data processing strategies, C-MSMPCA can respectively reach the accuracy of submillimeter- and millimeter-levels on the horizontal and vertical components of high-rate GNSS displacements. Spectral analysis on the 1 Hz coordinate time series from the Sichuan GNSS Continuous Observing Network of China suggests that Gauss-Markov process is an adequate noise model for the noise characteristics of the filtered GNSS data using MSMPCA and C-MSMPCA. The CME magnitudes of MSMPCA are 2.61, 2.55, and 6.72 mm (east, north, and vertical), and those of C-MSMPCA are 2.69, 2.80, and 7.80 mm, respectively.

The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator

We present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quantitative analysis of the above model based on the probabilistic limit theorem. If the predators’ gains and the time intervals between predator attacks are sufficiently small, our proposed method allows us to derive an approximative average dynamical system for mathematical expectations of population dynamics and the stochastic Ito differential equation for the random deviations from the average motion. Assuming that the averaged dynamical system is the classic Holling type II population model with asymptotically stable limit cycle, we prove that the dynamics of stochastic model may be approximated with a two-dimensional Gaussian Markov process with unboundedly increasing variances.

Stochastic Dynamics of and Collision Prediction for Low Altitude Earth Satellites

Air drag B∗ factors from Earth satellite element sets often show the characteristic near Gaussian distribution and autocorrelation exponential decay typical of a Gauss-Markov process. Assuming the “most current” set of orbital elements are correct, earlier elements can be used to construct covariance matrices as a function of prediction time into the future. If resolved in cylindrical orbit frame coordinates, these are remarkably structured, essentially showing only in-track error growth. Often the in-track position covariance element growth follows a fourth power in time rule, and is definitely forced by the uncertainty in the air drag factor. This observation is confirmed both by perturbation theory and by modeling stochastic state covariance propagation. Realizing that almost all error growth under the SGP4 model is in track, the Cosmos 2251/Iridium 33 event is reexamined. While a collision prediction from the last elements shows a minimum miss distance of about 700 m, those same elements show a closest approach distance of the orbits of only 32 m. Given large in-track uncertainty, minimum orbit separation may be a much more reliable metric for maneuver decisions.

On Optimizing Feedback Interval for Temporally Correlated MIMO Channels With Transmit Beamforming and Finite-Rate Feedback

A receiver with perfect channel state information (CSI) in a point-to-point multiple-input multiple-output (MIMO) channel can compute the transmit beamforming vector that maximizes the transmission rate. For frequency-division duplex, a transmitter is not able to estimate CSI directly and has to obtain a quantized transmit beamforming vector from the receiver via a rate-limited feedback channel. We assume that time evolution of MIMO channels is modeled as a Gauss-Markov process parameterized by a temporal-correlation coefficient. Since feedback rate is usually low, we assume rank-one transmit beamforming or transmission with single data stream. For given feedback rate, we analyze the optimal feedback interval that maximizes the average received power of the systems with two transmit or two receive antennas. For other system sizes, the optimal feedback interval is approximated by maximizing the rate difference in a large system limit. Numerical results show that the large system approximation can predict the optimal interval for finite-size system quite accurately. Numerical results also show that quantizing transmit beamforming with the optimal feedback interval gives larger rate than the existing Kalman-filter scheme does by as much as 10% and than feeding back for every block does by 44% when the number of feedback bits is small.

Asymptotics of Two-boundary First-exit-time Densities for Gauss-Markov Processes

The problem of escape times from a region confined by two time-dependent boundaries is considered for a class of Gauss-Markov processes. Asymptotic approximations of the first exit time probability density functions in case of asymptotically constant and asymptotically periodic boundaries are obtained firstly for the Ornstein-Uhlenbeck process and then extended to the class of Gauss-Markov processes that can be obtained by a specified transformation. Some examples of application to stochastic dynamics and estimations of involved parameters by using numerical approximations are provided.

Integrated stationary Ornstein–Uhlenbeck process, and double integral processes

We find a representation of the integral of the stationary Ornstein–Uhlenbeck (ISOU) process in terms of Brownian motion Bt; moreover, we show that, under certain conditions on the functions f and g, the double integral process (DIP) D(t)=∫βtg(s)∫αsf(u)dBuds can be thought as the integral of a suitable Gauss–Markov process. Some theoretical and application details are given, among them we provide a simulation formula based on that representation by which sample paths, probability densities and first passage times of the ISOU process are obtained; the first-passage times of the DIP are also studied.

A Truncated Prediction Framework for Streaming Over Erasure Channels

We propose a new coding technique for sequential transmission of a stream of Gauss–Markov sources over erasure channels under a zero decoding delay constraint. Our proposed scheme is a combination (hybrid) of predictive coding with truncated memory, and quantization-and-binning. We study the optimality of our proposed scheme using an information theoretic model. In our setup, the encoder observes a stream of source vectors that are spatially independent and identically distributed (i.i.d.) and temporally sampled from a first-order stationary Gauss–Markov process. The channel introduces an erasure burst of a certain maximum length $B$ , starting at an arbitrary time, not known to the transmitter. The reconstruction of each source vector at the destination must be with zero delay and satisfy a quadratic distortion constraint with an average distortion of $D$ . The decoder is not required to reconstruct those source vectors that belong to the period spanning the erasure burst and a recovery window of length $W$ following it. We study the minimum compression rate $R(B,W,D)$ in this setup. As our main result, we establish upper and lower bounds on the compression rate. The upper bound (achievability) is based on our hybrid scheme. It achieves significant gains over baseline schemes such as (leaky) predictive coding, memoryless binning, a separation-based scheme, and a group of pictures-based scheme. The lower bound is established by observing connection to a network source coding problem. The bounds simplify in the high resolution regime, where we provide explicit expressions whenever possible, and identify conditions when the proposed scheme is close to optimal. We finally discuss the interplay between the parameters of our burst erasure channel and the statistical channel models and explain how the bounds in the former model can be used to derive insights into the simulation results involving the latter. In particular, our proposed scheme outperforms the baseline schemes over the i.i.d. erasure channel and the Gilbert–Elliott channel, and achieves performance close to a lower bound in some regimes.

Generating random variates from PDF of Gauss–Markov processes with a reflecting boundary

Algorithms to generate random variates from probability density function of Gauss–Markov processes restricted by special lower reflecting boundary are formulated. They are essentially obtained by means of discretizations of stochastic equations or via acceptance–rejection methods. Particular attention is dedicated to restricted Wiener and Ornstein–Uhlenbeck processes.

Non-stationary sources separation based on maximum likelihood criterion using source temporal–spatial model

Abstract In this paper, we propose a new non-stationary sources separation algorithm, which is referred to as autoregressive hidden Markov Gaussian process (AR-HMGP), in which the sources are non-stationary and temporally correlated. For the proposed algorithm, a generative model is employed to track the non-stationarity of the source where the temporal dependencies of sources are represented by autoregressive model (AR) and the distribution of the associated innovation process is described using non-stationary Gaussian process with hidden Markov model (HMM). We further explore the maximum likelihood (ML) method to estimate the parameters of the source model by using the expectation maximum (EM) algorithm. Our important findings reveal that (a) AR-HMGP algorithm outperforms the other three BSS algorithms for non-stationary sources separation, the instantaneous mixture system is also well corroborated with the effectiveness of our algorithm; (b) both independent and dependent non-stationary sources have been successfully separated; (c) the proposed algorithm is robust with respect to noise, while the other three algorithms are not.

Colored noise and a stochastic fractional model for correlated inputs and adaptation in neuronal firing.

High variability in the neuronal response to stimulations and the adaptation phenomenon cannot be explained by the standard stochastic leaky integrate-and-fire model. The main reason is that the uncorrelated inputs involved in the model are not realistic. There exists some form of dependency between the inputs, and it can be interpreted as memory effects. In order to include these physiological features in the standard model, we reconsider it with time-dependent coefficients and correlated inputs. Due to its hard mathematical tractability, we perform simulations of it for a wide investigation of its output. A Gauss-Markov process is constructed for approximating its non-Markovian dynamics. The first passage time probability density of such a process can be numerically evaluated, and it can be used to fit the histograms of simulated firing times. Some estimates of the moments of firing times are also provided. The effect of the correlation time of the inputs on firing densities and on firing rates is shown. An exponential probability density of the first firing time is estimated for low values of input current and high values of correlation time. For comparison, a simulation-based investigation is also carried out for a fractional stochastic model that allows to preserve the memory of the time evolution of the neuronal membrane potential. In this case, the memory parameter that affects the firing activity is the fractional derivative order. In both models an adaptation level of spike frequency is attained, even if along different modalities. Comparisons and discussion of the obtained results are provided.