Additive Noise Process Research Articles

Unscented-Kalman-Filter-Based Remote State Estimation for Complex Networks With Quantized Measurements and Amplify-and-Forward Relays.

In this article, the remote estimation problem is addressed for a class of discrete-time complex networks under the influence of probabilistic quantization and amplify-and-forward (AF) relays. The underlying complex network model, which is inherently nonlinear and stochastic, is affected by additive process and measurement noises. Owing to the limited bandwidth of the transmission channel, the measurement outputs are quantized by a probabilistic quantizer prior to transmission. To enhance the signal quality over long-distance transmissions, the quantized measurements are sent to AF relays and subsequently forwarded to the estimator. Utilizing the unscented Kalman filter approach, a novel state estimator is designed to minimize an upper bound on the estimation error covariance. Moreover, sufficient conditions are derived to ensure that the estimation error is exponentially bounded in the mean-square sense. Lastly, the efficacy of the proposed scheme is illustrated through numerical simulations.

Denoising Non-Stationary Signals via Dynamic Multivariate Complex Wavelet Thresholding.

Over the past few years, we have seen an increased need to analyze the dynamically changing behaviors of economic and financial time series. These needs have led to significant demand for methods that denoise non-stationary time series across time and for specific investment horizons (scales) and localized windows (blocks) of time. Wavelets have long been known to decompose non-stationary time series into their different components or scale pieces. Recent methods satisfying this demand first decompose the non-stationary time series using wavelet techniques and then apply a thresholding method to separate and capture the signal and noise components of the series. Traditionally, wavelet thresholding methods rely on the discrete wavelet transform (DWT), which is a static thresholding technique that may not capture the time series of the estimated variance in the additive noise process. We introduce a novel continuous wavelet transform (CWT) dynamically optimized multivariate thresholding method (WaveL2E). Applying this method, we are simultaneously able to separate and capture the signal and noise components while estimating the dynamic noise variance. Our method shows improved results when compared to well-known methods, especially for high-frequency signal-rich time series, typically observed in finance.

Constrained profile estimation for distributed parameter system in one dimension using orthogonal collocation

Several industrial processes fall under the category of Distributed parameter systems (DPS) which are represented as coupled partial differential equations (PDEs). Obtaining accurate and physically meaningful profile estimates of states despite only a limited number of spatially distributed measurements, is a pre-requisite for adequately controlling such systems. In this work, we propose an online constrained state profile estimation approach for DPS. The approach is based on reduced-dimension model represented by differential and algebraic equations (DAE) which in turn is developed by discretization of the PDEs along the spatial coordinate using the orthogonal collocation (OC) method. A profile prediction error based approach is proposed for systematically fixing the dimension of the OC model to be used. Further, to compensate for the inevitable model plant mismatch (MPM) arising in the process of the dimensionality reduction, additive zero mean Gaussian white noise processes are introduced in both state dynamics as well as measurement model. Statistics of these noises are obtained based on simulation based realizations of approximation errors at collocation points and measurement locations. To incorporate the identified correlation between the resulting state and measurement noises in the estimation method, we propose a constrained DAE-EKF (extended Kalman Filter) extension to deal with such correlations. The estimation approach also involves an appropriately formulated optimization based update step to ensure constraint satisfaction over the entire spatial profile leading to physically meaningful state profile estimates. The constrained profile estimator is also integrated with an online moving window based parameter estimation, which enables tracking of both time varying parameters and state profiles. The efficacy of the proposed approach is demonstrated by conducting stochastic simulation studies using two tubular reactor systems. The results show that the proposed constrained online state estimation approach is able to obtain accurate profile estimates at low computational cost, while ensuring that the profile estimates satisfy the given constraints.

On the emergence of traffic jams in a stochastic traffic flow driven by additive and multiplicative white Gaussian noise processes

Understanding the influence of random phenomena on the emergence of jams in vehicular traffic flows is a fundamental question in the study of traffic flow dynamics. Motivated by the fact that additive and multiplicative white Gaussian noise processes represent fundamentally distinct sources of randomness that can engender qualitatively different patterns in the dynamics of traffic flow, here we compare and contrast the influences of those two distinct stochastic processes on the formation of traffic jams. Analyzing the macroscopic dynamics of spatially homogeneous single-lane traffic flow using a car distribution function that evolves according to a Fokker–Planck type equation, we first obtain the equilibrium distribution function in closed form for traffic flow driven by additive as well as multiplicative white Gaussian noise. We then derive the mean first passage time (MFPT) for both cases and also present results that elucidate the influence of varying noise strength on both the equilibrium distribution function and the MFPT. Additionally, we obtain fundamental diagrams corresponding to speed versus flux of vehicle flow for both cases that further highlight the distinct effects of the two types of noise. In summary, the results underscore the contrasting influences of additive and multiplicative noise processes on the emergence of traffic jams and thereby contribute to advancing our understanding of this aspect of stochastic traffic flow. The findings could also help improve our understanding of how randomness affects traffic flows that are a combination of human and self-driven vehicles.

Multivariate Estimator for Linear Dynamical Systems with Additive Laplace Measurement and Process Noises

Since uncertainties in many physical systems have impulsive properties poorly modeled by Gaussian distributions, heavier-tailed distributions, such as Laplace, may be used to improve model characteristics. From insights obtained through development of the scalar Laplace estimator, an algorithm is determined for the vector-state case. For a discrete-time vector linear system with scalar additive Laplace-distributed process and measurement noises, the a priori and a posteriori conditional probability density functions (pdfs) of the system states given the measurement history are propagated recursively and in closed form. The conditional pdfs are composed of signs and absolute values of affine functions, and a basis composed of signs of affine functions is constructed to simplify their representation. The a posteriori conditional mean and variance are derived analytically from the conditional pdf using characteristic functions. Generalization to independent vector measurement and process noises is straightforward. From the general method for deriving Laplace estimators in $n$-dimensions, a two-dimensional minimum variance estimator is explicitly developed, and a simulation is presented.

Laplace estimation for scalar linear systems

Uncertainties in many physical systems have impulsive properties poorly modeled by Gaussian distributions. Refocusing previous work, an estimator is derived for a scalar discrete-time linear system with additive Laplace measurement and process noises. The a priori and a posteriori conditional probability density functions (pdf) of the state given a measurement sequence are propagated recursively and in closed form, and the a posteriori conditional mean and variance are derived analytically from the conditional pdf. A simulation for an estimator is presented, demonstrating marked resilience to large, un-modeled spikes in the measurements.

Control-Based Continuation: A New Approach to Prototype Synthetic Gene Networks.

Control-Based Continuation (CBC) is a general and systematic method to carry out the bifurcation analysis of physical experiments. CBC does not rely on a mathematical model and thus overcomes the uncertainty introduced when identifying bifurcation curves indirectly through modeling and parameter estimation. We demonstrate, in silico, CBC applicability to biochemical processes by tracking the equilibrium curve of a toggle switch, which includes additive process noise and exhibits bistability. We compare the results obtained when CBC uses a model-free and model-based control strategy and show that both can track stable and unstable solutions, revealing bistability. We then demonstrate CBC in conditions more representative of an in vivo experiment using an agent-based simulator describing cell growth and division, cell-to-cell variability, spatial distribution, and diffusion of chemicals. We further show how the identified curves can be used for parameter estimation and discuss how CBC can significantly accelerate the prototyping of synthetic gene regulatory networks.

Adaptive Kalman Filtering for Recursive Both Additive Noise and Multiplicative Noise

This paper focuses on adaptive Kalman filtering problem for linear systems with unknown covariances of both dynamic multiplicative noise (multiplicative measurement noise) and additive noises (additive process and measurement noises). A recursive-noise adaptive Kalman filter is proposed to estimate both states and covariances of noises by using the varaitional Bayesian (VB) inference and an indirect method. First, we characterize inverse Wishart priors for both measurement noise covariance and process noise covariance and employ the Student's t-distribution to represent the likelihood function, which is non-Gaussian and affected by mixing multiplicative noise and additive measurement noise. Then, an adaptive Kalman filtering for recursive both noise covariance matrices and dynamic state, is proposed following VB inference. Performance analysis for VB procedures and the proposed filter is provided to ensure the convergence and stability. A target tracking example is provided to validate the effectiveness of the proposed filtering algorithm.

Consensus error analysis for linear continuous-time multi-agent systems with additive noises

This paper investigates the distributed consensus problem for linear continuous-time multi-agent systems subject to additive process noises. Due to the existence of noises, the asymptotic consensus cannot be achieved. Hence, a distributed consensus protocol is proposed such that all the agents can achieve consensus to some degree. Furthermore, based on Lyapunov theory, we characterize the consensus error under undirected and directed topologies. Numerical examples are presented to verify the effectiveness of two results.

Simplified augmented cubature information filtering and multi-sensor fusion for additive noise systems

For a highly nonlinear system with additive process and observation noises, the non-augmented sigma-point nonlinear filter induces a loss of odd-order moment information (skewness), thereby resulting in a degradation of state estimation accuracy. To address this problem, we present a novel filtering algorithm, namely augmented cubature information filter (ACIF). The adopted augmentation strategy can facilitate the estimator to capture and propagate higher odd-order moment information of random variables. In addition, the covariance resulting from linearization errors is compensated based on the proposed filtering framework. Further, the ACIF is extended to the decentralized multi-sensor system. The employed modified state augmentation strategy can eliminate the inconsistency of covariance estimation induced by the state-mean samples. And the propagation of cubature points is simplified to reduce the computational cost associated with state augmentation. To validate the proposed algorithm, a comparative study is performed via Monte Carlo simulation in the scenario of tracking a maneuvering target. The results show that the simplified ACIF can improve the filtering efficiency without dramatically increasing the computational burden.

An efficient and statistically accurate Lagrangian data assimilation algorithm with applications to discrete element sea ice models

Lagrangian data assimilation of complex nonlinear turbulent flows is an important but computationally challenging topic. In this article, an efficient data-driven statistically accurate reduced-order modeling algorithm is developed that significantly accelerates the computational efficiency of Lagrangian data assimilation. The algorithm starts with a Fourier transform of the high-dimensional flow field, which is followed by an effective model reduction that retains only a small subset of the Fourier coefficients corresponding to the energetic modes. Then a linear stochastic model is developed to approximate the nonlinear dynamics of each Fourier coefficient. Effective additive and multiplicative noise processes are incorporated to characterize the modes that exhibit Gaussian and non-Gaussian statistics, respectively. All the parameters in the reduced order system, including the multiplicative noise coefficients, are determined systematically via closed analytic formulae. These linear stochastic models succeed in forecasting the uncertainty and facilitate an extremely rapid data assimilation scheme. The new Lagrangian data assimilation is then applied to observations of sea ice floe trajectories that are driven by atmospheric winds and turbulent ocean currents. It is shown that observing only about 30 non-interacting floes in a 200 km×200 km domain is sufficient to recover the key multi-scale features of the ocean currents. The additional observations of the floe angular displacements are found to be suitable supplements to the center-of-mass positions for improving the data assimilation analysis. In addition, the observed large and small floes are more useful in recovering the large- and small-scale features of the ocean, respectively. The Fourier domain data assimilation also succeeds in recovering the ocean features in the areas where cloud cover obscures the observations. Finally, the multiplicative noise is shown to be crucial in recovering extreme events.

Multilevel RTN Removal Tools for Dynamic FBG Strain Measurements Corrupted by Peak-Splitting Artefacts.

Strain measurements using fibre Bragg grating (FBG) optical sensors are becoming ever more commonplace. However, in some cases, these measurements can become corrupted by sudden jumps in the signal, which manifest as spikes or step-like offsets in the data. These jumps are caused by a defect in the FBG itself, which is referred to as peak-splitting. The effects of peak splitting artefacts on FBG strain measurements show similarities with an additive multi-level telegraph noise process, in which the amplitudes and occurrences of the jumps are related to fibre deformation states. Whenever it is not possible to re-assess the raw spectral data with advanced peak tracking software, other means for removing the jumps from the data have to be found. The two methods presented in this article are aimed at removing additive multi-level random telegraph noise (RTN) from the raw data. Both methods are based on denoising the sample wise difference signal using a combination of an outlier detection scheme followed by an outlier replacement step. Once the difference signal has been denoised, the cumulative sum is used to arrive back at a strain time series. Two methods will be demonstrated for reconstructing severely corrupted strain time series; the data for this verification has been collected from sub-soil strain measurements obtained from an operational offshore wind-turbine. The results show that the proposed methods can be used effectively to reconstruct the dynamic content of the corrupted strain time series. It has been illustrated that errors in the outlier replacements accumulate and can cause a quasi-static drift. A representative mean value and drift correction are proposed in terms of an optimization problem, which maximizes the overlap between the reconstruction and a subset of the raw data; whereas a high-pass filter is suggested to remove the quasi static drift if only the dynamic band of the signal is of interest.

On the Capacity Enlargement of Gaussian Broadcast Channels With Passive Noisy Feedback

It is well known that the capacity region of an average transmit power constrained Gaussian Broadcast Channel (GBC) with independent noise realizations at the receivers is enlarged by the presence of causal noiseless feedback. When the noise variances at the receivers are identical, even passive feedback via independent memoryless Gaussian links can lead to a capacity region enlargement. The last fact remains true even when the feedback noise variance is very high, and available only from one of the receivers. While such capacity enlargements are feasible for several other feedback models in the Gaussian BC setting, it is also known that feedback does not change the capacity region for physically degraded broadcast channels. In this paper, we consider a two user GBC with independent noise realizations at the receivers, where the feedback links from the receivers are corrupted by independent additive Gaussian noise processes. We investigate the set of four noise variances, two forward and two feedback, for which no capacity enlargement is possible. A sharp characterization of this region is derived, i.e., any quadruple outside the presented region will lead to a capacity enlargement, whereas quadruples inside will leave the capacity region unchanged. Our results lead to the conclusion that when the forward noise variances are different, too noisy a feedback from one of the receivers alone is not always beneficial for enlarging the capacity region, be it from the stronger user or the weaker one, in sharp contrast to the case of equal forward noise variances.

Synchronization of stochastic lattice equations and upper semicontinuity of attractors

We consider a system of two coupled stochastic lattice equations driven by additive white noise processes, where the strength of the coupling is given by a parameter We show that these equations generate a random dynamical system which has a random pullback attractor. This attractor naturally depends on the parameter κ. When the intensity of the coupling becomes large, we observe that the components of the given system synchronize. To describe this phenomenon, we prove the upper semicontinuity of the family of attractors with respect to the attractor of a specific limiting system.

Reinforcement Learning for Robust Trajectory Design of Interplanetary Missions

This paper investigates the use of reinforcement learning for the robust design of low-thrust interplanetary trajectories in presence of severe uncertainties and disturbances, alternately modeled as Gaussian additive process noise, observation noise, and random errors in the actuation of the thrust control, including the occurrence of a missed thrust event. The stochastic optimal control problem is recast as a time-discrete Markov decision process to comply with the standard formulation of reinforcement learning. An open-source implementation of the state-of-the-art algorithm Proximal Policy Optimization is adopted to carry out the training process of a deep neural network, used to map the spacecraft (observed) states to the optimal control policy. The resulting guidance and control network provides both a robust nominal trajectory and the associated closed-loop guidance law. Numerical results are presented for a typical Earth–Mars mission. To validate the proposed approach, the solution found in a (deterministic) unperturbed scenario is first compared with the optimal one provided by an indirect technique. The robustness and optimality of the obtained closed-loop guidance laws is then assessed by means of Monte Carlo campaigns performed in the considered uncertain scenarios.

Estimation for heterogeneous traffic using enhanced particle filters

This article explores the state estimation problem for heterogeneous traffic (vehicles with distinct driving behaviours) using particle filtering (PF) approaches. We consider three variations of PF to enhance estimation. The benchmark PF utilises a deterministic partial differential equation and a state-independent additive process noise. We first consider a parameter-adaptive PF variation that also allows model parameters to be adjusted. The second variation is a PF with spatially-correlated noise. The last variation combines parameter-adaptive and the spatially-correlated-noise approaches. We compare the four filters in numerical experiments that represent heterogeneous traffic scenarios and on real-world heterogeneous traffic data. The results show that the enhanced filters can achieve up to an 80% and 46% of accuracy improvement as compared to an open loop simulation without measurement correction, with the synthetic settings and with real traffic data, respectively. Moreover, the enhanced filters outperform the standard PF in all the traffic scenarios based on accuracy.

Multiscale Optimal Filtering on the Sphere

We present a framework for the optimal filtering of spherical signals contaminated by realizations of an additive, zero-mean, uncorrelated and anisotropic noise process on the sphere. Filtering is performed in the wavelet domain given by the scale-discretized wavelet transform on the sphere. The proposed filter is optimal in the sense that it minimizes the mean square error between the filtered wavelet representation and wavelet representation of the noise-free signal. We also present a simplified formulation of the filter for the case when azimuthally symmetric wavelet functions are used. We demonstrate the use of the proposed optimal filter for denoising of an Earth topography map in the presence of additive, zero-mean, uncorrelated and white Gaussian noise, and show that the proposed filter performs better than the hard thresholding method and weighted spherical harmonic (weighted-SPHARM) signal estimation framework.

A model for Dansgaard–Oeschger events and millennial-scale abrupt climate change without external forcing

We propose a conceptual model which generates abrupt climate changes akin to Dansgaard-Oeschger events. In the model these abrupt climate changes are not triggered by external perturbations but rather emerge in a dynamic self-consistent model through complex interactions of the ocean, the atmosphere and an intermittent process. The abrupt climate changes are caused in our model by intermittencies in the sea-ice cover. The ocean is represented by a Stommel two-box model, the atmosphere by a Lorenz-84 model and the sea-ice cover by a deterministic approximation of correlated additive and multiplicative noise (CAM) process. The key dynamical ingredients of the model are given by stochastic limits of deterministic multi-scale systems and recent results in deterministic homogenisation theory. The deterministic model reproduces statistical features of actual ice-core data such as non-Gaussian $\alpha$-stable behaviour. The proposed mechanism for abrupt millenial-scale climate change only relies on the existence of a quantity, which exhibits intermittent dynamics on an intermediate time scale. We consider as a particular mechanism intermittent sea-ice cover where the intermittency is generated by emergent atmospheric noise. However, other mechanisms such as freshwater influxes may also be formulated within the proposed framework.

Mismatched Models to Lower Bound the Capacity of Optical Fiber Channels

A correlated phase-and-additive-noise (CPAN) mismatched model is developed for wavelength division multiplexing over optical fiber channels governed by the nonlinear Schr\"odinger equation. Both the phase and additive noise processes of the CPAN model are Gauss-Markov whereas previous work uses Wiener phase noise and white additive noise. Second order statistics are derived and lower bounds on the capacity are computed by simulations. The CPAN model characterizes nonlinearities better than existing models in the sense that it achieves better information rates. For example, the model gains 0.35 dB in power at the peak data rate when using a single carrier per wavelength. For multiple carriers per wavelength, the model combined with frequency-dependent power allocation gains 0.14 bits/s/Hz in rate and 0.8 dB in power at the peak data rate.

Bounds on the Sum-Rate of MIMO Causal Source Coding Systems with Memory under Spatio-Temporal Distortion Constraints.

In this paper, we derive lower and upper bounds on the OPTA of a two-user multi-input multi-output (MIMO) causal encoding and causal decoding problem. Each user’s source model is described by a multidimensional Markov source driven by additive noise process subject to three classes of spatio-temporal distortion constraints. To characterize the lower bounds, we use state augmentation techniques and a data processing theorem, which recovers a variant of rate distortion function as an information measure known in the literature as nonanticipatory -entropy, sequential or nonanticipative RDF. We derive lower bound characterizations for a system driven by an Gaussian noise process, which we solve using the SDP algorithm for all three classes of distortion constraints. We obtain closed form solutions when the system’s noise is possibly non-Gaussian for both users and when only one of the users is described by a source model driven by a Gaussian noise process. To obtain the upper bounds, we use the best linear forward test channel realization that corresponds to the optimal test channel realization when the system is driven by a Gaussian noise process and apply a sequential causal DPCM-based scheme with a feedback loop followed by a scaled ECDQ scheme that leads to upper bounds with certain performance guarantees. Then, we use the linear forward test channel as a benchmark to obtain upper bounds on the OPTA, when the system is driven by an additive non-Gaussian noise process. We support our framework with various simulation studies.