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We consider the problem of estimating the state of a linear time-invariant Gaussian system using $N$ sensors, where a subset of the sensors can potentially be compromised by an adversary. In this case, locating the compromised sensors is of crucial importance for obtaining an accurate state estimate. Inspired by the clustering algorithm in machine learning, we propose a Gaussian-mixture-model-based (GMM-based) detection mechanism. It clusters the local state estimate autonomously and provides a belief for each sensor, based on which measurements from different sensors can be fused accordingly. When a subset of the sensors are under the optimal innovation-based deception attacks, we derive the remote estimation error covariance recursions under different detection mechanisms, e.g., distributed $\chi ^2$ false-data detector, centralized $\chi ^2$ false-data detector, and GMM-based detection algorithm. The performance of the proposed GMM-based detection algorithm is further evaluated through average belief in the same attack scenario. Moreover, we discuss applications of GMM-based detection algorithm on other attack scenarios, e.g., false-data injection attack, replay attack, and $\epsilon$ -stealthy attack. Simulation examples are provided to demonstrate the developed results. [ABSTRACT FROM AUTHOR]

This paper analyzes the tracking performance of the family of adaptive filtering algorithms based on minimizing the ${{2}}L$ -th moment of the estimation error, with $L$ being an integer greater than 1. The analysis is done for a Markov plant without the usually used assumption of small degree of non-stationarity (DNS). General $L$ and DNS are considered in the analysis. Dependence of the minimum steady-state mean square weight deviation on the values of $L$ , DNS and type of the noise distribution is studied. A measure for evaluating the steady-state stability of the algorithm is introduced. Dependence of the steady-state stability on the values of $L$ , DNS, and type of the noise distribution is studied. Some results are surprising. Theoretical results are supported by simulations. [ABSTRACT FROM AUTHOR]

In Gaussian measurement models, the measurements are given by a known function of the unknown parameter vector, contaminated by additive zero-mean Gaussian noise. When the function is linear, the resulting maximum likelihood estimate (MLE) is well-known to be efficient [unbiased, with a mean square estimation error (MSE) matrix attaining the Cramér–Rao lower bound (CRLB)]. However, when the function is nonlinear, the MLE is only asymptotically efficient. The classical derivation of its asymptotic efficiency uses a first-order perturbation analysis, relying on a “small-errors” assumption, which under subasymptotic conditions turns inaccurate, rendering the MLE generally biased and inefficient. Although a more accurate (higher-order) performance analysis for such cases is of considerable interest, the associated derivations are rather involved, requiring cumbersome notations and indexing. Building on the recent assimilation of tensor computations into signal processing literature, we exploit the tensor formulation of higher-order derivatives to derive a tractable formulation of a higher (up to third-) order perturbation analysis, predicting the bias and MSE matrix of the MLE of parameter vectors in general nonlinear models under subasymptotic conditions. We provide explicit expressions depending on the first three derivatives of the nonlinear measurement function, and demonstrate the resulting ability to predict the “efficiency gap” (relative excess MSE beyond the CRLB) in simulation experiments. We also provide MATLAB code for easy computation of our resulting expressions. [ABSTRACT FROM AUTHOR]

Cascade dynamic nonlinear systems can describe a wide class of engineering problems, but little efforts have been devoted to the identification of such systems so far. One of the difficulties comes from its non-convex characteristic. In this paper, the identification of a general cascade dynamic nonlinear system is rearranged and transformed into a convex problem involving a double-input single-output nonlinear system. In order to limit the estimate error at the frequencies of interest and to overcome the singularity problem incurred in the least-square-based methods, the identification problem is, thereafter, decomposed into a multi-objective optimization problem, in which the objective functions are defined in terms of the spectra of the unbiased error function at the frequencies of interest and are expressed as a first-order polynomial of the model parameters to be identified. The coefficients of the first-order polynomial are derived in an explicit expression involving the system input and the measured noised output. To tackle the convergence performance of the multi-objective optimization problem, the bargaining game theory is used to model the interactions and the competitions among multiple objectives defined at the frequencies of interest. Using the game-theory-based approach, both the global information and the local information are taken into account in the optimization, which leads to an obvious improvement of the convergence performance. Numerical studies demonstrate that the proposed bargaining-game-theory-based algorithm is effective and efficient for the multi-objective optimization problem, and so is the identification of the cascade dynamic nonlinear systems. [ABSTRACT FROM AUTHOR]

This paper investigates the distributed federated Tobit Kalman filter fusion problem for a class of discrete-time systems subject to measurement censoring and packet delays. The censored measurement is characterized by the Tobit measurement model and the packet delay obeys the Poisson distribution. At the input terminal of each local estimator, a censoring detection device and a buffer with finite length are exploited to tackle the censored/delayed measurements. The filtering fusion is carried out in two steps: in the first step, every sensor in the network collects and sends its measurements to the corresponding local estimator via an unreliable network with probabilistic packet delays and, in the second step, the estimates from local estimators are further collected by the fusion center to form a fused estimate. The local estimator runs a modified Tobit Kalman filtering algorithm, which is designed based on the modified Tobit regression model, while the fusion center performs a distributed federated modified Tobit Kalman filtering algorithm, which is devised on the basis of the federated Kalman fusion rule. Given the buffer length, the prescribed bound on the estimation error covariance and the probability distribution of the packet delay, the stability performance of the proposed filtering fusion algorithm is evaluated via a probabilistic approach. Simulation results are provided to show the feasibility of the proposed filter. [ABSTRACT FROM AUTHOR]

This paper is concerned with the distributed fusion estimation problem for time-varying stochastic uncertain systems with multiple asynchronous sampling sensors of different fading measurement rates. Stochastic uncertainties of multiplicative noises exist in the state and measurement equations. The phenomena of fading measurements of different sensors are depicted by a group of stochastic variables with known statistics. Different sensors have different sampling rates. Sampling period of each sensor is uniform and an integer multiple of the state update period. By transforming the multiplicative noises into the additive noises, local estimators (LEs) and estimation error cross-covariance matrices between any two LEs are derived at the state update points. A real-time distributed fusion estimator (DFE) is obtained by using the matrix-weighted fusion estimation algorithm in the linear unbiased minimum variance sense. The asymptotic stability and steady-state property are analyzed. A new conception of $\boldsymbol{{period}}$ steady state is presented and the $\boldsymbol{{period}}$ steady-state properties of the LEs, cross-covariance matrices, and DFE are proven. Though DFE has lower accuracy than the centralized fusion estimator, it has better robustness and flexibility since it has a parallel structure. To avoid the calculation of cross-covariance matrices, a sequential covariance intersection fusion estimator is given by using two-sensor covariance intersection fusion algorithm. It has lower accuracy but smaller computational cost than DFE, and better accuracy than LEs. Two examples are given to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Novel physical insights are provided into the performances of strictly linear (SL) and widely linear (WL) estimators of the generality of complex-valued data, both proper (second order circular) and improper (second-order noncircular). This is achieved by first performing a novel complementary mean square error (CMSE) analysis, in order to quantify the degrees of improperness (second order noncircularity) of the SL and WL estimation errors. The exact bounds on the CMSE difference between the SL and WL estimators are investigated to show that only a joint consideration of the standard MSE analysis and the proposed CMSE analysis has enough degrees of freedom for a rigorous account of the performance of WL and SL estimators. This also makes it possible to rigourously quantify the contributions to the WL performance advantage from the individual real and imaginary channels, an important finding not possible to obtain by using the standard MSE analysis only. [ABSTRACT FROM AUTHOR]

Adaptive sampling theory has shown that, with proper assumptions on the signal class, algorithms exist to reconstruct a signal in \mathbb R^d with an optimal number of samples. We generalize this problem to the case of spatial signals, where the sampling cost is a function of both the number of samples taken and the distance traveled during estimation. This is motivated by our work studying regions of low oxygen concentration in the Great Lakes. We show that for one-dimensional threshold classifiers, a tradeoff between the number of samples taken and distance traveled can be achieved using a generalization of binary search, which we refer to as quantile search. We characterize both the estimation error after a fixed number of samples and the distance traveled in the noiseless case, as well as the estimation error in the case of noisy measurements. We illustrate our results in both simulations and experiments and show that our method outperforms existing algorithms in a large range of sampling scenarios. [ABSTRACT FROM AUTHOR]

Tightly coupled GNSS/INS has been widely approved as a promising substitute for standalone GNSS in urban areas navigation. However, due to the frequent GNSS signal outages, the filter used in GNSS/INS should be insensitive to the less informative observations. In this paper, a novel sigma-points update method is proposed to enhance the robustness of cubature Kalman filter (CKF) under the circumstance of unavailable observations. First, the problems of existing sampling-based filters are analyzed. Then, by transforming the posterior sigma-points error matrix from prediction phase of filtering to the posterior domain of update, the updated sigma-points are expected to capture the covariance more precisely than traditional sigma-points. Finally, an improved CKF (ICKF) is developed by embedding these points into the Bayesian estimation framework, and the upper bounds of error covariance matrices are analyzed theoretically. Signal outages with different durations are simulated and results demonstrate that ICKF outperforms state-of-the-art methods. [ABSTRACT FROM PUBLISHER]

We study the problem of identifying multiple rumor or infection sources in a network under the susceptible-infected model, and where these sources may start infection spreading at different times. We introduce the notion of an abstract estimator that, given the infection graph, assigns a higher value to each vertex in the graph it considers more likely to be a rumor source. This includes several of the single-source estimators developed in the literature. We introduce the concepts of a quasi-regular tree and a heavy center, which allows us to develop an algorithmic framework that transforms an abstract estimator into a two-source joint estimator, in which the infection graph can be thought of as covered by overlapping infection regions. We show that our algorithm converges to a local optimum of the estimation function if the underlying network is a quasi-regular tree. We further extend our algorithm to more than two sources, and heuristically to general graphs. Simulation results on both synthetic and real-world networks suggest that our algorithmic framework outperforms several existing multiple-source estimators, which typically assume that all sources start infection spreading at the same time. [ABSTRACT FROM PUBLISHER]

We consider deterministic parameter estimation and the situation where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. Unfortunately, in the general case, this marginalization is mathematically intractable, which prevents from using the known standard deterministic lower bounds (LBs) on the mean squared error (MSE). Actually the general case can be tackled by embedding the initial observation space in a hybrid one where any standard LB can be transformed into a modified one fitted to nonstandard deterministic estimation, at the expense of tightness however. Furthermore, these modified LBs (MLBs) appears to include the submatrix of hybrid LBs which is an LB for the deterministic parameters. Moreover, since in the nonstandard estimation, maximum likelihood estimators (MLEs) can be no longer derived, suboptimal nonstandard MLEs (NSMLEs) are proposed as being a substitute. We show that any standard LB on the MSE of MLEs has a nonstandard version lower bounding the MSE of NSMLEs. We provide an analysis of the relative performance of the NSMLEs, as well as a comparison with the MLBs for a large class of estimation problems. Last, the general approach introduced is exemplified, among other things, with a new look at the well-known Gaussian complex observation models. [ABSTRACT FROM PUBLISHER]

This paper proposes a novel method of estimating the Fourier transform (FT) of deterministic, continuous-time signals, from a finite number N of their samples taken from a fixed-length observation window. It uses alias-free hybrid-stratified sampling to probe the processed signal at a mixture of deterministic and random time instants. The FT estimator, specifically designed to work with this sampling scheme, is unbiased, consistent and fast converging. It is shown that if the processed signal has continuous third derivative, then the estimator’s rate of uniform convergence in mean square is N^-5. Therefore, in terms of frequency-independent upper bounds on the FT estimation error, the proposed approach significantly outperforms existing estimators that utilize alias-free sampling, such as total random, stratified sampling, and antithetical stratified whose rate of uniform convergence is N^-1. It is proven here that N^-1 is a guaranteed minimum rate for all stratified-sampling-based estimators satisfying four weak conditions formulated in this paper. Owing to the alias-free nature of the sampling scheme, no constraints are imposed on the spectral support of the processed signal or the frequency ranges for which the Fourier Transform is estimated. [ABSTRACT FROM AUTHOR]

In sparse target inference problems, it has been shown that significant gains can be achieved by adaptive sensing using convex criteria. We generalize this previous work on adaptive sensing to: a) include multiple classes of targets with different levels of importance and b) accommodate multiple sensor models. Optimization policies are developed to allocate a limited resource budget to simultaneously locate, classify and estimate a sparse number of targets embedded in a large space. Bounds on the performance of the proposed policies are derived by analyzing a baseline policy, which allocates resources uniformly across the scene, and an oracle policy which has a priori knowledge of the target locations/classes. These bounds quantify the potential benefit of adaptive sensing as a function of target frequency and importance. Numerical results indicate that the proposed policies perform close to the oracle bound when signal quality is sufficiently high. Moreover, the proposed policies improve on previous policies in terms of reducing estimation error, reducing misclassification probability, and increasing expected return. To account for sensors with different levels of agility, three sensor models are considered: global adaptive (GA), which can allocate different amounts of resource to each location in the space? global uniform (GU), which can allocate resources uniformly across the scene? and local adaptive (LA), which can allocate fixed units to a subset of locations. Policies that use a mixture of GU and LA sensors are shown to perform similarly to those that use GA sensors while being more easily implementable. [ABSTRACT FROM AUTHOR]

This paper considers the problem of resource-constrained and noise-limited localization and estimation of dynamic targets that are sparsely distributed over a large area. We generalize an existing framework [Bashan , 2008] for adaptive allocation of sensing resources to the dynamic case, accounting for time-varying target behavior such as transitions to neighboring cells and varying amplitudes over a potentially long time horizon. The proposed adaptive sensing policy is driven by minimization of a surrogate function for mean squared error within locations containing targets. We provide theoretical upper bounds on the performance of adaptive sensing policies by analyzing solutions with oracle knowledge of target locations, gaining insight into the effect of target motion and amplitude variation as well as sparsity. Exact minimization of the multistage objective function is infeasible, but myopic optimization yields a closed-form solution. We propose a simple non-myopic extension, the Dynamic Adaptive Resource Allocation Policy (D-ARAP), that allocates a fraction of resources for exploring all locations rather than solely exploiting the current belief state. Our numerical studies indicate that D-ARAP has the following advantages: (a) it is more robust than the myopic policy to noise, missing data, and model mismatch; (b) it performs comparably to well-known approximate dynamic programming solutions but at significantly lower computational complexity; and (c) it improves greatly upon nonadaptive uniform resource allocation in terms of estimation error and probability of detection. [ABSTRACT FROM PUBLISHER]

In this paper, we study the distributed blind equalization of networked single-input multi-output (SIMO) systems. An indirect distributed equalization framework is presented, which estimates the transfer functions followed by the associated equalizers. Two distributed indirect equalization algorithms are proposed: one depends on multiple average consensus operations, and the other relies on the combination of innovation and one average consensus operation. The former generates an approximate equalizer for which the associated estimation error is determined by the number of average consensus operations, while the latter can provide an accurate equalizer estimation under some mild conditions. The proposed algorithms estimate the desired equalizer recursively and recover the source signal in real time. Furthermore, the distributed equalization under a time-varying topology is investigated as well. Convergence properties of the proposed algorithms are established and numerical simulations are carried out to show the performances of the proposed algorithms. [ABSTRACT FROM PUBLISHER]

Measurements of a scalar linear Gauss–Markov process are sent over a fading channel. The fading channel is modeled as independent and identically distributed random variables with known realization at the receiver. The optimal estimator at the receiver is the Kalman filter. In contrast to the classical Kalman filter theory, given a random channel, the Kalman gain and the error covariance become random. Then, the probability distribution function of expected estimation error and its outage probability can be chosen for estimation quality assessment. In this paper and in order to get the estimation error outage, we provide means to characterize the stationary probability density function of the random expected estimation error. Furthermore and for the particular case of the i.i.d. Rayleigh fading channels, upper and lower bounds for the outage probability are derived that provide insight and simpler means for design purposes. We also show that the bounds are tight for the high SNR regime and that the outage probability decreases linearly with the inverse of the average channel SNR. [ABSTRACT FROM AUTHOR]