non-Gaussian Measurement Research Articles

Nonlinear Non-Gaussian Estimation Using Maximum Correntropy Square Root Cubature Information Filtering

This paper concerns the nonlinear filter designing methods in the information space of the nonlinear systems with non-Gaussian noises. Firstly, the prediction information vector is obtained by the traditional square root cubature information filtering algorithm. Then, under the maximum correntropy criterion, the prediction information vector is corrected with the contribution information vector obtained by the non-Gaussian measurement. The information filtering gain is obtained by utilizing the state information correntropy matrix and the measurement information correntropy matrix, in which, the state prediction is taken as the state value. In order to improve the advantage of the above nonlinear non-Gaussian information filter in filtering accuracy, with the help of fixed-point theory, an iterative computation method is further developed to update the estimation information vector and the state estimate. The effectiveness of the two proposed nonlinear non-Gaussian filtering methods is illustrated in final four simulation examples.

A New Robust Kalman Filter With Adaptive Estimate of Time-Varying Measurement Bias

To better model the non-Gaussian heavy-tailed measurement noise with unknown and time-varying bias, a new Student's t-inverse-Wishart (STIW) distribution is presented. The STIW distribution is firstly written as a Gaussian, inverse-Wishart and normal-Gamma hierarchical form, from which a new robust Kalman filter is then derived based on the variational Bayesian method. Simulation results illustrate the potentials of the new derived robust Kalman filter for addressing the above measurement noise.

A Novel Nonlinear Algorithm for Non-Gaussian Noises and Measurement Information Loss in Underwater Navigation

The ocean environment is complex and changeable because of all kinds of noise interferences, such as salt cliffs, ships around and other electromagnetic interferences, so the measurement information is prone to be lost. It is difficult to describe the complex noise and the acquisition probability of measurement information. In this paper, a continuous discrete variational Bayesian filter (CD VBF) is proposed to solve the problems of the heavy tailed noises and measurement random loss for state estimation. The variational Bayesian (VB) approach can effectively estimate the state vector, scale matrices, degree of freedom (DOF) parameters, Bernoulli random variables and the acquisition probability of measurement information. The performances of the proposed algorithm and traditional algorithms are tested in simulations and underwater experiments. The simulation results illustrate that the CD VBF has better localization accuracy and robustness. The experimental results demonstrate that the localization accuracy is improved by CD VBF and an optimal solution of iteration number is acquired.

Probabilistic learning on manifolds

This paper presents novel mathematical results in support of the probabilistic learning on manifolds (PLoM) recently introduced by the authors. An initial dataset, constituted of a small number of points given in an Euclidean space, is given. The points are independent realizations of a vector-valued random variable for which its non-Gaussian probability measure is unknown but is, a priori, concentrated in an unknown subset of the Euclidean space. A learned dataset, constituted of additional realizations, is constructed. A transport of the probability measure estimated with the initial dataset is done through a linear transformation constructed using a reduced-order diffusion-maps basis. It is proven that this transported measure is a marginal distribution of the invariant measure of a reduced-order Ito stochastic differential equation. The concentration of the probability measure is preserved. This property is shown by analyzing a distance between the random matrix constructed with the PLoM and the matrix representing the initial dataset, as a function of the dimension of the basis. It is further proven that this distance has a minimum for a dimension of the reduced-order diffusion-maps basis that is strictly smaller than the number of points in the initial dataset.

A Rao-Blackwellized Particle Filter With Variational Inference for State Estimation With Measurement Model Uncertainties

This paper develops a Rao-Blackwellized particle filter with variational inference for jointly estimating state and time-varying parameters in non-linear state-space models (SSM) with non-Gaussian measurement noise. Depending on the availability of the conjugate prior for the unknown parameters, the joint posterior distribution of the state and unknown parameters is approximated by using an auxiliary particle filter with a probabilistic changepoint model. The distribution of the SSM parameters conditionally on each particle is then updated by using variational Bayesian inference. Experiments are first conducted on a modified nonlinear benchmark model to compare the performance of the proposed approach with other state-of-the-art approaches. Finally, in the context of GNSS multipath mitigation, the proposed approach is evaluated based on data obtained from a measurement campaign conducted in a street urban canyon.

An Adaptive Gaussian Sum Kalman Filter Based on a Partial Variational Bayesian Method

In this article, we address the online state estimation problem of linear discrete-time systems in the presence of inaccurate and slowly time-varying non-Gaussian measurement noise (NGMN). Recently, the variational Bayesian (VB) method has been successfully used to jointly estimate the system state along with the statistics of the unknown Gaussian measurement noise. However, we prove that the original VB method for the non-Gaussian state-space models, modeled by the Gaussian mixture distributions, is analytically intractable. To overcome this problem, we propose a partial VB-based adaptive Gaussian sum Kalman filter, which uses a feedback-based filtering framework to independently calculate the posterior distribution of the state and posterior distribution of the NGMN. Experimental results demonstrate the effectiveness of the proposed filter.

Non-Gaussianity constraints using future radio continuum surveys and the multitracer technique

ABSTRACT Tighter constraints on measurements of primordial non-Gaussianity (PNG) will allow the differentiation of inflationary scenarios. The cosmic microwave background bispectrum – the standard method of measuring the local non-Gaussianity – is limited by cosmic variance. Therefore, it is sensible to investigate measurements of non-Gaussianity using the large-scale structure. This can be done by investigating the effects of non-Gaussianity on the power spectrum on large scales. In this study, we forecast the constraints on the local PNG parameter fNL that can be obtained with future radio surveys. We utilize the multitracer method that reduces the effect of cosmic variance and takes advantage of the multiple radio galaxy populations that are differently biased tracers of the same underlying dark matter distribution. Improvements on previous work include the use of observational bias and halo mass estimates, updated simulations, and realistic photometric redshift expectations, thus producing more realistic forecasts. Combinations of Square Kilometre Array simulations and radio observations were used as well as different redshift ranges and redshift bin sizes. It was found that in the most realistic case the 1σ error on fNL falls within the range 4.07–6.58, rivalling the tightest constraints currently available.

Quantum hypergraph states in continuous variables

The measurement based, or one-way, model of quantum computation for continuous variables uses a highly entangled state called a cluster state to accomplish the task of computing. Cluster states that are universal for computation are a subset of a class of states called graph states. These states are Gaussian states and therefore require that the homodyne detection (Gaussian measurement) scheme is supplemented with a non-Gaussian measurement for universal computation, a significant experimental challenge. Here we define a new non-Gaussian class of states based on hypergraphs which satisfy the requirements of the Lloyd-Braunstein criteria while restricted to a Gaussian measurement strategy. Our main result is to show that, taking advantage of the intrinsic multimode nonlinearity, a hypergraph consisting of 3-edges can be used to apply a three-mode operation to an input three-mode state. As a special case, this technique can be used to apply the cubic phase gate to a single mode.

Unscented Kalman Filter With Generalized Correntropy Loss for Robust Power System Forecasting-Aided State Estimation

Due to the existence of various anomalies such as non-Gaussian process and measurement noises, gross measurement errors, and sudden changes of system status, the robust forecasting-aided state estimation is pivotal for power system stability. This paper develops a novel unscented Kalman filter (UKF) with the generalized correntropy loss (GCL) (termed as GCL-UKF) to estimate power system state with forecasting aid. The GCL is used to replace the mean square error loss in the original UKF framework. The advantage of such an approach is that it combines the strength of the GCL developed in robust information theoretic learning for addressing the non-Gaussian interference and the strength of the UKF in handling strong model nonlinearities. In addition, we take into account the nontrivial influences of the bad data for the innovation vector. An enhanced GCL-UKF method is established by introducing an exponential function of the innovation vector to adjust a covariance matrix so as to improve the GCL-UKF-based state estimation accuracy under the change of gain matrix caused by bad factors. Numerical simulation results carried out on IEEE 14-bus, 30-bus, and 57-bus test systems validate the efficacy of the proposed methods for state estimation under various types of measurement.

Robust identification of non-stationary objects with nongaussian interference

The problem of identification of non-stationary parameters of a linear object, which can be described by the first-order Markov model, with non-Gaussian interference is considered. The identification algorithm is a gradient minimization procedure of the combined functional. The combined functional, in turn, consists of quadratic and modular functionals, the weights of which are set using the mixing parameter. Such a combination of functionals makes it possible to obtain estimates with robust properties. The identification algorithm does not require knowledge of the degree of non-stationarity of the investigated object. It is the simplest, since information about only one measurement cycle (step) is used in model construction. The use of the Markov model is quite effective, as it allows obtaining analytical estimates of the properties of the algorithm.Conditions of mean and mean-square convergence of the gradient algorithm in the estimation of non-stationary parameters and with non-Gaussian measurement interference are determined.The obtained estimates are quite general and depend both on the degree of object non-stationarity and statistical characteristics of useful signals and interference. In addition, expressions are determined for the asymptotic values of the parameter estimation error and asymptotic accuracy of identification. Since these expressions contain a number of unknown parameters (values of signal and interference dispersion, dispersion characterizing non-stationarity), estimates of these parameters should be used for their practical application. For this purpose, any recurrent procedure for evaluating these unknown parameters should be applied and the resulting estimates should be used to refine the parameters included in the algorithms. In addition, the asymptotic values of the estimation error and identification accuracy depend on the choice of mixing parameter

Probabilistic State Estimation Approach for AC/MTDC Distribution System Using Deep Belief Network With Non-Gaussian Uncertainties

The increasing complexity of distribution grids due to widespread deployment of renewable resources and/or power electronic devices, e.g., voltage source converters, has necessitated the needs of distribution system state estimation (DSSE) for efficient control relying on an accurate picture of the system states. This paper therefore explores the application of using the deep belief network (DBN) for pseudo measurements modeling in the context of DSSE. Two DBNs are trained respectively, for active and reactive power injection outputs, with load profiles and limited number of real measurements. Given the non-Gaussian behavior of the estimated pseudo measurements, we model the error by Gaussian mixture distribution and accordingly design a state estimator based on the Gaussian component combination method (GCCM). This method is able to handle the non-Gaussian measurement uncertainty while retaining the framework of the classic weighed least square (WLS) algorithm. The effectiveness of the proposed DSSE is demonstrated on a modified US PG&E69 distribution network in terms of the accuracy for both the estimated quantities and the associated uncertainty distributions.

The bispectrum of polarized galactic foregrounds

Understanding the properties of the galactic emission at millimetre wavelengths is important for studies of the cosmic microwave background (CMB). In this work we explore the bispectrum, the harmonic equivalent of the three point function, from galactic dust and synchrotron emission. We investigate these effects across a broad range of frequencies using the synchrotron dominated S-band Polarization All Sky Survey (SPASS) maps at 2.3 GHz, the Planck satellite maps (30-857 GHz) and dust dominated Infrared Astronomical Satellite (IRAS) maps at 3 THz. We measure bispectra of total intensity fields, T, as well as the gradient, E, and curl modes, B, of the polarization field. We find that the synchrotron and galactic dust have strong temperature bispectra with significant contributions in the squeezed limit, which probes the correlations between two small scale modes with a large scale mode. Additionally, we find that the dust also has strong polarised bispectra that also peak in the squeezed configuration. We explore parity odd bispectra, such as BTT bispectra, and find strong parity odd bispectra for the galactic dust notably in BTT, BTE and BEE configurations. After masking bright sources, we find no evidence for polarised synchrotron bispectra and no evidence for cross bispectra between the dust and synchrotron emission. The strong foreground bispectra discussed here need to be carefully controlled to avoid biasing measurements of primordial non-Gaussianity. Finally we use these bispectra tools to test for residual foregrounds in the component separated Planck maps and find no evidence of residual foregrounds. These tools will be useful for characterizing residual foregrounds in component separated maps, particularly for experiments with less frequency coverage than the Planck satellite.

A Decentralized H-Infinity Unscented Kalman Filter for Dynamic State Estimation Against Uncertainties

The widely used traditional Kalman filter-type power system dynamic state estimator is unable to address the unknown and non-Gaussian system process and measurement noise as well as dynamical model uncertainties. To this end, this paper proposes a decentralized H-infinity unscented Kalman filter that leverages the strength of the H-infinity criteria developed in robust control for handling system uncertainties with the advantage of the UKF for addressing strong model nonlinearities. Specifically, the statistical linerization approach is used to derive a linear-like batch-mode regression model similar to the linear Kalman filter. This allows us to resort to the linear H-infinity Kalman filter framework for the development of the proposed H-infinity UKF in the Krein space. An analytical form is also derived to tune the parameter of the H-infinity criterion. Two decoupled models are presented to enable the decentralized implementation of the H-infinity UKF using the local PMU measurements. Extensive simulation results carried out on the IEEE 39-bus system show that the proposed H-infinity UKF is able to bound the influences of various types of measurement and model uncertainties while obtaining accurate state estimates.

Efficient non-Gaussianity measure for single-mode fields by kurtosis of a Gaussian operator

We propose a method to faithfully probe and quantify the non-Gaussianity of single-mode fields in the coherent-state representations based on the averaged-cumulant technique. In comparison with the previous measure, S H Xiang et al (2018 Phys. Rev. A 97 042303), we characterize the non-Gaussianity of quantum states using the kurtosis of a Gaussian operator, rather than that of a quadrature operator. As applications, we investigate the non-Gaussianities of three different non-Gaussian states: a mixture of vacuum and Fock states, Schrödinger cat states, and photon-added coherent states. It is proven that such a kurtosis is a powerful and effective measure in appraising non-Gaussianity.

Optimized communication strategies with binary coherent states over phase noise channels

The achievable rate of information transfer in optical communications is determined by the physical properties of the communication channel, such as the intrinsic channel noise. Bosonic phase noise channels, a class of non-Gaussian channels, have emerged as a relevant noise model in quantum information and optical communication. However, while the fundamental limits for communication over Gaussian channels have been extensively studied, the properties of communication over Bosonic phase noise channels are not well understood. Here we propose and demonstrate experimentally the concept of optimized communication strategies for communication over phase noise channels to enhance information transfer beyond what is possible with conventional methods of modulation and detection. Two key ingredients are generalized constellations of coherent states that interpolate between standard on-off keying and binary phase-shift keying formats, and non-Gaussian measurements based on photon number resolving detection of the coherently displaced signal. For a given power constraint and channel noise strength, these novel strategies rely on joint optimization of the input alphabet and the measurement to provide enhanced communication capability over a non-Gaussian channel characterized in terms of the error rate as well as mutual information.

Faithful measure of quantum non-Gaussianity via quantum relative entropy

We introduce a measure of quantum non-Gaussianity (QNG) for those quantum states not accessible by a mixture of Gaussian states in terms of quantum relative entropy. Specifically, we employ a convex-roof extension using all possible mixed-state decompositions beyond the usual pure-state decompositions. We prove that this approach brings a QNG measure fulfilling the properties desired as a proper monotone under Gaussian channels and conditional Gaussian operations. As an illustration, we explicitly calculate QNG for the noisy single-photon states and demonstrate that QNG coincides with non-Gaussianity of the state itself when the single-photon fraction is sufficiently large.

Robust decentralised state estimation for formation flying spacecraft

In this study, a robust decentralised state estimation algorithm named decentralised Huber-based cubature filtering (DHCF) for formation flying spacecraft is proposed. In this algorithm, a decentralised architecture is designed, in which each deputy estimates its relative state with respect to the chief based on its own exteroceptive sensor measurement. The relative motion equation considering the non-spherical influence of the earth is derived. The relative measurements' information contain not only the line-of-sight between the deputy and the chief, but also the ranges among the deputies, which improves the redundancy of the relative navigation task. A set of cubature points are selected using spherical-radial rule to map the probability distribution, which is more accurate than the linearisation of the extended Kalman filter (EKF). The non-Gaussian noise in formation is approximated by Gaussian mixture models. To deal with the non-Gaussian noise, the Huber technique is used in measurement update of each deputy and the extra measurement uncertainty caused by linearisation is compensated. Simulation results indicate that the proposed DHCF provides better performance in state estimation accuracy and robustness when compared to decentralised EKF and decentralised cubature Kalman filtering in the presence of non-Gaussian measurement noise.

Cosmic variance mitigation in measurements of the integrated Sachs-Wolfe effect

The cosmic microwave background (CMB) is sensitive to the recent phase of accelerated cosmic expansion through the late-time integrated Sachs-Wolfe (ISW) effect, which manifests as secondary temperature fluctuations on large angular scales. However, the large cosmic variance from primary CMB fluctuations limits the usefulness of this effect in constraining dark energy or modified gravity. In this paper, we propose a novel method to separate the ISW signal from the primary signal using gravitational lensing, based on the fact that the ISW signal is, to a good approximation, not gravitationally lensed. We forecast how well we can isolate the ISW signal for different experimental configurations, and discuss various applications, including modified gravity, large-scale CMB anomalies, and measurements of local-type primordial non-Gaussianity. Although not within reach of current experiments, the proposed method is a unique way to remove the cosmic variance of the primary signal, allowing for better CMB-based constraints on late-time phenomena than previously thought possible.

A data-driven framework for sparsity-enhanced surrogates with arbitrary mutually dependent randomness

The challenge of quantifying uncertainty propagation in real-world systems is rooted in the high-dimensionality of the stochastic input and the frequent lack of explicit knowledge of its probability distribution. Traditional approaches show limitations for such problems, especially when the size of the training data is limited. To address these difficulties, we have developed a general framework of constructing surrogate models on spaces of stochastic input with arbitrary probability measure irrespective of the mutual dependencies between individual components of the random inputs and the analytical form. The present Data-driven Sparsity-enhancing Rotation for Arbitrary Randomness (DSRAR) framework includes a data-driven construction of multivariate polynomial basis for arbitrary mutually dependent probability measures and a sparsity enhancement rotation procedure. This sparsity enhancement method was initially proposed in our previous work (Lei et al., 2015) for Gaussian density distributions, which may not be feasible for non-Gaussian distributions due developed a new data-driven approach to construct orthonormal polynomials for arbitrary mutually dependent randomness, ensuring the constructed basis maintains the orthogonality/near-orthogonality with respect to the density of the rotated random vector, where directly applying the regular polynomial chaos including arbitrary polynomial chaos (aPC) Oladyshkin and Nowak (2012) shows limitations due to the assumption of the mutual independence between the components of the random inputs. The developed DSRAR framework leads to accurate recovery, with only limited training data, of a sparse representation of the target functions. The effectiveness of our method is demonstrated in challenging problems such as partial differential equations and realistic molecular systems within high-dimensional (O(10)) conformational spaces where the underlying density is implicitly represented by a large collection of sample data, as well as systems with explicitly given non-Gaussian probabilistic measures.

Cooperative Localisation of AUVs based on Huber-based Robust Algorithm and Adaptive Noise Estimation

In this paper, adaptive noise estimation is used along with a previously proposed Huber-based robust algorithm for cooperative localisation of Autonomous Underwater Vehicles (AUVs). The Huber-based robust cooperative localisation algorithm named Huber-based Iterative Divided Difference Filtering (HIDDF), proposed in our previous work, effectively achieved a robust result against abnormal measurement noise, enhanced the stability of the filtering algorithm and improved the performance of cooperative localisation state estimation. However, its performance could be significantly further improved if it could estimate the system's noise statistical properties online in real time and then adaptively adjust the filtering gain matrix accordingly. In this paper, a novel adaptive noise estimation algorithm is proposed based on a covariance matching method. The proposed algorithm is suitable for adaptively estimating Gaussian and non-Gaussian measurement as well as process noise. The efficacy of the proposed algorithm has been verified through simulation results. In order to further verify the effectiveness of the proposed algorithm in practical systems, lake tests were conducted. Then, based on offline test data, the performance of the cooperative positioning algorithm under dual-pilot and single-pilot schemes was simulated. The advantages and feasibility of the algorithm are analysed and compared through performance comparison. Cooperative localisation accuracy of the previously proposed Huber-based robust algorithm has been enhanced significantly when used with the proposed adaptive noise estimation algorithm.