Gaussian Assumption Research Articles

State estimation for dynamic systems with higher-order autoregressive moving average non-Gaussian noise

The classical Kalman filter is a very important state estimation approach, which has been widely used in many engineering applications. The Kalman filter is optimal for linear dynamic systems with independent Gaussian noises. However, the independence and Gaussian assumptions may not be satisfied in practice. On the one hand, modeling physical systems usually results in discrete-time state-space models with correlated process and measurement noises. On the other hand, the noise is non-Gaussian when the system is disturbed by heavy-tailed noise. In this case, the performance of the Kalman filter will deteriorate, or even diverge. This paper is devoted to addressing the state estimation problem of linear dynamic systems with high-order autoregressive moving average (ARMA) non-Gaussian noise. First, a triplet Markov model is introduced to model the system with high-order ARMA noise, since this model relaxes the independence assumption of the hidden Markov model. Then, a new filter is derived based on correntropy, instead of the commonly used minimum mean square error (MMSE), to deal with non-Gaussian noise. Unlike the MMSE, which uses only second-order statistics of error, correntropy can capture second-order and higher-order statistics. Finally, simulation results verify the effectiveness of the proposed algorithm.

Born to be wild: Second-to-fourth digit length ratio and risk preferences

The second-to-fourth digit length ratio of an individual’s hand (digit ratio) is a putative biomarker for prenatal exposure to testosterone. We examine the hypothesized negative association between the digit ratio and the preference for risk taking within a large U.S. population survey. Our statistical framework provides a cardinal proxy for the true digit ratio based on ordinal digit ratio measurements and accounts for measurement error under the assumptions of Gaussianity and time-invariant true digit ratios. Our empirical findings support the hypothesis and suggest a meaningful biological basis for risk preferences.

Gram-Charlier A Series Based Extended Rule-of-Thumb for Bandwidth Selection in Univariate Kernel Density Estimation

Bandwidth parameter estimation in univariate Kernel Density Estimation has traditionally two approaches. Rule(s)-of-Thumb (ROT) achieve ‘quick and dirty’ estimations with some specific assumption for an unknown density. More accurate solve-the-equation-plug-in (STEPI) rules have almost no direct assumption for the unknown density but demand high computation. This article derives a balancing third approach. Extending an assumption of Gaussianity for the unknown density to be estimated in \textit{normal reference} ROT (NRROT) to near Gaussianity, and then expressing the density using Gram-Charlier A (GCA) series to minimize the asymptotic mean integrated square error, it derives GCA series based Extended ROT (GCAExROT). The performance analysis using the simulated and the real datasets suggests to replace NRROT by a modified GCAExROT rule achieving a balancing performance by accuracy nearer to STEPI rules at computation nearer to NRROT, specifically at small samples.

A spatial copula interpolation in a random field with application in air pollution data.

Interpolating a skewed conditional spatial random field with missing data is cumbersome in the absence of Gaussianity assumptions. Copulas can capture different types of joint tail characteristics beyond the Gaussian paradigm. Maintaining spatial homogeneity and continuity around the observed random spatial point is also challenging. Especially when interpolating along a spatial surface, the boundary points also demand focus in forming a neighborhood. As a result, importing the concept of hierarchical clustering on the spatial random field is necessary for developing the copula model with the interface of the Expectation-Maximization algorithm and concurrently utilizing the idea of the Bayesian framework. This article introduces a spatial cluster-based C-vine copula and a modified Gaussian distance kernel to derive a novel spatial probability distribution. To make spatial copula interpolation compatible and efficient, we estimate the parameter by employing different techniques. We apply the proposed spatial interpolation approach to the air pollution of Delhi as a crucial circumstantial study to demonstrate this newly developed novel spatial estimation technique.

Covariance determination for improving uncertainty realism in orbit determination and propagation

The reliability of the uncertainty characterization, also known as uncertainty realism, is of the uttermost importance for Space Situational Awareness (SSA) services. Among the many sources of uncertainty in the space environment, the most relevant one is the inherent uncertainty of the dynamic models, which is generally not considered in the batch least-squares Orbit Determination (OD) processes in operational scenarios. A classical approach to account for these sources of uncertainty is the theory of consider parameters. In this approach, a set of uncertain parameters are included in the underlying dynamical model, in such a way that the model uncertainty is represented by the variances of these parameters. However, realistic variances of these consider parameters are not known a priori. This work introduces a methodology to infer the variance of consider parameters based on the observed distribution of the Mahalanobis distance of the orbital differences between predicted and estimated orbits, which theoretically should follow a chi-square distribution under Gaussian assumptions. Empirical Distribution Function statistics such as the Cramer-von-Mises and the Kolmogorov–Smirnov distances are used to determine optimum consider parameter variances. The methodology is presented in this paper and validated in a series of simulated scenarios emulating the complexity of operational applications.

Technical Note—The Elliptical Potential Lemma for General Distributions with an Application to Linear Thompson Sampling

A General Elliptical Potential Lemma In sequential learning and decision-making problems, the elliptical potential lemma is a key technique to quantify the decrease in the uncertainty of the model as more observations are obtained. However, it requires the observation noise and prior distribution of the unknown parameters to be Gaussian. In “The Elliptical Potential Lemma for General Distributions with an Application to Linear Thompson Sampling,” N. Hamidi and M. Bayati introduce a general version of the elliptical potential lemma that relaxes the Gaussian assumption. They also apply their general lemma to prove a minimax optimal Bayesian regret bound for the well-known Thompson sampling algorithm in stochastic linear bandits with changing action sets where prior and noise distributions are general.

A Tempered Particle Filter to Enhance the Assimilation of SAR-Derived Flood Extent Maps Into Flood Forecasting Models.

Data assimilation (DA) is a powerful tool to optimally combine uncertain model simulations and observations. Among DA techniques, the particle filter (PF) has gained attention for its capacity to deal with nonlinear systems and for its relaxation of the Gaussian assumption. However, the PF may suffer from degeneracy and sample impoverishment. In this study, we propose an innovative approach, based on a tempered particle filter (TPF), aiming at mitigating PFs issues, thus extending over time the assimilation benefits. Probabilistic flood maps derived from synthetic aperture radar data are assimilated into a flood forecasting model through an iterative process including a particle mutation in order to keep diversity within the ensemble. Results show an improvement of the model forecasts accuracy, with respect to the Open Loop: on average the root mean square error (RMSE) of water levels decrease by 80% at the assimilation time and by 60% 2 days after the assimilation. A comparison with the Sequential Importance Sampling (SIS) is carried out showing that although SIS performances are generally comparable to the TPF ones at the assimilation time, they tend to decrease more quickly. For instance, on average TPF‐based RMSE are 20% lower compared to the SIS‐based ones 2 days after the assimilation. The application of the TPF determines higher critical success index values compared to the SIS. On average the increase in performances lasts for almost 3 days after the assimilation. Our study provides evidence that the application of the variant of the TPF enables more persistent benefits compared to the SIS.

Safe motion planning with environment uncertainty

We present an approach for safe motion planning under robot state and environment (obstacle and landmark location) uncertainties. To this end, we first develop an approach that accounts for the landmark uncertainties during robot localization. Existing planning approaches assume that the landmark locations are well known or are known with little uncertainty. However, this might not be true in practice. Noisy sensors and imperfect motions compound to the errors originating from the estimate of environment features. Moreover, possible occlusions and dynamic objects in the environment render imperfect landmark estimation. Consequently, not considering this uncertainty can wrongly localize the robot, leading to inefficient plans. Our approach thus incorporates the landmark uncertainty within the Bayes filter estimation framework. We also analyze the effect of considering this uncertainty and delineate the conditions under which it can be ignored. Second, we extend the state-of-the-art by computing an exact expression for the collision probability under Gaussian distributed robot motion, perception and obstacle location uncertainties. We formulate the collision probability process as a quadratic form in random variables. Under Gaussian distribution assumptions, an exact expression for collision probability is thus obtained which is computable in real-time. In contrast, existing approaches approximate the collision probability using upper-bounds that can lead to overly conservative estimate and thereby suboptimal plans. We demonstrate and evaluate our approach using a theoretical example and simulations. We also present a comparison of our approach to different state-of-the-art methods.

Cancer Subtyping via Embedded Unsupervised Learning on Transcriptomics Data.

Cancer is one of the deadliest diseases worldwide. Accurate diagnosis and classification of cancer subtypes are indispensable for effective clinical treatment. Promising results on automatic cancer subtyping systems have been published recently with the emergence of various deep learning methods. However, such automatic systems often overfit the data due to the high dimensionality and scarcity. In this paper, we propose to investigate automatic subtyping from an unsupervised learning perspective by directly constructing the underlying data distribution itself, hence sufficient data can be generated to alleviate the issue of overfitting. Specifically, we bypass the strong Gaussianity assumption that typically exists but fails in the unsupervised learning subtyping literature due to small-sized samples by vector quantization. Our proposed method better captures the latent space features and models the cancer subtype manifestation on a molecular basis, as demonstrated by the extensive experimental results.

Moving horizon estimator for nonlinear and non-Gaussian stochastic disturbances

Over the last two decades, moving horizon estimation (MHE) has increasingly been used by researchers and industrial practitioners of nonlinear model predictive control as an alternative to recursive Bayesian estimation schemes. The MHE formulations available in the literature assume that uncertainties in the state dynamics can be modelled as additive and Gaussian stochastic processes. In practice, the unmeasured disturbances affect the system dynamics in a much more complex way and Gaussian assumption may prove inadequate to represent their behaviour. However, unlike the recursive Bayesian estimation area, handling of non-additive and non-Gaussian state disturbances has not received much attention in the MHE literature. In this work, a novel formulation of MHE is proposed in the Bayesian framework to solve state estimation problems associated with systems subjected to nonlinear non-Gaussian stochastic disturbances. To begin with, a Bayesian formulation that maximizes the joint posterior density function of the initial state and stochastic inputs is developed for a batch of data, which is further extended to the moving horizon framework. The proposed formulation associates the random fluctuations affecting the system dynamics with the unmeasured inputs entering the systems. The probabilistic formulation of MHE together with modelling of the state disturbances arising from physical sources enables us to work with any (non-Gaussian or Gaussian) probability density function (PDF) and systematically handle the stochastic inputs affecting the dynamics in a nonlinear manner without making any simplifying assumptions. We proceed to show that the disturbances/states with constraints can also be incorporated in our proposed formulation and can be interpreted as following truncated probability density functions. Efficacy of the proposed MHE approach is demonstrated using three benchmark simulation case studies and an experimental case study. Simulation studies show that the proposed MHE framework is able to perform similar to EnKF, a sampling based sequential Bayesian estimation approach which is capable of handling nonlinear non-Gaussian disturbances. It is further shown that the proposed MHE performs much better than the conventional MHE approach which assumes additive Gaussian noise in the state dynamics. Thus, the proposed MHE scheme provides an optimization based alternative to sampling based estimation schemes, such as Ensemble Kalman filtering, which can handle state estimation problems when state and measurement densities are non-Gaussian.

Entrywise Estimation of Singular Vectors of Low-Rank Matrices With Heteroskedasticity and Dependence

We propose an estimator for the singular vectors of high-dimensional low-rank matrices corrupted by additive subgaussian noise, where the noise matrix is allowed to have dependence within rows and heteroskedasticity between them. We prove finite-sample $\ell_{2,\infty}$ bounds and a Berry-Esseen theorem for the individual entries of the estimator, and we apply these results to high-dimensional mixture models. Our Berry-Esseen theorem clearly shows the geometric relationship between the signal matrix, the covariance structure of the noise, and the distribution of the errors in the singular vector estimation task. These results are illustrated in numerical simulations. Unlike previous results of this type, which rely on assumptions of gaussianity or independence between the entries of the additive noise, handling the dependence between entries in the proofs of these results requires careful leave-one-out analysis and conditioning arguments. Our results depend only on the signal-to-noise ratio, the sample size, and the spectral properties of the signal matrix.

Air-breathing hypersonic vehicle trajectory optimization with uncertain no-fly zones

The air-breathing hypersonic planes are considered to be the future of commercial airlines, which can fly from NYC to London in under an hour. For air-breathing vehicles, 3D trajectory planning will become extremely important due to its significant impact on flight performance. Past research on this issue was not comprehensive enough. Thus, we proposed a hybrid method for generating the optimal trajectories efficiently. No-fly zones are specified for geopolitical restrictions in air-breathing hypersonic vehicle missions. However, previous studies focused on no-fly zone constraints with fixed locations and boundaries. For robust execution, we must take into account no-fly zones’ uncertainties, which arise due to uncertain localization, measurement errors, and no-fly zones’ movement. A chance-constrained approach is presented here to deal with such uncertainties. The key idea of this method is controlling risk when the flight path is close to the uncertain no-fly zones. First, the trajectory planning of air-breathing hypersonic vehicles is modeled as a chance-constrained optimization (CCOPT) problem. With the help of the convexification and linearization techniques, we can approximate the non 3.2.2 CCOPT problem as a convex second-order cone programing under the Gaussian distribution assumption. It can be solved to global optimality by the interior point method. Finally, we give numerical simulation results to prove the effectiveness of this method.

Nonparametric probabilistic load forecasting based on quantile combination in electrical power systems

Probabilistic load forecasting (PLF) aims to predict the future uncertainties of loads to reduce the potential risks in power system planning and operation. In the increasingly complex power market environment, exploring advanced approaches to obtain more accurate PLF is still a significant topic. Optimizing individual forecasting method is no longer the only direction to improve the accuracy of load forecasting in recent years. Researchers started to focus on combination methods because of their better accuracy in most cases than a single model. There are existing combination methods designed for parametric environment, where some results are based on the certain assumption (e.g., Gaussian distribution assumption of single prediction). Combining probabilistic forecasts in nonparametric environment is rarely investigated, because modeling the combination problem without assuming distributions of parameters is hard. This paper proposes a novel combined model for probabilistic forecasting tailored to nonparametric environments, which combines multiple quantile-based models by minimizing the overall loss function composed of continuous ranked probability score (CRPS) under kernel density estimation (KDE). We define a multilayer Gaussian mixture distribution, which is an extended form of Gaussian mixture distribution that can simulate any distribution type in nonparametric environment. Based on the multilayer Gaussian mixture distribution, the combined model is further formulated into a quadratic programming problem with linear restrictions that can be solved efficiently. Case studies are performed using benchmark and competition datasets from the United States and China. The results show that our proposed method outperforms the best individual model and other existing combination methods. In summary, this paper constructs a complete theoretical framework of nonparametric probabilistic combination forecasting and proves its effectiveness in practical application.

Statistically Evolving Fuzzy Inference System for Non-Gaussian Noises

Non-Gaussian noises always exist in the nonlinear system, which usually lead to inconsistency and divergence of the regression and identification applications. The conventional evolving fuzzy systems (EFSs) in common sense have succeeded to conquer the uncertainties and external disturbance employing the specific variable structure characteristic. However, non-Gaussian noises would trigger the frequent changes of structure under the transient criteria, which severely degrades performance. Statistical criterion provides an informed choice of the strategies of the structure evolution, utilizing the approximation uncertainty as the observation of model sufficiency. The approximation uncertainty can be always decomposed into model uncertainty term and noise term, and is suitable for the non-Gaussian noise condition, especially relaxing the traditional Gaussian assumption. In this article, a novel incremental statistical evolving fuzzy inference system (SEFIS) is proposed, which has the capacity of updating the system parameters, and evolving the structure components to integrate new knowledge in the new process characteristic, system behavior, and operating conditions with non-Gaussian noises. The system generates a new rule based on the statistical model sufficiency which gives so insight into whether models are reliable and their approximations can be trusted. The nearest rule presents the inactive rule under the current data stream and further would be deleted without losing any information and accuracy of the subsequent trained models when the model sufficiency is satisfied. In our article, an adaptive maximum correntropy extend Kalman filter is derived to update the parameters of the evolving rules to cope with the non-Gaussian noises problems to further improve the robustness of parameter updating process. The parameter updating process shares an estimate of the uncertainty with the criteria of the structure evolving process to make the computation less of a burden dramatically. The simulation studies show that the proposed SEFIS has faster learning speed and is more accurate than the existing evolving fuzzy systems (EFSs) in the case of noise free and noisy conditions.

P-167 Male infertility and polycystic ovary syndrome (PCOS) affect absolute and relative embryo morphokinetics observed by time-lapse imaging

Abstract Study question This study aimed to test if morphokinetics are altered in embryos from patients with male infertility, polycystic ovary syndrome (PCOS) or recurrent pregnancy loss (RPL). Summary answer Morphokinetics differed significantly between embryos from patients with male infertility or PCOS compared to the control groups, possibly suggesting a negative impact on development potential. What is known already Time-lapse imaging technology allows continuous observation of embryonic morphology and developmental kinetics during the preimplantation period. The ESHRE Working group on Time-lapse technology recently summarised common absolute morphokinetic time points and intervals [DOI: 10.1093/hropen/hoaa008]. Additionally, four relative morphokinetic expressions were introduced by Cetinkaya et al.: Cleavage Synchronicity 2-8-, 4-8-, 2-4-cell stage (CS2-8, CS4-8, CS2-4) and DNA Replication time ratio (DR) [DOI: 10.1007/s10815-014-0341-x]. A study by Freis et al. observed worse CS2-8 and CS4-8 ratios in embryos from patients with endometriosis [DOI: 10.1177/1933719117741373]. Other studies have found that male infertility and PCOS may affect morphokinetic time points. Study design, size, duration In this retrospective observational study n = 1433 two-pronuclei oocytes from n = 433 patients between 18 and 45 years undergoing IVF/ICSI-treatment between 09/2016 and 12/2019 were examined. The use of a time-lapse incubator was obligatory. Exclusion criteria were the presence of endometriosis, history of chemotherapy, preimplantation genetic testing (PGT) cycles and genetic abnormalities. Every patient was included only once. Control groups were uterine, tubal factor and idiopathic infertility. Participants/materials, setting, methods Male infertility (n = 928 oocytes; 288 patients) versus control (n = 400; 115), PCOS (n = 48; 14) versus control (n = 400; 115) and RPL (n = 79; 21) versus control (n = 321; 94) were compared. Patient and treatment characteristics were compared using Mann-Whitney-U and Fisher’s exact test. Times normalised to fading of pronuclei (tPNf), intervals and ratios were analysed using a Generalised Linear Mixed Model (GLMM). Main results and the role of chance The male infertility group was significantly different (p < 0.05) from the control group regarding male age (p = 0.009), fertilisation rate (p = 0.008), number of embryos per patient (p = 0.033), treatment method (p < 0.001) and use of Calcium-Ionophore (p = 0.049) or SpermMobile (p = 0.003). DR was significantly higher in the male infertility group compared to the control group (p = 0.031). The start of blastulation (tSB-tPNf; p = 0.008) and start of expansion of the blastocyst (tEB-tPNf; p = 0.002) were significantly delayed. The PCOS group was significantly different concerning female age (p = 0.023) and AMH blood level (p < 0.001). CS2-8 was significantly lower (p = 0.003) and CS2-4 significantly higher (p = 0.001) in the PCOS group compared to the control group. The time intervals from three- to four-cell (s2; p = 0.013), from five- to eight-cell (s3; p = 0.032) and from four- to eight-cell stage (ECC3; p = 0.043) were significantly longer in the PCOS group. There was no significant difference in patient or treatment related characteristics between the RPL group and the control group. Also, none of the morphokinetic parameters analysed in this study was significantly different between the groups. Limitations, reasons for caution Small group sizes decreased further at later developmental time points. In general, time-lapse data show a high intra- and inter-observer variability. A GLMM was performed to take statistical dependencies between embryos of one patient into account. Its Gaussian assumption was partially violated by the true distribution of the data. Wider implications of the findings These data support the hypothesis that patients’ co-morbidities may affect embryo morphokinetics. The impact on clinical outcome should be examined further. Accordingly, our working group conducted another study about the predictive value of morphokinetics on blastocyst quality and implantation (DRKS-ID: DRKS00022983). The long-term goal is to improve embryo assessment. Trial registration number not applicable

Transfer learning in high-dimensional semiparametric graphical models with application to brain connectivity analysis.

Transfer learning has drawn growing attention with the target of improving statistical efficiency of one study (dataset) by digging up information from similar and related auxiliary studies (datasets). In this article, we consider transfer learning problem in estimating undirected semiparametric graphical model. We propose an algorithm called Trans-Copula-CLIME for estimating an undirected graphical model while uncovering information from similar auxiliary studies, characterizing the similarity between the target graph and each auxiliary graph by the sparsity of a divergence matrix. The proposed method relaxes the restrictive Gaussian distribution assumption, which deviates from reality for the fMRI dataset related to attention deficit hyperactivity disorder (ADHD) considered here. Nonparametric rank-based correlation coefficient estimators are utilized in the Trans-Copula-CLIME procedure to achieve robustness against normality. We establish the convergence rate of the Trans-Copula-CLIME estimator under some mild conditions, which demonstrates that if the similarity between the auxiliary studies and the target study is sufficiently high and the number of informative auxiliary samples is sufficiently large, the Trans-Copula-CLIME estimator shows great advantage over the existing non-transfer-learning ones. Simulation studies also show that Trans-Copula-CLIME estimator has better performance especially when data are not from Gaussian distribution. Finally, the proposed method is applied to infer functional brain connectivity pattern for ADHD patients in the target Beijing site by leveraging the fMRI datasets from some other sites.

On the Kurtosis of Modulation Formats for Characterizing the Nonlinear Fiber Propagation

Knowing only two high-order statistical moments of modulation symbols, often represented by the fourth moment called “kurtosis”, the overestimation of nonlinear interference (NLI) in a Gaussian noise (GN) model due to Gaussian signaling assumption can be corrected through an enhanced GN (EGN) model. However, in some modern optical communication systems where the transmitted modulation symbols are statistically correlated, such as in systems that use probabilistic constellation shaping (PCS) with finite-length sphere shaping, the kurtosis-based EGN model produces significant inaccuracies in analytical prediction of NLI. In this paper, we show that for correlated modulation symbols, the NLI can be more accurately estimated by substituting a statistical measure called windowed kurtosis into the EGN model, instead of the conventional kurtosis. Remarkably, the optimal window length for windowed kurtosis is found to be consistent with the self-phase modulation (SPM) and cross-phase modulation (XPM) characteristic times in various system configurations. The findings can be used in practice to analytically evaluate and design NLI-tolerant modulation formats.

On the Gaussian and Non-Gaussian Characteristics of Nonstationary Seismic Ground Motions

In earthquake engineering, seismic ground motions are most often modelled as a nonstationary Gaussian process. A few studies indicated that seismic ground motions should be treated as a nonstationary non-Gaussian process, by showing that the kurtosis coefficient of the historical ground motion records is much greater than three. These findings and conclusions are queried in the present study, which analyzes a large number of historical ground motion records. Our results indicate that while the mixture marginal distribution of the acceleration of the records is non-Gaussian with a heavy distribution tail, the mixture marginal distribution of the standardized record, defined by the time-varying record to its standard deviation, is only mildly non-Gaussian. We point out that the mixture marginal distribution of a nonstationary Gaussian process may not be Gaussian. The implication of these observations in simulating records is explained. The sampled nonstationary Gaussian/non-Gaussian records are used to compare the responses of single-degree-of-freedom systems. The results indicate that the error introduced by adopting the Gaussian assumption is small, suggesting that the ground motions could be assumed to be a nonstationary Gaussian process with sufficient accuracy, especially if structures that are not very stiff are considered.

Uncertainty estimation for ensemble particle image velocimetry

We present a novel approach to estimate the uncertainty in ensemble particle image velocimetry (PIV) measurements. The ensemble PIV technique is widely used when the cross-correlation signal-to-noise ratio is insufficient to perform a reliable instantaneous velocity measurement. Despite the utility of ensemble PIV, uncertainty quantification for this type of measurement has not been studied. Here, we propose a method for estimating the uncertainty directly from the probability density function of displacements found by deconvolving the ensemble cross-correlation from the ensemble autocorrelation. We then find the second moment of the probability density function and apply a scaling factor to report the uncertainty in the velocity measurement. We call this method the moment of probability of displacement (MPD). We assess MPD’s performance with synthetic and experimental images. We show that predicted uncertainties agree well with the expected root mean square (RMS) of the error in the velocity measurements over a wide range of image and flow conditions. MPD shows good sensitivity to various PIV error sources with around 86% accuracy in matching the RMS of the error in the baseline data sets. So, MPD establishes itself as a reliable uncertainty quantification algorithm for ensemble PIV. We compared the results of MPD against one of the existing instantaneous PIV uncertainty approaches, moment of correlation (MC). We adapted the MC approach for ensemble PIV, however, its primary limitations remain the assumption of the Gaussian probability density function of displacements and the Gaussian particles’ intensity profile. In addition, our analysis shows that ensemble MC consistently underestimates the uncertainty, while MPD outperforms that and removes the limiting Gaussian assumption for the particle and probability density function, thus overcoming the limitations of MC.

Direction of arrival estimation under Class A modelled noise in shallow water using variational Bayesian inference method

The shallow water noise shows obvious impulsive property, which greatly degrades the direction of arrival (DOA) performance due to the conventional design concept based on the Gaussian assumption. In this paper, DOA estimator in presence of impulsive noise utilising variational Bayesian inference is proposed. The Middleton's Class A noise model is considered as a typical underwater noise model to analyse the performance of DOA estimation. The DOA estimation problem is modelled as the sparse signal recovery problem, and the hierarchical Bayesian learning framework is formulated by considering the common sparsity of signal and the element-wise sparsity of the impulsive noise. The variational Bayesian inference realises the posterior estimation of signal and impulsive noise components. To mitigate the basis mismatches, the root sparse Bayesian learning method is applied to refine the steering vectors. Simulations verify the advantages of the proposed DOA method in terms of spatial resolution, root mean square error, accuracy, and robustness compared with the state-of-the-art benchmarks in the presence of Middleton's Class A noise.