Gaussian Assumption Research Articles

Robust low-rank covariance matrix estimation with a general pattern of missing values

This paper tackles the problem of robust covariance matrix estimation when the data is incomplete. Classical statistical estimation methodologies are usually built upon the Gaussian assumption, whereas existing robust estimation ones assume unstructured signal models. The former can be inaccurate in real-world data sets in which heterogeneity causes heavy-tail distributions, while the latter does not profit from the usual low-rank structure of the signal. Taking advantage of both worlds, a covariance matrix estimation procedure is designed on a robust (mixture of scaled Gaussian) low-rank model by leveraging the observed-data likelihood function within an expectation-maximization algorithm. It is also designed to handle general pattern of missing values. The proposed procedure is first validated on simulated data sets. Then, its interest for classification and clustering applications is assessed on two real data sets with missing values, which include multispectral and hyperspectral time series.

Simultaneous Estimation of Parameters and the State of an Optical Parametric Oscillator System

In this paper, we consider the filtering problem of an optical parametric oscillator (OPO). The OPO pump power may fluctuate due to environmental disturbances, resulting in uncertainty in the system modeling. Thus, both the state and the unknown parameter may need to be estimated simultaneously. We formulate this problem using a state-space representation of the OPO dynamics. Under the assumption of Gaussianity and proper constraints, the dual Kalman filter method and the joint extended Kalman filter method are employed to simultaneously estimate the system state and the pump power. Numerical examples demonstrate the effectiveness of the proposed algorithms.

Finite sample theory for high-dimensional functional/scalar time series with applications

Statistical analysis of high-dimensional functional times series arises in various applications. Under this scenario, in addition to the intrinsic infinite-dimensionality of functional data, the number of functional variables can grow with the number of serially dependent observations. In this paper, we focus on the theoretical analysis of relevant estimated cross-(auto)covariance terms between two multivariate functional time series or a mixture of multivariate functional and scalar time series beyond the Gaussianity assumption. We introduce a new perspective on dependence by proposing functional cross-spectral stability measure to characterize the effect of dependence on these estimated cross terms, which are essential in the estimates for additive functional linear regressions. With the proposed functional cross-spectral stability measure, we develop useful concentration inequalities for estimated cross-(auto)covariance matrix functions to accommodate more general sub-Gaussian functional linear processes and, furthermore, establish finite sample theory for relevant estimated terms under a commonly adopted functional principal component analysis framework. Using our derived non-asymptotic results, we investigate the convergence properties of the regularized estimates for two additive functional linear regression applications under sparsity assumptions including functional linear lagged regression and partially functional linear regression in the context of high-dimensional functional/scalar time series.

A geometric framework for covariance dynamics

Employing methods of differential geometry, we propose a new framework for covariance dynamics modeling. Our approach respects the intrinsic geometric properties of the space of covariance matrices and allows their natural evolution. We develop covariance models that exploit either asset returns or realized covariances and propose a new estimation method that minimizes the length of the geodesic between the forecast and the realization. The geodesic length is equivalent to the Fisher information metric under the Gaussian assumption and is deemed a proper measure of similarity between two covariance matrices. Empirical studies involving three data samples and various performance metrics suggest that our models outperform existing ones.

Bayesian optimization with informative parametric models via sequential Monte Carlo

AbstractBayesian optimization (BO) has been a successful approach to optimize expensive functions whose prior knowledge can be specified by means of a probabilistic model. Due to their expressiveness and tractable closed-form predictive distributions, Gaussian process (GP) surrogate models have been the default go-to choice when deriving BO frameworks. However, as nonparametric models, GPs offer very little in terms of interpretability and informative power when applied to model complex physical phenomena in scientific applications. In addition, the Gaussian assumption also limits the applicability of GPs to problems where the variables of interest may highly deviate from Gaussianity. In this article, we investigate an alternative modeling framework for BO which makes use of sequential Monte Carlo (SMC) to perform Bayesian inference with parametric models. We propose a BO algorithm to take advantage of SMC’s flexible posterior representations and provide methods to compensate for bias in the approximations and reduce particle degeneracy. Experimental results on simulated engineering applications in detecting water leaks and contaminant source localization are presented showing performance improvements over GP-based BO approaches.

Spectral-Spatial Deep Support Vector Data Description for Hyperspectral Anomaly Detection

Hyperspectral anomaly detection (HAD) aims to distinguish anomalies from background-by-background modeling. Deep learning has been applied to HAD and achieves promising detection results. However, there exist several issues that need to be addressed: 1) unrealistic Gaussian assumption on the latent representations may limit its application; 2) deep features are not well-suited to anomaly detection due to the separation between feature learning and anomaly detection; 3) lack of adequate exploitation of spectral-spatial features; 4) negative effect caused by spectral band redundancy. In this article, we propose an end-to-end trainable deep one-class classification network for HAD. Specifically, a minimal enclosing hypersphere is trained to involve the deep features of background samples. These background samples are selected by a density clustering-based method. In this way, feature learning and anomaly detection are incorporated into a unified framework. Meanwhile, there is no explicit Gaussian assumption on the background features. Moreover, due to the complementarity of spectral and spatial features, a novel feature fusion strategy is proposed to fuse spectral and spatial features extracted by a two-stream deep convolutional autoencoder network. Finally, a band attention module is used to automatically learn small weights for redundant bands and thus reduce the negative effect caused by redundant bands. Experimental results on five public datasets demonstrate the superiority of the proposed method compared to several state-of-the-art HAD methods in the detection performance.

Quantum-Inspired Secure Wireless Communication Protocol Under Spatial and Local Gaussian Noise Assumptions

Inspired by quantum key distribution, we consider wireless communication between Alice and Bob when the noise generated in the intermediate space between Alice and Bob is controlled by Eve. Our model divides the channel noise into two parts, the noise generated during the transmission and the noise generated in the detector. Eve is allowed to control the former, but is not allowed to do the latter. While the latter is assumed to be a Gaussian random variable, the former is not assumed to be a Gaussian random variable. In this situation, using backward reconciliation and random sampling, we propose a protocol to generate secure keys between Alice and Bob under the assumption that Eve’s detector has a Gaussian noise and Eve is not so close to Alice’s transmitting device. In our protocol, the security criteria are quantitatively guaranteed even with finite block-length code based on the evaluation of error of the estimation of the channel.

Deep Neural Network Correlators for GNSS Multipath Mitigation

Machine learning and, more precisely, data-driven models are providing solutions where physics-based models are intractable. This article discusses the use of deep learning models to characterize the intricate effects of multipath propagation on GNSS correlation outputs. Particularly, we aim at substituting standard correlation schemes, optimal under single-ray Gaussian noise assumptions, with Neural Network (NN)-based correlation schemes, that are able to learn the otherwise challenging to model multipath channels. The paper shows that Deep Neural Networks (DNNs), as applied to tracking loops, can provide enhanced performance as compared to standard correlation schemes in <inline-formula><tex-math notation="LaTeX">$i)$</tex-math></inline-formula> Line-of-Sight (LOS) scenarios, by filtering out more noise thanks to strong prior regularization through knowledge of correlation characteristics and Gaussian noise during training process; and <inline-formula><tex-math notation="LaTeX">$ii)$</tex-math></inline-formula> at the same time, the DNN can adjust its behavior to better disentangle multipath signals from LOS signals. This article provides results showing the superiority of the proposed DNN trained models, with focus on time-delay tracking in a variety of realistic scenarios.

Hyperspectral Image Denoising With Weighted Nonlocal Low-Rank Model and Adaptive Total Variation Regularization

Hyperspectral image (HSI) is always corrupted by various types of noise during image capturing, such as Gaussian noise, stripe noise, deadline noise, impulse noise, and more. Such complicated noise significantly degrades imaging quality and thus limits the performance of downstream vision tasks. Current HSI denoising methods tackle this problem by modeling either the spectral-spatial prior of HSI or the noise characteristic of HSI, and few work consider the two aspects simultaneously. In this paper, we propose a new HSI denoising method by simultaneously modeling the HSI prior and the HSI noise characteristic. Specifically, we firstly utilize the non independent and identically distributed (non i.i.d.) mixture of Gaussian (MoG) assumption to characterize the complex noise, which corresponds to optimize a weighted fidelity function. Secondly, we exploit HSI’s non-local similarity and spatial-spectral correlation priors by applying non-local low rank model. Thirdly, we design an adaptive edge preserving total variation regularization term to characterize the non-local smooth property of HSI. Finally, we propose a new denoising model and develop effective ADMM algorithm to solve it. Extensive experiments on simulated data and real data substantiate the superiority of the proposed method beyond state-of-the-arts.

A variational method for Abel inversion tomography with mixed Poisson-Laplace-Gaussian noise

&lt;p style='text-indent:20px;'&gt;Abel inversion tomography plays an important role in dynamic experiments, while most known studies are started with a single Gaussian assumption. This paper proposes a mixed Poisson-Laplace-Gaussian distribution to characterize the noise in charge-coupled-device (CCD) sensed radiographic data, and develops a multi-convex optimization model to address the reconstruction problem. The proposed model is derived by incorporating varying amplitude Gaussian approximation and expectation maximization algorithm into an infimal convolution process. To solve it numerically, variable splitting and augmented Lagrangian method are integrated into a block coordinate descent framework, in which anisotropic diffusion and additive operator splitting are employed to gain edge preserving and computation efficiency. Supplementarily, a space of functions of adaptive bounded Hessian is introduced to prove the existence and uniqueness of solution to a higher-order regularized, quadratic subproblem. Moreover, a simplified algorithm with higher order regularizer is derived for Poisson noise removal. To illustrate the performance of the proposed algorithms, numerical tests on synthesized and real digital data are performed.&lt;/p&gt;

SPECULATIVE BUBBLE AND ITS EXISTENCE IN STOCK MARKET OF PAKISTAN

This empirical study is an attempt to detect speculative bubbles in different indexes of the Pakistan stock market (KSE-100, KSE-30 &amp; KMI-30) using monthly time series data from (2005– 2020) to evaluate all four Rtadf (Recursive, Right tail Augmented Dickey-Fuller) tests. This includes Standard ADF, Rolling-window ADF, Supreme ADF (SADF) and Generalized SADF (GSADF) with Monte-Carlo simulation, under Gaussian assumptions. The findings of this study confirm the two prominent episodes of exuberance and collapse of multiple speculative bubbles in the KSE-100 index while only a single (multiple) bubbles in KSE-30 and KMI-30 indexes using RADF &amp; GSADF tests. The results, in other words, suggest among all four tests, RADF and GSADF are better models to detect the past bubbles in PSX. These results contribute to scarce literature through testing and date stamping bubbles in different indexes of (PSX) with the use of (all) right-tailed tests. For future research, advanced techniques can be employed to predict future bubbles along with their direction.

Optimal stealthy attack against stochastic event‐based scheduling for remote state estimation in cyber‐physical systems

AbstractThis article concentrates on security issues of stochastic event‐based (SEB) sensor scheduling in cyber‐physical systems (CPSs). An SEB attack mechanism is proposed to compromise remote state estimation, in which the attacker can jam the feedback channel and inject false acknowledgments. Based on the Gaussian assumption, the minimum mean square error estimator under the SEB attack is obtained utilizing Bayes' theorem. In this article, the SEB attack stealthiness is composed of communication ratio invariance and anomaly detection invisibility with the forward channel. Furthermore, a double stealthy attack condition is given to complete persistent attacks from the perspective of the attacker. Aiming at maximizing average error covariances on the remote estimator, we design a generation algorithm of optimal SEB stealthy attack sequence and prove its optimality by rearrangement inequality and mathematical induction. Finally, the scalar and networked dc motor vector system simulations illustrate the corresponding theoretical results.

The Causal Interaction between Complex Subsystems.

Information flow provides a natural measure for the causal interaction between dynamical events. This study extends our previous rigorous formalism of componentwise information flow to the bulk information flow between two complex subsystems of a large-dimensional parental system. Analytical formulas have been obtained in a closed form. Under a Gaussian assumption, their maximum likelihood estimators have also been obtained. These formulas have been validated using different subsystems with preset relations, and they yield causalities just as expected. On the contrary, the commonly used proxies for the characterization of subsystems, such as averages and principal components, generally do not work correctly. This study can help diagnose the emergence of patterns in complex systems and is expected to have applications in many real world problems in different disciplines such as climate science, fluid dynamics, neuroscience, financial economics, etc.

Generalization of the power‐law rating curve using hydrodynamic theory and Bayesian hierarchical modeling

AbstractThe power‐law rating curve has been used extensively in hydraulic practice and hydrology. It is given by , where is discharge, is water elevation, , , and are unknown parameters. We propose a novel extension of the power‐law rating curve, referred to as the generalized power‐law rating curve. It is constructed by linking the physics of open channel flow to a model of the form . The function is referred to as the power‐law exponent and it depends on the water elevation. The proposed model and the power‐law model are fitted within the framework of Bayesian hierarchical models. By exploring the properties of the proposed rating curve and its power‐law exponent, we find that cross‐sectional shapes that are likely to be found in nature are such that the power‐law exponent will usually be in the interval . This fact is utilized for the construction of prior densities for the model parameters. An efficient Markov chain Monte Carlo sampling scheme, that utilizes the lognormal distributional assumption at the data level and Gaussian assumption at the latent level, is proposed for the two models. The two statistical models were applied to four datasets. In the case of three datasets the generalized power‐law rating curve gave a better fit than the power‐law rating curve while in the fourth case the two models fitted equally well and the generalized power‐law rating curve mimicked the power‐law rating curve. We developed an R package, bdrc, for fitting the power‐law and generalized power‐law rating curve models. It is available on CRAN.

Disparity-filtered differential correlation network analysis: a case study on CRC metabolomics

Differential network analysis has become a widely used technique to investigate changes of interactions among different conditions. Although the relationship between observed interactions and biochemical mechanisms is hard to establish, differential network analysis can provide useful insights about dysregulated pathways and candidate biomarkers. The available methods to detect differential interactions are heterogeneous and often rely on assumptions that are unrealistic in many applications. To address these issues, we develop a novel method for differential network analysis, using the so-called disparity filter as network reduction technique. In addition, we propose a classification model based on the inferred network interactions. The main novelty of this work lies in its ability to preserve connections that are statistically significant with respect to a null model without favouring any resolution scale, as a hard threshold would do, and without Gaussian assumptions. The method was tested using a published metabolomic dataset on colorectal cancer (CRC). Detected hub metabolites were consistent with recent literature and the classifier was able to distinguish CRC from polyp and healthy subjects with great accuracy. In conclusion, the proposed method provides a new simple and effective framework for the identification of differential interaction patterns and improves the biological interpretation of metabolomics data.

Nonparametric Functional Graphical Modeling Through Functional Additive Regression Operator

In this article, we develop a nonparametric graphical model for multivariate random functions. Most existing graphical models are restricted by the assumptions of multivariate Gaussian or copula Gaussian distributions, which also imply linear relations among the random variables or functions on different nodes. We relax those assumptions by building our graphical model based on a new statistical object—the functional additive regression operator. By carrying out regression and neighborhood selection at the operator level, our method can capture nonlinear relations without requiring any distributional assumptions. Moreover, the method is built up using only one-dimensional kernel, thus, avoids the curse of dimensionality from which a fully nonparametric approach often suffers, and enables us to work with large-scale networks. We derive error bounds for the estimated regression operator and establish graph estimation consistency, while allowing the number of functions to diverge at the exponential rate of the sample size. We demonstrate the efficacy of our method by both simulations and analysis of an electroencephalography dataset. Supplementary materials for this article are available online.

Point set registration via stochastic particle flow filter

I present an innovative registration algorithm using the particle flow filter. The particle flow filter is a Bayesian filter that uses particles to represent probability densities. The particle flow filter is not constrained to the highly restrictive unimodal, linear, and Gaussian assumptions of many Bayesian filters such as the Kalman filter. Additionally, the particle flow filter is computationally more efficient than other multimodal filters such as the better-known particle filter (PF). Unlike the PF, the particle flow filter does not require particle resampling or importance weight updates. Rather, the proposal density is formed by flowing the prior probability to the posterior using the Fokker–Planck equation. The particle flow filter algorithms were implemented using MATLAB. Both 2D and 3D rigid body point-set registration were conducted using the Gromov particle flow filter variant. Additionally, the PF method and iterative closest point (ICP) algorithms were implemented for comparison. For the same alignment accuracy, the new particle flow filter approach was 244% faster than the PF for certain challenging problems. For the same alignment time, the particle flow filter reduced misalignment by as much as 35% over that of the PF. The particle flow filter achieved 100% alignment with enough particles, and reduced misalignment by as much as 75% over that of ICP. These results demonstrate that image registration via the particle flow filter significantly outperforms the PF and ICP algorithms in the presence of noise and for a high degree of initial misalignment.

Invariant tests for functional data with application to an earthquake impact study

Motivated by an earthquake impact study, this paper develops new tests with several invariant properties for functional data. For multi-sample functional ANOVA (mfANOVA), based on local polynomial regression, exact local and global ANOVA decompositions into within- and between-group of variations are obtained. A local mfANOVA test is developed to examine differences in a local neighborhood, and combining local quantities, a global mfANOVA test statistic is formed. We show that both the local and global mfANOVA test statistics are location, scale, and translation invariant, allow interchanging the order of smoothing and ANOVA projection, and have asymptotic F-distributions under the null hypotheses with the Gaussian assumption. This paper contributes to the literature by being the first, to our knowledge, to study mfANOVA tests with several invariant properties. Simulation studies are presented to compare the proposed global mfANOVA test with some existing procedures. Application to an earthquake impact study in Taiwan reveals that when an earthquake in 2016 resulted in closed highways, the patterns of traffic flows were significantly different between three time periods, before the earthquake, during, and after the repair period. The information could be useful in planning for disaster preparedness.

Variational Bayesian Kalman filter using natural gradient

We propose a technique based on the natural gradient method for variational lower bound maximization for a variational Bayesian Kalman filter. The natural gradient approach is applied to the Kullback-Leibler divergence between the parameterized variational distribution and the posterior density of interest. Using a Gaussian assumption for the parametrized variational distribution, we obtain a closed-form iterative procedure for the Kullback-Leibler divergence minimization, producing estimates of the variational hyper-parameters of state estimation and the associated error covariance. Simulation results in both a Doppler radar tracking scenario and a bearing-only tracking scenario are presented, showing that the proposed natural gradient method outperforms existing methods which are based on other linearization techniques in terms of tracking accuracy.

Bayesian Hamiltonian Monte Carlo method for the estimation of pyrolysis parameter S1

A Gaussian mixture Hamiltonian Monte Carlo (HMC) Bayesian method has been developed for the inversion of petrophysical parameters such as pyrolysis parameter S1, which is driven by a statistical shale rock-physics model. Pyrolysis parameter S1 can be used to indicate the content of free or adsorbed hydrocarbons in source rock, and it is an important indicator to evaluate the production of shale oil reservoirs. However, most studies on pyrolysis parameters are based on pyrolysis experiments and there is no relevant study to inverse pyrolysis parameter S1 from seismic data. In addition, compared to the total organic carbon content, pyrolysis S1 is more accurate for evaluating gas and oil in shale. In particular, high values of pyrolysis S1 can directly indicate the content of shale oil. We have developed a strategy for assessing shale oil sweet spots through estimating pyrolysis S1 and other petrophysical parameters. Based on the Gaussian mixture assumptions for the prior distribution of the model, we build a joint distribution to link the pyrolysis parameter S1 with elastic attributes, and then we derive a formulation to inverse S1 with the Bayesian model. Due to the components of the Gaussian mixture, the HMC method has been used to sample the posterior distribution. Our study finds that the HMC method for sampling can improve the efficiency and allow a more robust quantification of the uncertainty; also, application to real seismic data sets indicates that the delineation of sweet spots is more accurate combined with pyrolysis S1.