Degrees Of Freedom Parameter Research Articles

Higgs decay and CP violation phase in the CPV TNMSSM

In this study, we calculate the Higgs mass matrix and explore the limitations of the minimum conditions of the scalar potential on parameter degrees of freedom in the CP violation TNMSSM. We discuss the contributions of some parameters to Higgs mass, and their impact on the strength of Higgs decay signals in different decay channels h→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h\\rightarrow $$\\end{document}γγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \\gamma $$\\end{document}, h→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h\\rightarrow $$\\end{document}VV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$VV$$\\end{document}(V=W,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(V=W,Z)$$\\end{document} and h→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h\\rightarrow $$\\end{document}ff¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\bar{f}$$\\end{document}(f=b,c,τ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(f=b,c,\ au )$$\\end{document}.

A novel robust adaptive Kalman filter with application to urban vehicle integrated navigation systems

This study addresses the formidable challenge presented by non-Gaussian, time-varying, heavy-tailed process noise in integrated vehicle navigation systems. While robust adaptive filtering methods exist, their efficacy is constrained, and certain algorithms exhibit suboptimal usability due to an excess of a priori parameters. We deduce an adaptive robust Kalman filter, characterizing process noise with a Student's t distribution and measurement noise adhering to a Gaussian distribution. To enhance robustness against unknown or varying noise statistics, our method simultaneously estimates the scale matrix, degrees of freedom parameters, state vector, and measurement noise covariance through the expectation–maximization (EM) algorithm and Bayesian variational inference theory. Rigorous simulations and empirical experiments validate its superior accuracy and stability, requiring minimal a priori parameter information. Vehicle-based experiments further demonstrate positioning error below 2.78 m, velocity error below 0.34 m/s, and heading error less than 1.28° after achieving a steady state. This approach exhibits significant potential for advancing the precision and reliability of vehicle navigation systems.

An adaptive outlier-robust Kalman filter based on sliding window and Pearson type VII distribution modeling

Due to inaccurate noise modeling and only using current measurement, existing robust filtering algorithms cannot obtain good state estimation results when measurement is interfered by outliers. In this article, a Pearson type VII (PTV) distribution-based adaptive sliding window outlier-robust Kalman filter (PTVSWAKF) is proposed. According to the characteristics of the parameters, the PTV and inverse Wishart distributions are respectively modeled as prior distributions for heavy-tailed measurement noises (HMN) and covariance. Two different Gamma distributions are modeled as priori distributions for the two degree of freedom (DOF) parameters to achieve adaptive updating of them. By the smooth posterior distribution of the sliding window state values, the state, measurement noise covariance, auxiliary random variable and DOF parameters are jointly iterated and updated to solve the outlier interference problem through variational Bayesian. Finally, the simulation experiments verify the proposed PTVSWAKF can obtain more accurate state estimation than existing filters in the environments with varying degrees of HMN and unknown non-stationary HMN.

A Gaussian–Pearson type VII adaptive mixture distribution-based outlier-robust Kalman filter

Considering that existing robust filtering algorithms rely on the selection of initial values of degree of freedom (DOF) parameters in outlier interference environments and cannot effectively cope with unknown non-stationary heavy-tailed measurement noises (HMN), a Gaussian–Pearson type VII (PTV) adaptive mixture distribution-based outlier-robust Kalman filter (GPTVMAKF) is proposed. In order to determine whether the current measurement is a normal value or an outlier, a judgment factor subject to the Beta-Bernoulli distribution is introduced. PTV distribution is used to model HMN caused by outliers, and two Gamma distributions are used to model the two different DOF parameters, which can make the PTV distribution have the adaptive adjustment ability. By introducing the inverse Wishart distribution as the prior distribution of the measurement noise covariance, which is adaptively estimated to cope with the unknown time-varying measurement noises. The state and parameters are jointly estimated by variational Bayesian. Finally, the simulation experiments verify that the proposed GPTVMAKF can obtain more accurate state estimation than existing filters in the environments with varying degrees of HMN and unknown non-stationary HMN.

Kernel Regression Residual Decomposition-Based Polynomial Frequency Modulation Integral Algorithm to Identify Physical Parameters of Time-Varying Systems under Random Excitation

The physical parameters (stiffness, damping) of time-varying (TV) systems under random excitation provide valuable information for their working condition but they are often overwhelmed by noise interference. To overcome this problem, this paper presents a novel multi-level kernel regression residual decomposition method, which can not only effectively separate each modal component from the raw vibration acceleration signal, but also eliminate noise interference. Additionally, the multiple degree-of-freedom (DOF) parameter identification problem is transformed into a single DOF parameter identification problem. Combined with the derived polynomial frequency modulation integral algorithm and the cross-correlation theory based on the fractional Fourier ambiguity function, a physical parameter identification method is proposed. The method provides a new idea in modeling TV systems and identifying physical parameters under random excitation. To demonstrate the effectiveness of the proposed method, numerical simulations are conducted with three different cases of variation (variation, quadratic variation, and periodic variation) in time. Moreover, its robustness is evaluated by adding different signal-to-noise ratio levels of noise (20 dB, 50 dB, 100 dB) to the input vibration acceleration signal. The analysis results confirm the performance of the proposed method for the parameter identification of TV systems under random excitation.

A novel robust IMM filter for jump Markov systems with heavy-tailed process and measurement noises

In some tracking application scenarios, the performances of the traditional interacting multiple model (IMM) approach may degrade dramatically since the presence of outliers in process and measurement noises. Aiming at solving the filtering problem for jump Markov systems containing heavy-tailed process and measurement noises, a new robust student's t-based Gaussian approximate filter utilizing the IMM filtering framework is presented in this paper. Firstly, the heavy-tailed process and measurement noises are assumed to obey student's t-distributions. Secondly, the degree of freedom parameters and the scale matrices are modeled as Gamma distributions and inverse-Wishart distributions respectively. Finally, the system state, degree of freedom parameters, scale matrices and the mode probabilities are inferred simultaneously by introducing the variational Bayesian (VB) theory. Maneuvering target tracking simulation verifies that the designed algorithm achieves superior estimation performances among the state-of-the-art filters.

A Bayesian Regression Model for the Non-standardized t Distribution with Location, Scale and Degrees of Freedom Parameters

A Bayesian Regression Model for the Non-standardized t Distribution with Location, Scale and Degrees of Freedom Parameters

Comparison of perturbative and non-perturbative methods in f(R) gravity

In this work, we investigate the structure and properties of neutron stars in R^2 gravity using two approaches, viz: the perturbative and non-perturbative methods. For this purpose, we consider NS with several nucleonic, as well as strange EoS generated in the framework of relativistic mean field models. The strange particles in the core of NS are in the form of Lambda hyperons and quarks, in addition to the nucleons and leptons. In both the approaches, we obtain mass–radius relation for a wide range of values of the extra degree of freedom parameter a arising due to modification of gravity at large scales. The mass–radius relation of the chosen equation of states lies well within the observational limit in the case of GR. We identify the changes in the property of neutron star in the background of f(R) gravity, and compare the results in both the methods. We also identify the best suited method to study the modified gravity using the astrophysical observations.

A variational Bayesian based robust cubature Kalman filter under dynamic model mismatch and outliers interference

This paper proposes a variational Bayesian-based cubature Kalman filter to solve the state estimation problem of nonlinear discrete-time systems under dynamic model mismatch and outliers interference, which occurs in the maneuvering target tracking system. In the proposed filter, the measurement noise is divided into the normal measurement noise and the outlier noise, and two inverse Wishart distributions with different degree of freedom parameters are used to model the unknown normal measurement noise covariance and the unknown outlier noise covariance, respectively. To overcome outliers interference, the judgment factor which follows the Bernoulli-beta distribution is designed to identify the type of measurement noise. Meanwhile, the input vector which follows the Gaussian distribution is introduced to realize the dynamic model revision. The state, the judgment factor, the normal measurement noise covariance, the outlier noise covariance, and the input vector are jointly estimated under the variational Bayesian framework. The numerical simulation is illustrated to verify the performance of the proposed filter and the corresponding results show that the proposed filter has good state estimation performance and robustness.

A flexible robust Student's t-based multimodel approach with maximum Versoria criterion

The performance of the state estimation for Gaussian state space models can be degraded if the models are affected by the non-Gaussian process and measurement noises with uncertain degree of non-Gaussianity. In this paper, we propose a flexible robust Student's t-based multimodel approach. More specifically, the degrees of freedom parameter from the Student's t-distribution is assumed unknown and modelled by a Markov chain of state values. In order to capture more information of the Student's t-distributions propagated through multiple models, we establish a model-based Versoria cost function in the form of a weighted mixture rather than the original form, and maximize the function to interact and fuse the multiple models. Simulated results prove the flexibility of the robustness of the proposed Student's t-basedmultimodel approach when the existence probability of the outliers is uncertain.

Laplace ℓ1 robust Student’s T-filter for attitude estimation of satellites

A Laplace ℓ1 robust student’s T-filter is presented for satellites to estimate their attitude state despite severe measurement noise and modeling error. Although the student’s T-filter (STF) has the capability of handling measurement noise, it cannot address the unknown modeling error. It is further sensitive to the degree of freedom (DOF). Hence, the measurement covariance is updated by using the maximum correntropy criterion to accommodate the covariance of unknown modeling error in robust filtering design. Moreover, the Laplace distribution is derived to reduce the influence of the DOF parameter by forming a surrogate function of an optimization problem. Then, the majorization minimization approach is formulated to solve such an optimization problem and present the proposed filtering algorithm in the STF framework. Numerical simulation of applying the proposed attitude estimation scheme is performed by comparing it with the third/fifth-order cubature Kalman filters.

Bayesian Poroelastic Aquifer Characterization From InSAR Surface Deformation Data. Part I: Maximum A Posteriori Estimate

AbstractCharacterizing the properties of groundwater aquifers is essential for predicting aquifer response and managing groundwater resources. In this work, we develop a high‐dimensional scalable Bayesian inversion framework governed by a three‐dimensional quasi‐static linear poroelastic model to characterize lateral permeability variations in groundwater aquifers. We determine the maximum a posteriori (MAP) point of the posterior permeability distribution from centimeter‐level surface deformation measurements obtained from Interferometric Synthetic Aperture Radar (InSAR). The scalability of our method to high parameter dimension is achieved through the use of adjoint‐based derivatives, inexact Newton methods to determine the MAP point, and a Matérn class sparse prior precision operator. Together, these guarantee that the MAP point is found at a cost, measured in number of forward/adjoint poroelasticity solves, that is independent of the parameter dimension. We apply our methodology to a test case for a municipal well in Mesquite, Nevada, in which InSAR and GPS surface deformation data are available. We solve problems with up to320,824state variable degrees of freedom (DOFs) and16,896parameter DOFs. A consistent treatment of noise level is employed so that the aquifer characterization result does not depend on the pixel spacing of surface deformation data. Our results show that the use of InSAR data significantly improves characterization of lateral aquifer heterogeneity, and the InSAR‐based aquifer characterization recovers complex lateral displacement trends observed by independent daily GPS measurements.

Alternatives to the EM algorithm for ML estimation of location, scatter matrix, and degree of freedom of the Student t distribution

In this paper, we consider maximum likelihood estimations of the degree of freedom parameter ν, the location parameter μ and the scatter matrix Σ of the multivariate Student t distribution. In particular, we are interested in estimating the degree of freedom parameter ν that determines the tails of the corresponding probability density function and was rarely considered in detail in the literature so far. We prove that under certain assumptions a minimizer of the negative log-likelihood function exists, where we have to take special care of the case nu rightarrow infty , for which the Student t distribution approaches the Gaussian distribution. As alternatives to the classical EM algorithm we propose three other algorithms which cannot be interpreted as EM algorithm. For fixed ν, the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix ν, we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for ν. We show how the objective function behaves for the different updates of ν and prove for all three algorithms that it decreases in each iteration step. We compare the algorithms as well as some accelerated versions by numerical simulation and apply one of them for estimating the degree of freedom parameter in images corrupted by Student t noise.

Revisit the Wishart Distributionm

If S_pxp can be written as S=X' X , where X_nxp is a data matrix from N_p(0,V) , then S is said to have a Wishart distribution with scale matrix V of degree of freedom parameter n. We write S~W_p(V,n). When V=I,  the distribution is said to be in standard form. When p=1, the W_1(σ^2, n)  distribution is found to be Σ^n_i=1(x^2_i) , where the elements of x_i  are identically independently distributed unit normal variables; being the σ^2(x_n)^2 distribution. Although Anderson (1984, p248~249) has presented two theorems for the Wishart distribution. In the following we give an alternative proof.

Variational Bayesian with Embedded Laplace Approximation for AR Model with Outliers

The student-t distributed innovation is robust to Auto regressive (AR) observations with outliers. However, the characteristics of student-t distribution lead to unclosed solution while using Variational Bayesian (VB) to estimate the model parameters. This paper proposed to embed the Laplace approximation in VB framework to get closed form solution. Markov Chain Monte Carlo (MCMC) is a typical method to unclosed form solutions, while it is known that MCMC has large computation costs. The degree of freedom parameter (DOF) changes greatly before and after the outlier points has been deleted, so the estimation can be improved. Experimental results show that, more accurate estimation coefficients estimation are obtained using the proposed methods compared with using the Gaussian innovation, and more accurate estimation is obtained after the additive outlier is deleted.

Outlier-Robust Student's-$t$-Based IMM-VB Localization for Manned Aircraft Using TDOA Measurements

This article proposes a novel outlier-robust localization algorithm that is based on time difference of arrival measurements at an airport for multilateration surveillance. An outlier-robust filtering scheme is derived based on Student's- $t$ -distribution, where the state, a scale matrix, and a degree of freedom parameter are estimated simultaneously using variational Bayesian (VB) inference. An interacting multiple model (IMM) filter with different system models is implemented to handle the multimodal dynamics of the aircraft, yielding the IMM-VB algorithm. Specifically, the likelihood function is newly derived using VB inference for the combination procedure in the proposed IMM-VB algorithm. The experimental results obtained from a flight test using a commercial aircraft at an airport demonstrate that the proposed IMM-VB algorithm has better localization accuracy and robustness to outlier measurements than the existing state-of-the-art approaches.

Distributed Consensus Student-t Filter for Sensor Networks With Heavy-Tailed Process and Measurement Noises

In the estimation of distributed sensor networks, process noise and measurement noise may have outliers which have heavy-tailed characteristics. To solve this problem, this paper proposes a distributed consensus estimating method for sensor networks based on Student-t distribution. In the state space model, both process noise and measurement noise are modeled as Student-t distributions with heavy-tailed characteristics. First, for the assumption that the process noise and measurement noise have the same degree of freedom parameters, an exact distributed consensus Student-t filtering algorithm is derived. In practical applications, this assumption is often not true, and due to the increasing degrees of freedom, the method will quickly converge to the traditional distributed consensus Kalman filter. Therefore, it is necessary to relax the assumption of the same degree of freedom and keep the degree of freedom unchanged within a certain range. Based on this, an approximate distributed consensus Student-t filter algorithm is proposed. Simulation results verify the effectiveness of the proposed algorithm.

Controlling the degrees of freedom in metasurface designs for multi-functional optical devices.

This review focuses on the control over the degrees of freedom (DOF) in metasurfaces, which include the input DOF (the polarization, wavelength and incident angle of the input light and some dynamic controls), parameter DOF (the complex geometric design of metasurfaces) and output DOF (the phase, polarization and amplitude of the output light). This framework could clearly show us the development process of metasurfaces, from single-functional to multi-functional ones. Advantages of the multi-functional metasurfaces are discussed in the context of various applications, including 3D holography, broadband achromatic metalenses and multi-dimensional encoded information. By combining all the input and output DOF together, we can realize ideal optical meta-devices with deep subwavelength thickness and striking functions beyond the reach of traditional optical components. Moreover, new research directions may emerge when merging different DOF in metasurfaces with other important concepts, such as parity-time symmetry and topology, so that we can have the complete control of light waves in a prescribed manner.

Robust FIR State Estimation of Dynamic Processes Corrupted by Outliers

The outlier is a common issue in the design of state estimators for an industrial process. In this paper, a robust finite impulse response (FIR) filter is proposed for time-invariant state-space models with its noise following the Student's $t$ distributions. A batch solution is first derived by maximizing the likelihood, and then, an equivalent iterative realization is given to provide a clearer insight into the FIR structure. It shows that the essence of the proposed approach is the convergence of the maximum likelihood estimates in horizon scale through iterations, and the state estimate at each sampling instant is independent of the degree-of-freedom (DOF) parameter of the Student's $t$ distribution. Based on this, a modified algorithm that updates the DOF parameter in each iteration is further proposed. Applications to a moving target tracking example and a 3-DOF helicopter system demonstrate that the proposed methods can exhibit good immunity against outliers during the filtering.

Robust Identification of LPV Time-Delay System With Randomly Missing Measurements

The robust parameter and output estimation for linear parameter varying (LPV) time-delay system with output data contaminated with outliers and subjected to randomly missing measurements are considered in this paper. The outliers, missing data, and the time-delay are widely existed in practical industry and have imposed extra difficulties on complex process modeling. The robust probability model to describe the LPV time-delay system is constructed with the student’s ${t}$ -distribution and the estimation problems are formulated in the framework of generalized expectation-maximization algorithm. The time-delay and parameter varying process properties, the outliers, and randomly missing measurements are taken into consideration comprehensively in the derivations of proposed algorithm and the unknown model parameters, scale parameter, degree of freedom parameter, the time-delay, and the noise-free output data are estimated simultaneously. The numerical example and a practical chemical process are used to present the efficacy of proposed algorithm.