Order Estimation Research Articles

Model order estimation based on the correntropy of observation eigenvalues

This paper presents a novel approach for estimating the model order in the presence of observation errors. The proposed method is based on the correntropy estimation of eigenvalues in the observation space. By utilizing the bootstrap method to resample the observations, more precise estimations can be obtained. The next step in the algorithm involves partitioning the observation space, generated by the covariance matrix of mixtures, into two subspaces: the signal subspace and the noise subspace. The order of the model, corresponding to the dimension of the signal subspace, is determined using a correntropy estimator based on kernel functions. Theoretical analysis demonstrates the consistency of the proposed algorithm. Comparative evaluations against existing methods in the literature highlight the superior performance of this information-theoretic approach.

MIXED VOLUME ELEMENT-CHARACTERISTIC MIXED VOLUME ELEMENT OF NUMERICAL SIMULATION OF NUCLEAR CONTAMINATION TREATMENT AND ITS CONVERGENCE ANALYSIS

We consider a nonlinear system with boundary-initial value conditions of convection-diffusion partial differential equations describing nuclear waste disposal contamination in porous media. The flow pressure is determined by an elliptic equation, the concentrations of brine and radionuclide are formulated by convection-diffusion equations, and the transport of temperature is defined by a heat equation. The transport pressure appears in the concentration equations and heat equation accompanying with Darcy velocity, and controls their processes. The flow equation is solved by the conservative method of mixed volume element and the accuracy of Darcy velocity is improved one order. The method of characteristic mixed volume element is applied to solve the concentrations and the heat, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm high computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while it has smaller time-truncation error and higher order of accuracy. The mixed volume element method has law of conservation on every element to treat the diffusion and it can obtain numerical solution is of the concentration and adjoint vectors. Using the theory and technique of priori estimate of differential equations, we derive an optimal second order estimate in l2 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve such an international famous problem.

X-ray polarization properties of partially ionized equatorial obscurers around accreting compact objects

ABSTRACT We present the expected X-ray polarization signal resulting from distant reprocessing material around black holes. Using a central isotropic power-law emission at the centre of the simulated model, we add distant equatorial and axially symmetric media that are covering the central accreting sources. We include partial ionization and partial transparency effects, and the impact of various polarization and steepness of the primary radiation spectrum. The results are obtained with the Monte Carlo code STOKES that considers both line and continuum processes and computes the effects of scattering and absorption inside static homogenous wedge-shaped and elliptical toroidal structures, varying in relative size, composition and distance to the source. We provide first order estimates for parsec-scale reprocessing in Compton-thin and Compton-thick active galactic nuclei, as well as winds around accreting stellar-mass compact objects, for observer’s inclinations above and below the grazing angle. The resulting reprocessed polarization can reach tens of per cent with either parallel or perpendicular orientation with respect to the axis of symmetry, depending on subtle details of the geometry, density, and ionization structure. We also show how principal parameters constrained from X-ray spectroscopy or polarimetry in other wavelengths can lift the shown degeneracies in X-ray polarization. We provide an application example of the broad modelling discussion by revisiting the recent IXPE 2–8 keV X-ray polarimetric observation of the accreting stellar-mass black hole in Cygnus X-3 from the perspective of partial transparency and ionization of the obscuring outflows.

Dimensionless parameter of structural ordering and excess entropy of metallic and tellurite glasses

Using a notion on the excess entropy of glass with respect to the counterpart crystal, we introduce a simple dimensionless order parameter ξ, which changes from ξ→0 to ξ→1. The former case corresponds to a strongly disordered liquid-like structure while the latter instance describes a highly ordered crystal-like glass. This approach is applied to 13 metallic and 2 tellurite glasses. We found that ξ is strongly sensitive to structural state and/or chemical composition. This parameter can be also used for a comparison of the order in glasses belonging to different classes and appears to represent a new way of structural order estimate. It is important that the ordering parameter ξ has a strong correlation with the glass-forming ability.

Nonlinear approximation of functions by sets of finite pseudo-dimension in the probabilistic and average case settings

This paper aims to explore the optimal nonlinear approximation on the class of functions in Sobolev space in the probabilistic and average case settings. Specifically, we investigate the optimal approximation through the utilization of a set with finite pseudo-dimension, measured by the Kolmogorov probabilistic nonlinear [Formula: see text]-width (or say, the Kolmogorov probabilistic pseudo-[Formula: see text]-width). Furthermore, we provide an estimation of the exact asymptotic order of the Kolmogorov probabilistic nonlinear [Formula: see text]-width and [Formula: see text]-average nonlinear [Formula: see text]-width on the class of functions in Sobolev space.

Order estimation for a fractional Brownian motion model of glucose control

When a subject is at rest and meals have not been eaten for a relatively long time (e.g. during the night), presumably near-constant, zero-order glucose production occurs in the liver. Glucose elimination from the bloodstream may be proportional to glycemia, with an apparently first-order, linear elimination rate. Besides glycemia itself, unobserved factors (insulinemia, other hormones) may exert second and higher order effects. Random events (sleep pattern variations, hormonal cycles) may also affect glycemia. The time-course of transcutaneously, continuously measured glycemia (CGM) thus reflects the superposition of different orders of control, together with random system error. The problem may be formalized as a fractional random walk, or fractional Brownian motion. In the present work, the order of this fractional stochastic process is estimated on night-time CGM data from one subject.

Notes on a paper of Pokhozhaev

We prove a second order identity for the Kirchhoff equation which yields, in particular, a simple and direct proof of Pokhozhaev’s second order conservation law when the nonlinearity has the special form . As applications, we give: an estimate of order ɛ −4 for the lifespan T ɛ of the solution of the Cauchy problem with initial data of size ɛ in Sobolev spaces when the nonlinearity is given by any C 2 function ; a necessary and sufficient condition for boundedness of a second order energy of the solutions.

A refined long time asymptotic bound for 3D axially symmetric Boussinesq system with zero thermal diffusivity

In this paper, we obtain a refined temporal asymptotic upper bound of the global axially symmetric solution to the Boussinesq system with no thermal diffusivity. We show the spacial W1,p-Sobolev (2≤p<∞) norm of the velocity can only grow at most algebraically as t→+∞. Under a signed potential condition imposed on the initial data, we further derive that the aforementioned norm is uniformly bounded at all times. Higher order estimates are also given: We find the H1 norm of the temperature fluctuation grows sub-exponentially as t→+∞. Meanwhile, for any m≥1, we deduce that the Hm-temporal growth of the solution is slower than a double exponential function. As a result, these improve the results in [11] where the authors only provided rough temporal asymptotic upper bounds while proving the global well-posedness.

Parameter and order estimation algorithms and convergence analysis for lithium‐ion batteries

AbstractThe fractional‐order equivalent circuit model can reflect the internal reaction mechanism of a lithium‐ion battery well. This article aims to design an effective model and optimization method to describe and analyze the operating characteristics of the lithium‐ion battery based on online measurement data. The fractional‐order controlled autoregressive model is derived by exploiting the memory superiorities of the fractional‐order, which comprises the electrochemical impedance spectroscopy and the ‐RC equivalent circuit model as the special cases. The utilization of polynomial properties reduces the difficulty of identification while preserving the ability of the controlled autoregressive model to fit the characteristics of the battery. To realize simultaneous parameter and the order estimation, a weighted gradient descent algorithm is proposed. The approach designs a new gradient direction and fully utilizes the data from the lithium‐ion battery by adding a suitable weighted factor. In addition, the forgetting factor is introduced to speed up convergence and produce more accurate parameter estimation. Furthermore, in order to analyze the convergence of the proposed algorithms, the convergence properties are proved by using martingale convergence theory and stochastic principle. Finally, the experimental simulation result shows the performance of the proposed algorithms.

The H content of aubrites: An evaluation of bulk versus in situ methods for quantifying water in meteorites

Aubrites and enstatite chondrites (ECs) are isotopically similar to the Earth and therefore may resemble the primary materials that accreted to form our planet. Recent bulk H elemental and isotopic analyses of ECs and the Norton County aubrite suggest that enstatite-rich materials are H-rich and may represent a significant source of terrestrial water, with measured values of 3000±2000 μg/g H2O and 5300±900 μg/g H2O in the bulk and enstatite fractions of Norton County (Piani et al., Science, 2020). Here, we present a detailed investigation of in situ H2O concentrations in enstatite, diopside, forsterite, and plagioclase from a suite of main group aubrites, including Norton County, and Shallowater. We find that enstatite (4±2 μg/g H2O), diopside (4.8±0.5 μg/g H2O), and forsterite (5±3 μg/g H2O) have similar H2O concentrations, and all are significantly lower than plagioclase (24±3 μg/g H2O). We combine our in situ analyses of H2O contents with equilibrium partition coefficients and bulk mineralogies to estimate the bulk H2O content of our samples. We compare these first order estimates with bulk volatile analyses conducted using sample pyrolysis and find that the previous bulk H2O analyses of aubrites predominantly reflect terrestrial contamination and alteration. If our conclusion that the reported bulk H2O analyses of Norton County primarily reflect terrestrial contamination and alteration extends to bulk analyses of ECs, then EC-like material may not be a significant source of terrestrial water. Our results support the hypothesis that thermal metamorphism, melting, and differentiation leads to efficient desiccation of planetesimals relative to chondrites, and that differentiated planetesimals contributed, at most, trace amounts to Earth's water budget.

A generalized weak Galerkin method for Oseen equation

In this work, the authors introduce a generalized weak Galerkin (gWG) finite element method for the evolutionary Oseen equation. The gWG method is based on a new framework for approximating the gradient operator. Both a semi-discrete and a fully-discrete numerical schemes are developed and analyzed for their convergence, stability, and error estimates. The backward Euler discretization is employed in the design of the fully-discrete scheme. Error estimates of optimal order are established mathematically, and they are validated numerically with some benchmark examples.

Macroevolutionary constraints on global microbial diversity.

Biologists have long sought to quantify the number of species on Earth. Often missing from these efforts is the contribution of microorganisms, the smallest but most abundant form of life on the planet. Despite recent large-scale sampling efforts, estimates of global microbial diversity span many orders of magnitude. It is important to consider how speciation and extinction over the last 4 billion years constrain inventories of biodiversity. We parameterized macroevolutionary models based on birth-death processes that assume constant and universal speciation and extinction rates. The models reveal that richness beyond 1012 species is feasible and in agreement with empirical predictions. Additional simulations suggest that mass extinction events do not place hard limits on modern-day microbial diversity. Together, our study provides independent support for a massive global-scale microbiome while shedding light on the upper limits of life on Earth.

Simulated Gravity Field Estimation for the Main Belt Comet 133P/Elst-Pizarro Based on a Satellite-to-satellite Tracking Mode

Most asteroids and comets are formed in the early stages of the solar system and therefore contain a wealth of information about their birth. The asteroid exploration mission planned in the coming years by China will likely target the celestial body named 133P/Elst-Pizarro (estimated diameter of about 4 km). The orbit of this asteroid stays within the asteroid belt, but nevertheless, it displays a comet-like dust tail. In this study, we used differential tracking data between two simulated probes and the data from an Earth station to estimate 133P gravity field model. This observation mode is similar to how the gravity field was estimated for large celestial objects in the GRAIL and GRACE missions, but here the object is the very small 133P asteroid. We compared the estimated gravity fields obtained for 133P from the satellite-to-satellite combined with the Earth-based two-way range-rate observation mode, with only the Earth-based two-way range rate mode. The results show that the accuracy of the low-degree (4 degree and order) estimate of the gravity field is improved by one order of magnitude by using the satellite-to-satellite combined with the Earth-based two-way range-rate observation mode with respect to the Earth-only tracking. Furthermore, another order of magnitude improvement in the gravity field solution is gained by decreasing the orbit altitude from 12 to 8 km.

Model order estimation for low-frequency oscillations in power systems by an advanced K-mean clustering approach

The estimation of low frequency modes present in the complex inter connected power system is very much crucial for taking remedial action to maintain power system stability and security. Modified total least square estimation of signal parameters via rotational invariance techniques (TLS-ESPRIT) is a popular mode estimation technique used for signal analysis, even in noisy conditions. But this technique requires a preliminary idea about the number of modes (Model order) present in the signal and its accuracy of estimation degrade when noise level increases. So the key contribution of this work is to develop a robust model order estimation technique, which will work efficiently in all range of signal and noise level. So, in this proposed work, an unsupervised sequential K-Mean++ clustering technique is implemented for the estimation of model order of low-frequency Phasor Measurement Unit (PMU) data by separating the significant eigenvalues of autocorrelation matrix (ACM) iteratively into signal and noise subspace. This two-layer K-Means++ algorithm promises a valid estimation of model order so that the insignificant eigenvalues are not clustered into the signal subspace. To validate the efficiency and robustness of the proposed method, a comparative study is carried out with the other techniques. Finally, the method is validated on two area data, real probing data obtained from Western Electricity Coordinating Council (WECC) network and oscillatory power data obtained for WSCC 9 and IEEE 39 bus system simulated on Real Time Digital Simulator (RTDS).

Sharp endpoint estimates for Schrödinger groups on Hardy spaces

Let L be a non-negative self-adjoint operator acting on L2(X) where X is a space of homogeneous type with a dimension n. Suppose that the heat kernel of L satisfies the Davies-Gaffney estimates of order m≥2. Let HL1(X) be the Hardy space associated with L. In this paper we obtain the sharp endpoint estimate for the Schrödinger group eitL associated with L such that‖(I+L)−n/2eitLf‖L1(X)+‖(I+L)−n/2eitLf‖HL1(X)≤C(1+|t|)n/2‖f‖HL1(X),t∈R for some constant C=C(n,m)>0 independent of t. We further apply our result to provide the sharp estimate for Schrödinger group of the Kohn Laplacian □b on polynomial model domains treated by Nagel–Stein [41], where e−t□b satisfies only the second order Davies-Gaffney estimates.Moreover, when the heat kernel of L satisfies a Gaussian upper bound, by a duality and interpolation argument, it gives a new proof of a recent result of [13] for sharp endpoint Lp-Sobolev bound for eitL:‖(I+L)−seitLf‖Lp(X)≤C(1+|t|)s‖f‖Lp(X),t∈R,s≥n|12−1p| for every 1<p<∞, which extends the classical results due to Miyachi ([39,40]) for the Laplacian on the Euclidean space Rn.

Unified Filter Order Estimate for Minimax-Designed Linear-Phase FIR Wideband and Lowpass Digital Differentiators

Digital differentiators enable the computation of the derivative of a continuous-time signal at discrete time instances, and they are used in many signal processing applications. This paper derives a unified filter order estimate for digital differentiators that are realized with linear-phase finite-length impulse response filters and designed in the minimax sense. The estimate is useful at the high-level system design when assessing the implementation complexity and it enables fewer designs when finding the minimal filter order required to satisfy a prescribed tolerable approximation error. The proposed unified estimate covers both wideband and lowpass differentiators of integer degrees up to ten. Furthermore, degree-individual filter order estimates are derived which improve and extend previous results. The performance of both the unified and degree-individual order estimates is evaluated through simulation examples and compared with previous estimates.

Development of a semi-empirical wake formation and dissipation prediction model for HAHT placed in channel flow

Hydrokinetic (HK) energy production has been primarily developed for use in tidal energy applications. However, where inland water infrastructure systems with sufficient velocities and spatial requirements exist, HK energy may hold great potential. A first order estimate of the wake length and dissipation rate behind a device is necessary for installation design and analysis. Some analytical approximations have been developed to estimate the wake field, although the majority of these approximations do not consider operational conditions in confined flow settings. This paper focuses on the development of a new semi-empirical model for the prediction of the wake formation, dissipation, and flow recovery. Various HK turbines are modelled, and benchmark validated using commercially available computational fluid dynamics software. The developed semi-empirical wake model adequately predicts wake behaviour over a range of performance conditions (linked to the specific turbine thrust), ambient turbulence conditions as well as blockage ratios, which are all important parameters in inland flow applications. The model enables an approximation of the wake behaviour with an accuracy of within 10% over the tested range of turbines. This approximation is valuable for facilitating the planning of turbine placement and determining the spatial requirements for inland hydrokinetic (HK) schemes.

Midway reassignment of Runge–Kutta step length

At the end of each Runge–Kutta (RK) step, a new step‐size is predicted, and later, during the evaluations of the next step, if we feel that the prediction was not correct, nothing can be done. Here, we suggest an intermediate error estimation of low order, at the beginning of the integration, that can be used for the reconsideration of the step length before this is completed. This requires the evaluation of a variable coefficient RK. These coefficients depend on the value , where is the new prediction of the step‐size. In addition, a new control for the step length at the end of each integration is designed taking advantage from the extra estimator. The basic concept of this idea is to produce general purpose RK methods that have the capability to reduce the number of step rejections for difficult problems (such as orbits with high eccentricities and Van der Pole equation).

Initial value compensation of an adaptive fractional-order unscented Kalman filter for estimating the state of charge of lithium-ion batteries

This study indicates an adaptive fractional-order unscented Kalman filter (UKF) with unknown parameters and order, which provides a higher accuracy of SOC estimation. In order to solve the problem of reduced estimation accuracy caused by the low order in the adaptive estimation, this paper adopts the augmented vector method for the initial value compensation (IVC) to reduce the impact of initial values on the estimation accuracy. Because the parameters of the LIB change with the charge and discharge of the battery use and the working conditions, and to increase the ability of the adaptive SOC estimation, this paper studies an adaptive unscented Kalman filter (AUKF) algorithm for the joint estimation of state, parameters, and order of model based on IVC. The algorithm can achieve the adaptive estimation of SOC under different working conditions. Finally, several sets of experiments are conducted under different working conditions to verify that the AUKF algorithm with IVC can achieve the effective estimation of SOC, and the adaptive effect and the estimation error of IVC on the algorithm are better than those without IVC.

Characteristics of linear and nonlinear approximation of isotropic classes of periodic multivariate functions

Exact order estimates for some characteristics of linear and nonlinear approximation of the isotropic Nikol'skii-Besov classes $\mathbf{B}^r_{p,\theta}$ of periodic multivariate functions in the spaces $B_{q,1}$, $1\leq q \leq \infty$, are obtained. Among them are the best orthogonal trigonometric approximations, best $m$-term trigonometric approximations, Kolmogorov, linear and trigonometric widths.&#x0D; For all considered characteristics, their estimates coincide in order with the corresponding estimates in the spaces $L_{q}$. Moreover, the obtained exact in order estimates (except the case $1&lt;p&lt;2\leq q &lt; \frac{p}{p-1}$) are realized by the approximation of functions from the classes ${\mathbf{B}}^r_{p,\theta}$ by trigonometric polynomials with the spectrum in cubic regions. In any case, they do not depend on the smoothness parameter $\theta$.