First Approximation Research Articles

Understanding stunting and its determinant among children in the most populous state of India using estimation of population mean under ranked set sampling in the presence of missing data

ABSTRACT In real-life scenarios, encountering data with missing values is common, and if not managed carefully from the outset of a study, it can lead to significant biases in survey estimates. Various methods exist for imputing missing values in sampling procedures. Ranked set sampling (RSS) is widely recognized for its superior efficiency compared to simple random sampling. However, limited research has been conducted on ranked set sampling in the presence of missing data. This article introduces novel imputation methods designed to estimate population means in the context of missing data under RSS. These innovative estimators are developed by integrating ratio, exponential, and logarithmic estimators judiciously. Expressions for the bias and mean squared error of the proposed estimators are derived up to the first-order approximation. Through simulation studies and an application to stunting and its determinants among children in Uttar Pradesh, India’s most populous state, the effectiveness of the suggested estimators in handling missing data is demonstrated. Numerical examples involving stunting in Uttar Pradesh, as well as simulated data generated using R software, confirm the superior performance of the proposed estimators over existing methods, as evidenced by comparisons of percentage relative efficiency and mean squared error. The results are promising, indicating improvement over all existing imputation methods. Additionally, pertinent recommendations are provided for survey professionals regarding future research.

Improved memory-type ratio estimator for population mean in stratified random sampling under linear and non-linear cost functions

This paper offers an improved memory-type ratio estimator in stratified random sampling under linear and non-linear cost functions. The issue is given as all integer non-linear programming problems (AINLPPs). The sampling properties mainly the bias and the mean squared error of the introduced estimator are derived up to the first order of approximation. The optimum value of the characterizing scalar is obtained by the Lagrange method of maxima–minima. The least value of the MSE of the suggested estimator is also obtained for this optimum value of the charactering constant. The suggested estimator is compared both theoretically and empirically with the competing estimators. Under this setup, the optimum allocation with mean square error of the suggested estimator is attained, and the estimator is compared to other comparable estimators. The AINLPP is solved using the genetic programming approach, which is applied to both actual and simulated data sets from a bivariate normal distribution.

Unveiling the True Nature of Plasma Dynamics from the Reference Frame of a Superpenumbral Fibril

The magnetic geometry of the solar atmosphere, combined with projection effects, makes it difficult to accurately map the propagation of ubiquitous waves in fibrillar structures. These waves are of interest due to their ability to carry energy into the chromosphere and deposit it through damping and dissipation mechanisms. To this end, the Interferometric Bidimensional Spectrometer at the Dunn Solar Telescope was employed to capture high-resolution Hα spectral scans of a sunspot, with the transverse oscillations of a prominent superpenumbral fibril examined in depth. The oscillations are reprojected from the helioprojective Cartesian frame to a new frame of reference oriented along the average fibril axis through nonlinear force-free field extrapolations. The fibril was found to be carrying an elliptically polarized, propagating kink oscillation with a period of 430 s and a phase velocity of 69 ± 4 km s−1. The oscillation is damped as it propagates away from the sunspot with a damping length of approximately 9.2 Mm, resulting in the energy flux decreasing at a rate on the order of 460 W m−2/Mm. The Hα line width is examined and found to increase with distance from the sunspot, a potential sign of a temperature increase. Different linear and nonlinear mechanisms are investigated for the damping of the wave energy flux, but a first-order approximation of their combined effects is insufficient to recreate the observed damping length by a factor of at least 3. It is anticipated that the reprojection methodology demonstrated in this study will aid with future studies of transverse waves within fibrillar structures.

Robust design optimization using a non-intrusive second-order approximation of stochastic moments

This paper presents a new formulation of the second-order fourth-moment method (sometimes referred to as second-order perturbation method or second-order method of moments). The method allows to efficiently predict the stochastic moments of a response function and is therefore often used within robust design optimization. The new approach allows a non-intrusive implementation at the same cost as existing, highly intrusive formulations. Therefore, the new approach can be applied to any objective function without significant implementation effort. It is based on a few finite difference steps into special directions and hence is dependent on the corresponding step sizes. An automatic step size procedure is supplied beside a detailed convergence analysis. The advantages of the new formulation are demonstrated by robust design optimizations of a 2D and a 3D example using the geometrically nonlinear finite element method.

New classes of difference cum-ratio-type exponential estimators for a finite population variance in stratified random sampling

The problem of estimating the variance of a finite population is an important issue in practical situations where controlling variability is difficult. During experiments conducted in the fields of agriculture and biology, researchers often face this issue, resulting in outcomes that appear uncontrollable for the desired results. Using auxiliary information effectively has the potential to enhance the precision of estimators. This article aims to introduce improved classes of efficient estimators that are specifically designed to estimate the study variable's finite population variance. When stratified random sampling is used, these estimators are particularly efficient when the minimum and maximum values of the auxiliary variable are known. The bias and mean squared error (MSE) of the proposed classes of estimators are determined by a first-order approximation. In order to evaluate their performance and verify the theoretical results, we performed simulation research. The proposed estimators show higher percent relative efficiencies (PREs) in all simulation scenarios compared to other existing estimators, according to the results. Three datasets are utilized in the application section, which are used to further validate the effectiveness of the proposed estimators.

An improved approximate integral method for nonlinear reliability analysis

In order to evaluate the failure probability corresponding to the Limit State Function (LSF) in structural reliability, the First Order Reliability Method (FORM) linearizes the LSF and directly calculates the failure probability based on the Most Probable Point (MPP). But this method is unable to effectively handle nonlinear problems. The Second Order Reliability Method (SORM), on the other hand, utilizes the Hessian matrix to obtain curvature information at the MPP and computes the failure probability with Breitung's second-order approximation formula. However, SORM is unable to maintain satisfactory computational accuracy in highly nonlinear problems due to its lack of detailed analysis on the shape of the limit state surface. To address this issue, this paper proposes an Improved Approximation Integration Method (IAIM). The initial approximation of the limit state surface is based on the intersection region between the hypersphere and the n-dimensional paraboloid. Subsequently, a statistical analysis of the deviation between the aforementioned region and its projected area is conducted to construct a corresponding fitting function, which is then utilized for dimensionality reduction in the multidimensional integration of the approximation area. Finally, several examples are presented to demonstrate the accuracy and feasibility of the proposed IAIM.

Higher-Order Network Interactions Through Phase Reduction for Oscillators with Phase-Dependent Amplitude

Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction—the reduction of the dynamics onto an invariant torus—captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart–Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators.

On some robust imputation methods in presence of correlated measurement errors with real data applications

Missing data occurs frequently in sample surveys. Numerous imputation techniques have been proposed to tackle the problem of missing data, and, in fact, limited works are available in the literature to deal with the issue of missingness while the data are ridden with measurement errors (ME). In addition, no work has been done on the robustness to handle the issue of missing data when it is contaminated with correlated measurement errors (CME). This article proposes some robust imputation methods (RIM) to impute the missing data in the presence of CME. The mean square error (MSE) of the proposed RIM is derived from the first-order approximation and examined with the MSE of the conventional imputation methods. The results are theoretically established and explained using a vast simulation study. Two applications of real data sets are presented to illustrate the efficiency and superiority of the suggested estimators relative to some estimators considered in this study.

Biological direct-shortcut deep residual learning for sparse visual processing

We show, based on the following three grounds, that the primary visual cortex (V1) is a biological direct-shortcut deep residual learning neural network (ResNet) for sparse visual processing: (1) We first highlight that Gabor-like sets of basis functions, which are similar to the receptive fields of simple cells in the primary visual cortex (V1), are excellent candidates for sparse representation of natural images; i.e., images from the natural world, affirming the brain to be optimized for this. (2) We then prove that the intra-layer synaptic weight matrices of this region can be reasonably first-order approximated by identity mappings, and are thus sparse themselves. (3) Finally, we point out that intra-layer weight matrices having identity mapping as their initial approximation, irrespective of this approximation being also a reasonable first-order one or not, resemble the building blocks of direct-shortcut digital ResNets, which completes the grounds. This biological ResNet interconnects the sparsity of the final representation of the image to that of its intra-layer weights. Further exploration of this ResNet, and understanding the joint effects of its architecture and learning rules, e.g. on its inductive bias, could lead to major advancements in the area of bio-inspired digital ResNets. One immediate line of research in this context, for instance, is to study the impact of forcing the direct-shortcuts to be good first-order approximations of each building block. For this, along with the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\ell}}_{1}$$\\end{document}-minimization posed on the basis function coefficients the ResNet finally provides at its output, another parallel one could e.g. also be posed on the weights of its residual layers.

A Compact Coupling Interface Method with Second-Order Gradient Approximation for Elliptic Interface Problems

A Compact Coupling Interface Method with Second-Order Gradient Approximation for Elliptic Interface Problems

Evaluating biases in Penman and Penman–Monteith evapotranspiration rates at different timescales

The Penman and Penman–Monteith equations are widely used for estimating surface evapotranspiration (ET) at regional and global scales. These nonlinear equations were derived from the turbulent transport of heat fluxes and, in theory, need to be applied to a temporal scale ranging from half hour to an hour. However, these equations have been frequently applied with hydrometeorological variables averaged at daily, monthly, and even decadal time intervals, resulting in biases due to their nonlinearities. In this study, we used global reanalysis data and Taylor expanded Penman and Penman–Monteith equations to explore their nonlinear components and the biases associated with the timescale mismatches. We found that global average biases for approximating Penman equation range from 0.72 to 1.31 mm day−1 from daily to annual timescales, which mainly stem from the temperature–radiation, temperature–vapor pressure deficit (VPD), and aerodynamic conductance–VPD covariances. For Penman–Monteith equation, the corresponding biases vary from 0.47 to 0.53 mm day−1, which may be associated with the addition of stomatal conductance–VPD covariances. As a reference, the global averages from Penman and Penman–Monteith at hourly timescale over one year are 7.1 and 1.7 mm day−1. Large biases also exist around the world across various climate zones, where one or multiple covariances between meteorological variables makes the first-order approximations of Penman and Penman–Monteith equations less accurate. This analysis serves as a reminder of nonlinearities in Penman and Penman–Monteith equations, hence the requirement of data at high temporal resolution for estimating potential or actual evapotranspiration.

Autonomous Vision-Based Algorithm for Interplanetary Navigation

The surge of deep-space probes makes it unsustainable to navigate them with standard radiometric tracking. Autonomous interplanetary satellites represent a solution to this problem. In this work, a vision-based navigation algorithm is built by combining an orbit determination method with an image processing pipeline suitable for interplanetary transfers of autonomous platforms. To increase the computational efficiency of the algorithm, an extended Kalman filter is selected as state estimator, fed by the positions of the planets extracted from deep-space images. An enhancement of the estimation accuracy is performed by applying an optimal strategy to select the best pair of planets to track. Moreover, a novel analytical measurement model for deep-space navigation is developed providing a first-order approximation of the light-aberration and light-time effects. Algorithm performance is tested on a high-fidelity, Earth–Mars interplanetary transfer, showing the algorithm applicability for deep-space navigation.

Dynamic stability of the Mindlin-Reissner plate using a time-modulated axial force

A time-periodic axial force acting on a continuous structure can be destabilized by parametric excitation. Moreover, it is capable of increasing the equivalent damping and shortening the transient vibrations. In this work, this concept is extended, and dynamic stability of a Mindlin-Reissner plate is analyzed. The FE formulation for the four-node plate element is derived by employing the variational approach including the action of an axial force. The stability of the time-periodic system is evaluated both numerically and analytically, using Floquet theory and first-order approximation based on the averaging method, respectively. The influence of plate length, thickness, and aspect ratio on dynamic stability is highlighted. It is demonstrated that the instability tongue at a parametric combination resonance frequency of the summation type is shifted toward higher axial force amplitudes by increasing the plate width and decreasing the plate length. Alternatively, the axial force acting in the vicinity of a parametric combination frequency of the difference type shows a parametric antiresonance which is maximized by exactly the opposite tuning of the plate design. This guideline is important when plate design under the action of a harmonic axial force is considered.

The impact of transformations on the performance of variance estimators of finite population under adaptive cluster sampling with application to ecological data

This paper aims to investigate the impact of transformed auxiliary variables on the performance of variance estimators of finite population under adaptive cluster sampling scheme. Further, the formulation of an efficient variance estimator of a finite population is also under consideration in this article. Specifically, we explore the gain in efficiency obtained through various transformations and define dominance space for each transformation. These dominance regions provide valuable insights into the circumstances under which one transformation prevails over another regarding precision and accuracy. The theoretical properties of the suggested estimators have been discussed along with the dominance region under each transformation. The bias and Mean Square Error (MSE) have been derived up to the first order of approximation. To evaluate and empirically validate our methodology, we conduct a numerical analysis using real-life ecological data of blue-winged teal. The finding reflects the superior performance of the suggested variance estimators over the competing estimators, thereby substantiating its importance in making informed decisions in real-world applications.

Double Exponential Ratio Estimator of a Finite Population Variance under Extreme Values in Simple Random Sampling

This article presents an improved class of efficient estimators aimed at estimating the finite population variance of the study variable. These estimators are especially useful when we have information about the minimum/maximum values of the auxiliary variable within a framework of simple random sampling. The characteristics of the proposed class of estimators, including bias and mean squared error (MSE) under simple random sampling are derived through a first-order approximation. To assess the performance and validate the theoretical outcomes, we conduct a simulation study. Results indicate that the proposed class of estimators has lower MSEs as compared to other existing estimators across all simulation scenarios. Three datasets are used in the application section to emphasize the effectiveness of the proposed class of estimators over conventional unbiased variance estimators, ratio and regression estimators, and other existing estimators.

Improved Product-type Estimator for Estimating Population Mean using Harmonic Mean

In this paper, we have developed a new product-type estimator using harmonic mean of known auxiliary variable. The bias and mean square error of proposed newly product -type estimator has been derived up to first order of approximation. An empirical study has been carried out to show the performance of proposed estimator along with existing estimators. It is observed that the proposed product-type estimator is more efficient than the competing estimators. For theoretical support a general study is also carried out using normal and Weibull distribution.. KEYWORDS :Bias, Mean square error, Product estimator, Auxiliary variable, Normal distribution, Weibull distribution.

Enhanced Estimation of Population Variance under Simple Random Sampling with an Application to Real Data

In this paper, we present an enhanced class of estimators for finite population variance under simple random sampling that incorporates an auxiliary information. The equations for bias and mean square error of the proposed estimators are produced up to the first order of approximation. Some well-known estimators are found as a special case of the proposed estimators for suitably chosen values of the scalars. The effectiveness of the proposed estimators is calculated in relation to the existing estimators both theoretically and empirically using five real data sets. The empirical results exhibit the superiority of the suggested estimators over to the existing estimators in each population.. KEYWORDS :Auxiliary variable, Mean square error, Variance estimation, Class of estimators.

Refined batch of Estimators for Estimating Population Mean with the help of known Auxiliary Parameters

In the presented work, we have instituted a batch of ratio type estimators for estimating the population mean under simple random sampling with the help of known auxiliary variables or function of auxiliary variables, for more refined result. The expression of bias and mean square error of the instituted class are induced up to the first order of approximation. The strength of the proposed batch of estimators is compared with some existing estimators in terms of MSEs, first theoretically and then a numerical study is also carried out by using real data set to support the findings. It is concluded that the proposed class outperforms the various prevailing estimators in terms of minimum MSE and higher percentage relative efficiency.. KEYWORDS :Estimator, Mean, Bias, Mean square error, Percentage relative efficiency, Auxiliary information.

Efficient Classes of Robust Ratio Type Estimators of Mean and Variance in Adaptive Cluster Sampling

This paper proposes two classes of robust ratio type estimators of finite population mean and two classes of robust ratio type estimators of finite population variance using a single auxiliary variable under the adaptive cluster sampling design. Seven robust ratio type estimators have been developed from each class. The generalized expressions of bias and mean square error for each class have been derived up to the first order of approximation. The exact expressions of bias and MSE for all the developed estimators have been presented. Using simulation studies conducted in R programming, the high efficiency of all the developed estimators over similar existing estimators in adaptive cluster sampling has been demonstrated. . KEYWORDS :Adaptive cluster sampling, Classes of robust ratio type estimators, Hidden clustered data, Estimation of mean, Estimation of variance.

MATRICS: The implicit matrix-free Eulerian hydrodynamics solver

Context. There exists a zoo of different astrophysical fluid dynamics solvers, most of which are based on an explicit formulation and hence stability-limited to small time steps dictated by the Courant number expressing the local speed of sound. With this limitation, the modeling of low-Mach-number flows requires small time steps that introduce significant numerical diffusion, and a large amount of computational resources are needed. On the other hand, implicit methods are often developed to exclusively model the fully incompressible or 1D case since they require the construction and solution of one or more large (non)linear systems per time step, for which direct matrix inversion procedures become unacceptably slow in two or more dimensions. Aims. In this work, we present a globally implicit 3D axisymmetric Eulerian solver for the compressible Navier–Stokes equations including the energy equation using conservative formulation and a fully simultaneous approach. We use the second-order-in-time backward differentiation formula for temporal discretization as well as the κ scheme for spatial discretization. We implement different limiter functions to prohibit the occurrence of spurious oscillations in the vicinity of discontinuities. Our method resembles the well-known monotone upwind scheme for conservation laws (MUSCL). We briefly present efficient solution methods for the arising sparse and nonlinear system of equations. Methods. To deal with the nonlinearity of the Navier–Stokes equations we used a Newton iteration procedure in which the required Jacobian matrix-vector product was reconstructed with a first-order finite difference approximation to machine precision in a matrix-free way. The resulting linear system was solved either completely matrix-free with a combination of a sufficient Krylov solver and an approximate Jacobian preconditioner or semi-matrix-free with the incomplete lower upper factorization technique as a preconditioner. The latter was dependent on a standalone approximation of the Jacobian matrix, which was optionally calculated and needed solely for the purpose of preconditioning. Results. We show our method to be capable of damping sound waves and resolving shocks even at Courant numbers larger than one. Furthermore, we prove the method’s ability to solve boundary value problems like the cylindrical Taylor-Couette flow (TC), including viscosity, and to model transition flows. To show the latter, we recover predicted growth rates for the vertical shear instability, while choosing a time step orders of magnitude larger than the explicit one. Finally, we verify that our method is second order in space by simulating a simplistic, stationary solar wind.