Hyperbolic Tangent Transformation Research Articles

An energy estimate and a stabilized Lagrange–Galerkin scheme for a multiphase flow model

Multiphase flow models are commonly employed for understanding complex fluid flows, while few mathematical discussions exist. For a general multiphase flow model in Gidaspow (1994), an energy decay property is proved. A stabilized Lagrange–Galerkin scheme for the model and its stability property are presented. Here, a hyperbolic tangent transformation is employed to preserve the boundedness of the volume fraction. A novel artificial term is added to obtain the stability property. Two-dimensional numerical examples exhibit the experimental order of convergence and applicability in modelling sedimentation phenomena.

Multiple test procedures of disease prevalence based on stratified partially validated series in the presence of a gold standard

ABSTRACT This paper discusses the problem of disease prevalence in clinical studies, focusing on multiple comparisons based on stratified partially validated series in the presence of a gold standard. Five test statistics, including two Wald-type test statistics, the inverse hyperbolic tangent transformation test statistic, likelihood ratio test statistic, and score test statistic, are proposed to conduct multiple comparisons. To control the overall type I error rate, several adjustment procedures are developed, namely the Bonferroni, Single-step adjusted MaxT, Single-step adjusted MinP, Holm’s Step-down, and Hochberg’s step-up procedures, based on these test statistics. The performance of the proposed methods is evaluated through simulation studies in terms of the empirical type I error rate and empirical power. Simulation results show that the Single-step adjusted MaxT procedure and Single-step adjusted MinP procedure generally outperform the other three procedures, and these two test procedures based on all test statistics have satisfactory performance. Notably, the Single-step adjusted MinP procedure tends to exhibit higher empirical power than the Single-step adjusted MaxT procedure. Furthermore, the Step-down and Step-up procedures show greater power compared to the Bonferroni method. The study also observes that as the validated ratio increases, the empirical type I errors of all test procedures approach the nominal level while maintaining higher power. Two real examples are presented to illustrate the proposed methods.

The Unit Folded Normal Distribution: A New Unit Probability Distribution with the Estimation Procedures, Quantile Regression Modeling and Educational Attainment Applications

In this paper, we develop a continuous distribution on the unit interval characterized by the distribution of the absolute hyperbolic tangent transformation of a random variable following the normal distribution. The lack of research on the prospect of hyperbolic transformations providing flexible distributions on the unit interval is a motivation for the study. First, we study it theoretically and discuss its properties of interest from a modeling point of view. In particular, it is shown that the proposed distribution accommodates various levels of skewness and kurtosis. Then, some statistical work is performed. We investigate diverse estimation methods for the involved parameters and evaluate their performance through two simulation studies. Subsequently, the quantile regression model derived from the proposed distribution is developed. Two real-world data applications of interest are provided. The first application is about the univariate modeling of the percentage of the educational attainment of some countries, which is one indicator of the education topic of the Better Life Index (BLI) of the Organization for Economic Co-operation and Development (OECD) countries. The second application is to explain the relationship between the percentage of educational attainment of some countries with one indicator of the work-life balance, safety, and health topics of BLI via median quantile regression modeling. For the considered data sets, the proposed distribution and quantile regression models show that they have better modeling abilities than competitive models under some comparison criteria. The results also indicate that covariates are (statistically) significant at any ordinary level of significance for the median response.

A new alternative unit- Lindley distribution with increasing failure rate

In this paper, a new one-parameter distribution is proposed by unitizing the Lindley distribution through the hyperbolic tangent transformation. The goal is to map the functionality of the Lindley distribution on the unit interval, with the perspective of offering a new modeling option for treating unit data. In the first part, we provide the motivations and some mathematical properties of the new distribution. Two truncated moments and hazard rate functions are used to characterize the distribution. The emphasis is then switched to its statistical characteristics. Several methods are used to discuss the point estimation of the parameter. The related bias and mean squared error behavior is tested using Monte Carlo simulations for a range of sample sizes. To demonstrate the ability of the model to fit real data, distributional analyses are given.

Artificial neural networks in the prediction of fraud in integral milk powder by adding whey powder

ABSTRACT: This research was performed to ascertain the most suitable Artificial Neural Network (ANN) model to quantify the degree of fraud in powdered milk through the addition of powdered whey via regular standard physicochemical analyses. In this study, an evaluation was done on 103 samples with different quantities of added whey powder to whole milk powder. Using Fourier Transform Infrared Spectroscopy the fat, cryoscopy, total solids, defatted dry extract, lactose, protein and casein were analyzed. The hyperbolic tangent transformation function was used with 45 topologies, and the Holdback and K-fold validation methods were tested. In the Holdback method, 75% of the database was employed for training, while 25% was used for validation. In the K-fold method, the database was categorized into five equal sized subsets, which alternated between training and validation. Of the two methods, the K-fold method was proven to have superior efficiency. Next, analysis was done on three models of multilayer perceptron networks with feedforward architecture. In Model 1, the input layer contained all the physicochemical analyses conducted, in model 2 the casein analysis was excluded, and in model 3 the routine analyses performed for dairy products was done (fat, defatted dry extract, cryoscopy and total solids). From Model 3 an ANN was derived which could satisfactorily predict fraud calculated from using the routine and standard analyses for dairy products, containing 64 nodes in the hidden layer, with R2 of 0.9935 and RMSE of 0.5779 for training, and R2 of 0.9964 and RMSE of 0.4358 for validation.

Confidence intervals of proportion differences for stratified combined unilateral and bilateral data

In otolaryngologic studies, subjects may contribute either unilateral observations from only one ear or bilateral observations from a pair of ears. For bilateral subjects, each person contributes information from two ears, the values of which are generally highly correlated. To avoid the confounding effect in stratified otolaryngologic studies, stratified data analysis is an important research topic. Based on the dependence model in stratified designs, this article presents five simultaneous confidence intervals (CIs) and two bootstrap simultaneous CIs for proportion differences with combined unilateral and bilateral data. Six approximate CIs and two bootstrap CIs for the common proportion difference are also developed. The proposed CIs are evaluated by empirical coverage probability, empirical coverage width and the ratio of mesial non-coverage probability to the non-coverage probability. Simulation results show that the simultaneous CIs based on Wilson method, the inverse hyperbolic tangent transformation, score statistic and the bootstrap percentile simultaneous CIs perform well and hence be recommended for applications, score CI and the CI based on Cochran statistic for the common proportion difference behave well even under small sample sizes, other CIs produce good performance when sample size is not small. A real otolaryngology study data is used to illustrate the proposed methodologies.

Efficient self-rotation perception and its relativistic consequences

An event occurring within a stationary environment, in the direction toward which an observer self-rotates, is perceived to precede a simultaneous event, in the direction away from which she moves. When self-rotation results from angular acceleration in the dark, the perception of space is also distorted, such that the subjective straight ahead shifts in the opposite direction to motion and temporal event promotion. A reference frameshift theory, based on the Special Theory of Relativity, is proposed to explain these findings. Here, a hyperbolic tangent transformation of objective angular velocity constrains subjective self-rotation velocity within finite bounds, consistent with it being a limited perceptual resource. Identifying this subjective variable with vestibular nystagmus slow-phase angular velocity, the asymptotic perceived self-rotation velocity is estimated at 200°/s. When included in the Lorentz transformations of the new formalism, this value predicts experimental simultaneity distortion. Hypothetically, the hyperbolic tangent objective-to-subjective transfer function would maximize the differential entropy of the percept, and thereby also the stimulus/percept mutual information, if angular velocities of body rotation encountered in naturalistic environmental interaction have a logistic probability density distribution of scale 100°/s, a proposed experimental test of the scheme.

Relation between stress, time management, and academic achievement in preclinical medical education: A systematic review and meta-analysis.

Identifying the learners' problems is important. Besides, many factors are associated with academic failure, among which time management and stress are more important than any others based on evidence. By using a systematic review and meta-analysis, this study aims to synthesize the findings of studies about the correlation of time management and stress with academic failure to suggest a more in-depth insight into the effect of these two factors on academic failure. Four databases were searched from the inception of January 2018. Publication bias was evaluated visually using funnel plots and sized up by Egger's test. Ninety-four articles were found to be qualified for inclusion after full-text review and additional manual reference made. Of these, 8 were studies of educational interventions that were reviewed in this paper. Regarding the relation of stress and academic performance, the Funnel plot (results not shown) and Egger's test showed no publication bias in the studies (P = 0.719). Based on this result, the estimated pooled correlation (reverted by hyperbolic tangent transformation) between stress and academic performance was found to be -0.32 (95% confidence interval: -0.38–-0.25). In conclusion, the review recognized a series of potentially mutable medium-to-large correlates of academic achievement, time management, and stress. It would be essential to have experimental data on how easily such self-regulatory capacities can be altered, and these interventions could help students enhance their potential, providing empirical tests for offered process models of academic achievement.

A new multistage Parker-Sochacki method for solving the Troesch’s problem

In this article, we introduce a new method to obtain an approximate analytical solution of the highly unstable Troesch’s problem. In the proposed method, without recourse to any hyperbolic tangent transformation or finite term approximation of the hyperbolic sine function, the problem is recast as a system of projectively polynomials which allows straightforward computation of the series solution of the problem. The radius of convergence of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Using a step length ; the problem domain is divided into subintervals, where corresponding subproblems are defined and solved with Parker-Sochacki method with very high accuracy. Highly accurate piecewise continuous approximate solution is thus obtained on the entire integration interval. The obtained solution, which is valid for every choice of the Troesch parameter , showed comparable accuracy to known numerical solutions in the literature. In particular, new results are presented for large values of in the range [20;500].

Tests for homogeneity of risk differences in stratified design with correlated bilateral data

ABSTRACTCorrelated bilateral data arise from stratified studies involving paired body organs in a subject. When it is desirable to conduct inference on the scale of risk difference, one needs first to assess the assumption of homogeneity in risk differences across strata. For testing homogeneity of risk differences, we herein propose eight methods derived respectively from weighted-least-squares (WLS), the Mantel-Haenszel (MH) estimator, the WLS method in combination with inverse hyperbolic tangent transformation, and the test statistics based on their log-transformation, the modified Score test statistic and Likelihood ratio test statistic. Simulation results showed that four of the tests perform well in general, with the tests based on the WLS method and inverse hyperbolic tangent transformation always performing satisfactorily even under small sample size designs. The methods are illustrated with a dataset.

A correlation of magnetic properties with material density for soft magnetic composite cores

Purpose – The purpose of this paper is to examine the relationships between processing conditions and magnetic properties of cores made of Soft Magnetic Composite (SMC) Somaloy 500. Design/methodology/approach – The effects of compaction pressure and hardening temperature may be combined considering SMC density. This quantity may be chosen for optimization of properties of ready-made cores. In order to describe hysteresis loops the phenomenological model based on hyperbolic tangent transformation is applied. Findings – SMC density affects substantially the shape of hysteresis loop. The paper provides a number of charts useful for checking how the parameters of the hysteresis model are affected. Research limitations/implications – The present study considers just one composition of the SMC and one type of lubricant. Future research shall be devoted to verification of the approach on a wider class of SMCs. Practical implications – Material density may be a relevant quantity in optimization of magnetic prope...

Estimation of the probability distribution of the random bearing capacity of cohesionless soil using the random finite element method

Soil mathematical modelling is a complex task when considering random soil parameters. To consider soil spatial variability, one can assume that soil properties are random variables or random fields. This study considered the estimation of the probability distribution of the bearing capacity for shallow footings in cohesionless soil using the random finite element method. In the deterministic case, the FEM code was calibrated to receive capacity evaluations similar to the results obtained using Hill's mechanism. Final capacity simulations were conducted under the assumption that the friction angle of the investigated subsoil constitutes a random field characterised by a bounded distribution. The random field was obtained by applying a hyperbolic tangent transformation to a Gaussian random field with an ellipsoidal (anisotropic) correlation function. The final outcome of the numerical analysis was an evaluation of the probability distribution that fits the bearing capacity for various foundation widths and depths. In all the cases considered, the empirical probability distribution of the bearing capacity closely resembled the Weibull distribution. Moreover, the estimated distribution of the bearing capacity was used as the basis for a foundation safety assessment.

Confidence Interval Estimation for a Difference between Two Dependent Intraclass Correlation Coefficients with Variable Class Sizes

Abstract Four confidence intervals for a difference between two dependent intraclass correlation coefficients (ICCs) are developed, focusing on applications to family studies. The basic idea adopted here is that confidence intervals for a difference between two parameters can be obtained from confidence limits for each parameter. Among the four confidence procedures considered, the one based on the inverse hyperbolic tangent transformation for a single ICC performed best. Francis Galton's data on human stature are used to illustrate the methodology.

An Interval-Scaled Scoring Algorithm for Visual Function Questionnaires

The primary purpose of the study is to present and test a simple algorithm for scoring visual function questionnaires (VFQs) that approximates person measure estimates from Rasch analysis, does not introduce nonlinearities at extreme scores, and is insensitive to missing data. A secondary purpose is to test the hypothesis that all VFQs measure the same visual ability variable and can be calibrated to a common measurement scale. Each of 407 consecutively recruited low vision patients were administered two of four visual function questionnaires: Activities of Daily Living Scale (ADVS), National Eye Institute Visual Functioning Questionnaire (NEI VFQ), 14-item Visual Functioning Index (VF-14), Visual Activities Questionnaire (VAQ). Separate Rasch analyses, using the Andrich rating scale model, were performed on responses to each of the four VFQs and again on the merged data of all instruments. An approximation of visual ability, based on average functional reserve and an inverse hyperbolic tangent transformation, is presented and tested by comparing visual ability estimates from the Rasch analyses to corresponding estimates from the approximations. Relative to person measure estimates from Rasch analysis, the approximations were observed to be linear and highly reliable (intraclass correlations ranged from 0.97 to 0.997). The measurement scale of each of the four instruments was observed to be a linear transformation of the measurement scale estimated from the merged responses of all four instruments. The approximation algorithm transforms rating scale responses for each instrument to a common measurement scale. By randomly censuring item responses for each subject, it was demonstrated that the approximation algorithm is robust and insensitive to missing data. A simple scoring algorithm based on an inverse hyperbolic tangent transformation of average functional reserve produces highly reliable approximations of visual ability estimated from Rasch analysis for the ADVS, NEI VFQ, VAQ, and VF-14. All four instruments measure the same visual ability variable in units that can be calibrated to a common measurement scale.

Estimation of the correlation coefficient using the Bayesian Approach and its applications for epidemiologic research

BackgroundThe Bayesian approach is one alternative for estimating correlation coefficients in which knowledge from previous studies is incorporated to improve estimation. The purpose of this paper is to illustrate the utility of the Bayesian approach for estimating correlations using prior knowledge.MethodsThe use of the hyperbolic tangent transformation (ρ = tanh ξ and r = tanh z) enables the investigator to take advantage of the conjugate properties of the normal distribution, which are expressed by combining correlation coefficients from different studies.ConclusionsOne of the strengths of the proposed method is that the calculations are simple but the accuracy is maintained. Like meta-analysis, it can be seen as a method to combine different correlations from different studies.

Collocation Methods for Distillation Design. 1. Model Description and Testing

Fast and accurate distillation design requires a model that significantly reduces the problem size while accurately approximating a full-order distillation column model. This collocation model builds on the concepts of past collocation models for design of complex real-world separation systems. Two variable transformations make this method unique. Polynomials cannot accurately fit trajectories which flatten out. In columns, flat sections occur in the middle of large column sections or where concentrations go to 0 or 1. With an exponential transformation of the tray number which maps zero to an infinite number of trays onto the range 0−1, four collocation trays can accurately simulate a large column section. With a hyperbolic tangent transformation of the mole fractions, the model can simulate columns which reach high purities. Furthermore, this model uses multiple collocation elements for a column section, which is more accurate than a single high-order collocation section.

A robust nonparametric procedure to estimate response functions for binary choice models

This paper introduces a promising two-phase nonparametric procedure for estimating response functions for binary choice models. The ℓ p -metric estimation in Phase 1 of the procedure serves to obtain preliminary estimates of group membership of the observations, while in Phase 2 a hyperbolic tangent transformation and simple maximum likelihood procedure result in a response function which closely resembles the logistic function. The advantages of our proposed method over the logistic are that: (1) the transformation in Phase 2 is flexible in that it can accomodate distributions of group membership probabilities with tails of various degrees of thickness, (2) due to the flexible ℓ p -metric estimation criterion in Phase 1, the procedure gives good classificatory results for a variety of distributional properties of the independent variables, and (3) the procedure does not suffer from the potential convergence problem associated with maximum likelihood estimation of the logistic function. The performance of our procedure is evaluated against the logistic method using two published data sets. These preliminary results suggest that our procedure may be a useful alternative to the logistic under certain data conditions.

A Concordance Correlation Coefficient to Evaluate Reproducibility

A new reproducibility index is developed and studied. This index is the correlation between the two readings that fall on the 45 degree line through the origin. It is simple to use and possesses desirable properties. The statistical properties of this estimate can be satisfactorily evaluated using an inverse hyperbolic tangent transformation. A Monte Carlo experiment with 5,000 runs was performed to confirm the estimate's validity. An application using actual data is given.

Some New Results on the Distribution of the Sample Correlation Coefficient

Summary A simple approximate normalization for the correlation coefficient in normal samples is obtained which is shown to compare favourably with previous normalization approximations (the usual large-sample normalization, Fisher's hyperbolic tangent transformation and Hotelling's improvement of that transformation). A second approximation, based on F and valid for large n | ρ | (n = sample size, ρ = population correlation), is also obtained. The results follow from a representation of the correlation in normal samples as the ratio of a linear combination of a standardized normal deviate and a Xn-1 to a Xn-2, where the three latter variates are independent.