Transformation Of Independent Variables Research Articles

The Overlooked Importance of Constants Added in Log Transformation of Independent Variables with Zero Values: A Proposed Approach for Determining an Optimal Constant

ABSTRACTTransformation of variables has been recommended for addressing deviations from regression assumptions of normality, constant variance, and linearity. Though the log transformation has become the most commonly used transformation in biomedical, public health, and psychosocial research, challenges arise when the variable to be transformed has zero values. One suggested solution is adding constants. The constants added are usually chosen based on what is commonly used in a given field and have been shown to affect statistical inference. In this article, we examine the effect of the added constants in log transformation of independent variables and propose an approach to improve the choice of the added constant by considering it as a parameter to be estimated simultaneously with other model parameters. We reveal that the constants that are added to deal with zero values when log-transforming independent variables have profound effects on the goodness-of-fit of regression models. Arbitrary chosen values for constants may therefore result in poor fitting models. In contrast, considering the added constant as a model parameter optimizes the model fit.

How Many Zolotarëv Fractions are There?

Known properties of Chebyshev polynomials are the following: they have simple critical points with only two (finite) critical values. Those properties uniquely determine the named polynomials modulo affine transformations of dependent and independent variables. A similar property of Zolotarev fractions: simple critical points and only four critical values generates already many classes of rational functions modulo projective transformations of their dependent and independent variables. They are listed in this note.

N-Rotating Loop-Soliton Solution of the Coupled Integrable Dispersionless Equation

In this paper, we investigate the Rotating N Loop-Soliton solution of the coupled integrable dispersionless equation (CIDE) that describes a current-fed string within an external magnetic field in 2D-space. Through a set of independent variable transformation, we derive the bilinear form of the CIDE Equation. Based on the Hirota’s method, Perturbation technique and Symbolic computation, we present the analytic N-rotating loop soliton solution and proceed to some illustrations by presenting the cases of three- and four-soliton solutions.

A comparative study of regression methods to predict forest structure and canopy fuel variables from LiDAR full-waveform data

<p>Los métodos de regresión se utilizan ampliamente en el ámbito forestal para la predicción y el cartografiado de las variables de estructura y combustibilidad. En este artículo se evalúan diferentes modelos de regresión (lineal, no lineal, árboles de regresión y ensemble). Como variables independientes se utilizaron métricas extraídas de datos LiDAR full-waveform, mientras que los valores de las variables dependientes se generaron a partir de modelos basados en datos de campo obtenidos para 78 parcelas de 16 m de radio. Se llevaron a cabo transformaciones de las variables dependientes e independientes con selección de atributos para evaluar su influencia en la predicción de la variable respuesta. Con el fin de verificar diferencias significativas y ordenar los modelos de regresión se emplearon los tests no paramétricos de Wilcoxon y Friedman, y el análisis post-hoc o los tests de comparación post-hoc por pares, como el de Nemenyi, para el test de Friedman. Las regresiones basadas en la transformación de la variable dependiente, como raíz cuadrada o logaritmo, o en la transformación de las variables independientes, obtuvieron un incremento de la R<sup>2</sup> de hasta un 6% con respecto a la regresión lineal. Mediante el método CART (Classification and Regression Tree) se obtuvieron resultados discretos, si bien su uso puede estar indicado para la categorización o estratificación. Con el método basado en la transformación de la variable dependiente mediante raíz cuadrada se consiguieron los mejores resultados comparativos en la predicción de variables forestales, excepto para el volumen. Sin embargo, su uso no siempre implica una mejora significativa con respecto a los otros métodos de regresión usados en este trabajo.</p>

MRT lattice Boltzmann method for 2D flows in curvilinear coordinates

Objective: The objective of the presented paper is establishing the multi-relaxation-time lattice Boltzmann method (MRT-LBM) for solving 2D flow equations transformed in a curvilinear coordinate system.Method: Using the complete transformation approach – which includes transformation of both dependent and independent variables between the physical and computational domain – corresponding forms of the equilibrium function and of the force term for the 2D Navier–Stokes equations and the shallow water equations have been derived. The physical flow domain of arbitrary geometry in the horizontal plane, is covered with adequate curvilinear mesh, while the calculation procedure is carried out in the D2Q9 square lattice, applying the basic form of the boundary condition method on water-solid and open boundaries as well.Test cases: The method is tested using four different examples: Couette flow in a straight inclined channel, Taylor–Couette flow between two cylinders, a non-prismatic channel in a 180° bend, and a segment of irrigation channel with a parabolic cross section in a 90° bend. In the cases of the bent channels, previously available velocity measurements have been used for validation of the model. In addition, the procedure employs a mathematical model based on traditional CFD procedures.Results: The remarkable agreement between the results obtained by the proposed model and the corresponding analytical values and measurements shows that the presented curvilinear form of the LBM is capable of solving very complex environmental problems, maintaining the order of accuracy, simplicity and efficiency of the basic LBM.

New exact double periodic wave and complex wave solutions for a generalized sinh–Gordon equation

In this paper, dependent and independent variable transformations are introduced to solve a generalized sinh–Gordon equation by using the binary F-expansion method and the knowledge of elliptic equation and Jacobian elliptic functions. Many different new exact solutions such as double periodic wave and complex wave solutions are obtained. Some previous results are extended.

Model selection in kernel ridge regression

Kernel ridge regression is a technique to perform ridge regression with a potentially infinite number of nonlinear transformations of the independent variables as regressors. This method is gaining...

Symmetries and conserved quantities of discrete wave equation associated with the Ablowitz—Ladik—Lattice system

In this paper, we present a new method to obtain the Lie symmetries and conserved quantities of the discrete wave equation with the Ablowitz—Ladik—Lattice equations. Firstly, the wave equation is transformed into a simple difference equation with the Ablowitz—Ladik—Lattice method. Secondly, according to the invariance of the discrete wave equation and the Ablowitz—Ladik—Lattice equations under infinitesimal transformation of dependent and independent variables, we derive the discrete determining equation and the discrete restricted equations. Thirdly, a series of the discrete analogs of conserved quantities, the discrete analogs of Lie groups, and the characteristic equations are obtained for the wave equation. Finally, we study a model of a biological macromolecule chain of mechanical behaviors, the Lie symmetry theory of discrete wave equation with the Ablowitz—Ladik—Lattice method is verified.

Equivalence Transformations of Quasilinear First Order Systems and Reduction to Autonomous and Homogeneous Form

Classes of 2×2 first order quasilinear partial differential equations involving arbitrary continuously differentiable functions that can be mapped into autonomous and homogeneous form through equivalence transformations are considered. Equivalence transformations are point transformations of independent and dependent variables of differential equations involving arbitrary elements. The transformations act on the arbitrary elements as point transformations of an augmented space of independent, dependent variables and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space determined by the independent and dependent variables, we determine some finite transformations mapping the system into autonomous and homogeneous form. Some physical applications are considered and a comparison with reduction of quasilinear first order systems to autonomous and homogeneous form through Lie point symmetries is discussed.

Marginal Distributions of Random Vectors Generated by Affine Transformations of Independent Two-Piece Normal Variables

Marginal probability density and cumulative distribution functions are presented for multidimensional variables defined by nonsingular affine transformations of vectors of independent two-piece normal variables, the most important subclass of Ferreira and Steel's general multivariate skewed distributions. The marginal functions are obtained by first expressing the joint density as a mixture of Arellano-Valle and Azzalini's unified skew-normal densities and then using the property of closure under marginalization of the latter class.

Novel Simple Representations for Gaussian Class Multivariate Distributions With Generalized Correlation

Novel single-integral representations for the multivariate probability density functions and cumulative distribution functions of Gaussian class distributions (Rayleigh, Rician, Weibull, Nakagami- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> , and generalized Rician) are derived. The solutions are expressed in terms of well-known functions which are available in common mathematical software. The marginal random variables are not necessarily identically distributed as is the case for some past solutions. A generalized correlation structure based on a special linear transformation of independent Gaussian random variables is used in this study. The advantage of the new representation is that only a single-integral computation is needed to compute an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -dimensional distribution.

Some Traveling Wave Solutions of a Hyperbolic Model for Chemotaxis in 1-D

The traveling wave solution of a hyperbolic model for chemotaxis in one space dimension is studied in this paper. By using some transformations of dependent variables and independent variables, we apply the tanh method and improved tanh method to the model, from which some traveling wave solutions in explicit form are presented.

New Representations for the Gaussian Class Multivariate Weibull Distribution with Constant Correlation and Applications

Novel single-integral representations for the multivariate probability density functions (PDFs) and cumulative distribution functions (CDFs) of the Gaussian class Weibull distribution are derived. The solutions are expressed in terms of familiar mathematical functions which are available in common mathematical software. The well known equal (constant) correlation model is considered. A special linear transformation of independent Gaussian random variables is used to generate correlated Weibull random variables. The advantage of the new representation is that only a single integral computation is needed to compute a N-dimensional distribution. The new representation of the CDF is used for the performance evaluation of a selection diversity combiner operating in equally correlated Weibull fading channels. The new PDF representation is also used for an analysis of the moments of the output signal-to-noise ratio of an equal-gain diversity combiner operating in equally correlated Weibull fading channels.

Loop soliton solutions of a short wave model for a degasperis-procesi equation

An analytic method, i.e. the homotopy analysis method, was applied for constructing the solutions of the short waves model equations associated with the Degasperis-Procesi (DP) shallow water waves equation. The explicit analytic solutions of loop soliton governing the propagation of short waves were obtained. By means of the transformation of independent variables, an analysis one-loop soliton solution expressed by a series of exponential functions was obtained, which agreed well with the exact solution. The results reveal the validity and great potential of the homotopy analysis method in solving complicated solitary water wave problems.

Transformation of independent variables in polynomial regression models

In representing a relationship between a response and a number of independent variables, it is preferable when possible to work with a simple functional form in transformed variables rather than with a more complicated form in the original variables. In this paper, it is shown that linear transformations applied to independent variables in polynomial regression models affect the t ratio and hence the statistical significance for certain parameters of the polynomial regression models.

New exact explicit peakon and smooth periodic wave solutions of the [formula omitted] equation

Based on an independent variable transformation, the K ( 3 , 2 ) equation is investigated by the bifurcation method of planar systems and qualitative theory of polynomial differential system. In different regions of the parametric space, some new exact explicit peakon and smooth periodic wave solutions are obtained. The results presented in this paper improve the previous results.

Breather Lattice Solutions to Negative mKdV Equation

In this paper, dependent and independent variable transformations are introduced to solve the negative mKdV equation systematically by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the negative mKdV equation, including breather lattice solution and periodic wave solution.

Generalized functions asymptotically homogeneous along special transformation groups

Distributions having (quasi)asymptotics in the asymptotic scale of regularly varying functions along special groups of transformation of independent variables are said to be asymptotically homogeneous along these transformation groups. In particular, all 'quasihomogeneous' distributions have this property. A complete description of asymptotically homogeneous distributions along a transformation group determined by a vector is obtained, including in the case of critical orders. Special distribution spaces are introduced and investigated to this end. The results obtained are used for the analysis of singularities of holomorphic functions in the tube domains over coordinate sectors.Bibliography: 10 titles.

New peakon, solitary wave and periodic wave solutions for the modified Camassa–Holm equation

In this paper, an independent variable transformation is introduced to solve the modified Camassa–Holm equation using the bifurcation theory and the method of phase portrait analysis. Some peakons, solitary waves and periodic waves are found and their exact parametric representations in explicit form and in implicit form are obtained.

Systems of linear differential equations symmetric with respect to transformations nonlinear in a function

We show that if a system of first order linear differential equations with constant coefficients admits an operator whose coordinates responsible for the transformation of independent variables depend on functions then it admits an operator whose coordinates responsible for the transformation of functions depend nonlinearly on functions.