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Linearization and Approximation

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...

Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...

Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

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A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...

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One-Compartment Open Model: Wagner-Nelson and Loo Riegelman Method for ka Estimation

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Updated: Jun 9, 2026

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Published on: July 3, 2020

On Estimation of Partially Linear Transformation Models.

Wenbin Lu¹, Hao Helen Zhang

¹( wlu4@stat.ncsu.edu ), Department of Statistics, North Carolina State University, Raleigh, NC 27695.

Journal of the American Statistical Association

|August 31, 2010

Summary

This study introduces a new method for analyzing survival data using partially linear transformation models. The approach effectively estimates both linear and nonlinear covariate effects, improving survival data analysis.

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Last Updated: Jun 9, 2026

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Published on: July 3, 2020

Area of Science:

Biostatistics
Survival Analysis
Statistical Modeling

Background:

Linear transformation models are widely used but may not fully capture complex covariate effects.
Incorporating nonlinear effects is crucial for accurate survival data analysis.
Existing methods may struggle to estimate parametric and nonparametric effects simultaneously.

Purpose of the Study:

To develop a unified approach for estimating parametric and nonparametric covariate effects in survival data.
To extend linear transformation models by incorporating nonlinear covariate effects.
To provide a robust statistical framework for survival data analysis with complex covariate relationships.

Main Methods:

A novel martingale-based estimating equation approach is proposed.
The method integrates global and kernel-weighted local estimation equations.
A resampling method is introduced for variance estimation of linear effects.

Main Results:

Consistent and asymptotically normal parameter estimates are achieved for linear effects with appropriate bandwidth selection.
Asymptotic properties of the estimated nonlinear effects are established.
The proposed resampling method effectively estimates asymptotic variance.

Conclusions:

The developed method offers a unified and effective way to analyze survival data with both linear and nonlinear covariate effects.
The iterative algorithm facilitates practical implementation of the procedure.
Numerical examples demonstrate the method's good finite-sample performance.