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Related Concept Videos

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
Linear Approximations01:23

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...
Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...
Linearization and Approximation01:26

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...
Application of Linearization and Approximation01:29

Application of Linearization and Approximation

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...
Mechanistic Models: Compartment Models in Individual and Population Analysis01:23

Mechanistic Models: Compartment Models in Individual and Population Analysis

Mechanistic models are utilized in individual analysis using single-source data, but imperfections arise due to data collection errors, preventing perfect prediction of observed data. The mathematical equation involves known values (Xi), observed concentrations (Ci), measurement errors (εi), model parameters (ϕj), and the related function (ƒi) for i number of values. Different least-squares metrics quantify differences between predicted and observed values. The ordinary least squares (OLS)...

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Related Experiment Video

Updated: Jun 22, 2026

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
04:35

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

Published on: July 3, 2020

Piecewise linear models with guaranteed closeness to the data.

Longin Jan Latecki1, Marc Sobel, Rolf Lakaemper

  • 1Department of Computer and Information Sciences, Temple University, Philadelphia, PA, USA. latecki@temple.edu

IEEE Transactions on Pattern Analysis and Machine Intelligence
|June 23, 2009
PubMed
Summary

This study introduces a novel method for piecewise linear approximation of point sets, overcoming limitations of traditional maximum likelihood estimates (MLE). The approach accurately fits models to data and automatically determines the optimal number of components.

Related Experiment Videos

Last Updated: Jun 22, 2026

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
04:35

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

Published on: July 3, 2020

Area of Science:

  • Computational geometry
  • Statistical modeling
  • Data analysis

Background:

  • Traditional maximum likelihood estimation (MLE) for piecewise linear approximation of point sets suffers from poor data proximity and difficulty in determining the optimal number of model components.
  • Existing methods struggle to assess model fit and require predefined component counts, limiting practical application.

Purpose of the Study:

  • To develop a robust method for piecewise linear approximation of point sets that addresses the drawbacks of classical MLE.
  • To ensure inferred models are close to data points and to automatically determine the optimal number of model components without additional computation.

Main Methods:

  • Fitting a nonparametric density to data points to define an informative neighborhood.
  • Optimizing Kullback-Leibler Divergence (KLD) between the nonparametric density and a mixture of parametric models within the defined neighborhood.
  • Utilizing a nonparametric density estimation to identify informative data points and uninformative ones.

Main Results:

  • The proposed method ensures inferred model components are close to their ground-truth counterparts, even with incorrect initial component assumptions.
  • The approach provides a reliable way to detect when models fail to fit data properly.
  • The number of significant model components is estimated as part of the inference process, without extra computational cost.

Conclusions:

  • The novel approach offers a significant improvement over classical MLE for piecewise linear approximation, providing accurate model fitting and automatic component selection.
  • This method enhances the reliability and applicability of piecewise linear models in various data analysis scenarios.
  • The technique effectively handles point sets without order constraints and accurately estimates the number of significant model components.