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Related Concept Videos

Survival Tree01:19

Survival Tree

Survival trees are a non-parametric method used in survival analysis to model the relationship between a set of covariates and the time until an event of interest occurs, often referred to as the "time-to-event" or "survival time." This method is particularly useful when dealing with censored data, where the event has not occurred for some individuals by the end of the study period, or when the exact time of the event is unknown.
 Building a Survival Tree
Constructing a survival tree begins...
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...
Calibration Curves: Linear Least Squares01:20

Calibration Curves: Linear Least Squares

A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
For data that follow a straight line, the standard method for fitting is the linear...
Quadratic Models01:23

Quadratic Models

Quadratic models are mathematical representations used to describe relationships in which the rate of change changes at a constant rate. These models appear in a wide variety of natural and engineered systems, especially those involving motion, forces, and optimization. One common application is analyzing the vertical motion of objects influenced by gravity, such as a ball thrown into the air.In such scenarios, the object's height changes over time in a curved pattern, rising to a maximum point...
Systems of Linear Equations in Two Variables01:25

Systems of Linear Equations in Two Variables

Solving a system of linear equations is a fundamental concept in algebra. A system of equations consists of two or more linear equations involving the same set of variables. One of the most efficient algebraic methods for solving such systems is the substitution method. This technique involves expressing one variable in terms of the other from one equation and substituting it into the second equation. This method is particularly useful when one of the equations is easily rearranged.Consider the...
Graphs of Equations in Two Variables01:30

Graphs of Equations in Two Variables

An equation with two variables, typically written in the form y = f(x) or Ax + By = C, describes a relationship between quantities represented by x and y. Each solution to such an equation is an ordered pair (x, y) that satisfies the equation when substituted. These pairs can be represented graphically to understand the variables' relationship visually.A common technique for constructing the graph of a two-variable equation is to create a value table. Begin by choosing several values for the...

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Related Experiment Video

Updated: Jul 1, 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

Combinatorics of least-squares trees.

Radu Mihaescu1, Lior Pachter

  • 1Departments of Mathematics and Computer Science, University of California, Berkeley, CA 94704, USA. mihaescu@berkeley.edu

Proceedings of the National Academy of Sciences of the United States of America
|September 10, 2008
PubMed
Summary

Researchers discovered a new condition for weighted least-squares methods in phylogenetics. This finding simplifies estimating evolutionary tree edge lengths and improves computational algorithms for phylogenetic analysis.

Related Experiment Videos

Last Updated: Jul 1, 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:

  • Phylogenetics
  • Computational Biology
  • Statistical Genetics

Background:

  • Least-squares methods are crucial for estimating evolutionary relationships (phylogenies).
  • Combinatorial formulas for edge lengths have advanced phylogenetic algorithm development.
  • Understanding connections between algorithms like neighbor-joining and least-squares is vital.

Purpose of the Study:

  • To identify a desirable property for weighted least-squares phylogenetic methods.
  • To fully characterize methods satisfying this property.
  • To generalize and unify existing work in least-squares phylogenetics.

Main Methods:

  • Characterizing methods based on a multiplicative four-point condition for the variance matrix.
  • Utilizing the tree-additivity of Lagrange multipliers from the Gauss-Markov theorem.
  • Generalizing previous findings on ordinary least squares and related models.

Main Results:

  • A complete characterization of weighted least-squares methods satisfying the desired phylogenetic property.
  • The necessary and sufficient condition is a multiplicative four-point condition on the variance matrix.
  • The proof reveals tree-additive properties of Lagrange multipliers.

Conclusions:

  • The identified condition provides a unified framework for least-squares phylogenetics.
  • This work extends and completes prior research in the field.
  • A time-optimal algorithm for computation is a direct outcome of these findings.