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Related Concept Videos

Oriented Surfaces01:30

Oriented Surfaces

A surface is called orientable if a consistent choice of unit normal vector can be made at every point on the surface. A thin soap film stretched across a wire loop provides a familiar example. The film separates the air on one side from the air on the other, so one side can be selected as positive and the opposite side as negative. Once this choice is made, a unit normal vector can be assigned smoothly across the entire surface.At each point on the soap film, a unit normal vector points...
Area Between Curves: Problem Solving01:27

Area Between Curves: Problem Solving

A region can be enclosed by three curves: a square root function, a reflected cube root function, and a linear function. The linear function intersects each of the other two curves, and these intersection points determine where the boundary of the enclosed region changes. Because different curves serve as the upper and lower boundaries in different parts of the graph, the area cannot be found using a single setup over the entire interval.To compute the area, the region is first divided into two...
Tangent Planes to a Parametric Surface01:22

Tangent Planes to a Parametric Surface

A tangent plane provides a linear approximation to a curved surface at a specific point, capturing the local behavior of the surface. It can be understood as the plane that just touches the surface at that point and is defined by the tangent directions of curves lying on the surface. These tangent directions arise naturally when the surface is described parametrically, allowing systematic construction of the plane.For a surface expressed in parametric form, the position of any point is...
Parametric Surfaces01:30

Parametric Surfaces

A parametric surface in three-dimensional space is defined through a vector-valued function\begin{equation*}\mathbf{r}(u, v) = x(u, v)\mathbf{i} + y(u, v)\mathbf{j} + z(u, v)\mathbf{k}\end{equation*}where u and v are parameters within a specified domain D in the uv-plane. The functions x(u, v), y(u, v), and z(u, v) define the coordinates of points on the surface. As u and v vary over D, the position vector r(u, v) traces a continuous surface in space. This parametric representation is essential...
Quadric Surfaces01:28

Quadric Surfaces

Quadric surfaces are three-dimensional surfaces characterized by second-degree equations in the variables x, y, and z. These surfaces are smooth and continuous, and specific combinations of squared and linear terms define their shapes. The main types of quadric surfaces include ellipsoids, cones, paraboloids, and hyperboloids. Each type exhibits distinct geometric features depending on how the variables are arranged and related within the equation.Ellipsoids are closed surfaces formed when all...
Lagrange Multipliers: Two Constraints01:28

Lagrange Multipliers: Two Constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space.

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A Real-Time Interactive System for Studying Confrontational Pursuit Behavior in Rodents
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A Real-Time Interactive System for Studying Confrontational Pursuit Behavior in Rodents

Published on: May 16, 2025

Grouping pursuit through a regularization solution surface.

Xiaotong Shen, Hsin-Cheng Huang

    Journal of the American Statistical Association
    |August 7, 2010
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces a novel homotopy method for grouping pursuit in high-dimensional regression. The method enhances predictive performance and model interpretability by adaptively grouping predictors for better outcome prediction.

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    Area of Science:

    • Statistics
    • Computational Biology
    • Machine Learning

    Background:

    • High-dimensional data analysis requires identifying predictor subgroups for improved regression models.
    • Grouping pursuit enhances predictive performance and model interpretability.
    • Applications include gene network analysis to understand disease progression.

    Purpose of the Study:

    • To develop a novel method for computing the entire solution surface in grouping pursuit.
    • To address challenges in adaptive grouping and nearly unbiased estimation.
    • To improve the efficiency and accuracy of predictor subgroup identification.

    Main Methods:

    • A novel homotopy method is introduced for grouping pursuit.
    • A piecewise linear penalty is used, which is nonconvex and overcomplete.
    • Grouped subdifferentials and difference convex programming enable efficient computation.

    Main Results:

    • The proposed method achieves high performance in numerical analyses.
    • The method demonstrates desired optimality for grouping pursuit and prediction.
    • Adaptive grouping and nearly unbiased estimation are facilitated.

    Conclusions:

    • The novel homotopy method effectively addresses challenges in grouping pursuit for high-dimensional data.
    • The method offers enhanced predictive performance and interpretability.
    • Theoretical results support the optimality and practical utility of the approach.