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Related Concept Videos

Hyperbolas01:30

Hyperbolas

A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse axis is...
Ellipses01:30

Ellipses

An ellipse is formed when a right circular cone is intersected by an inclined plane that does not cut through its base. This intersection yields a closed, symmetric curve characterized by distinctive geometric properties. Most notably, an ellipse is defined as the collection of all points in a plane for which the combined distances to two fixed points—called the foci—remain constant.The ellipse features two principal axes: the major and the minor axes. The major axis is the longest diameter,...
Geometry of Hyperbolas01:30

Geometry of Hyperbolas

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first column of the Routh...
Routh-Hurwitz Criterion I01:15

Routh-Hurwitz Criterion I

Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.
To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the...
Eccentricity of an Ellipse01:27

Eccentricity of an Ellipse

An ellipse is a fundamental conic section defined by the constant sum of distances from any point on its curve to two fixed points, known as the foci. This geometric property can be physically demonstrated using a pencil, string, and two pins. By anchoring the string at both ends and maintaining it taut with a pencil, one can trace the outline of an ellipse.The shape and extent of the ellipse are determined by its eccentricity, e, defined as the ratio of the distance between the center and a...

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Related Experiment Video

Updated: Jun 2, 2026

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
08:12

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

Published on: March 1, 2022

Hyperellipsoidal statistical classifications in a reproducing kernel Hilbert space.

Xun Liang1, Zhihao Ni

  • 1School of Information, Remin University of China, Beijing, China. xliang@ruc.edu.cn

IEEE Transactions on Neural Networks
|May 10, 2011
PubMed
Summary
This summary is machine-generated.

This study introduces support vector machines (SVMs) using Mahalanobis distance kernels, which offer improved performance over standard Euclidean distance kernels. These novel hyperellipsoidal kernels slightly outperform hyperspherical ones with fewer support vectors.

Related Experiment Videos

Last Updated: Jun 2, 2026

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
08:12

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

Published on: March 1, 2022

Area of Science:

  • Machine Learning
  • Statistical Learning Theory

Background:

  • Standard Support Vector Machines (SVMs) utilize kernels based on Euclidean distance.
  • Euclidean distance kernels result in hyperspherical decision boundaries.

Purpose of the Study:

  • To extend standard SVMs by incorporating kernels based on the Mahalanobis distance.
  • To systematically develop Mahalanobis distance-based kernels within a reproducing kernel Hilbert space framework.

Main Methods:

  • Developed SVMs with Mahalanobis distance kernels, creating hyperellipsoidal kernels.
  • Compared Mahalanobis distance kernels against Euclidean distance kernels in a reproducing kernel Hilbert space.
  • Investigated the condition where Mahalanobis distance becomes a special case of Euclidean distance (identity covariance matrix).

Main Results:

  • Mahalanobis distance-based kernels lead to hyperellipsoidal shapes.
  • Extensive experiments showed hyperellipsoidal kernels slightly outperform hyperspherical kernels.
  • The proposed method resulted in a reduction in the number of support vectors (SVs).

Conclusions:

  • SVMs with Mahalanobis distance kernels offer a viable and slightly superior alternative to standard Euclidean distance kernels.
  • The hyperellipsoidal kernels demonstrate enhanced performance and efficiency, indicated by fewer support vectors.