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Related Concept Videos

Dimensional Analysis02:19

Dimensional Analysis

The concept of dimension is important because every mathematical equation linking physical quantities must be dimensionally consistent, implying that mathematical equations must meet the following two rules. The first rule is that, in an equation, the expressions on each side of the equal sign must have the same dimensions. This is fairly intuitive since we can only add or subtract quantities of the same type (dimension). The second rule states that, in an equation, the arguments of any of the...
Dimensional Analysis03:40

Dimensional Analysis

Dimensional analysis, also known as the factor label method, is a versatile approach for mathematical operations. The main principle behind this approach is: the units of quantities must be subjected to the same mathematical operations as their associated numbers. This method can be applied to computations ranging from simple unit conversions to more complex and multi-step calculations involving several different quantities and their units.
Conversion Factors and Dimensional Analysis
The unit...
Dimensional Analysis01:23

Dimensional Analysis

Dimensional analysis is a powerful tool that is used in physics and engineering to understand and predict the behavior of physical systems. The basic idea behind dimensional analysis is to express physical quantities in terms of fundamental dimensions such as the mass, length, and time. Derived dimensions like the velocity, acceleration, and force are derived from the combinations of these fundamental dimensions.
Dimensional analysis allows us to analyze and compare physical quantities on a...
Dimensional Analysis01:27

Dimensional Analysis

Dimensional analysis is a valuable technique in fluid mechanics for simplifying complex problems by reducing them into dimensionless groups. These groups capture the essential relationships between the variables involved, allowing researchers and engineers to analyze fluid flow without dealing with each variable individually. This approach reduces the number of independent variables, allowing for easier analysis and better understanding of physical phenomena.
In fluid mechanics, dimensional...
Problem Solving: Dimensional Analysis01:08

Problem Solving: Dimensional Analysis

Every mathematical equation that connects separate distinct physical quantities must be dimensionally consistent, which implies it must abide by two rules. For this reason, the concept of dimension is crucial. The first rule is that an equation's expressions on either side of an equality must have the exact same dimension, i.e., quantities of the same dimension can be added or removed. The second rule stipulates that all popular mathematical functions, such as exponential, logarithmic, and...
Collisions in Multiple Dimensions: Introduction01:05

Collisions in Multiple Dimensions: Introduction

It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a problem,...

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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
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Bayesian supervised dimensionality reduction.

Mehmet Gönen

    IEEE Transactions on Cybernetics
    |June 13, 2013
    PubMed
    Summary

    This study introduces a Bayesian supervised dimensionality reduction method that improves prediction performance by coupling dimensionality reduction and supervised learning. The novel approach enhances classification and retrieval in benchmark datasets and image recognition tasks.

    Area of Science:

    • Machine Learning
    • Statistical Modeling
    • Computer Vision

    Background:

    • Dimensionality reduction is a standard preprocessing step for supervised learning.
    • Coupling dimensionality reduction with supervised learning may enhance predictive accuracy.
    • Existing methods often treat these steps separately, potentially limiting performance.

    Purpose of the Study:

    • To introduce a novel Bayesian supervised dimensionality reduction method.
    • To integrate linear dimensionality reduction and linear supervised learning.
    • To improve prediction performance in multiclass classification tasks.

    Main Methods:

    • Developed a principled Bayesian probabilistic model for supervised dimensionality reduction.
    • Implemented Gibbs sampling and variational approximation for model learning.

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  • Extended the formulation for model selection using automatic relevance determination to find intrinsic dimensionality.

    • Main Results:

    • The proposed Bayesian method significantly outperforms seven baseline linear dimensionality reduction algorithms.
    • Superior generalization performance was observed on benchmark datasets, especially at very low dimensions.
    • Achieved top results in classification and retrieval for an image recognition task.

    Conclusions:

    • The coupled Bayesian approach offers a principled and effective way to perform supervised dimensionality reduction.
    • The method demonstrates strong performance gains over traditional separate approaches.
    • This technique is beneficial for tasks requiring high accuracy in low-dimensional spaces, such as image recognition.