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Related Concept Videos

Geometric Mean01:15

Geometric Mean

The mean is a measure of the central tendency of a data set. In some data sets, the data is inherently multiplicative, and the arithmetic mean is not useful. For example, the human population multiplies with time, and so does the credit amount of financial investment, as the interest compounds over successive time intervals.
In cases of multiplicative data, the geometric mean is used for statistical analysis. First, the product of all the elements is taken. Then, if there are n elements in the...
Trimmed Mean01:10

Trimmed Mean

While measuring the mean of a data set, care needs to be taken when associating the mean to its central tendency. The same goes for the arithmetic mean, the geometric mean, or the harmonic mean. This is because the presence of a single outlier data value can significantly affect the mean. That is, the mean is sensitive to fluctuations in the data set.
Although certain measures of central tendency are not sensitive to outliers, there are alternative versions of the mean that get around the...
Harmonic Mean01:09

Harmonic Mean

The arithmetic mean is usually skewed towards the larger values in the data set. Therefore, to avoid this inherent bias towards smaller values, the harmonic mean is used.
Take the example of the speed of a car, which is the measure of the rate of distance traveled. If the vehicle traverses the same distance back-and-forth, its average speed equals the total distance traveled divided by the total time taken. However, if the car moves with varying speeds, then the arithmetic mean is more skewed...
Root Mean Square00:57

Root Mean Square

If in an experiment, data values have a probability of being both positive and negative, neither the arithmetic mean, the geometric mean, nor the harmonic mean can be used to calculate the central tendency of the data set. In particular, if the positive and negative values are equally likely, the arithmetic mean is close to zero.
For example, consider the velocity of gas molecules in a container. The gas molecules are moving in different directions, which might impart positive and negative...
Cross Product and Its Geometry01:27

Cross Product and Its Geometry

In three-dimensional space, any two non-zero vectors that are not parallel define a unique plane and geometrically outline a parallelogram. The cross product of these vectors results in a third vector that is orthogonal to the plane formed by the initial two. This vector not only encodes information about direction but also reflects important physical quantities in applied contexts.The orientation of the cross product vector is determined using the Right-Hand Rule. When the fingers of the right...
Arithmetic Mean01:08

Arithmetic Mean

The arithmetic mean is the most commonly used measure of the central tendency of a data set. It is defined as the sum of all the elements constituting the data set, divided by the total number of elements. It is sometimes loosely referred to as the “average.”
When all the values in a data set are not unique, the sum in the numerator can be calculated by multiplying each distinct value by its frequency.
Sometimes, the arithmetic mean of a sample can be affected by a few data points that are...

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Related Experiment Video

Updated: Jun 26, 2026

Measuring 3D In-vivo Shoulder Kinematics using Biplanar Videoradiography
06:09

Measuring 3D In-vivo Shoulder Kinematics using Biplanar Videoradiography

Published on: March 12, 2021

Geometric mean for subspace selection.

Dacheng Tao1, Xuelong Li, Xindong Wu

  • 1School of Computer Engineering, Nanyang Technological University, Singapore. dacheng.tao@gmail.com

IEEE Transactions on Pattern Analysis and Machine Intelligence
|December 27, 2008
PubMed
Summary
This summary is machine-generated.

This study introduces a new geometric mean approach for subspace selection, improving upon Fisher's linear discriminant analysis (FLDA). The proposed method effectively reduces class merging issues in pattern classification tasks.

Related Experiment Videos

Last Updated: Jun 26, 2026

Measuring 3D In-vivo Shoulder Kinematics using Biplanar Videoradiography
06:09

Measuring 3D In-vivo Shoulder Kinematics using Biplanar Videoradiography

Published on: March 12, 2021

Area of Science:

  • Pattern Classification
  • Dimensionality Reduction
  • Data Visualization

Background:

  • Fisher's linear discriminant analysis (FLDA) is a key subspace approach in pattern classification.
  • FLDA's linear dimensionality reduction can merge classes when the subspace dimension is less than c-1.
  • This merging occurs because FLDA maximizes the mean Kullback-Leibler (KL) divergence between classes under specific Gaussian distribution assumptions.

Purpose of the Study:

  • To investigate geometric mean criteria for subspace selection.
  • To address the class merging drawback of FLDA's linear dimensionality reduction.
  • To propose a more effective discriminative subspace selection method.

Main Methods:

  • Analysis of three geometric mean criteria for subspace selection.
  • Criterion 1: Maximization of the geometric mean of KL divergences.
  • Criterion 2: Maximization of the geometric mean of normalized KL divergences.
  • Criterion 3: Combination of criteria 1 and 2.

Main Results:

  • Preliminary experiments were conducted using synthetic data, the UCI Machine Learning Repository, and handwriting digit datasets.
  • The third criterion (combination of geometric means) showed promising results.
  • This method significantly reduced the class separation problem compared to FLDA and its extensions.

Conclusions:

  • The proposed geometric mean-based subspace selection method, particularly the combined criterion, is a potential advancement.
  • It offers a significant improvement in reducing class merging issues in pattern classification.
  • This approach enhances discriminative subspace selection capabilities.