Exploration of distributions of ratio of partial sum of sample eigenvalues when all population eigenvalues are the same

Research output: Contribution to journalArticle

Abstract

This paper explores empirically the first two moments of ratio of the partial sum of the first two sample eigenvalues to the sum of all eigenvalues when the population eigenvalues of a covariance matrix are all the same. Estimation of the first two moments can be practically crucial in assessing non-randomness of observed patterns on planar graphical displays based on lower rank approximations of data matrices. For derivation of the moments, exact and large sample asymptotic distributions of the sample ratios are reviewed but neither can be applicable to derivation of the moments. Therefore, I rely on simulations, where data matrices X with order nxm element-wise independent normal distribution with mean 0 and variance σ2 are assumed, that is, X ∼ N(0, σ2Inm), and then derive formulas for estimates of means and standard deviations of the sample ratios within a range of order of the data matrix. The derivations are based on the biplot graphical diagnostic methods proposed by Bradu and Gabriel (1976).

Original languageEnglish (US)
Pages (from-to)420-427
Number of pages8
JournalJournal of Modern Applied Statistical Methods
Volume1
Issue number2
StatePublished - 2002
Externally publishedYes

Fingerprint

Partial Sums
Moment
Eigenvalue
Biplot
Low-rank Approximation
Graphical Display
Element Order
Mean deviation
Standard deviation
Asymptotic distribution
Covariance matrix
Gaussian distribution
Diagnostics
Eigenvalues
Estimate
Range of data
Simulation

Keywords

  • Bias
  • Biplot
  • Eigenvalues
  • Multivariate Gaussian; Schönemann-Lingoes- Gower coefficient

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Statistics and Probability

Cite this

@article{63b5a952e1714083a542e85d9a4ddf9c,
title = "Exploration of distributions of ratio of partial sum of sample eigenvalues when all population eigenvalues are the same",
abstract = "This paper explores empirically the first two moments of ratio of the partial sum of the first two sample eigenvalues to the sum of all eigenvalues when the population eigenvalues of a covariance matrix are all the same. Estimation of the first two moments can be practically crucial in assessing non-randomness of observed patterns on planar graphical displays based on lower rank approximations of data matrices. For derivation of the moments, exact and large sample asymptotic distributions of the sample ratios are reviewed but neither can be applicable to derivation of the moments. Therefore, I rely on simulations, where data matrices X with order nxm element-wise independent normal distribution with mean 0 and variance σ2 are assumed, that is, X ∼ N(0, σ2Inm), and then derive formulas for estimates of means and standard deviations of the sample ratios within a range of order of the data matrix. The derivations are based on the biplot graphical diagnostic methods proposed by Bradu and Gabriel (1976).",
keywords = "Bias, Biplot, Eigenvalues, Multivariate Gaussian; Sch{\"o}nemann-Lingoes- Gower coefficient",
author = "Moonseong Heo",
year = "2002",
language = "English (US)",
volume = "1",
pages = "420--427",
journal = "Journal of Modern Applied Statistical Methods",
issn = "1538-9472",
publisher = "Wayne State University",
number = "2",

}

TY - JOUR

T1 - Exploration of distributions of ratio of partial sum of sample eigenvalues when all population eigenvalues are the same

AU - Heo, Moonseong

PY - 2002

Y1 - 2002

N2 - This paper explores empirically the first two moments of ratio of the partial sum of the first two sample eigenvalues to the sum of all eigenvalues when the population eigenvalues of a covariance matrix are all the same. Estimation of the first two moments can be practically crucial in assessing non-randomness of observed patterns on planar graphical displays based on lower rank approximations of data matrices. For derivation of the moments, exact and large sample asymptotic distributions of the sample ratios are reviewed but neither can be applicable to derivation of the moments. Therefore, I rely on simulations, where data matrices X with order nxm element-wise independent normal distribution with mean 0 and variance σ2 are assumed, that is, X ∼ N(0, σ2Inm), and then derive formulas for estimates of means and standard deviations of the sample ratios within a range of order of the data matrix. The derivations are based on the biplot graphical diagnostic methods proposed by Bradu and Gabriel (1976).

AB - This paper explores empirically the first two moments of ratio of the partial sum of the first two sample eigenvalues to the sum of all eigenvalues when the population eigenvalues of a covariance matrix are all the same. Estimation of the first two moments can be practically crucial in assessing non-randomness of observed patterns on planar graphical displays based on lower rank approximations of data matrices. For derivation of the moments, exact and large sample asymptotic distributions of the sample ratios are reviewed but neither can be applicable to derivation of the moments. Therefore, I rely on simulations, where data matrices X with order nxm element-wise independent normal distribution with mean 0 and variance σ2 are assumed, that is, X ∼ N(0, σ2Inm), and then derive formulas for estimates of means and standard deviations of the sample ratios within a range of order of the data matrix. The derivations are based on the biplot graphical diagnostic methods proposed by Bradu and Gabriel (1976).

KW - Bias

KW - Biplot

KW - Eigenvalues

KW - Multivariate Gaussian; Schönemann-Lingoes- Gower coefficient

UR - http://www.scopus.com/inward/record.url?scp=18444412528&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18444412528&partnerID=8YFLogxK

M3 - Article

VL - 1

SP - 420

EP - 427

JO - Journal of Modern Applied Statistical Methods

JF - Journal of Modern Applied Statistical Methods

SN - 1538-9472

IS - 2

ER -