Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

¹H NMR: Interpreting Distorted and Overlapping Signals01:02

¹H NMR: Interpreting Distorted and Overlapping Signals

Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are slanted or...
IR Spectrum Peak Splitting: Symmetric vs Asymmetric Vibrations01:08

IR Spectrum Peak Splitting: Symmetric vs Asymmetric Vibrations

Identical bonds within a polyatomic group can stretch symmetrically (in-phase) or asymmetrically (out-of-phase). Similar to hydrogen bonding, these vibrations also influence the shape of the IR peak. Generally, asymmetric stretching frequencies are higher than symmetric stretching frequencies. For example, primary amines exhibit two distinct IR peaks between 3300–3500 cm−1 corresponding to the symmetric and asymmetric N-H stretching, while secondary amines exhibit a single stretching vibration...
Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
In many applications, the magnitudes and directions of...
¹H NMR Signal Multiplicity: Splitting Patterns01:13

¹H NMR Signal Multiplicity: Splitting Patterns

When protons A and X are coupled, their nuclear spin energy levels are slightly modified. This is because the energy required to excite proton A to a spin state parallel to proton X is slightly different from the energy required for it to become anti-parallel to spin X. Consequently, there are two possible excitation frequencies for A (A1 and A2), depending on the spin state of X, and vice versa. The mutual nature of coupling implies that the difference between frequencies A1 and A2, indicated...
Mass Spectrometry: Complex Analysis01:21

Mass Spectrometry: Complex Analysis

Mass spectrometry is an important technique for the identification of pure compounds. However, it has some limitations for the analysis of complex mixtures, often due to excessive fragmentation making the spectrum too complicated to decipher. Mass spectrometry can be combined with suitable separation methods in sequence, forming hyphenated methods, which are useful in the analysis of complex mixtures.
GC–MS is a powerful hyphenated method commonly used in forensics and environmental...
Symmetry Elements in a Crystal01:27

Symmetry Elements in a Crystal

Crystal symmetry operations are isometric transformations that map objects onto indistinguishable copies while preserving distances, angles, and volumes. The simplest symmetry operation is translation, which shifts the entire infinite crystal lattice parallelly by a translation vector.Crystallographic rotations involve rotations by an angle of 2π/n around an axis without changing the positions of points on the axis. It is called the rotational axis of the symmetry, denoted by n. The combination...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

An exploration of the impact of SARS-CoV-2 (COVID-19) restrictions on marginalised groups in the UK.

Public health·2021
Same author

A qualitative exploration of patients' experiences with and perceptions of recommencing feeding after colorectal surgery.

Journal of human nutrition and dietetics : the official journal of the British Dietetic Association·2018
Same author

A systematic review of feeding practices among postoperative patients: is practice in-line with evidenced-based guidelines?

Journal of human nutrition and dietetics : the official journal of the British Dietetic Association·2017
Same author

A paediatric palliative care programme in development: trends in referral and location of death.

Archives of disease in childhood·2009
Same author

Statistical mechanics of learning multiple orthogonal signals: asymptotic theory and fluctuation effects.

Physical review. E, Statistical, nonlinear, and soft matter physics·2007
Same author

Rat brain serotonin neurones that express neuronal nitric oxide synthase have increased sensitivity to the substituted amphetamine serotonin toxins 3,4-methylenedioxymethamphetamine and p-chloroamphetamine.

Neuroscience·2005

Related Experiment Video

Updated: Jul 11, 2026

Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data
14:27

Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data

Published on: June 26, 2013

Principal-component-analysis eigenvalue spectra from data with symmetry-breaking structure.

D C Hoyle1, M Rattray

  • 1Department of Computer Science, University of Manchester, Kilburn Building, Oxford Road, Manchester M13 9PL, United Kingdom. david.c.hoyle@man.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 5, 2004
PubMed
Summary

This study analyzes the eigenvalue distribution in principal component analysis (PCA) with structured data. It reveals phase transitions and universal eigenvalue distributions, regardless of data distribution, when symmetry is broken.

More Related Videos

Analysis of SEC-SAXS data via EFA deconvolution and Scatter
10:59

Analysis of SEC-SAXS data via EFA deconvolution and Scatter

Published on: January 28, 2021

ARL Spectral Fitting as an Application to Augment Spectral Data via Franck-Condon Lineshape Analysis and Color Analysis
07:11

ARL Spectral Fitting as an Application to Augment Spectral Data via Franck-Condon Lineshape Analysis and Color Analysis

Published on: August 19, 2021

Related Experiment Videos

Last Updated: Jul 11, 2026

Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data
14:27

Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data

Published on: June 26, 2013

Analysis of SEC-SAXS data via EFA deconvolution and Scatter
10:59

Analysis of SEC-SAXS data via EFA deconvolution and Scatter

Published on: January 28, 2021

ARL Spectral Fitting as an Application to Augment Spectral Data via Franck-Condon Lineshape Analysis and Color Analysis
07:11

ARL Spectral Fitting as an Application to Augment Spectral Data via Franck-Condon Lineshape Analysis and Color Analysis

Published on: August 19, 2021

Area of Science:

  • Multivariate Statistics
  • Statistical Physics

Background:

  • Principal Component Analysis (PCA) is a core multivariate statistical technique.
  • It analyzes eigenvalues and eigenvectors of the sample covariance matrix.
  • Existing studies often assume data with independent and identically distributed elements.

Purpose of the Study:

  • To investigate the expected eigenvalue distribution rho(lambda) in PCA.
  • To analyze the impact of symmetry-breaking structures in the covariance matrix C.
  • To explore phase transitions in eigenvalue distributions as data dimensionality increases.

Main Methods:

  • Utilized the replica method to calculate the expected eigenvalue distribution.
  • Considered N-dimensional data vectors xi with a covariance matrix C.
  • Introduced symmetry-breaking terms into the covariance matrix: C=sigma^2I + sigma^2 * Sum_{m=1}^S A_m B_m B_m^T.

Main Results:

  • The bulk of eigenvalues follow the distribution of independent and identically distributed data.
  • Observed phase transitions at specific alpha values (alpha = A_m^{-2}).
  • A delta function separates from the bulk distribution at each transition point, lambda_u(A) = sigma^2[1+A][1+(alphaA)^{-1}].

Conclusions:

  • The replica analysis results are universal, independent of the underlying data distribution (given the fourth moment exists).
  • Symmetry-breaking directions in the covariance matrix lead to distinct eigenvalue behaviors and phase transitions.
  • The findings provide deeper insights into the statistical properties of PCA with structured data.