Principal Component Analysis¶
Jolliffe, I. T. (2002). Principal Component Analysis. Springer.
Cited by¶
8 citations across 8 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Correlation
- The correlation matrix lets an analyst survey hundreds of variables at once, spotting clusters of co-movement that merit investigation, and feeds directly into dimensionality-reduction techniques such as principal component analysis that re-express the data along its axes of greatest shared variance.
This sourceStandard reference on PCA: re-expresses high-dimensional data along the axes of greatest shared variance derived from the covariance/correlation matrix — the canonical correlation-driven dimensionality reduction.
Supported in partVerified against the publisher's abstract
The book's abstract supports PCA as the eigen-decomposition of a correlation/covariance matrix, but not the claim's survey-of-hundreds or co-movement-cluster assertions.
“Principal component analysis is the empirical manifestation of the eigen value-decomposition of a correlation or covariance matrix.”
- The correlation matrix lets an analyst survey hundreds of variables at once, spotting clusters of co-movement that merit investigation, and feeds directly into dimensionality-reduction techniques such as principal component analysis that re-express the data along its axes of greatest shared variance.
- Dimensionality Reduction
- Dimensionality reduction transforms high-dimensional data into a lower-dimensional representation that preserves the structural properties most important for downstream tasks — variance, pairwise distances, neighborhood relationships, or predictive information — while discarding redundant, noisy, or low-information dimensions
Supported in partVerified against the publisher's abstract
The book's own abstract identifies it as the definitive treatment of PCA, the eigenvalue decomposition of a covariance matrix, which covers the variance half of the claim but not the rest of the enumerated list.
“Principal component analysis is the empirical manifestation of the eigen value-decomposition of a correlation or covariance matrix.”
- Dimensionality reduction transforms high-dimensional data into a lower-dimensional representation that preserves the structural properties most important for downstream tasks — variance, pairwise distances, neighborhood relationships, or predictive information — while discarding redundant, noisy, or low-information dimensions
- Eigenvalue And Eigenvector
- The data scientist's PCA is the same shape once more: the top eigenvector of the covariance matrix is the principal component, its eigenvalue the variance captured, reorganizing high-dimensional data around its dominant axes.
This sourcePrincipal components are the top eigenvectors of the covariance matrix, with eigenvalues reporting the variance captured along each axis.
Supported in partVerified against the publisher's abstract
Jolliffe's book is the standard reference establishing that PCA is the eigenvalue decomposition of a covariance (or correlation) matrix, which is the frame the claim rests on.
“Principal component analysis is the empirical manifestation of the eigen value-decomposition of a correlation or covariance matrix.”
- The data scientist's PCA is the same shape once more: the top eigenvector of the covariance matrix is the principal component, its eigenvalue the variance captured, reorganizing high-dimensional data around its dominant axes.
- Linear Independence
- The prime's intervention guidance follows directly: the failure of independence is a diagnosis, not a nuisance — the fix is not "collect more rows of the same predictors" but "drop the redundant column, or construct genuinely orthogonal predictors" (principal components, built precisely to be uncorrelated).
This sourcePrincipal components constructed precisely to be uncorrelated, the standard remedy for collinear predictors.
Supported in partVerified against the publisher's abstract
Jolliffe's abstract confirms PCA as the eigen-decomposition-based multivariate technique, but says nothing about uncorrelated components or relieving collinear predictors, so only the technique's frame is backed.
“Principal component analysis is the empirical manifestation of the eigen value-decomposition of a correlation or covariance matrix.”
- The prime's intervention guidance follows directly: the failure of independence is a diagnosis, not a nuisance — the fix is not "collect more rows of the same predictors" but "drop the redundant column, or construct genuinely orthogonal predictors" (principal components, built precisely to be uncorrelated).
- Projection
- In statistics and signal processing dimensionality reduction, feature selection, and sufficient statistics are projections of a high-dimensional sample onto a representation chosen to retain the inferentially relevant variation.
This sourceTreats dimensionality reduction as projection onto leading eigendirections that retain the load-bearing variation.
Supported in partVerified against the publisher's abstract
Jolliffe's abstract supplies PCA as an eigendecomposition-based reduction of multivariate data, which backs only the statistics/dimensionality-reduction half of the claim, not signal processing, feature selection, or sufficient statistics.
“Principal component analysis is the empirical manifestation of the eigen value-decomposition of a correlation or covariance matrix.”
- In statistics and signal processing dimensionality reduction, feature selection, and sufficient statistics are projections of a high-dimensional sample onto a representation chosen to retain the inferentially relevant variation.
- Span
- The span concept moved from linear algebra into numerical computation and ML — PCA's principal components span the data's effective subspace, compressed sensing relies on sparse representation in a chosen span, and expressiveness theorems characterize a network's span.
This sourceEstablishes that principal components span the effective subspace of the data — the linear-algebra span concept carried into dimensionality reduction.
- The span concept moved from linear algebra into numerical computation and ML — PCA's principal components span the data's effective subspace, compressed sensing relies on sparse representation in a chosen span, and expressiveness theorems characterize a network's span.
Mechanisms¶
- Data-Adapted Basis Learning
- The basis can overfit, mistaking sampling noise for structure; the learned axes can be unstable, shifting under a fresh draw of data
This sourceShows that principal axes estimated from sample data have sampling variability and can shift under a fresh sample.
- The basis can overfit, mistaking sampling noise for structure; the learned axes can be unstable, shifting under a fresh draw of data
- PCA-like Projection
- Variance is not relevance. A big nuisance — a sensor drift, a batch effect — lands in the top components and masquerades as signal, while a small but crucial target hides in the discarded tail.
This sourceWarns that a low-variance principal component can nevertheless be important for prediction, so discarding components solely by variance can lose useful signal.
- Variance is not relevance. A big nuisance — a sensor drift, a batch effect — lands in the top components and masquerades as signal, while a small but crucial target hides in the discarded tail.
Verification¶
Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.
Does it back the claim? Read against the text for 5 of 8 citations: 5 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.
See how references were verified.
Links previously used in the corpus¶
Before the registry existed this work was also linked 1 other way.
Registry ID ref:e257f0b5ed45 · see in the full table