Principal Component Analysis¶
Jolliffe, I. T. (2002). Principal Component Analysis. Springer.
Cited by¶
7 citations across 7 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.
- 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
This sourceStandard PCA reference; supports markers 061 (preservation of variance/structure while discarding low-information dimensions), 064 (PCA constructs new features vs feature selection), 068 (PCA exposing tumor-subtype clusters), and 074 (linear methods have closed-form, interpretable loadings).
- 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.
- 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.
- 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.
- 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
Verification¶
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