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name: op-mat-pca
description: Principal component analysis of a battery — loadings and eigenvectors (p × k, rectangular), every eigenvalue with explained / cumulative shares, communalities, KMO (overall and per member) and Bartlett's sphericity test; correlation or covariance input, listwise or pairwise, weighted.
kind: operator
category: MAT
operator: MAT_PCA
type: reference
applies_to: process, compose, predict
examples_tags: [matrix, correlation-analysis]

As op-mat-covariance.

Use when

How many underlying dimensions a set of related measures covers and which measures belong to each, so a few summaries can stand in for many.

Questions it answers:

  • How many distinct things do these twelve rating questions actually measure?
  • Is this set of measures correlated enough to be worth summarising into components?

Use something else:

  • MAT_CORRELATION when you only want to see which measures go together, pair by pair.

Params

  • on — correlation (default) or covariance. rotation — none, varimax (k ≥ 2). repair — nearest.
  • components — int k (≤ p), kaiser (λ > 1, default) or {variance: s}; required on covariance.
  • missing, max_drop_share, weight, encoding: op-mat-correlation.

Inputs

Integer / float members.

Output

primary: loadings (eigenvector · √λ), PC1…PCk. auxiliary: eigenvectors, n (pairwise), unrotated. vectors: eigenvalues, explained_variance, cumulative, communalities, kmo_msa, rotated_variance. scalars: kmo, bartlett_*, components_retained.

Reading the output

  • primary.values: Each cell is a loading: how strongly the row’s measure is tied to the column’s component, the correlation between the two on a correlation input. A component is named by the measures with the largest loadings; components are listed from the one that summarises the most spread down.
  • scalars.kmo: The Kaiser-Meyer-Olkin measure of sampling adequacy: how much of the measures’ correlation is shared across the whole set rather than tied to single pairs, from 0 to 1. Higher means a component summary is more worthwhile.

Components

n, n_null, n_listwise_dropped; pairwise min_pair_n / max_pair_n; weighted sum_weights, n_eff, n_weight_invalid.

Gotchas

  • Signs are free. Rotated: read rotated_variance.
  • Bartlett’s n: rows, Σw or n_eff, pairwise the smallest pair.
  • Singular: KMO, Bartlett null (PULSE_MATRIX_SINGULAR). Non-PSD pairwise: PULSE_MATRIX_NOT_PSD unless repair.

See