Principal components of numeric external features

Express numeric attributes as principal components: new variables ordered by how much variation they describe. For example, summarize rater experience, training and workload together. Choose components when you want to retain fewer variables; the default retains all nonzero components. PCA does not estimate rater quality, ability or groups. Identifiers and omitted rows are retained.

Usage

mfrm_pca(
  x,
  components = NULL,
  scale = TRUE,
  weights = NULL,
  missing = c("error", "omit")
)

# S3 method for class 'mfrm_pca'
print(x, ...)

# S3 method for class 'mfrm_pca'
summary(object, ...)

Arguments

x

A table reviewed with mfrm_features(). For mfrm_cluster_kmeans(), a result from mfrm_pca() is also accepted.

components

Number of leading principal components to retain. NULL retains the numerical rank, using sqrt(.Machine$double.eps) relative to the largest singular value. A requested count cannot exceed that rank.

scale

Divide centered numeric features by their sample standard deviations (TRUE, default). With FALSE, distances retain the original feature units. Numeric coding must have a meaningful Euclidean interpretation; factors, ordered factors, characters and logical features are refused.

weights

Optional named, strictly positive finite numeric vector with one weight per selected feature. Defaults to equal weights. Names, not vector order, identify features. Remove a feature to exclude it.

missing

Either "error" (default) or "omit". The latter excludes incomplete entities explicitly, retaining their IDs, reasons, and missing group membership in the result. No values are imputed.

...

Reserved for method compatibility.

object

An object returned by mfrm_pca().

Value

An mfrm_pca object with ID-aligned scores (omitted rows are NA), loadings, a variance table (variance, proportion, cumulative proportion and retained status), transformation (centers, divisors and weight factors), original feature_data and settings. summary() returns variance.

Details

Uses stats::prcomp() on centered, optionally standardized features, multiplied by sqrt(weights / max(weights)). Thus weights apply to squared Euclidean distances. Weight names identify original features. Constant features, nonfinite transformations and numerically unrepresentable variance are refused. Missingness stops by default; explicit omission retains every excluded ID and its original reason.

PCA finds directions of feature variance, not clusters or latent MFRM abilities. All nonzero components preserve distances in the transformed feature space, within numerical precision. Retaining fewer components changes the clustering objective and can remove informative directions. High explained variance does not establish valid groups. Signs of loadings and scores are arbitrary; tied eigenvalues can also rotate their subspace. Loadings are eigenvector coefficients in the transformed feature space, not correlations in the original units.

Use the same fitted PCA to review scores, loadings and groups. Passing its result to mfrm_cluster_kmeans() uses the retained scores without further standardization or whitening. PCA is not a required step for k-means. Group comparison and imputation sensitivity retain their descriptive scope.

See also

mfrm_cluster_kmeans(), mfrm_cluster_compare(), plot.mfrm_pca(), stats::prcomp()

Examples

raters <- data.frame(Rater = paste0("R", 1:8),
  ExperienceYears = c(1, 2, 3, 4, 10, 12, 13, 15),
  WorkshopHours = c(4, 12, 8, 20, 12, 28, 16, 32),
  AnnualRatings = c(80, 120, 90, 160, 280, 350, 300, 400))
features <- mfrm_features(raters, "Rater", names(raters)[-1])
pca <- mfrm_pca(features, components = 2)
summary(pca)
#>   Component   Variance  Proportion Cumulative Retained
#> 1       PC1 2.71834456 0.906114853  0.9061149     TRUE
#> 2       PC2 0.26968565 0.089895216  0.9960101     TRUE
#> 3       PC3 0.01196979 0.003989931  1.0000000    FALSE
pca$loadings
#>                       PC1        PC2
#> ExperienceYears 0.5831713 -0.5122424
#> WorkshopHours   0.5485673  0.8207647
#> AnnualRatings   0.5991537 -0.2528894
groups <- mfrm_cluster_kmeans(pca, k = 2, seed = 17, silhouette = FALSE)
plot(pca, type = "scores", groups = groups)

plot(groups, type = "profile", feature = "ExperienceYears")