Two-step Baseline Algorithm for PCA/FA

This function implements a two-step baseline approach for comparison with mixture-based methods:

1. Step 1: Individual Decomposition - Apply PCA or FA to each subject's data separately to extract r basis vectors per subject.

2. Step 1.5: Basis Alignment - Align basis vectors across subjects using the Hungarian algorithm to ensure corresponding components are in the same order, improving clustering accuracy.

3. Step 2: Clustering - Normalize the basis vectors, concatenate them across all subjects, and apply k-means clustering. Assign each subject to a cluster based on majority vote of their basis vectors.

Usage

fit_two_step_baseline(
  data_list,
  r,
  K,
  method = c("PCA", "FA"),
  seed = 123,
  z_true = NULL,
  refine_assignments = FALSE,
  align_basis = TRUE
)

Arguments

data_list

A list of (T_i x M) matrices, one per subject.

r

Number of components (factors or principal components) to extract per subject.

K

Number of clusters for k-means.

method

Either "PCA" or "FA" for the decomposition method.

seed

Random seed for reproducibility.

z_true

Optional true cluster labels for computing ARI. If NULL, ARI is set to NA.

refine_assignments

Logical. If TRUE, performs a single reassignment pass after fitting cluster-specific models in Step 4. Default is FALSE.

align_basis

Logical. If TRUE (default), applies Hungarian algorithm alignment in Step 1.5 to ensure corresponding components across subjects are in the same order before clustering. Set to FALSE to disable alignment.

Value

A list with:

z_est

Integer vector of estimated cluster assignments (1..K).

W_list

List of normalized basis matrices, one per subject.

W_centroids

Matrix of cluster centroids in basis space.

mu_list

List of centering vectors used for standardization, one per subject.

sd_list

List of scaling vectors used for standardization, one per subject.

cluster_models

List of cluster-specific models (each with W and n_members).

method

Character string indicating the method used (e.g., "TwoStep_PCA").

BIC

Set to NA (k-means does not have a canonical BIC).

ARI

Adjusted Rand Index if z_true is provided, otherwise NA.

Details

The algorithm addresses sign indeterminacy by normalizing each basis vector so that its maximum absolute value element is positive. Tie-breaking in majority vote is handled by computing the mean distance from a subject's basis vectors to each tied cluster centroid and selecting the closer one.

Step 1.5 (basis alignment) uses the Hungarian algorithm to align basis vectors across subjects. This ensures that corresponding components (e.g., the first component of each subject) represent similar patterns, which is crucial for accurate clustering. Without alignment, the arbitrary ordering of components from individual PCA/FA can lead to poor clustering performance.

Step 4 (re-optimization) fits cluster-specific models in standardized space to ensure consistency with Step 1. This improves reconstruction metrics (SSE/VAF) but does not change ARI unless refine_assignments = TRUE.

If refine_assignments = TRUE, a single reassignment pass is performed after fitting cluster-specific models. Each subject is reassigned to the cluster that minimizes reconstruction error. This can improve ARI while maintaining the two-step nature of the baseline (unlike iterative EM which would drift toward the mixture model).

For FA decomposition, the function uses psych::fa() if available. If the psych package is not installed or if FA fails for a subject, the function falls back to PCA with a warning.

Examples

if (FALSE) { # \dontrun{
sim <- simulate_mixture_data(N = 50, K = 3, r = 2, M = 6, T_each = 100)
result <- fit_two_step_baseline(
  sim$list_of_data, r = 2, K = 3, method = "PCA",
  seed = 123, z_true = sim$z_true
)
print(result$ARI)
} # }