LCPA (version 1.0.0)

Kmeans.LCA: Initialize LCA Parameters via K-means Clustering

Description

Performs hard clustering of observations using K-means algorithm to generate initial parameter estimates for Latent Class Analysis (LCA) models. This provides a data-driven initialization strategy that often outperforms random starts when the number of observed categorical variables \(I\) is large (i.e., \(I > 50\)).

Usage

Kmeans.LCA(response, L, nrep = 10)

Value

A list containing:

params

List of initialized parameters:

par

An \(L \times I \times K_{\max}\) array of initial conditional probabilities, where \(K_{\max}\) is the maximum number of categories across items. Dimension order: latent classes (1:L), items (1:I), response categories (1:K_max).

P.Z

Numeric vector of length \(L\) containing initial class prior probabilities derived from cluster proportions.

P.Z.Xn

An \(N \times L\) matrix of posterior class probabilities. Contains hard assignments (0/1 values) based on K-means cluster memberships.

Arguments

response

A numeric matrix of dimension \(N \times I\), where \(N\) is the number of observations and \(I\) is the number of observed categorical variables. Each column must contain nominal-scale discrete responses (e.g., integers representing categories). Non-sequential category values are automatically re-encoded to sequential integers starting from 1.

L

Integer specifying the number of latent classes. Must be \(2 \leq L < N\).

nrep

Integer specifying the number of random starts for K-means algorithm (default: 10). The solution with the lowest within-cluster sum of squares is retained.

Details

The function executes the following steps:

• Data preprocessing: Automatically adjusts non-sequential category values to sequential integers (e.g., categories {1,3,5} become {1,2,3}) using internal adjustment routines.

• K-means clustering: Scales variables to mean=0 and SD=1 before clustering. Uses Lloyd's algorithm with Euclidean distance.

• Parameter estimation:

  • For each cluster \(l\), computes empirical response probabilities \(P(X_i=k|Z=l)\) for all items \(i\) and categories \(k\).

  • Handles singleton clusters by assigning near-deterministic probabilities (e.g., \(1-10^{-10}\) for observed category, \(10^{-10}\) for others).

• Posterior probabilities: Constructs hard-classification matrix where \(P(Z=l|\mathbf{X}_n)=1\) for the assigned cluster and 0 otherwise.

Examples

# Simulate response data
set.seed(123)
response <- matrix(sample(1:4, 200, replace = TRUE), ncol = 5)

# Generate K-means initialization for 3-class LCA
init_params <- Kmeans.LCA(response, L = 3, nrep = 5)

# Inspect initial class probabilities
print(init_params$params$P.Z)