Penalised Spline-Based Multinomial Restricted Unfolding: Methodology, Simulation, and Application

Multinomial Restricted Unfolding (MRU) is a geometric framework for modelling categorical outcomes by embedding observations and response categories within a shared low-dimensional latent space, where class membership probabilities are determined by Euclidean distances. Although this formulation captures nonlinear structure in the latent space, the mapping from observed predictors to latent coordinates is linear, limiting the model’s ability to represent predictor–class relationships. This study extends the MRU framework by introducing B-spline transformations of the predictors, enabling flexible modelling of covariate effects. A ridge penalty is incorporated into the negative log-likelihood objective function to control model complexity and reduce over-fitting. The proposed approach is evaluated through a simulation study manipulating sample size and latent class probability sharpness, alongside a comparative application to the lmap liver data set. Simulation results show that the...Show moreMultinomial Restricted Unfolding (MRU) is a geometric framework for modelling categorical outcomes by embedding observations and response categories within a shared low-dimensional latent space, where class membership probabilities are determined by Euclidean distances. Although this formulation captures nonlinear structure in the latent space, the mapping from observed predictors to latent coordinates is linear, limiting the model’s ability to represent predictor–class relationships. This study extends the MRU framework by introducing B-spline transformations of the predictors, enabling flexible modelling of covariate effects. A ridge penalty is incorporated into the negative log-likelihood objective function to control model complexity and reduce over-fitting. The proposed approach is evaluated through a simulation study manipulating sample size and latent class probability sharpness, alongside a comparative application to the lmap liver data set. Simulation results show that the spline-based penalised MRU can recover non-linear predictor–class relationships to a moderate degree, with performance improving as sample size increases. In the comparison analysis, spline-based model achieved a fit and predictive accuracy comparable to the original MRU. Overall, the results demonstrate that penalised spline extension can enhance the representational capacity of MRU though effectiveness is moderated by their distributional properties and homogeneity of predictor effects.Show less

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Penalised Spline-Based Multinomial Restricted Unfolding: Methodology, Simulation, and Application