Applying non-negative matrix factorization with covariates to structural equation modeling for blind input-output analysis

Structural equation modeling (SEM) describes directed dependence and feedback, whereas non-negative matrix factorization (NMF) provides interpretable, parts-based representations for non-negative data. We propose NMF-SEM, a unified non-negative framework that embeds NMF within a simultaneous-equation structure, enabling latent feedback loops and a reduced-form input-output mapping when intermediate flows are unobserved. The mapping separates direct effects from cumulative propagation effects and summarizes reinforcement using an amplification ratio. We develop regularized multiplicative-update estimation with orthogonality and sparsity penalties, and introduce structural evaluation metrics for input-output fidelity, second-moment (covariance-like) agreement, and feedback strength. Applications show that NMF-SEM recovers the classical three-factor structure in the Holzinger-Swineford data, identifies climate- and pollutant-driven mortality pathways with negligible feedback in the Los Angeles system, and separates deprivation, general morbidity, and deaths-of-despair components with weak feedback in Mississippi health outcomes.