Applying Non-negative Matrix Factorization with Covariates to the Longitudinal Data as Growth Curve Model

Using Non-negative Matrix Factorization (NMF), an observed matrix is approximated by a basis matrix times a coefficient matrix. When each individual's coefficient vector is explained by covariates, the coefficient matrix factorizes into a parameter matrix and a covariate matrix -- a tri-factorization whose mean structure coincides with that of the Growth Curve Model (GCM) for longitudinal data. This correspondence has been noted but not examined. We make three contributions. First, we compare NMF with covariates and the GCM: the basis is prescribed in the GCM but optimized in NMF, and the NMF-optimized basis can be used within the GCM and may improve its fit, the two agreeing when covariate effects are non-negative. Second, the main contribution, we develop statistical inference for the parameter matrix linking covariates to basis components: conditional on the optimized basis we provide standard errors, Wald-type tests, and one-sided confidence intervals, with a simulation study confirming good calibration for covariate-effect contrasts. Third, we compare NMF with principal component analysis (PCA) and functional PCA (FPCA): its non-negative coefficients are membership probabilities giving a soft clustering directly, whereas signed PCA/FPCA scores require a downstream classifier. Illustrations use growth data and a kernel-based varying-coefficient model.