Using Non-negative Matrix Factorization (NMF), the observed matrix can be approximated by the product of the basis and coefficient matrices. Moreover, if the coefficient vectors are explained by the covariates for each individual, the coefficient matrix can be written as the product of the parameter matrix and the covariate matrix, and additionally described in the framework of Non-negative Matrix tri-Factorization (tri-NMF) with covariates. Consequently, this is equal to the mean structure of the Growth Curve Model (GCM). The difference is that the basis matrix for GCM is given by the analyst, whereas that for NMF with covariates is unknown and optimized. In this study, we applied NMF with covariance to longitudinal data and compared it with GCM. We have also published an R package that implements this method, and we show how to use it through examples of data analyses including longitudinal measurement, spatiotemporal data and text data. In particular, we demonstrate the usefulness of Gaussian kernel functions as covariates.

翻译：使用非负矩阵分解（NMF），观测矩阵可近似为基矩阵与系数矩阵的乘积。此外，若系数向量可由每个个体的协变量解释，则系数矩阵可写为参数矩阵与协变量矩阵的乘积，并进一步在含协变量的非负矩阵三因子分解（tri-NMF）框架下描述。因此，这等价于增长曲线模型（GCM）的均值结构。不同之处在于，GCM的基矩阵由分析者预先设定，而含协变量的NMF中的基矩阵未知且需优化。本研究将含协变量的NMF应用于纵向数据，并将其与GCM进行比较。我们还发布了一个实现该方法的R包，并通过包含纵向测量、时空数据和文本数据的分析示例展示其用法。特别地，我们证明了高斯核函数作为协变量的有效性。