Order-Preserving Factor Analysis—Application to Longitudinal Gene Expression

We present a novel factor analysis method that can be applied to the discovery of common factors shared among trajectories in multivariate time series data. These factors satisfy a precedence-ordering property: certain factors are recruited only after some other factors are activated. Precedence-ordering arise in applications where variables are activated in a specific order, which is unknown. The proposed method is based on a linear model that accounts for each factor's inherent delays and relative order. We present an algorithm to fit the model in an unsupervised manner using techniques from convex and nonconvex optimization that enforce sparsity of the factor scores and consistent precedence-order of the factor loadings. We illustrate the order-preserving factor analysis (OPFA) method for the problem of extracting precedence-ordered factors from a longitudinal (time course) study of gene expression data.