Analysis of Feedback Mechanisms with Unknown Delay Using Sparse Multivariate Autoregressive Method

This paper discusses the study of two interacting processes in which a feedback mechanism exists between the processes. The study was motivated by problems such as the circadian oscillation of gene expression where two interacting protein transcriptions form both negative and positive feedback loops with long delays to equilibrium. Traditionally, data of this type could be examined using autoregressive analysis. However, in circadian oscillation the order of an autoregressive model cannot be determined a priori. We propose a sparse multivariate autoregressive method that incorporates mixed linear effects into regression analysis, and uses a forward-backward greedy search algorithm to select non-zero entries in the regression coefficients, the number of which is constrained not to exceed a pre-specified number. A small simulation study provides preliminary evidence of the validity of the method. Besides the circadian oscillation example, an additional example of blood pressure variations using data from an intervention study is used to illustrate the method and the interpretation of the results obtained from the sparse matrix method. These applications demonstrate how sparse representation can be used for handling high dimensional variables that feature dynamic, reciprocal relationships.