Estimation of Parameters Subject to Order Restrictions on a Circle With Application to Estimation of Phase Angles of Cell Cycle Genes.

Motivated by a problem encountered in the analysis of cell cycle gene expression data, this article deals with the estimation of parameters subject to order restrictions on a unit circle. A normal eukaryotic cell cycle has four major phases during cell division, and a cell cycle gene has its peak expression (phase angle) during the phase that may correspond to its biological function. Because the phases are ordered along a circle, the phase angles of cell cycle genes are ordered unknown parameters on a unit circle. The problem of interest is to estimate the phase angles using the information regarding the order among them. We address this problem by developing a circular version of the well-known isotonic regression for Euclidean data. Because of the underlying geometry, the standard pool adjacent violator algorithm (PAVA) cannot be used for deriving the circular isotonic regression estimator (CIRE). However, PAVA can be modified to obtain a computationally efficient algorithm for deriving the CIRE. We illustrate the CIRE by estimating the phase angles of some of well-known cell cycle genes using the unrestricted estimators obtained in the literature.