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.

A cell division cycle is a well-coordinated process in eukaryotes with cell cycle genes exhibiting a periodic expression over time. There is considerable interest among cell biologists to determine genes that are periodic in multiple organisms and whether such genes are also evolutionarily conserved in their relative order of time to peak expression. Interestingly, periodicity is not well-conserved evolutionarily. A conservative estimate of a number of periodic genes common to fission yeast (Schizosaccharomyces pombe) and budding yeast (Saccharomyces cerevisiae) (‘core set FB’) is 35, while those common to fission yeast and humans (Homo sapiens) (‘core set FH’) is 24. Using a novel statistical methodology, we discover that the relative order of peak expression is conserved in ∼80% of FB genes and in ∼40% of FH genes. We also discover that the order is evolutionarily conserved in six genes which are potentially the core set of signature cell cycle genes. These include ace2 (a transcription factor) and polo-kinase plo1, which are well-known hubs of early M-phase clusters, cdc18 a key component of pre-replication complexes, mik1 which is critical for the establishment and maintenance of DNA damage check point, and histones hhf1 and hta2.

A considerable amount of papers has been published in recent years proposing supervised classifiers to diagnose the health of a machine. The usual procedure with these classifiers is to train them using data acquired through controlled experiments, expecting them to perform well on new data, classifying correctly the condition of a motor. But, obviously, the new motor to be diagnosed cannot be the same that has been used during the training process; it may be a motor with different characteristics and fed from a completely different source. These different conditions between the training process and the testing one can deeply influence the diagnosis. To avoid these drawbacks, in this paper, a new method is proposed, which is based on robust statistical techniques applied in quality control applications. The proposed method is based on the online diagnosis of the operating motor and can detect deviations from the normal operational conditions. A robust approach has been implemented using high-breakdown statistical techniques, which can reliably detect anomalous data that often cause an unexpected overestimation of the data variability, reducing the ability of standard procedures to detect faulty conditions in earlier stages. A case study is presented to prove the validity of the proposed approach. Motors of different characteristics, fed from the power line and several different inverters, are tested. Three different fault conditions are provoked: a broken bar, a faulty bearing, and mixed eccentricity. Experimental results prove that the proposed approach can detect incipient faults.