Larriba, Y, Rueda, C, Fernández, MA, Peddada, SD.  Statistics in Medicine. 2020; 39: 265– 278. https://doi.org/10.1002/sim.8397

Oscillatory systems are commonly encountered in biological, physical and social sciences. For example, hormones such as progesterone, estrogen and luteinizing hormone are components of female endocrine system that display rhythmic/oscillatory temporal patterns in a premenopausal women’s monthly cycle. Similarly, all living creatures experience various physiological changes in their bodies during a 24 hour clock with day and night cycles. Such oscillatory systems are made up of components that display temporal rhythmic patterns for the system to function. Using time course experiments, researchers are interested in identifying components of an oscillatory system that display rhythmic patterns and estimate various parameters characterizing the shape of such patterns. Often researchers use parametric models (such as a sinusoidal function) to describe rhythmic patterns. However, such models can be very rigid/restrictive because in many cases true patterns are not necessarily symmetric, let alone sinusoidal. This paper provides a model free approach to describe shape using mathematical inequalities among the means at various time points. In some applications, such as tissues obtained from people whose time to death is unknown, researchers are interested in identifying rhythmic genes and various biological and statistical parameters associated with such genes. Thus, before one can identify rhythmic genes, one needs to first determine temporal order among the biological specimens. Formulating the problem as a traveling salesman problem, in this paper we develop a model free approach to determine temporal order among the biological specimens and then determine rhythmic genes using the estimated temporal order among the specimens.