Motivated by applications in physical and biological sciences, we developed a Frequency Modulated Möbius (FMM) model to describe rhythmic patterns in oscillatory systems. Unlike standard symmetric sinusoidal models, FMM is a flexible parametric model that allows deformations to sinusoidal shape to accommodate commonly seen asymmetries in applications. FMM model parameters are easy to estimate and the model is easy to interpret complex rhythmic data. We illustrate FMM model in three disparate applications, namely, circadian clock gene expression, corticoptropin levels in depressed patients and the temporal light intensity patterns of distant stars. In each case, FMM model is demonstrated to be flexible, scientifically plausible and easy to interpret. Analysis of synthetic data derived from patterns of real data, suggest that FMM model fits the data very well both visually as well as in terms of the goodness of fit measure total mean squared error. An R language based software for implementing FMM model is available.

A novel approach for analysing cardiac rhythm data is presented in this paper. Heartbeats are decomposed into the five fundamental P, Q, R, S and T waves plus an error term to account for artifacts in the data which provides a meaningful, physical interpretation of the heart’s electric system. The morphology of each wave is concisely described using four parameters that allow all the different patterns in heartbeats to be characterized and thus differentiated This multi-purpose approach solves such questions as the extraction of interpretable features, the detection of the fiducial marks of the fundamental waves, or the generation of synthetic data and the denoising of signals. Yet the greatest benefit from this new discovery will be the automatic diagnosis of heart anomalies as well as other clinical uses with great advantages compared to the rigid, vulnerable and black box machine learning procedures, widely used in medical devices. The paper shows the enormous potential of the method in practice; specifically, the capability to discriminate subjects, characterize morphologies and detect the fiducial marks (reference points) are validated numerically using simulated and real data, thus proving that it outperforms its competitors.

This paper is motivated by applications in oscillatory systems where researchers are typically interested in discovering components of those systems that display rhythmic temporal patterns. The contributions of this paper are twofold. First, a methodology is developed based on a circular signal plus error model that is defined using order restrictions. This mathematical formulation of rhythmicity is simple, easily interpretable and very flexible, with the latter property derived from the nonparametric formulation of the signal. Second, we address various commonly encountered problems in the analysis of oscillatory systems data. Specifically, we propose a methodology for (a) detecting rhythmic signals in an oscillatory system and (b) estimating the unknown sampling time that occurs when tissues are obtained from subjects whose time of death is unknown. The proposed methodology is computationally efficient, outperforms the existing methods, and is broadly applicable to address a wide range of questions related to oscillatory systems.

Motivation: Many biological processes, such as cell cycle, circadian clock, menstrual cycles, are governed by oscillatory systems consisting of numerous components that exhibit rhythmic patterns over time. It is not always easy to identify such rhythmic components. For example, it is a challenging problem to identify circadian genes in a given tissue using time-course gene expression data. There is a great potential for misclassifying non-rhythmic as rhythmic genes and vice versa. This has been a problem of considerable interest in recent years. In this article we develop a constrained inference based methodology called Order Restricted Inference for Oscillatory Systems (ORIOS) to detect rhythmic signals. Instead of using mathematical functions (e.g. sinusoidal) to describe shape of rhythmic signals, ORIOS uses mathematical inequalities. Consequently, it is robust and not limited by the biologist's choice of the mathematical model. We studied the performance of ORIOS using simulated as well as real data obtained from mouse liver, pituitary gland and data from NIH3T3, U2OS cell lines. Our results suggest that, for a broad collection of patterns of gene expression, ORIOS has substantially higher power to detect true rhythmic genes in comparison to some popular methods, while also declaring substantially fewer non-rhythmic genes as rhythmic. Availability and Implementation: A user friendly code implemented in R language can be downloaded from http://www.niehs.nih.gov/research/atniehs/labs/bb/staff/peddada/index.cfm. Contact: peddada@niehs.nih.gov

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