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Department of Mathematical Sciences


We introduce an ordinate method for noisy data analysis, based solely on rank information and thus insensitive to outliers. The method is nonparametric and objective, and the required data processing is parsimonious. The main ingredients include a rank-order data matrix and its transform to a stable form, which provide linear trends in excellent agreement with least squares regression, despite the loss of magnitude information. A group symmetry orthogonal decomposition of the 2D rank-order transform for iid (white) noise is further ordered by principal component analysis. This two-step procedure provides a noise “etalon” used to characterize arbitrary stationary stochastic processes. The method readily distinguishes both the Ornstein-Uhlenbeck process and chaos generated by the logistic map from white noise. Ranking within randomness differs fundamentally from that in deterministic chaos and signals, thus forming the basis for signal detection. To further illustrate the breadth of applications, we apply this ordinate method to the canonical nonlinear parameter estimation problem of two-species radioactive decay, outperforming special-purpose least squares software. We demonstrate that the method excels when extracting trends in heavy-tailed noise and, unlike the Thiele-Sen estimator, is not limited to linear regression. A simple expression is given that yields a close approximation for signal extraction of an underlying, generally nonlinear signal.

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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Publisher’s version of record:

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Physical Review X

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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


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