  We introduce an ordinate method for noisy data analysis, based solely on rank
information and thus insensitive to outliers. The method is nonparametric,
objective, and the required data processing is parsimonious. Main ingredients
are 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 square
software. It is demonstrated that the method excels when extracting trends in
heavy-tailed noise and, unlike the Thiele-Sen estimator, is not limited to
linear regression. Lastly, a simple expression is given that yields a close
approximation for signal extraction of an underlying generally nonlinear
signal.
