(b) Same data but with only the daily rank recorded in rows. (a) Daily high-temperature values displayed as a day (row) and year (column) matrix. Moreover, for “heavy-tailed” noise, least-squares methods fail, overwhelmed by “black swan” events, while the proposed method, without modification, matches or beats the performance of specially designed estimators.ĭata processing, illustrated on weather station SZ000009480 (GHCN) Lugano, Switzerland. While the loss of data magnitude might be thought to preclude quantitative deduction, the rank-estimated slope for a linear signal in the presence of common white noise matches that from traditional least-squares regression. This same rank approach readily distinguishes noise from chaos. In the context of ever-changing climate, for example, this characterization offers tools to quantitatively define (with confidence limits) an intermediate category of natural variability that is neither signal nor noise.
Unexpectedly, we find that a special matrix transform of the rank expression for white noise is fully characterized by planar symmetries of reflection and rotation, each of which is associated with a unique transient pattern. Here, we use a whole-number description of data by ranking a sequence of numbers-sorting by value-as the basis for answering two fundamental questions: With what confidence can one assert that rank information rises above the level of that for pure noise, thus indicating the presence of a signal? And when a signal is present, what form does that signal take?Įven in featureless white noise, patterns arise by chance. Many methods have been developed to detect and extract signals in the presence of noise, but most are tailored to a particular signal or application. Science advances by comparison against experimental data, but neither instruments nor measurements are perfect, and noise is ever present. A simple expression is given that yields a close approximation for signal extraction of an underlying, generally nonlinear signal. 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. 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. Ranking within randomness differs fundamentally from that in deterministic chaos and signals, thus forming the basis for signal detection. The method readily distinguishes both the Ornstein-Uhlenbeck process and chaos generated by the logistic map from white noise.
This two-step procedure provides a noise “etalon” used to characterize arbitrary stationary stochastic processes. A group symmetry orthogonal decomposition of the 2D rank-order transform for iid (white) noise is further ordered by principal component analysis. 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. The method is nonparametric and objective, and the required data processing is parsimonious. We introduce an ordinate method for noisy data analysis, based solely on rank information and thus insensitive to outliers.
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