Statistics

One sigma Rule : The Smallest Detectable Step in Gaussian Time Series

Authors: Michael Zot
Comments: 4 Pages.

We present a symbolic regression framework for detecting the smallest resolvable step change (∆) in noisy time series, approaching the detection boundary where ∆ ≈ σ (noise standard deviation). Unlike classical methods such as the t-test, this approach uses residual structure analysis and entropy collapse in symbolic space to identify shifts without assumptions of distributional normality or fixed window sizes. The detector is interpretable, model-free, and does not require labels, training, or convolutional priors. It works as a symbolic microscope, surfacing structural inflection points buried under stochastic variance.This tool was constructed from scratch to isolate step changes near the theoretical limit of detectability. It outperforms standard hypothesis tests in edge-case scenarios and reveals hidden transitions in behavioral, financial, biological, and simulated data. The method operates using a single-pass symbolic signature engine, relying solely on residual breakdowns and phase transitions in model fit error.The tool is open-source for academic and personal use under a restricted license: scientific invention or derivative work based on this detection method requires attribution and explicit permission. This constraint ensures credit to foundational symbolic techniques while enabling researchers to explore novel applications and insights using the method.Code repository: url{https://github.com/mikecreation/symbolic-step-detection}
Category: Statistics