Extracting Signal from Noise: Wiener’s Filter, Cybernetics, and the Statistical Face of the Inverse Problem (the Origin in Markets).
To the corpus’s family of inverse problems (scattering=GLM 89 / boundary-value=Calderon 99 / spectral=Kac 102), this paper adds a FOURTH member — the STATISTICAL face: Norbert Wiener. His WIENER FILTER optimally estimates a hidden signal s from an observation y=s+n by minimizing mean-square error; in the frequency domain H(f)=S_ss/(S_ss+S_nn) — given the power spectra of signal and noise, one best extracts signal from noise. Born from WWII anti-aircraft fire control (predicting an aircraft’s future position from noisy tracking), later broadened into CYBERNETICS (control, communication, feedback). Two cores: (1) the STATISTICAL version of J4 (the inverse-problem stance) — where Zero-Tension (89) reads the hidden cause from deviations off a perfect null, Wiener best-estimates the signal (the deviation) within the noise (the null), probabilistically; this crosses directly with the author’s origin (fishing = signal->hidden cause; quantitative FX). (2) DISTINCT ROOTS — deterministic inverse problems (89/99/102, exact recovery) vs statistical estimation (Wiener, optimal under uncertainty); and Wiener (stationary, Gaussian, linear, optimal) vs Mandelbrot (non-stationary, fat-tailed, descriptive, 94) are distinct roots for the same “markets,” indeed in TENSION (Mandelbrot criticized exactly the Wiener-type stationarity/Gaussianity). HONESTLY: the filter’s optimality is CONDITIONAL (stationarity, known spectra, linearity), which markets violate — so it is NOT sold as a prediction tool / edge (discipline of 94/101; “caught by luck != skill of catching”). Also: the Wiener process is the math identity of 78’s fluctuation, but physical thermal noise and signal estimation are distinct roots. IMPORTANT (scope): a Layer-2 structural interpretation of established estimation theory; no new mathematics and no new prediction. The Universe OS is a Layer-3 non-science lens.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Extracting Signal from Noise [Paper 106]. Zenodo.
DOI: 10.5281/zenodo.21836984
ORCID: 0009-0006-8587-3450