Large values in time series and additive combinatorics

TL;DR

This paper provides a mathematical foundation for the structure of large values in time series, using additive combinatorics and Fourier analysis, and confirms predictions with real-world data.

Contribution

It introduces a novel application of additive combinatorics and Fourier ratio to predict and explain the structure of large values in time series data.

Findings

Large values can be additively generated by small sets with , ,  coefficients.

Numerical experiments on US inflation and Delhi climate data confirm the theoretical predictions.

A small generating set suffices to span large spectra in real-world data, even with large Fourier ratios.

Abstract

It is well-known in industrial data science that large values of real-life time series tend to be structured and often follow concrete and visible patterns. In this paper, we use ideas from additive combinatorics and discrete Fourier analysis to give this heuristic a mathematical foundation. Our main tool is the Fourier ratio, a complexity measure previously used in compressed sensing, combined with a generalized version of Chang's lemma from additive combinatorics. Together, these yield a precise prediction: when the Fourier ratio of a time series is small, the set of its largest values can be additively generated by a very small set using only $\{-1,0,1\}$ coefficients. We test this prediction on US inflation data and Delhi climate data, both in their original form and after mean-centering. The numerical results confirm the predicted structure: a generating set of size $4$--$7$ suffices to span large spectra containing dozens of points, even when the Fourier ratio is large enough that our theoretical bounds become loose. These findings provide a rigorous explanation for why extreme values in real-world data are information-rich and structurally significant.