Large values in time series and additive combinatorics

2026-04-23

AI Summary

This study uncovers the mathematical mechanism underlying the structural patterns of extreme values in real-world time series. By integrating additive combinatorics with discrete Fourier analysis, and leveraging Fourier sparsity complexity together with a generalized Chang’s lemma, the authors prove that when a time series exhibits low Fourier complexity, its set of maxima can be exactly reconstructed via integer linear combinations with coefficients in \{-1,0,1\} from a minimal generating set containing only 4–7 elements. This work provides the first rigorous theoretical explanation for the structural regularity of extreme values in time series, establishing a quantitative link between Fourier spectral properties and additive generative capacity. The theory is validated on both raw and mean-centered U.S. inflation and Delhi climate data, highlighting the informational richness and pronounced internal structure of extreme events.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Time-Series/Data Streams

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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.

Problem

• time series
• large values
• additive combinatorics
• Fourier analysis
• structured extremes

Innovation

• Fourier ratio
• additive combinatorics
• Chang's lemma
• time series extremes
• discrete Fourier analysis