Large values in time series and additive combinatorics
For data scientists and time series analysts, this offers a rigorous explanation for the structure of extreme values, potentially improving anomaly detection and data compression.
This paper provides a mathematical foundation for the heuristic that large values in real-world time series are structured, using additive combinatorics and Fourier analysis. The authors prove that when the Fourier ratio is small, the set of largest values can be additively generated by a very small set, and confirm this on US inflation and Delhi climate data, where generating sets of size 4-7 suffice for large spectra.
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.