Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fourier minimization and imputation of time series

Will Burstein, Alex Iosevich, Azita Mayeli, Hari Sarang Nathan

TL;DR

This work studies imputing missing values in time series by formulating imputation as a convex L1-minimization problem in the Fourier domain: given f: $Z_N\to C$ with missing indices M, find g that matches f on M^c while minimizing $||\widehat{g}||_{L^1}$. The approach extends Logan's phenomenon from exact recovery to robust imputation under concentration and noise, providing quantitative bounds (Theorems 'quantitativeLogan', 'looseLogan', 'quantitativerandomLogan', 'quantitativerandomnoisyLogan') that hold under both deterministic and random missingness, including random perturbations and noisy observations. The analysis leverages Bourgain-type results and concentration inequalities (e.g., Talagrand) to establish high-probability recovery for random missingness, complemented by an appendix revisiting the classic DS Logan89 proof. Numerical experiments on a real-world dataset (Australian beer production) show that L1-minimization-imputed values yield lower MAE compared to linear interpolation and ANN baselines, demonstrating practical robustness. Overall, the paper provides a convex, Fourier-domain imputation framework with provable guarantees and demonstrates its effectiveness for time-series data with incomplete observations.

Abstract

One of the most common procedures in modern data analytics is filling in missing values in times series. For a variety of reasons, the data provided by clients to obtain a forecast, or other forms of data analysis, may have missing values, and those values need to be filled in before the data set can be properly analyzed. Many freely available forecasting software packages, such as the sktime library, have built-in mechanisms for filling in missing values. The purpose of this paper is to adapt the classical $L^1$ minimization method for signal recovery to the filling of missing values in times. The theoretical justifications of these methods leverage results by Bourgain (\cite{Bourgain89}), Talagrand (\cite{Talagrand98}), the second and the third listed authors (\cite{IM24}), and the result by the second listed author, Kashin, Limonova and the third listed author (\cite{IKLM24}). Brief numerical tests for these algorithms are given but more extensive will be discussed in a companion paper.

Fourier minimization and imputation of time series

TL;DR

This work studies imputing missing values in time series by formulating imputation as a convex L1-minimization problem in the Fourier domain: given f: with missing indices M, find g that matches f on M^c while minimizing . The approach extends Logan's phenomenon from exact recovery to robust imputation under concentration and noise, providing quantitative bounds (Theorems 'quantitativeLogan', 'looseLogan', 'quantitativerandomLogan', 'quantitativerandomnoisyLogan') that hold under both deterministic and random missingness, including random perturbations and noisy observations. The analysis leverages Bourgain-type results and concentration inequalities (e.g., Talagrand) to establish high-probability recovery for random missingness, complemented by an appendix revisiting the classic DS Logan89 proof. Numerical experiments on a real-world dataset (Australian beer production) show that L1-minimization-imputed values yield lower MAE compared to linear interpolation and ANN baselines, demonstrating practical robustness. Overall, the paper provides a convex, Fourier-domain imputation framework with provable guarantees and demonstrates its effectiveness for time-series data with incomplete observations.

Abstract

One of the most common procedures in modern data analytics is filling in missing values in times series. For a variety of reasons, the data provided by clients to obtain a forecast, or other forms of data analysis, may have missing values, and those values need to be filled in before the data set can be properly analyzed. Many freely available forecasting software packages, such as the sktime library, have built-in mechanisms for filling in missing values. The purpose of this paper is to adapt the classical minimization method for signal recovery to the filling of missing values in times. The theoretical justifications of these methods leverage results by Bourgain (\cite{Bourgain89}), Talagrand (\cite{Talagrand98}), the second and the third listed authors (\cite{IM24}), and the result by the second listed author, Kashin, Limonova and the third listed author (\cite{IKLM24}). Brief numerical tests for these algorithms are given but more extensive will be discussed in a companion paper.

Paper Structure

This paper contains 16 sections, 14 theorems, 115 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Notation and Conventions
  3. $L^1$-minimization algorithm and Logan's phenomenon
  4. Connections with the theory of exact signal recovery
  5. Numerical example
  6. Proofs of theorems
  7. Proof of Theorem \ref{['theorem:quantitativeLogan']}
  8. Proof of Theorem \ref{['theorem:looseLogan']}
  9. Proof of Theorem \ref{['theorem:quantitativerandomLogan']}
  10. Proof of Theorem \ref{['theorem:quantitativerandomnoisyLogan']}
  11. Proof of Theorem \ref{['theorem:hoeffding_M_vs_Z_N']}
  12. Appendix: Proof of Theorem \ref{['theorem:DSLogan89Orig']}
  13. Statements and Declarations
  14. Competing Interests
  15. Data Availability
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $f: {\mathbb Z}_N \to {\mathbb C}$ be supported in $E \subset {\mathbb Z}_N$. Suppose that $\widehat{f}$ is transmitted but the frequencies ${\{\widehat{f}(m)\}}_{m \in S}$ are unobserved, where $S \subset {\mathbb Z}_N$, with $|E| \cdot |S|<\frac{N}{2}$. Then $f$ can be recovered exactly and un

Figures (2)

  • Figure 1: Linear vs $L^1$-minimization Results on Monthly Beer Production Data in Australia. The left figure shows the original missing values and the results of filling in the missing values with linear interpolation and $L^1$-minimization. The right figure shows difference in absolute errors with linear interpolation and $L^1$-minimization. Values above zero indicate that $L^1$-minimization improved results and values below zero indicate the opposite.
  • Figure 2: ANN vs $L^1$-minimization Results on Monthly Beer Production Data in Australia. The left figure shows the original missing values and the results of filling in the missing values with the ANN and $L^1$-minimization. The right figure shows difference in absolute errors with the ANN and $L^1$-minimization. Values above zero indicate that $L^1$-minimization improved results and values below zero indicate the opposite.

Theorems & Definitions (29)

  • Theorem 1.1: DS89
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9: Talagrand's theorem
  • Remark 1.10
  • ...and 19 more