Beyond average warming: Two-sample inference for dense-sparse functional data reveals changes in intraday temperature patterns

2026-05-26

AI Summary

This study addresses the challenge of comparing functional data arising from disparate observation frequencies—hourly historical versus 10-minute modern temperature records—by proposing an inference framework for the difference between two mean functions under mixed dense-sparse sampling. The authors construct a novel estimator achieving optimal convergence rates, establish a functional central limit theorem in continuous function space, and develop a multiplier bootstrap procedure to construct uniform confidence bands. The methodology is extended to functional time series and enhanced via transfer learning to improve practical applicability. Application to German meteorological station data reveals that climate warming not only elevates daily average temperatures but also substantially reshapes intraday temperature patterns: in Berlin, for instance, warming from morning to early afternoon far exceeds the increase in daily mean, while nighttime warming remains comparatively muted.

Abstract

Modern weather stations in Germany record daily temperatures every 10 minutes, whereas measurements from historical reference periods are often only available at much coarser temporal resolutions, typically hourly. This discrepancy must be accounted for when comparing historical and current daily temperature patterns. Motivated by this problem, we develop two-sample inference procedures for functional data under sampling schemes where one sample is densely observed while the other is relatively sparse. Building on recent ideas from transfer learning for functional data, we derive estimators of the difference of the mean functions that attain optimal convergence rates in the supremum norm. We further establish a functional central limit theorem in the space of continuous functions and develop multiplier bootstrap methods for constructing uniform confidence bands. Extensions to functional time series are also discussed. Applying the proposed methodology to daily temperature curves from German weather stations, analyzed separately by month, reveals that climate change has altered not only average temperatures but also intraday temperature patterns. In particular, for stations such as Berlin, warming from morning to early afternoon exceeds the daily average increase, whereas evening and nighttime temperatures exhibit comparatively smaller increases.

Problem

• functional data
• two-sample inference
• dense-sparse sampling
• intraday temperature patterns
• climate change

Innovation

• dense-sparse functional data
• two-sample inference
• functional central limit theorem
• multiplier bootstrap
• transfer learning for functional data