Detection of mean changes in partially observed functional data

TL;DR

The paper develops a permutation-based mean-change test for functional data that are partially observed on subsets of the domain, accommodating both abrupt and gradual changes within a unified AMOC framework. It introduces a robust test statistic built on partially observed CUSUM-type processes with data-dependent weighting to handle missingness, and offers two permutation-based inference schemes: a plain fixed-sample approach and a sequential Monte Carlo approach with bounded resampling risk. The methodology is complemented by a detailed implementation plan on discretized grids, integral approximations, and efficient computation, along with extensive simulations and a real-data application to butterfly counts demonstrating practical usefulness in small samples and under various missingness patterns. The results show controlled size under $H_0$ and competitive power under $H_1$, with guidance on weight choices and robustness to misspecification of the change shape. Overall, the approach provides a versatile, statistically principled tool for detecting mean changes in partially observed functional data across diverse applications.

Abstract

We propose a test for a change in the mean for a sequence of functional observations that are only partially observed on subsets of the domain, with no information available on the complement. The framework accommodates important scenarios, including both abrupt and gradual changes. The significance of the test statistic is assessed via a permutation test. In addition to the classical permutation approach with a fixed number of permutation samples, we also discuss a variant with controlled resampling risk that relies on a random (data-driven) number of permutation samples. The small sample performance of the proposed methodology is illustrated in a Monte Carlo simulation study and an application to real data.

Figures (5)

• Figure 1: Weekly butterfly counts at Drayton site, UK, in 1993--2011 from butterfly. If the weather conditions did not meet the protocol, the measurement for that week is missing, so the number of missing observations within a year ranges between 3 and 18.
• Figure 2: An example of a partially observed data $X_1,\dots,X_n$ for $n=80$ and $D=[0,1]$ with an abrupt change at $k=n/2$ (left plot) and with a linear gradual change with $\kappa=1/2$ (right plot).
• Figure 3: An example of the functional data $X_1,\dots,X_n$ with the three missingness patterns (M1)--(M3), under the null hypothesis. Each plot contains the same $n=20$ functions generated as described in Section \ref{['sec:simul:1']} with observation sets (M1), (M2), and (M3), respectively. Complete profiles are in gray and incomplete ones are black.
• Figure 4: Estimated probability of observing $X_i(u)$ computed for each $u\in [0,1]$ for the three scenarios (M1)--(M3) as a relative proportion in a sample of size $n=10\,000$.
• Figure 5: Boxplots of the estimator $\widehat{\kappa}_{\gamma}$,computed for significant outputs only, for the alternative $\delta(u)=0.7$, sample size $n=80$, tuning parameter $\gamma\in\{0,1/4,1/2\}$ and the three missingness patterns (M1)--(M3) and complete data (C).

Theorems & Definitions (4)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Remark 3.3