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Gradual changes in functional time series

Patrick Bastian, Holger Dette

TL;DR

The paper develops a statistically principled framework for detecting gradual changes in the mean function of non-stationary functional time series by measuring the sup-norm distance $d_\infty$ to a benchmark function $g_\mu$. It introduces a bootstrap-based test using the statistic $\hat{T}_{n,\Delta}=\sqrt{nh_n}(\hat{d}_{\infty,n}-\Delta)$ and proves its asymptotic validity under mild dependence via Gaussian approximation, without requiring second-order stationarity. It also proposes consistent estimators for the first time a deviation of size at least $\Delta$ occurs, $t^*(\Delta)$, and its functional variant $t^*(s,\Delta)$, with rates governed by a local modulus of continuity. The methods are demonstrated through simulations and a real climate-data example, showing accurate detection of gradual changes and early identification of relevant deviations, and are supported by extensive theoretical proofs including block bootstrap guarantees and dependent-data Gaussian approximations. Overall, the work offers practical, interpretable tools for gradual-change analysis in functional data with non-stationary errors, with potential impact on climate studies and related fields.

Abstract

We consider the problem of detecting gradual changes in the sequence of mean functions from a not necessarily stationary functional time series. Our approach is based on the maximum deviation (calculated over a given time interval) between a benchmark function and the mean functions at different time points. We speak of a gradual change of size $Δ$, if this quantity exceeds a given threshold $Δ>0$. For example, the benchmark function could represent an average of yearly temperature curves from the pre-industrial time, and we are interested in the question if the yearly temperature curves afterwards deviate from the pre-industrial average by more than $Δ=1.5$ degrees Celsius, where the deviations are measured with respect to the sup-norm. Using Gaussian approximations for high-dimensional data we develop a test for hypotheses of this type and estimators for the time where a deviation of size larger than $Δ$ appears for the first time. We prove the validity of our approach and illustrate the new methods by a simulation study and a data example, where we analyze yearly temperature curves at different stations in Australia.

Gradual changes in functional time series

TL;DR

The paper develops a statistically principled framework for detecting gradual changes in the mean function of non-stationary functional time series by measuring the sup-norm distance to a benchmark function . It introduces a bootstrap-based test using the statistic and proves its asymptotic validity under mild dependence via Gaussian approximation, without requiring second-order stationarity. It also proposes consistent estimators for the first time a deviation of size at least occurs, , and its functional variant , with rates governed by a local modulus of continuity. The methods are demonstrated through simulations and a real climate-data example, showing accurate detection of gradual changes and early identification of relevant deviations, and are supported by extensive theoretical proofs including block bootstrap guarantees and dependent-data Gaussian approximations. Overall, the work offers practical, interpretable tools for gradual-change analysis in functional data with non-stationary errors, with potential impact on climate studies and related fields.

Abstract

We consider the problem of detecting gradual changes in the sequence of mean functions from a not necessarily stationary functional time series. Our approach is based on the maximum deviation (calculated over a given time interval) between a benchmark function and the mean functions at different time points. We speak of a gradual change of size , if this quantity exceeds a given threshold . For example, the benchmark function could represent an average of yearly temperature curves from the pre-industrial time, and we are interested in the question if the yearly temperature curves afterwards deviate from the pre-industrial average by more than degrees Celsius, where the deviations are measured with respect to the sup-norm. Using Gaussian approximations for high-dimensional data we develop a test for hypotheses of this type and estimators for the time where a deviation of size larger than appears for the first time. We prove the validity of our approach and illustrate the new methods by a simulation study and a data example, where we analyze yearly temperature curves at different stations in Australia.
Paper Structure (15 sections, 15 theorems, 122 equations, 4 figures, 1 table)

This paper contains 15 sections, 15 theorems, 122 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Gradual changes in functional data
  3. Assumptions
  4. A Bootstrap test for gradual changes
  5. The first time of a relevant deviation
  6. Finite sample properties
  7. Simulation study
  8. Real Data Example
  9. Proofs
  10. Proof of Theorem \ref{['pt2']}
  11. Block Bootstrap Approximation
  12. Proof of Theorem \ref{['pt4']}
  13. Proof of Theorem \ref{['pt5']}
  14. Gaussian approximation for dependent data
  15. Further technical details

Key Result

Theorem 2.3

Let Assumptions (A1)-A(6) be satisfied and denote by $\rho_n$ any sequence such that $\rho_n^{-1}=o ((nh_n)^{1/2}h_n^{(3+\alpha^{-1})/J} )$. We then have with probability converging to $1$ that where $\mathcal{E}_{\rho_n}$ is defined in p201 and the kernel $K^*$ is given by $K^*=2\sqrt{2}K(\sqrt{2}x)-K(x)$.

Figures (4)

  • Figure 1: Empirical rejection probabalities of the test \ref{['p92']} for the hypotheses \ref{['p3']} for different $\Delta$ and sample sizes $n=100, 250, 500$. The mean function is given by \ref{['p80']} and the error processes by \ref{['p81']} (left) and \ref{['p82']} (right). The benchmark function $g_\mu$ is given by \ref{['det10']} and $d_\infty=2$ in all cases.
  • Figure 2: Empirical rejection probabalities of the test \ref{['p92']} for the hypotheses \ref{['p3']} for different $\Delta$ and sample sizes $n=100, 250, 500$. The mean function is given by \ref{['p80a']} and the error processes by \ref{['p81']} (left) and \ref{['p82']} (right). The benchmark function $g_\mu$ is given by \ref{['det11']} with $x_0=0.25$ and $d_\infty=0.4585$ in all cases.
  • Figure 3: Minimal temperature measured by a weather station in Boulia between 1888 and 2013.
  • Figure 4: Plots of the estimator $\hat{t}^*(s,\Delta)$ defined in \ref{['p60']} for the Boulia weather station data for different choices of $\Delta$.

Theorems & Definitions (34)

  • Example 2.1
  • Example 2.2: Continuation of Example \ref{['pex1']}
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 3.1
  • Remark 3.2
  • Remark 4.1
  • Lemma 5.1
  • ...and 24 more