Multiscale detection of practically significant changes in a gradually varying time series

TL;DR

This paper tackles detecting practically significant deviations in a gradually varying time series after a reference time $t_0$. It develops a multiscale test based on local means across scales, with a centering term $\Gamma_n(c)$ and a threshold $\Delta$, and proves a Brownian-limit–type result $\hat{T}_n \Rightarrow T_{d_\infty}$ that depends on the underlying mean function via extremal sets; it also provides a conservative testing approach using a long-run variance estimate and a refined approach using estimated extremal sets to achieve nominal level. The work further derives the rate at which the first time of a relevant deviation $t^*$ can be estimated, with distinct rates for smooth changes versus jumps, and demonstrates both finite-sample performance improvements in simulations and practical usefulness in a real climate data application. Overall, the methods enable detection of local alternatives at the parametric rate $n^{-1/2}$ without smoothing parameters and are applicable to piecewise-smooth means with no strict geometric constraints. The framework offers robust, multiscale inference for practically meaningful changes in slowly evolving processes with broad applicability to climate, economics, and biomedical time series.

Abstract

In many change point problems it is reasonable to assume that compared to a benchmark at a given time point $t_0$ the properties of the observed stochastic process change gradually over time for $t >t_0$. Often, these gradual changes are not of interest as long as they are small (nonrelevant), but one is interested in the question if the deviations are practically significant in the sense that the deviation of the process compared to the time $t_0$ (measured by an appropriate metric) exceeds a given threshold, which is of practical significance (relevant change). In this paper we develop novel and powerful change point analysis for detecting such deviations in a sequence of gradually varying means, which is compared with the average mean from a previous time period. Current approaches to this problem suffer from low power, rely on the selection of smoothing parameters and require a rather regular (smooth) development for the means. We develop a multiscale procedure that alleviates all these issues, validate it theoretically and demonstrate its good finite sample performance on both synthetic and real data.

Figures (2)

• Figure 1: Plot of the regression function $\mu_a(x)$ in \ref{['meanfuncs']} for $a=2$. The dotted line is given by $\mu_0^{1/4} = 4\int_0^{1/4}\mu_2(s)ds$.
• Figure 2: Histograms for the estimator $\hat{t}$ defined in \ref{['Def:LocEst']} (left) and the estimator proposed in equation (5.2) of *buecher21 (right).

Theorems & Definitions (21)

• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 2.6
• Remark 2.7
• Theorem 3.1
• Theorem 3.2
• Lemma 5.1
• Lemma 5.2