Change Point Detection in the Frequency Domain with Statistical Reliability

TL;DR

This work tackles change point detection in the frequency domain with statistical reliability by extending Selective Inference to frequency-domain CPs. It introduces a two-stage framework: a heuristic CP candidate selection across multiple frequencies using dynamic programming and simulated annealing, followed by SI-based computation of selective $p$-values to declare final CPs. The approach leverages the discrete Fourier transform and STFT structure to form a provably valid testing procedure, including a truncated chi distribution for the conditional null, with guarantees on false positive rates. Empirical results on synthetic and real bearing data demonstrate controlled type I error and competitive power, offering a principled tool for root-cause analysis in frequency-domain signals while acknowledging current limitations and avenues for extending to multi-sensor settings.

Abstract

Effective condition monitoring in complex systems requires identifying change points (CPs) in the frequency domain, as the structural changes often arise across multiple frequencies. This paper extends recent advancements in statistically significant CP detection, based on Selective Inference (SI), to the frequency domain. The proposed SI method quantifies the statistical significance of detected CPs in the frequency domain using $p$-values, ensuring that the detected changes reflect genuine structural shifts in the target system. We address two major technical challenges to achieve this. First, we extend the existing SI framework to the frequency domain by appropriately utilizing the properties of discrete Fourier transform (DFT). Second, we develop an SI method that provides valid $p$-values for CPs where changes occur across multiple frequencies. Experimental results demonstrate that the proposed method reliably identifies genuine CPs with strong statistical guarantees, enabling more accurate root-cause analysis in the frequency domain of complex systems.

Figures (55)

• Figure 3: Illustration of the local search operation which merges two adjacent CP locations. In this figure, CP for frequency $d_2$ at time point $68$ and CP for $d_3$ at $74$ are merged at $70$.
• Figure 4: Schematic illustration of the SI framework. A point in the data space $\mathbb{R}^N$ corresponds to a sequence with length $N$. The darkly shaded regions in the data space indicate that, if we input a point in these regions into the algorithm $\mathcal{A}$, the CP candidates are the same as $\mathcal{A}(\bm{x})$ obtained from the observed sequence $\bm x$. By conditioning on these regions and $\mathcal{Q}(\bm{X}) = \mathcal{Q}(\bm{x})$, the conditional sampling distribution of the test statistic $T_k(\bm X)$ is represented as a truncated $\chi$-distribution. Thus, selective $p$-values are defined based on the tail probability of such a truncated $\chi$-distribution.
• Figure 9: Schematic illustration of the proposed line search method for the identification of the truncation region. We first compute the over-conditioned region where the process of the algorithm $\mathcal{A}$ remains unchanged. Then, we identify the truncation region $\mathcal{Z}$ by removing the redundant conditioning using parametric programming.
• Figure : (a) Time series signal.
• Figure : (a) Adding at $57$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2