OPTICS: Order-Preserved Test-Inverse Confidence Set for Number of Change-Points

Abstract

Determining the number of change-points is a first-step and fundamental task in change-point detection problems, as it lays the groundwork for subsequent change-point position estimation. While the existing literature offers various methods for consistently estimating the number of change-points, these methods typically yield a single point estimate without any assurance that it recovers the true number of changes in a specific dataset. Moreover, achieving consistency often hinges on very stringent conditions that can be challenging to verify in practice. To address these issues, we introduce a unified test-inverse procedure to construct a confidence set for the number of change-points. The proposed confidence set provides a set of possible values within which the true number of change-points is guaranteed to lie with a specified level of confidence. We further proved that the confidence set is sufficiently narrow to be powerful and informative by deriving the order of its cardinality. Remarkably, this confidence set can be established under more relaxed conditions than those required by most point estimation techniques. We also advocate multiple-splitting procedures to enhance stability and extend the proposed method to heavy-tailed and dependent settings. As a byproduct, we may also leverage this constructed confidence set to assess the effectiveness of point-estimation algorithms. Through extensive simulation studies, we demonstrate the superior performance of our confidence set approach. Additionally, we apply this method to analyze a bladder tumor microarray dataset. Supplementary Material, including proofs of all theoretical results, computer code, the R package, and extended simulation studies, are available online.

Figures (6)

• Figure 1: Left figure: Piechart for ranks of $K^*$ in an ascending order according to prediction error. Right figure: Histogram of the difference in the out-of-sample prediction errors between the models fitted with the true number of change-points $K^*$ and those fitted with the minimizer $\hat{K}$.
• Figure 2: Left Panel: The $k$th moment plots of $10$ individuals, where the black curve represents $y = k^{k/2}$. The y-axis is rescaled using $\log_{10}$. Right Panel: Boxplots of Hausdorff distances between the confidence sets obtained by OPTICS and those generated randomly.
• Figure 3: Detected change-points for each individual separately.
• Figure 4: Detected change-points for 10 individuals analyzed jointly. The bold dashed lines (dark red) represent change-points selected by the OPTICS FWER control criteria, while the solid light lines (light red) indicate additional change-points identified by the default stopping criteria in the wbs R package.
• Figure S.1: The boxplot of cardinalities of OPTICS and the line plot of coverage rates in univariate mean change-point model: $N(0,1)$ error.

Theorems & Definitions (16)

• Theorem 3.1
• Theorem 3.2
• Definition 3.1: Signal-to-noise ratio
• Theorem 3.3
• Lemma S.1
• proof
• Lemma S.2
• proof
• Lemma S.3
• proof