High-Dimensional Change Point Analysis for Temporally Dependent Data

Technical Summary: High-Dimensional Change Point Analysis for Temporally Dependent Data

Problem Formulation
The paper addresses the statistical inference of mean changes in high-dimensional time series where the dimension $p$ may be comparable to or larger than the sample size $n$. The data are modeled as $X_i = \mu_0 + \delta I\{i > \tau\} + \epsilon_i$, where $\epsilon_i$ is a strictly stationary, centered vector process exhibiting temporal dependence. The challenge lies in detecting and locating the change point $\tau$ and the jump vector $\delta$ under two distinct signal regimes:

1. Dense alternatives: Many coordinates change by small amounts.
2. Sparse alternatives: Few coordinates change by large amounts.

Existing methods often assume independence or fail to jointly calibrate quadratic (dense) and maximum (sparse) statistics under general non-Gaussian vector dependence. Furthermore, standard CUSUM processes require adjustments for long-run covariance structures and edge effects when serial dependence is present.

Methodology
The authors propose a unified framework combining quadratic CUSUM statistics (for dense signals) and coordinatewise maximum statistics (for sparse signals), adapted for temporally dependent data.

1. Test Statistics and Weighting:

   • Two temporal weightings are employed to handle changes at different locations: an unweighted scan ($\gamma=0$) over the full sample and a variance-standardized scan ($\gamma=1/2$) over a trimmed interval to mitigate boundary effects.
   • Quadratic Statistic ($S_\gamma$): Aggregates information across all coordinates. The authors derive a dependence-adjusted centering term and a long-run scale estimator. Crucially, they account for the non-symmetry of lag covariance matrices by estimating two distinct trace orientations: the Frobenius product $tr\{\Gamma(h)\Gamma(k)^\top\}$ and the crossed product $tr\{\Gamma(h)\Gamma(k)\}$.
   • Maximum Statistic ($M_\gamma$): Takes the maximum absolute value of coordinatewise standardized CUSUMs. Marginal long-run variances are estimated using third-order difference-based estimators to handle serial dependence without direct long-run covariance matrix inversion.

2. Adaptive Combination:

   • Recognizing that the sparsity level is unknown, the paper employs a Cauchy combination test. It combines the $p$-values of the matched quadratic and maximum statistics (sharing the same $\gamma$) using the Cauchy distribution function. This relies on establishing the asymptotic independence between the quadratic and maximum statistics under the null hypothesis.

3. Localization and Multiple Change Points:

   • Single Change: Estimators are defined as the maximizers of the centered scores. The dense estimator maximizes the first-order centered score (avoiding the second-order correction used in testing to ensure global monotonicity), while the sparse estimator maximizes the squared coordinatewise score.
   • Multiple Changes: The procedures are extended via Wild Binary Segmentation (WBS). The algorithm generates random intervals, applies the adaptive tests, and employs a "shortest-significant" selection rule to isolate changes. A separation guard prevents duplicate detection of the same change.

Theoretical Contributions and Results
The paper establishes a rigorous asymptotic theory under general non-Gaussian vector dependence, covering strong mixing, $m$-dependence, and physical dependence frameworks.

• Limiting Distributions: The authors prove that the normalized quadratic CUSUM process converges to a centered Gaussian process with a specific covariance structure. For the boundary-weighted statistic, the limit is shown to be equivalent to a stationary Ornstein–Uhlenbeck process, leading to a Darling–Erdős extreme-value limit. The coordinatewise maximum statistics converge to a Gumbel distribution.
• Estimator Consistency: Uniform consistency is established for the feasible centering and scaling estimators, including the edge-corrected centering and the orientation-complete scale estimator.
• Asymptotic Independence: A key theoretical result is the proof of asymptotic independence between the matched quadratic and maximum statistics under the null hypothesis. This justifies the validity of the Cauchy combination test for adaptivity.
• Consistency and Rates: Under dense and sparse alternatives, the proposed tests are shown to be consistent. Localization rates are derived for both single and multiple change points, demonstrating that the estimators converge to the true change points at optimal rates depending on the signal strength and sparsity.

Empirical Results

• Simulations: The proposed methods demonstrate accurate size control (close to the nominal 5% level) under both Gaussian and non-Gaussian (Gamma) innovations, even with strong serial dependence. In contrast, methods assuming independence (Jin et al., 2016; Wang et al., 2019) suffer severe size distortions. The adaptive Cauchy tests ($T_{CC, \gamma}$) successfully adapt to unknown sparsity, maintaining high power across the spectrum from sparse to dense alternatives.
• Applications:
  • FRED-MD Panel: Applied to U.S. macroeconomic data, the methods detect multiple regime shifts corresponding to historical economic events (e.g., the 2008 financial crisis, the 2020 pandemic shock). The analysis reveals that dense and sparse components often identify distinct structural breaks, highlighting the value of the adaptive approach.
  • Electricity Load Data: Applied to daily electricity consumption, the methods identify seasonal transitions and regime shifts. Diagnostic tests confirm that significant serial dependence remains even after removing the estimated mean changes, validating the necessity of the dependence-adjusted calibration.

Significance and Claims
The paper claims to provide the first unified framework for high-dimensional change point analysis that simultaneously handles:

1. Unknown signal sparsity via Cauchy combination of quadratic and maximum statistics.
2. General temporal dependence (non-Gaussian, non-linear) through dependence-adjusted centering, orientation-complete scale estimation, and valid asymptotic independence results.
3. Multiple change point recovery using guarded wild binary segmentation with consistent estimation of both the number and locations of changes.

The authors emphasize that their approach avoids the invalidity of independence-based calibrations in dependent settings and offers a robust, adaptive tool for detecting structural breaks in complex, high-dimensional time series. The theoretical results are specific to the proposed statistics and assumptions (fixed number of changes, bounded long-run spectrum), and the paper does not claim to solve problems involving diverging numbers of changes or shrinking spacings without further multiscale extensions.