Two-Window Inspection Scheme (TWIN)

• TWIN is a sequential change point detection method that aggregates nearly symmetric past and recent samples through a multiscale weighting strategy.
• It achieves rapid detection with logarithmic delay scaling and robust performance even for late-occurring changes, outperforming traditional CUSUM approaches.
• Underpinned by new exponential moment bounds and self-normalized extensions, TWIN demonstrates effective real-time monitoring in applications like COVID-19 testing.

The Two-Window Inspection Scheme (TWIN) is a sequential testing methodology for online change point detection, designed for both mean shifts and broader distributional changes in time series data. TWIN aggregates observations into nearly symmetric past and recent samples, applying a multiscale weighting strategy. This enables pronounced improvements in both statistical power and detection speed compared to traditional CUSUM-type approaches, with logarithmic detection delays and robust performance even for late-occurring changes. The scheme is theoretically supported by new exponential moment bounds for the modulus of continuity in partial sum processes, and is demonstrated in real-world epidemiological monitoring, notably for real-time detection of structural breaks in COVID-19 data.

1. Symmetric Window Aggregation and Detector Construction

TWIN operates in a sequential monitoring context, beginning with a training sample of size $N$ where the in-control mean is assumed constant but unknown. Subsequent observations arrive one by one, and at each monitoring step $k$ (total sample size $N + k$), the scheme computes a CUSUM-type statistic over a range of scales $\ell$ ($1 \leq \ell \leq (N+k)/2$ and $\ell \leq k$). For each $\ell$, TWIN compares:

• An aggregate of early observations (either from the training period or early monitoring data), and
• An equally sized "recent window" of the most recent $\ell$ observations.

For mean change detection, the TWIN CUSUM statistic is defined as: $$\hat{\gamma}^{TC}(\ell, k) = \Bigl|\min\Bigl(1, \frac{\ell}{N}\Bigr)S_{\max(\ell, N)} - \bigl(S_{N+k} - S_{N+k-\ell}\bigr)\Bigr|,$$ where $S_j = \sum_{i=1}^j X_i$.

This nearly symmetric construction aligns the comparison scales between past and recent data, minimizing statistical bias and facilitating rapid detection.

TWIN applies an explicit multiscale weight function: $$w^{TC,\beta}(\ell,k) = \ell^{-1/2}\,\log^{-\beta}\Bigl(C_0+\frac{N}{\ell}\Bigr) \log^{-\beta}\Bigl(C_0+\frac{N+k}{N}\Bigr),$$ where $\beta>1/2$ and $C_0>1$ control the decay rate. This weighting strategy gives near-equal importance to small windows, only discounting larger windows logarithmically, ensuring stability and sensitivity across a wide spectrum of scales.

The complete TWIN detector is given by: $$\widehat{\Gamma}^{TC}(k) = \max_{\substack{1 \leq \ell \leq (N+k)/2,\,\ell \leq k}} w^{TC,\beta}(\ell,k)\,\hat{\gamma}^{TC}(\ell, k).$$

2. Detection Delay Analysis

A principal contribution of TWIN is attaining detection delays that scale logarithmically in the monitoring horizon. Traditional methods (e.g., standard CUSUM, Page CUSUM, MOSUM) exhibit polynomial or linear delay growth with sample size and monitoring time, especially for late-stage changes. TWIN, by contrast, demonstrates:

$$\tilde{\tau} = O_P\left(\frac{\log^{2\beta}(N)\,\log^{2\beta}(C_0 + k^\star/N)}{\Delta_N^2}\right),$$

where $k^\star$ is the change time and $\Delta_N$ the jump size.

This logarithmic scaling ensures uncommonly short vigilance gaps, notably preserving low false alarm rates under the null. The improvement remains substantial for late change points—scenarios in which classical techniques perform particularly poorly.

3. Statistical Power and Performance Characterization

TWIN achieves enhanced statistical power through its nearly symmetric sample aggregation and rigorous multiscale weighting scheme. The consistency condition under local alternatives is: $$\Delta_N \gg \frac{\log^\beta(C_0+N/k^\star)\,\log^\beta(C_0 + k^\star/N)}{\sqrt{N+k^\star}},$$ up to logarithmic factors.

This relation reveals that for sufficiently late change points ($k^\star \gg N$), detectable changes of amplitude $\Delta_N \ll N^{-1/2}$ become feasible, far below the classical threshold. The methodology incorporates information across a variety of scales, operating against a wide class of local alternatives, while conventional detectors are optimized for early-stage or particular change profiles only.

4. Self-Normalized Extension for Mean Changes

Temporally dependent noise complicates estimation of nuisance parameters such as long-run variance $\sigma_{LR}$. TWIN addresses this by introducing a self-normalized version for mean change detection. The self-normalizing term, computed using only training data, is: $$V_N = \frac{1}{N^{3/2}} \sum_{i=1}^N \Big| S_i - \frac{i}{N}S_N \Big|,$$ with $S_i$ as cumulative sums.

The self-normalized TWIN statistic is: $$\widehat{\Gamma}^{SNTC}(k) = \frac{\widehat{\Gamma}^{TC}(k)}{V_N}.$$ $V_N$ is asymptotically proportional to $\sigma_{LR}$, ensuring the statistic converges to a pivotal limit ($L_{SN}$), which obviates the need to estimate long-run variance and enables determination of rejection thresholds based only on training data. This construction automatically cancels out temporal dependence effects, increasing robustness to realistic timeseries noise.

5. Empirical Applications

The methodology is substantiated through applications to epidemiological data, notably monitoring daily cycle threshold (Ct) values for COVID-19 testing. These Ct values serve as proxies for viral load and are typically bounded (subgaussian). The emergence of new virus variants is often associated with distributional shifts in Ct values.

In practice, an initial training period (e.g., May 2020) is used, followed by sequential application of TWIN (and its nonparametric extension NP-TWIN) in monitoring. Empirical results demonstrate nearly immediate detection of structural breaks in median Ct values upon emergence of new variants. Competing methods exhibit detection delays of up to one year or more. TWIN’s short delay profiles position it as particularly valuable for real-time monitoring where prompt action is necessary.

6. Probabilistic Foundations and Exponential Moment Bounds

The theoretical advancement underlying TWIN is the derivation of new exponential moment bounds for the global modulus of continuity of the partial sum process. For i.i.d. subgaussian noise $\varepsilon_i$ satisfying $\|\varepsilon_1\|_{\Psi_2} \leq C_\varepsilon$, define the partial sum process: $$P_N(t) = \frac{1}{\sqrt{N}} \left( \sum_{i=1}^{\lfloor Nt \rfloor} \varepsilon_i + (Nt - \lfloor Nt \rfloor)\,\varepsilon_{\lfloor Nt \rfloor + 1} \right).$$ The key bound is on

$$M_{P_N} := \sup_{0<s<t<\infty,\, h=|t-s|} \frac{|P_N(t) - P_N(s)|}{\sqrt{h\left(1 + \log(t/h) + \delta|\log(t)|\right)}},$$

with $\mathbb{E} \exp(M_{P_N}^2) \leq C_P$ for some $C_P$ independent of $N$.

Equivalent bounds are provided for the empirical distribution function in the nonparametric NP-TWIN variant. These moment bounds are essential for establishing uniform control in weak convergence arguments. A plausible implication is that such results may find utility in controlling the oscillation of partial sum and empirical processes across multiscale statistical tasks beyond change point detection.

7. Summary and Directions

TWIN constitutes a comprehensive change point detection approach combining symmetric sample aggregation with rigorous multiscale weighting. Its main technical innovations result in drastically reduced (logarithmic) detection delays and increased power—especially for late-occurring changes—relative to established CUSUM-type detectors. The model is extended via a self-normalizing variant for dependent noise and nonparametric versions robust to heavy-tailed data, with theoretical underpinnings grounded in new exponential moment bounds for process continuity. Empirical evaluations in COVID-19 monitoring underscore TWIN's capacity for immediate structural break detection, highlighting its utility for time-critical monitoring applications and suggesting broad future applicability where multiscale statistical vigilance is required.