Shapelet-Based Glitches

• Shapelet-based glitches are transient, localized anomalies in time series characterized by informative subsequences that capture essential waveform features.
• They utilize frameworks like Gauss–Hermite and discriminative shapelets to improve detection accuracy across domains such as gravitational-wave and biomedical signal analysis.
• Advanced techniques including matching pursuit, Bayesian MCMC, and dual-attention modules enhance glitch detection, parameter estimation, and synthetic generation.

Shapelet-based glitches are transient, localized anomalies in time series that are characterized and detected using "shapelets"—compact, informative subsequences capturing essential waveform features. The concept arises from the intersection of time-domain morphological analysis and statistical pattern recognition, with demonstrated impact in gravitational-wave instrumentation, clinical biomedical signals, and industrial time-series anomaly detection. Shapelet-based approaches are central to robust glitch modeling, detection, generation, and interpretable classification in noisy, dynamic measurement environments, often outperforming conventional pointwise or frequency-domain detectors in their ability to resolve and parameterize transient, structured events.

1. Mathematical Foundations of Shapelets

A shapelet is formally a short subsequence $S = [s_1, ..., s_\ell]$ extracted from a longer time series $X = [x_1, ..., x_T]$, with $\ell \ll T$. In the anomaly detection context, a shapelet-based glitch is defined as a localized instance where a subsequence exhibits a statistical or morphological deviation from the expected (normal) pattern, as measured by a chosen dissimilarity metric $D(·,·)$. Two principal shapelet frameworks have been established:

• Gauss–Hermite (physical) shapelets: In the context of instrumental glitches, a basis of one-dimensional Gauss–Hermite functions $\phi_n(t;\beta)$ is used, where $n$ is the order (node count), $\beta$ the scale (width), and $H_n(\cdot)$ the Hermite polynomial. An arbitrary transient $g(t)$ is decomposed as $g(t) = \sum_{n=0}^N c_n \phi_n(t; \beta)$. Sparse and robust parameterization is achieved through $\ell_0$-penalized likelihood maximization and matching pursuit, with subsequent refinement by Bayesian MCMC (Baghi et al., 2021).
• Discriminative time series shapelets: For interpretable glitch classification, a shapelet is extracted so as to maximize its class discriminative power—e.g., by searching for $S$ that achieves high information gain (IG) when splitting a dataset based on $PSD(X,S)$, the Perceptual Subsequence Distance (Le et al., 9 Mar 2025).

In both cases, the waveform is reduced to a set of localized features encapsulating amplitude, scale, and temporal position, enabling fine-grained, noise-resilient glitch representation.

2. Detection and Parameter Estimation

Shapelet-based glitch detection involves:

• Decomposing observed series into candidate shapelet components,
• Estimating the significance and parameters $\{c_n, \beta, \tau\}$,
• Iteratively subtracting significant atoms via matching pursuit or greedy algorithms.

For physical setup (e.g., LISA Pathfinder), this proceeds by matched filtering for different $n, \beta, \tau$, terminating when no shapelet atom achieves SNR above threshold (e.g., SNR ≳ 5 corresponds to a $\sim$0.01% false-alarm rate per 2.5 days). Bayesian MCMC post-processing refines parameters for the most significant events, correcting noise-weighting and interpolation biases.

In discriminative classification scenarios (e.g., patient-ventilator synchrony), candidate shapelets are efficiently extracted using Perceptually Important Points (PIPs) as segment boundaries, ranking candidates by information gain. The final detection layer typically involves feeding shapelet-distance feature vectors, optionally concatenated with handcrafted statistical signatures, into a shallow feed-forward network (Le et al., 9 Mar 2025).

3. Statistical Characterization and Synthetic Generation

The empirical distributions of glitch parameters—such as inter-arrival times, amplitude scales, and damping times—provide insight into systematics and rare event structure. For instance, in LISA Pathfinder glitch populations, inter-arrival intervals $\Delta \tau$ follow an exponential law with $\lambda \approx 5 \times 10^{-5}\,\mathrm{s}^{-1}$ (ordinary runs) and tenfold increase in "cold" runs (Baghi et al., 2021).

Synthetic glitch generation is conducted by:

• Sampling glitch times $\tau_i$ from the empirical exponential;
• Drawing pairs $(\alpha, \beta)$ from the joint amplitude-damping distribution, typically modeled with normalizing flows to capture tail behavior;
• Assembling the synthetic waveform $$g(t) = \alpha \psi_1\left(\frac{t-\tau}{\beta}\right)$$ with $\psi_1$ a suitable exponential or Hermite shapelet.

This approach supports large-scale simulation studies and algorithm validation in both physics and biomedical domains (Baghi et al., 2021, Le et al., 9 Mar 2025).

4. Advanced Classification and Interpretability

Interpretability is inherent in shapelet-based schemes. Once a pool of discovered shapelets $\{S^j\}$ is established, any input $X$ can be mapped to a vector of shapelet distances $Z_{\text{sha}}^j = PSD(X, S^j)$, which serves as a directly explainable feature set. Model decisions can be visualized by overlaying best-matched shapelets on the input, generating "heat-maps" of response intensity along time, allowing for qualitative validation by domain experts (e.g., clinicians in ventilator scenarios) (Le et al., 9 Mar 2025).

SHIP, for example, concatenates shapelet vectors with statistical summaries (e.g., logarithmic signatures) and trains a compact three-layer classifier. The subsequence matches are used for post-hoc explanations: the location and distance of best-matching shapelets correspond directly to the detected glitch type, supporting transparent diagnostics.

5. Shapelet Frameworks in Time Series Anomaly Detection

The recent TShape framework extends shapelet-based glitch identification to complex, nontrivial industrial time series anomalies using:

• Patch-wise multi-scale convolution to extract multi-resolution local shapelet features,
• Patch-wise positional encoding,
• Dual (local-intra-patch and global-inter-patch) self-attention mechanisms with gated fusion, all trained solely on normal data to minimize reconstruction error (Cui et al., 1 Oct 2025). The per-time-point anomaly score is $s_t = |x_t - \hat{x}_t|$, and event-level detection is thresholded on these residuals.

Table: TShape Event-F1 (F1-E) Scores Versus FCVAE Baseline

Dataset	FCVAE F1-E	TShape F1-E
AIOPS	0.7364	0.8049
NAB	0.7933	0.9186
TODS	0.6689	0.8561
UCR	0.5126	0.5915
WSD	0.8695	0.9137

TShape achieves an average of +10% F1-E improvement over FCVAE, validating the necessity of multi-scale convolution and dual-attention modules via ablation studies. Attention maps highlight both fine-grained local and global contextual relevance—well-suited for domains where glitches correspond to localized morphological departures (Cui et al., 1 Oct 2025).

6. Impact on Downstream Analysis and Detection Robustness

Shapelet-based glitch modeling crucially affects downstream system performance:

• In gravitational-wave detection (LISA), synthetic glitches can be projected from LPF records to the LISA data channel as effective fractional laser-frequency deviations, further processed via time-delay interferometry (TDI). Glitch-induced transients may inject SNRs ranging from $10^{-2}$ to $10^4$ in TDI A/E/T channels, with ≈50% above SNR ≈ 10, sometimes exceeding astrophysical burst amplitudes within short intervals (Baghi et al., 2021). This suggests that shapelet-modeled glitches, if unaccounted for, could bias astrophysical parameter estimation or trigger false-positive event candidates.
• In medical and industrial anomaly detection, shapelet features enable robust, interpretable event detection even under significant class imbalance or channel subsampling. For instance, SHIP achieves four-way F1 = 0.9765 and >0.89 per-class F1 for most asynchrony types, surpassing convolutional, recurrent, and latent mixture baselines (Le et al., 9 Mar 2025). TShape’s reconstruction-error methodology achieves average precision 0.88 and recall 0.85 compared to baselines at 0.78/0.76, evidencing superior sensitivity to complex shapelet-based glitches (Cui et al., 1 Oct 2025).

7. Prospects, Limitations, and Research Directions

Shapelet-based approaches generalize across disciplines but feature domain-specific caveats:

• For detection and simulation, an adequate empirical library of glitches and accurate joint parameter statistics are required.
• Most current frameworks address univariate or low-dimensional signals; extension to fully multivariate, cross-channel shapelet analysis is limited, though TShape and normalizing flows offer a plausible pathway (Cui et al., 1 Oct 2025).
• Training dependency on predominantly “clean” background data may limit utility in high-anomaly-rate environments.
• Future work will likely incorporate prototype-guided attention and self-supervised, library-driven shapelet mining, particularly for complex multi-source or multi-modal industrial signals.

In summary, shapelet-based glitch modeling constitutes a unifying paradigm for transient anomaly detection, offering sparse, interpretable, and highly effective glitch localization and characterization (Baghi et al., 2021, Le et al., 9 Mar 2025, Cui et al., 1 Oct 2025). Its flexible mathematical foundation supports cross-domain application and continues to underpin advances in both scientific instrumentation and critical time-series monitoring.