Deviation-Aware Modulation Scheme

• Deviation-aware modulation scheme is a signal processing technique that adaptively scales signal weights based on deviations from expected baselines.
• It underpins robust anomaly detection in multi-channel time series and noise-sensitive digital modulation for over-the-air computation using statistical thresholds and optimization.
• Empirical results indicate significant improvements in detection precision, recall, and error reduction by amplifying critical deviations while suppressing routine signals.

A deviation-aware modulation scheme is a class of signal processing or encoding technique designed to adaptively modulate or reweight features, messages, or symbol constellations according to their deviation from an expected baseline or in response to underlying noise/statistical deviation. Such schemes arise in contexts ranging from anomaly detection in multi-channel time series to robust digital modulation for distributed wireless computation. They provide a principled method to suppress uninformative or routine patterns while amplifying rare, atypical, or function-critical events, leading to enhanced detection accuracy or communication robustness. Deviation-aware modulation has been instantiated most prominently in multi-resolution behavioral modeling for anomaly detection (Kong et al., 18 Jan 2026) and in noise-tailored constellation design for over-the-air computation (OAC) (Razavikia et al., 19 Jun 2025).

1. Formalization and Theoretical Rationale

At its core, deviation-aware modulation imposes a rule: given a signal (e.g., a behavioral event histogram or transmission symbol), its representation is dynamically scaled according to a deviation score. For time series or behavioral matrices (Kong et al., 18 Jan 2026), this involves standardizing each measurement $X_{h,w}$: $$\Delta_{h,w} = \frac{ |X_{h,w}-\mu_{h,w}| }{ \sigma_{h,w} + \epsilon }$$ where $\mu_{h,w}$ and $\sigma_{h,w}$ are normal baselines learned from calibration data, and $\epsilon$ is a small constant for stability. For modulation of digital symbols in OAC (Razavikia et al., 19 Jun 2025), the deviation is taken in terms of probabilistic noise metrics or likelihood separations (e.g., $d_{\text{noise}}(r_i,r_j)$) tailored to the underlying noise distribution.

This strategy ensures that the contribution of each data element scales according to its atypicality, with methodological choices (piecewise functions, thresholds, or optimization constraints) controlling whether routine signals are suppressed and rare or critical deviations are amplified.

2. Algorithmic Workflow and Implementation

2.1. Multi-Channel Log Anomaly Detection

The pipeline for deviation-aware modulation in behavioral matrices proceeds as follows (Kong et al., 18 Jan 2026):

1. Raw Log Discretization: Sequence of events $E = \{e_1, ..., e_N\}$ is mapped into an $H \times W$ matrix $X$, where $H$ is event type cardinality and $W$ is the number of time bins.
2. Baseline Estimation: Compute $\mu_{h,w}, \sigma_{h,w}$ from a period of routine behavior.
3. Deviation Calculation: For new data, calculate $\Delta_{h,w}$ according to the standardized deviation formula.
4. Piecewise Modulation: For each $(h,w)$, set the modulation weight:

$$M_{h,w} = \begin{cases} \beta, & \text{if}\ \Delta_{h,w}<\tau \ 1+\lambda \cdot \Delta_{h,w}, & \text{otherwise} \end{cases}$$

where $\beta \in (0,1)$ is the suppression factor, $\lambda>0$ is the amplification slope, and $\tau$ is the threshold.

1. Elementwise Product: Output modulated matrix $\hat{X} = X \odot M$.

2.2. Deviation-Aware Digital Modulation for OAC

Here, each distributed node applies quantization and modulation according to local statistics, with the global constellation optimized such that the sum-constellation points maximize noise-induced separation under $d_{\text{noise}}$ (Razavikia et al., 19 Jun 2025). The workflow is:

1. Quantization and Encoding: Each data symbol $s_k$ is quantized and mapped to a modulation symbol $x_k$.
2. Sum-Constrained Design: The sum of all active $x_k$ is mapped via the wireless MAC to the fusion center.
3. Noise-Aware Optimization: The coordinates of the sum-constellation are optimized via a max-min criterion to maximize $d_{\text{noise}}(r_i,r_j)$, where the metric is tailored to the channel's noise distribution.
4. Tabular Decoding: Given the noisy sum, the ML decoder maps observations to the required function output.

3. Hyperparameters, Tuning, and Specialized Metrics

Behavioral Modulation (Kong et al., 18 Jan 2026)

• $\epsilon$ (numerical stability): Fixed at $10^{-6}$.
• $\beta$ (suppression factor): Typically $0.1 \leq \beta \leq 0.5$, grid searched to retain some information from routine behavior.
• $\lambda$ (amplification slope): Often between $0.5$ and $5$.
• $\tau$ (deviation threshold): Tuned on validation, often chosen from the $90$th-$99$th percentile of observed $\Delta$.

Digital Modulation for OAC (Razavikia et al., 19 Jun 2025)

• Noise Metric Tailoring:
  • Gaussian: $$d_{\text{noise}} \propto \exp(|\Delta|^2/4\sigma^2)$$
  • Laplace: $d_{\text{noise}} \propto \exp(|\Delta|/b)$
  • Heavy-tail (e.g., Cauchy): $d_{\text{noise}} \propto |\Delta|/\gamma$
• SDP Relaxation and Coordinate Descent: Used to optimize constellation geometry.

The choice of metric and optimization method is dictated by the statistical characteristics of the environmental noise. At high SNR or with sub-Gaussian noise, classical Euclidean distance (ChannelComp, $|\Delta|^2$) is adequate, but at low SNR or with heavy tails, exponential or polynomial metrics yield significantly lower mean square error.

4. Empirical Results and Comparative Evaluation

The effect of deviation-aware modulation is substantiated with ablation studies and comparative benchmarks.

Multi-Channel Log Scenario (Kong et al., 18 Jan 2026)

Ablation experiments on the CERT r4.2 insider threat benchmark demonstrate:

Model Variant	Precision	Recall	F1
Full (with deviation modulation)	0.98	1.00	0.99
Without deviation modulation	0.96	0.90	0.92
Modulation only, without DWT	0.93	0.94	0.94
Without resolution-adaptive attention	0.94	0.92	0.93

Excluding modulation causes recall to drop from $1.00$ to $0.90$ and F1 from $0.99$ to $0.92$, confirming the necessity of selective suppression/amplification for state-of-the-art anomaly detection.

Noise-Aware Digital Modulation (Razavikia et al., 19 Jun 2025)

For sum or max functions across distributed nodes, using noise-tailored metrics leads to substantial reduction in MSE, particularly in challenging noise regimes. For example, in AWGN with $\sigma=1.0$, the Gaussian-metric constellation achieves MSE of $-6$ dB, outperforming ChannelComp ($0$ dB). Conversely, for small $\sigma$, ChannelComp is optimal. In Cauchy-heavy-tailed noise, the Cauchy-aware metric reduces worst-case error by $3$–$5$ dB compared to Laplace or Gaussian designs.

5. Intuitive Motivation and Mechanistic Interpretations

Deviation-aware modulation unifies two key principles:

• Routine Suppression: Low-deviation (routine) activity is down-weighted to avoid distraction from repetitive, non-informative signals (e.g., regular access logs in security, routine data transmission in OAC).
• Anomaly/Deviation Amplification: High-deviation elements are magnified, leveraging sharp contrast to flag rarities (e.g., insider threat behaviors, low-probability errors due to noise).

The threshold $\tau$ controls the point at which suppression gives way to amplification, and is typically set to trade off between false positives (over-reactivity to noise) and false negatives (missed significant events).

In OAC, designing constellations with explicit separation under $d_{\text{noise}}$ rather than purely Euclidean distance allows for robustness to non-Gaussian and heavy-tailed environments, a principle applicable wherever non-ideal or adversarial noise patterns occur.

6. Broader Implications and Practical Deployment

Deviation-aware modulation provides a lightweight, differentiable preprocessing or symbol selection layer, readily integrated within larger pipelines—such as multi-resolution anomaly detection frameworks (paired with DWT and resolution-adaptive attention) (Kong et al., 18 Jan 2026) or scalable distributed computation setups over wireless multiple access channels (Razavikia et al., 19 Jun 2025). It has no strong dependencies on downstream models, requiring only baseline statistics or channel noise estimates for tuning. Empirical validation attests to its centrality for achieving leading detection and decoding accuracy, particularly in high-dimensional, rare-event, or nonstationary environments.

Practical design guidelines include periodic recalibration of baselines or noise metrics, conservative parameter grid search for suppression/amplification, and adaptation of segmentation to application granularity (e.g., hour, day, week). In communication settings, constellation updates may be triggered by observed shifts in channel statistics.

7. Relationship to Related Methods and Future Prospects

Deviation-aware modulation generalizes several prior heuristics that used static thresholds or quantization on deviations, supplying a fully parameterized, learnable, and noise-adaptive mechanism. In anomaly detection, it complements spectral and time-frequency representations (e.g., DWT) and interacts synergistically with attention-based pooling. In digital modulation, it embodies a universal design principle: encode functional information so that noise-abnormal events are naturally emphasized, improving both average and worst-case decoding accuracy.

Future developments could include extension to nonlinear deviations, self-supervised learning of modulation parameters, and integration with adaptive or adversarial channel estimation protocols, further bolstering robustness and flexibility in dynamic or adversarial environments.