FAR-HED Pareto Frontier: Temporal Trade-Offs

• The FAR-HED Pareto Frontier formalizes the trade-off between false alarm rates and early detection, using a measure-theoretic and axiomatic framework.
• It employs a robust temporal decay mechanism and thresholding method to compute HED scores and construct the Pareto-optimal operating curve.
• The framework enables calibration across diverse domains by adjusting the Hiremath decay constant, ensuring practical, timely detection in dynamic environments.

The FAR-HED Pareto frontier formalizes the trade-off between the false alarm rate (FAR) and the Hiremath Early Detection (HED) score in the evaluation of early-warning systems operating over non-stationary stochastic processes with abrupt regime transitions. It provides a rigorous, algorithmically explicit characterization of the attainable balance between “timeliness” and “spurious alarms” for detection algorithms, rooted in a measure-theoretic and axiomatic foundation distinct from ROC/AUC paradigms (Hiremath, 5 Apr 2026).

1. Formal Framework and Measure-Theoretic Definitions

The HED Score $H[P; t_0, \lambda_H]$ quantifies the acuity and latency of a detection profile $P(\cdot)=\Pr(\mathcal R_1|F_t)$ with respect to a known regime transition at onset time $t_0$. It is defined as

$$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$

with $$P_{\mathrm{base}} = \frac1{t_0} \int_0^{t_0} P(t)\,dt$$ and decay $\lambda_H > 0$. This structure penalizes delayed detections—later triggers contribute exponentially less to the aggregate score.

For binary detection streams induced by thresholds $\theta \in [0,1]$ (i.e., $D_\theta(t) = \mathbf1\{P(t) \geq \theta\}$), the False Alarm Rate is

$$\mathrm{FAR}(\theta) = \frac{1}{t_0} \sum_{t=0}^{t_0-1} D_\theta(t).$$

The HED Score corresponding to this thresholded stream is

$$H(\theta) = \frac{1}{T-t_0} \sum_{t=t_0}^T \max\{0, D_\theta(t) - \mathrm{FAR}(\theta)\} e^{-\lambda_H (t-t_0)}.$$

Pairs $P(\cdot)=\Pr(\mathcal R_1|F_t)$0 over all thresholds $P(\cdot)=\Pr(\mathcal R_1|F_t)$1 yield candidate operating points for performance evaluation.

2. Axiomatic Characterization and Interpretation

The HED Score satisfies three axiomatic properties:

1. Temporal Monotonicity (A1): Earlier detection after $P(\cdot)=\Pr(\mathcal R_1|F_t)$2 strictly increases $P(\cdot)=\Pr(\mathcal R_1|F_t)$3, for fixed detection shapes.
2. Invariance to Pre-Attack Bias (A2): Raising all probabilities uniformly (including pre-onset) does not increase $P(\cdot)=\Pr(\mathcal R_1|F_t)$4—the score is immune to “trigger-happy” baseline elevation.
3. Sensitivity Decomposability (A3): $P(\cdot)=\Pr(\mathcal R_1|F_t)$5 can be decomposed over any partition $P(\cdot)=\Pr(\mathcal R_1|F_t)$6, supporting phase-by-phase diagnostics.

These principles enforce calibration, latency sensitivity, and decomposability in contrast to temporally agnostic metrics such as AUC.

3. Construction and Computation of the Pareto Frontier

The FAR–HED Pareto frontier is constructed by thresholding the posterior sequence at a grid $P(\cdot)=\Pr(\mathcal R_1|F_t)$7, computing $P(\cdot)=\Pr(\mathcal R_1|F_t)$8 for each threshold, and extracting the non-dominated points: $P(\cdot)=\Pr(\mathcal R_1|F_t)$9

Algorithmic steps:

1. For each $t_0$0:
   • Form binary detections $t_0$1.
   • Compute $t_0$2 and $t_0$3.
2. Construct a list of candidate operating points.
3. Sort by increasing FAR.
4. Retain only those $t_0$4 improving on previous $t_0$5 at each $t_0$6—this set is the Pareto frontier.

This procedure is computationally efficient and robust for practical datasets.

4. Calibration of Temporal Trade-off: The Hiremath Decay Constant

The trade-off between early penalty and late detection credit is governed by $t_0$7, interpreted via the “information half-life” $t_0$8. Domain specificity is imposed by calibrating $t_0$9 according to the minimum effective response window $$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$0: $$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$1 The “Hiremath Standard Table” provides domain-calibrated defaults for $$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$2, e.g., higher values for HFT, lower for seismic detection.

Domain	Representative System	$$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$3	$$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$4 (steps)
HFT engines	Ultra-high-frequency	0.50	1.39
Network Security IDS	Intrusion Detection	0.14	4.95
Bio-Refinery Control	Industrial SCADA	0.05	13.86
Epidemiological Surveillance	Pandemic Onset	0.02	34.66
Seismic Early Warning	P-wave Discrimination	0.01	69.31

This calibration ensures that timeliness is quantified in a manner consistent with operational exigencies.

5. Trade-off Visualization and Operating Point Selection

The FAR–HED curve, plotting $$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$5 versus $$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$6 as threshold $$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$7 varies, typically begins at $$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$8 (high threshold, no detections), rises to a maximum (“knee”), and returns to $$H\bigl[P(\cdot);\;t_0,\lambda_H\bigr] = \frac{1}{T-t_0} \int_{t_0}^T \max\{0, P(t)-P_{\mathrm{base}}\} \exp\bigl[-\lambda_H\,(t-t_0)\bigr] dt$$9 (low threshold, all spurious alarms, no informative credit). The “knee point” generally represents an optimal balance between spurious alarms and timeliness.

Operating points may be selected via application-tailored utility,

$$P_{\mathrm{base}} = \frac1{t_0} \int_0^{t_0} P(t)\,dt$$0

with $$P_{\mathrm{base}} = \frac1{t_0} \int_0^{t_0} P(t)\,dt$$1 encoding the relative weight of timely detection versus avoidance of false alarms.

Visualization is typically performed via $$P_{\mathrm{base}} = \frac1{t_0} \int_0^{t_0} P(t)\,dt$$2 plots; LaTeX/TikZ code is provided for stylized frontiers.

6. Context and Relationship to General Pareto-Frontier Fairness–Utility Models

In broader algorithmic fairness contexts, Pareto frontiers have been used to characterize the optimal trade-off between fairness (e.g., exposure parity, group disparity) and utility (e.g., predictive accuracy, aggregate reward) (Xu et al., 2022, Kletti et al., 2022, Little et al., 2022). The FAR–HED frontier specializes this paradigm to early-warning (temporal detection) domains, quantifying the intrinsic cost of spurious alarms against the reward for rapid regime change identification within a rigorously calibrated, measure-theoretic framework.

While traditional fairness–utility trade-offs (e.g., in group fairness) are often handled via convex hull methods or exposure polytopes (expohedra), the FAR–HED procedure operates in the space of detection thresholds and posterior dynamics, with the Pareto frontier reflecting the achievable operational envelope under the HED axioms and temporal discounting.

7. Empirical and Practical Impact

Empirical results on benchmarks such as NSL-KDD show that algorithms optimizing for the HED score outperform baselines (e.g., Random Forests), especially in terms of early detection acuity and resilience to pre-onset bias. The block-bootstrap procedure confirms statistical significance of gains (Hiremath, 5 Apr 2026). Practical adoption mandates domain-specific choice of $$P_{\mathrm{base}} = \frac1{t_0} \int_0^{t_0} P(t)\,dt$$3 and careful selection of operation point on the FAR–HED curve according to risk tolerance and downstream action lags.

The FAR–HED Pareto frontier provides a complete, principled toolset for evaluating, visualizing, and selecting detection algorithms in time-critical, non-stationary environments where the timing of alarms is as crucial as their overall statistical accuracy.