The Hiremath Early Detection (HED) Score: A Measure-Theoretic Evaluation Standard for Temporal Intelligence

Abstract: We introduce the Hiremath Early Detection (HED) Score, a principled, measure-theoretic evaluation criterion for quantifying the time-value of information in systems operating over non-stationary stochastic processes subject to abrupt regime transitions. Existing evaluation paradigms, chiefly the ROC/AUC framework and its downstream variants, are temporally agnostic: they assign identical credit to a detection at t + 1 and a detection at t + tau for arbitrarily large tau. This indifference to latency is a fundamental inadequacy in time-critical domains including cyber-physical security, algorithmic surveillance, and epidemiological monitoring. The HED Score resolves this by integrating a baseline-neutral, exponentially decaying kernel over the posterior probability stream of a target regime, beginning precisely at the onset of the regime shift. The resulting scalar simultaneously encodes detection acuity, temporal lead, and pre-transition calibration quality. We prove that the HED Score satisfies three axiomatic requirements: (A1) Temporal Monotonicity, (A2) Invariance to Pre-Attack Bias, and (A3) Sensitivity Decomposability. We further demonstrate that the HED Score admits a natural parametric family indexed by the Hiremath Decay Constant (lambda_H), whose domain-specific calibration constitutes the Hiremath Standard Table. As an empirical vehicle, we present PARD-SSM (Probabilistic Anomaly and Regime Detection via Switching State-Space Models), which couples fractional Stochastic Differential Equations (fSDEs) with a Switching Linear Dynamical System (S-LDS) inference backend. On the NSL-KDD benchmark, PARD-SSM achieves a HED Score of 0.0643, representing a 388.8 percent improvement over a Random Forest baseline (0.0132), with statistical significance confirmed via block-bootstrap resampling (p < 0.001). We propose the HED Score as the successor evaluation standard to ROC/AUC.

• The paper introduces the HED Score as a measure-theoretic evaluation standard that factors in detection latency to reward prompt regime change identification.
• It employs an integral formulation with exponential temporal discounting and baseline-corrected posteriors to ensure robust, time-sensitive evaluation.
• Empirical validation using PARD-SSM on the NSL-KDD dataset demonstrates a significant 3.8-fold improvement over conventional methods, underscoring its practical advantages.

The Hiremath Early Detection (HED) Score: Measure-Theoretic Evaluation for Temporal Intelligence

Introduction

The paper introduces the Hiremath Early Detection (HED) Score as a rigorous, measure-theoretic metric for evaluating temporal detection systems tasked with identifying regime shifts in non-stationary stochastic processes. Recognizing the fundamental inadequacy of prevailing evaluation standards such as ROC/AUC—which are agnostic to detection latency—the HED Score explicitly encodes the time-value of information, assigning maximal credit to prompt, well-calibrated detections following a regime shift, and penalizing delayed or falsely anticipatory responses.

Formalization of the HED Score

The HED Score is formulated as an integral (or summation in discrete domains) over the excess, baseline-corrected posterior probability assigned to the anomalous regime, exponentially discounted by the time-lag since the regime shift. This temporal discount is parametrized by the Hiremath Decay Constant ($\lambda$), providing direct control over the effective information half-life and enabling domain calibration (e.g., cyber-physical systems, epidemiological monitoring, high-frequency trading).

Formally, the HED Score is defined by

$$H[P, \tau ; \lambda] = \frac{1}{T - \tau} \int_{\tau}^T \max(0, P(t) - \bar{P})\, e^{-\lambda (t - \tau)} \, dt$$

where $P(t)$ denotes the posterior of the targeted regime at time $t$, $\bar{P}$ is the empirical pre-onset mean (baseline correction), $\tau$ is the ground-truth onset of the regime change, and $T$ the observation horizon. The discrete estimator uses summation and is amenable to practical time-series settings.

The HED Score integrates three critical components: baseline-neutral detection lift, exponential temporal discount, and horizon normalization, ensuring robustness to pre-attack bias and evaluation window variability. The decoupling of baseline and post-onset activity prevents inflation of credit by overconfident or "trigger-happy" detectors that maintain persistent high posteriors regardless of ground-truth transitions.

Axiomatic Foundations

The HED Score is validated against three axiomatic requirements:

• A1: Temporal Monotonicity—Earlier detections (closer to regime onset) always accrue higher scores, strictly enforcing the time-criticality principle.
• A2: Invariance to Pre-Attack Bias—Global additive shifts (e.g., constant false positive bias) do not alter the HED Score, preserving evaluation integrity against threshold manipulation.
• A3: Sensitivity Decomposability—The score naturally decomposes across non-overlapping sub-intervals, supporting granular, phase-resolved analysis of temporal detection performance.

Theoretically, the HED Score is proven to be strictly bounded, non-negative, and sensitive only to genuine differential lift in response to regime changes.

Empirical Validation: PARD-SSM and Benchmark Results

To exemplify and operationalize the HED framework, the paper introduces PARD-SSM (Probabilistic Anomaly and Regime Detection via Switching State-Space Models), integrating fractional Stochastic Differential Equations (fSDEs) with a Switching Linear Dynamical System (S-LDS). The fSDE component, leveraging Hurst exponent $H > 1/2$, captures long-range dependency and rich temporal structure, while the S-LDS supports sharp, well-calibrated regime discrimination in posterior space.

Empirical evaluation on the NSL-KDD intrusion detection benchmark establishes the clear superiority of PARD-SSM under the HED metric. A HED Score of $0.0643$ is achieved, more than a 3.8-fold improvement over a Random Forest baseline ($0.0132$), with the difference attaining high statistical significance ($$H[P, \tau ; \lambda] = \frac{1}{T - \tau} \int_{\tau}^T \max(0, P(t) - \bar{P})\, e^{-\lambda (t - \tau)} \, dt$$0 via moving block bootstrap). This result robustly demonstrates the advantage of leveraging temporally acute models in conjunction with an evaluation framework that properly rewards temporal acuity.

Advanced Evaluation: FAR-HED Pareto Frontier

The paper introduces the FAR-HED Pareto frontier, tracing trade-offs between False Alarm Rate (FAR) and HED as a function of the decision threshold. The area between these curves (ABC) constitutes a scalar summary analogous to ROC/AUC but conveys time-sensitive performance. Pareto-dominance framework allows comprehensive, threshold-free model comparison in temporal settings.

Extensions, Implications, and Future Directions

The HED Score is designed as a model-agnostic framework, compatible with any probabilistic stream yielding calibrated posteriors. The paper outlines several avenues for further research:

• HED-Aware Loss Functions: Replacing the hard threshold in the HED integrand with a smooth surrogate enables direct optimization of lead-time-sensitive objectives during model training.
• Adaptive Decay Scheduling: Adapting $$H[P, \tau ; \lambda] = \frac{1}{T - \tau} \int_{\tau}^T \max(0, P(t) - \bar{P})\, e^{-\lambda (t - \tau)} \, dt$$1 over time (e.g., in response to changing system latency budgets) is theoretically sound as long as the adaptation remains $$H[P, \tau ; \lambda] = \frac{1}{T - \tau} \int_{\tau}^T \max(0, P(t) - \bar{P})\, e^{-\lambda (t - \tau)} \, dt$$2-measurable.
• Multi-Regime Generalization: For systems with multiple latent regimes, the HED can be generalized to a matrix quantifying pairwise transition detection acuity.

Practical adoption of the HED Score implies a paradigm shift in benchmarking and deployment of early warning systems in domains where detection latency incurs nonlinear costs—examples include cyber-physical security, pandemic onset, and high-frequency trading environments. By formalizing the time-value of information, the HED Score sets a new standard for the evaluation of models in non-stationary, real-time detection contexts, with direct implications for both academic benchmarking and operational deployment.

Conclusion

The HED Score provides a principled, axiomatic, and practically calibrated standard for quantifying temporal intelligence in early warning systems operating on non-stationary processes. By resolving the epistemic limitations of temporally agnostic metrics and enabling robust evaluation across diverse application domains, the HED framework advances the scientific rigor and practical utility of detection system evaluation. Its integration with advanced state-space models such as PARD-SSM demonstrates significant empirical benefit in temporal lead-time, and the outlined methodological extensions offer fertile ground for theoretical and applied advancement in next-generation temporal detection architectures.