Cost per Correct Detection Analysis

• Cost per correct detection analysis is the quantitative evaluation of costs incurred by detection systems to achieve true positive outcomes through embedded cost matrices.
• It integrates explicit cost frameworks into detection algorithms to balance false positives against false negatives, ensuring optimized operational performance.
• Empirical findings in areas like intrusion and tuberculosis detection demonstrate that cost-sensitive methods can significantly reduce overall system expenses.

Cost per correct detection analysis refers to the quantitative evaluation of the operational or financial expense incurred by a detection system to achieve successful (true positive) outcomes. This measure is critical in applications where different error types (false positives and false negatives) lead to heterogeneous and highly asymmetric costs—common in fields such as intrusion detection, screening systems, and critical decision-making domains. Cost-sensitive frameworks seek to directly minimize the expected or average cost per detection by integrating explicit cost matrices, trade-off ratios, or operational constraints into both the detection algorithms and their empirical evaluation.

1. Fundamentals of Cost-Sensitive Detection

The cost per correct detection is formalized by embedding a cost matrix into the classification or detection process, allowing the model to prioritize minimizing the most consequential errors. In intrusion detection, for instance, the cost matrix C(i, j) quantifies the penalty of predicting class i when the true class is j, leading to an expected cost of:

$L(x, i) = \sum_{j} P(Y = j | X = x) \, C(i, j)$

where $L(x, i)$ is the expected cost of selecting label $i$ for observation $x$. The classifier then selects the decision $f(x) = \arg\min_i L(x, i)$ that minimizes expected cost, as opposed to maximizing raw classification probability. This cost-aware approach generalizes across detection tasks, including feature selection (where the benefit-cost ratio is applied) and deep learning-based detectors, where the loss function itself is reweighted to reflect cost asymmetry.

2. Formulations and Evaluation Metrics

Detection systems are commonly evaluated using both traditional performance metrics and those tailored for cost analysis. The expected cost over a dataset D is given by:

$$E[C] = \frac{1}{|D|} \sum_{(x, y) \in D} C(f(x), y)$$

while cost-sensitive metrics such as Detection Cost Function (DCF) in screening tasks are expressed as:

$$C_{\mathrm{DCF}}(t) = C_\mathrm{miss} \, P_\mathrm{tar} \, P_\mathrm{miss}(t) + C_\mathrm{fa} \, (1 - P_\mathrm{tar}) \, P_\mathrm{fa}(t)$$

Here $P_\mathrm{miss}$ and $P_\mathrm{fa}$ are the miss and false alarm rates; $C_\mathrm{miss}$ and $C_\mathrm{fa}$ are their costs, and $P_\mathrm{tar}$ is the target prior. For outlier evaluation of models, cost per correct detection (CCD) is computed as total system cost divided by the number of true positive detections:

$$CCD = \frac{\mathrm{Total \; Cost}}{\# \; \mathrm{Correct \; Detections}}$$

Advanced object detection frameworks introduce the Optimal Correction Cost (OC-cost), formulating the problem as an optimal transport between detections and ground-truth objects, with correction costs defined for mislocalization, misclassification, false positives, and false negatives (Otani et al., 2022).

3. Model Designs and Algorithmic Strategies

Cost minimization may be achieved by integrating cost structures into the inference algorithms. In classification-based detection, thresholds are directly tuned to minimize total cost rather than maximize accuracy. In sequential and online detection, cost constraints shape experiment design: the two-threshold quickest change detection framework (Banerjee et al., 2011) employs separate thresholds to independently control false alarm probability and average number of observations, minimizing resource use while maintaining detection speed.

In domains with multiple diagnostic actions (experiments), the 2E-CUSUM algorithm (Lubenia et al., 17 Sep 2025) and its extensions swap between high-quality, high-cost and low-quality, low-cost experiments to meet a pre-specified cost constraint (e.g., limiting the proportion of time that costly experiments are performed):

Parameter	Meaning	Control Aspect
$A$ (upper thr.)	CUSUM stop decision threshold	False alarm probability
$N_X$, $a_Y$	X-experiment limit, scaling factor	Cost constraint (POR_Y)

This design yields detection delays asymptotically matching those of fully informed procedures, but with cost strictly bounded.

In feature selection under cost, the benefit-cost ratio (BCR) selects features by maximizing performance gain per unit cost. However, BCR can overstress low-cost noise features if costs are highly skewed or hyperparameters are not tuned (Jagdhuber et al., 2020).

4. Empirical Findings and Application-Specific Analysis

Empirical studies consistently demonstrate that cost-sensitive frameworks reduce the cost per correct detection compared to cost-agnostic alternatives, albeit with varying effect size depending on the model class and data regime:

• In intrusion detection, linear and Gaussian Mixture Model classifiers with embedded cost matrices exhibit marked reductions in expected cost, especially when the cost of false negatives is set higher (0807.2043). The false alarm rate increases only marginally as costs are shifted, highlighting efficient trade-offs.
• In contour (edge) detection, cost-sensitive loss or sampling increases the cost-weighted accuracy, especially for rare edge pixels (Hwang et al., 2014).
• CAD4TB v6 (tuberculosis detection) at 90% sensitivity achieves a specificity of 76% and reduces screening cost to \$5.95 per subject (from \$8.38 in v3), with cost per TB case detected \$38.75—showing that improved specificity directly reduces per-detection costs (Murphy et al., 2019).
• In machine-learning-based malware detection, fast, high-precision detectors paired with ML models detect more zero-day threats, reducing the average monetary loss per malware event compared to signature-only scanning (Bridges et al., 2020).
• In object detection, cost-aware evaluation (controlling for inference latency or resource use) exposes that performance gains accredited to architectural novelty may, in fact, stem from increased compute budgets. Scaling simpler models (e.g., SECOND with high BEV resolution) matches the detection accuracy of more complex models at the same cost (Wang et al., 2022).
• Zero-shot Vision-LLMs (e.g., Gemini Flash 2.5) attain a substantially lower CCD (\$0.00050 vs \$0.143 for supervised YOLO) at operationally relevant inference scales (e.g., 100,000 inferences), though with lower accuracy. The break-even point for supervised systems (where total cost matches and surpasses per-inference cost of a VLM) is only justified at extremely high inference volumes, such as 55 million images (Al-Hamadani, 13 Oct 2025).

5. Pitfalls, Mitigations, and Design Guidance

Empirical and simulation-based works identify several risks:

• In benefit-cost ratio–based feature selection, excessive scaling can cause selection mechanisms to favor very cheap features that are statistically noise, missing true (but costlier) predictive features. Mitigations include avoiding near-zero costs, rescaling costs, or introducing hyperparameterized cost–performance trade-offs (Jagdhuber et al., 2020).
• Cost-sensitive thresholds in object detection must be carefully selected to avoid cases where the benefit from discarding false positives does not outweigh the possible loss of correct detections. Algebraic conditions derived for threshold admissibility ensure that increases in missed detections do not outpace the cost savings from reducing false detections (Sbeyti et al., 26 Apr 2024).
• In evaluation, using metrics like OC-cost provides more stable and human-aligned detector rankings and penalizes mislocalization and misclassification more fairly than global measures like mAP (Otani et al., 2022).

Recommendations include modular allocation of thresholds for false alarms versus observation cost (or operational budgets), explicit control of model selection via cost-aware metrics (e.g., Cscore (Marwah et al., 19 Jul 2024)), and always reporting both classical and cost-sensitive performance metrics for transparency.

6. Future Perspectives and Open Questions

Future research is expected to further integrate real-world operational constraints into both the design and evaluation of detection systems. Directions noted in the literature include:

• Generalization to scenarios with more complex and shifting cost structures, such as category-dependent misclassification costs or evolving deployment domains, requiring adaptive or meta-learned cost models (Al-Hamadani, 13 Oct 2025).
• Extension of cost-aware evaluation methods (e.g., OC-cost, PCR (Yoo et al., 16 Aug 2025)) to account for annotation uncertainty and dynamically changing detection landscapes.
• Development of automated, hybrid, and resource-adjustable detection systems that couple robust (but costly) methods with lightweight, cost-efficient prefilters or decision cascades, as demonstrated in two-tiered agentic approaches for large multimodal agents in phishing detection (Trad et al., 3 Dec 2024).
• Broader adoption of simulation-based, data-efficient threshold calibration strategies in sequential and streaming change detection (Cobb et al., 2021), reducing both statistical and computational costs per detection event.

Overall, the cost per correct detection analysis underscores the necessity of aligning technical advances in detection accuracy with deployment-scale economic constraints, ensuring that resource allocation and model choice remain commensurate with operational requirements and risk.