Early Rejection Mechanism

• Early Rejection Mechanism is a decision-theoretic approach that defers predictions when confidence is low, balancing the trade-off between timely decisions and accuracy.
• It uses an ensemble of classifiers with a reject option to assess sequential data, leveraging consensus to trigger early deferral of uncertain predictions.
• Applied in systems like the Forefront-Nose for odor detection, it demonstrates high accuracy with limited data by minimizing misclassification risks.

An early rejection mechanism is a decision-theoretic approach that allows a model or system to abstain from making a prediction or take corrective action as soon as a sufficiently low confidence, insufficient consensus, or adversarial/novel evidence is detected, often prior to full data or complete computation. In time-series classification, this mechanism is crucial for online settings where decisions must be made rapidly and accurately based on partial or streaming data. It is particularly relevant in applications such as biosignal processing, fault detection, and real-time monitoring, where delaying the decision for more evidence must be carefully balanced against the risks of premature or incorrect classification.

1. Classifier Architecture with Reject Option

The mechanism is implemented within a classifier architecture denoted as "Classifier With a Reject Option" (CWRO), structured as a serial ensemble of experts. Each expert/classifier operates on an increasing prefix of the input time series. For each incoming data patch, the expert can output a label or a special reject symbol (⊘), triggering deferral to the next expert that waits for more of the time series. Unlike approaches that rely on a single classifier thresholded by posterior probability, CWRO leverages ensemble agreement ("ensemble consensus") to determine whether to accept or reject an early label.

The experts themselves are chosen for both high individual accuracy and diversity, quantified by the double fault (DF) measure,

$\mathrm{DF}_{(i,j)} = \frac{\#~\text{of samples both $i$and$j$ misclassify}}{\text{total number of samples}}$

Ensemble members with low DF are selected to maximize joint reliability while ensuring statistical independence.

2. Operational Principle of Early Rejection

The CWRO operates online, ingesting the time-series segment by segment. At each stage, the ensemble's discriminant function $f$ computes a candidate output. Rather than directly thresholding a posterior probability $P(\omega_i|x)$, the system estimates confidence from the ensemble's agreement. If ensemble agreement is insufficient—i.e., consensus among classifiers does not exceed a threshold—the model issues a rejection (⊘) and defers the decision. Formally, a label $f(x)\in\{1,\dots,N_c\}$ is issued if $|f(x)|>\tau$ (with $\tau\in[0,1)$), else $f(x) = \bigcirc$ (reject). The process continues with the appended time-series prefix until consensus is achieved or the maximum signal length is reached.

The rejection decision is not only a matter of calibrated confidence, but also an explicit risk-tradeoff: the cost function incorporates both misclassification and the cost $d > 0$ of rejection itself:

$$E[l(Y, f(X))] = P\{\text{misclassification}\} + d\,P\{\text{rejection}\}$$

with the loss function

$$l(z) = \begin{cases} 1 & \text{if}\ z\ \text{is incorrect}\ d & \text{if}\ z\ \text{is a rejection}\ 0 & \text{otherwise} \end{cases}$$

3. Application: Odor/Gas Signal Time-Series and the Forefront-Nose

The early rejection mechanism is operationalized in the "Forefront-Nose"—an Electronic-Nose system for odor and gas discrimination. The system comprises an array of 8 chemo-resistive metal-oxide sensors sampling the environment in real time. Rapid and reliable gas identification is essential for environmental safety, industrial process control, and confined-habitat monitoring. In practical deployment, processing the entire gas-sensor time-series offline is often infeasible due to the requirement for low-latency recognition. The CWRO enables the device to issue reliable (accept/reject) decisions using only initial signal segments, with the option to defer for additional evidence in ambiguous cases.

4. Experimental Performance and Trade-Offs

Extensive experiments conducted on wind-tunnel gas-plume data illustrate the efficacy of the early rejection mechanism. Under controlled but highly dynamic and spatially varying test conditions, the Forefront-Nose outperformed standard classifiers on both earliness and recognition accuracy. Notably, using only the first 10 seconds of data yielded mean accuracy rates up to approximately 94%, whereas competing methods (without reject option) exhibited both lower accuracy and increased error volatility as earliness increased. The system demonstrates robust performance in the presence of sensor noise and environmental variability, substantiating the claim that ensemble- and consensus-based early rejection confers superior resilience to uncertainty.

A core performance metric is the accuracy conditioned on accepted classifications,

$$\text{Accuracy} = \frac{P(\text{correct})}{P(\text{correct}) + P(\text{error})}$$

The experimental results consistently show that early rejection leads to a favorable trade-off: it maintains high conditional accuracy at the cost of occasionally deferring difficult decisions to later stages.

5. Theoretical Formulations

The mathematical underpinnings of the mechanism include:

• Bayes Decision with Reject Option: Given posterior $P(\omega_i|x)$, output a label $i$ if $P(\omega_i|x) > d$, and otherwise reject:

$$f_o(x) = \begin{cases} i & P(\omega_i|x) > d\ 0 & P(\omega_i|x) \leq d \end{cases}$$

• Double Fault Diversity: For classifiers $i$ and $j$,

$$\mathrm{DF}_{i,j} = \frac{\#(\text{samples misclassified by both } i, j)}{\text{total samples}}$$

This is essential for optimizing ensemble selection toward robust consensus.

• Cost-Minimized Thresholds: The reject threshold $\tau$ and risk weighting $d$ are analytically chosen using empirical risk to optimize the earliness-accuracy trade-off and to calibrate the cost/benefit of early rejection.

6. Implications, Generalization, and Future Directions

The early rejection mechanism enables real-time, reliable classification on incomplete time-series, prioritizing system reliability over forced early prediction. In the context of dynamic, safety-critical applications, such as chemical hazard monitoring, the explicit modeling of rejection avoids costly or dangerous misclassifications. The adoption of ensemble consensus (rather than single-classifier probabilistic thresholds) affords additional robustness, especially under nonstationary noise and drift.

Future research directions outlined in the paper include (a) source localization (combining label prediction with origin estimation using the same sequential rejection principle); (b) generalization to varying sensor topologies and deployment geometries, with the prospect of location-invariant recognition; and (c) exploration of alternative diversity metrics and online ensemble adaptation schemes. Broadly, there is scope to extend early rejection to other real-time multimodal sensor fusion and streaming classification tasks.

7. Summary Table: Core Mechanism Elements

Mechanism Element	Implementation	Mathematical Expression
Ensemble agreement	Serial experts, diversity-optimized	$\mathrm{DF}_{i,j}$; $f(x)$
Reject threshold	Empirical CDF/statistics	$|f(x)|>\tau$, else ⊘
Cost-risk trade-off	Explicit cost of misclassification and rejection	$E[l(Y,f(X))]$
Conditional accuracy	Acceptance-based metric	$P(\text{correct}|~\text{accepted})$

In conclusion, early rejection mechanisms as formalized in CWRO architectures allow for dynamic and verifiable decision-making from partial observations, avoiding premature or unreliable predictions in time-sensitive, data-streaming environments. The mathematical formulations, empirical validation, and proposed generalizations provide a foundation for application in a broad class of online time-series classification problems.