Fractionally Supervised Classification with Maxima Nominated Samples

Abstract: Fractionally supervised classification (FSC) offers a flexible framework for combining labeled and unlabeled data in model-based classification, but existing formulations assume simple random sampling. In many applications, however, the retained observation is an extreme order statistic from a set rather than a randomly selected unit. This is particularly appealing when the target population is rare, since maxima nomination sampling (NS) can enrich the sample with the most informative observations, as in screening, environmental monitoring, repeated testing, and reliability studies. Under such designs, the likelihood function changes fundamentally, and the usual FSC EM construction is no longer valid. We develop FSC for nominated samples by introducing a latent representation that accounts for both the class membership of the observed maximum and the latent composition of the remaining units in the set. The resulting method yields a proper EM algorithm and a coherent weighted-likelihood FSC procedure for NS data. We present the methodology in general form, illustrate it for a rare-event contamination normal mixtures, and show through simulation that it substantially improves on the misspecified alternative by ignoring the extra rank information of such data. A real-data analysis demonstrates its practical value.

• The paper introduces a novel FSC approach using latent-pair augmentation to accurately model nomination sampling bias in mixture models.
• It demonstrates that incorporating NS data yields stable parameter estimates and enhanced classification metrics such as ARI and F1 scores.
• Simulation and real data studies reveal that ignoring the NS design results in significant bias and deteriorated classification performance.

Fractionally Supervised Classification with Maxima Nominated Samples

Introduction and Problem Formulation

This work establishes a rigorous framework for fractionally supervised classification (FSC) in settings where unlabeled samples are obtained via maxima nomination sampling (NS), rather than simple random sampling (SRS). Classical FSC leverages a weighted likelihood to integrate labeled and unlabeled data, interpolating between supervised, semi-supervised, and unsupervised paradigms. However, standard FSC implicitly assumes SRS. In numerous applications—screening for rare diseases, environmental monitoring, reliability studies, and financial tail risk analysis—data are ascertained through selection of extreme (e.g., maximal) responses in small sets to enrich the sample with the rare class. This nomination mechanism alters the likelihood structure and invalidates direct application of the standard EM-based FSC.

The authors introduce a new latent-variable augmentation for FSC under NS, explicitly modeling both the generating process for observed maxima and the unmeasured members of each set. The resulting EM algorithm accounts for the induced sample selection bias, enabling proper estimation and classification for mixture models under nomination-enriched designs.

Methodology

Likelihood under Nomination Sampling

In NS, each observed data point is the maximum of a latent, i.i.d. set of size $k$. For a two-component mixture model with density $f(x) = \pi f_1(x) + (1-\pi) f_2(x)$, the NS likelihood for an observed maximum $x$ is:

$g_k(x) = k\, f(x)\, F(x)^{k-1}$

where $F(x)$ denotes the mixture CDF. Thus, the observed maxima are biased toward the upper tail, concentrating the rare class if its corresponding density dominates the right tail.

Latent Variables for the EM Algorithm

The standard EM augmentation (single binary indicator for mixture component membership) is shown to be improper for NS data. It fails to recover the true observed likelihood; the marginalization yields a sum-of-powers in $\pi$ rather than a power-of-sums required by $g_k(x)$. The paper proposes a correct latent structure consisting of a pair $(Z_i, V_i)$ for each nominated sample:

• $Z_i$: component membership indicator for the observed maximum,
• $V_i$: count of the $f(x) = \pi f_1(x) + (1-\pi) f_2(x)$0 unmeasured set members from component 1.

This latent pair enables the complete-data likelihood to marginalize back to the observed NS likelihood. The E-step computes the expected values $f(x) = \pi f_1(x) + (1-\pi) f_2(x)$1 and $f(x) = \pi f_1(x) + (1-\pi) f_2(x)$2, reflecting the density and cumulative distribution contributions of component 1, respectively.

Figure 1: Log-likelihood comparison for the correct NS likelihood (solid blue) and the misspecified SRS-based objective (dashed red) as a function of $f(x) = \pi f_1(x) + (1-\pi) f_2(x)$3.

Figure 2: Normalized log-likelihood surfaces for $f(x) = \pi f_1(x) + (1-\pi) f_2(x)$4: correct NS objective (blue) versus misspecified SRS (red), demonstrating bias and surface distortion.

This construction restores the proper connection between the EM procedure and likelihood maximization under NS.

FSC-NS Weighted Likelihood

With $f(x) = \pi f_1(x) + (1-\pi) f_2(x)$5 in place, the FSC-NS weighted likelihood for labeled and unlabeled (nominated) samples is:

$f(x) = \pi f_1(x) + (1-\pi) f_2(x)$6

where each $f(x) = \pi f_1(x) + (1-\pi) f_2(x)$7 is the appropriate NS density for that group/component. The M-step update for the mixture proportion, for instance, is

$f(x) = \pi f_1(x) + (1-\pi) f_2(x)$8

significantly, the denominator accounts for the total number of latent draws across all sets.

For parametric mixtures (e.g., normal mixtures and rare-event contamination mixtures), the method yields closed-form or efficiently computable M-steps. The EM implementation is straightforward, using latent-pair-based updates for all sufficient statistics.

Simulation Studies

Rare Event Mixture and Enrichment

Simulation experiments focus on a rare-event mixture model, where the rare class is well-separated:

$f(x) = \pi f_1(x) + (1-\pi) f_2(x)$9

with known background parameters and $x$0 small. NS enriches the rare component—for example, set sizes $x$1 amplify the effective prevalence of the rare class by several fold.

Key performance metrics include ARI for clustering, rare-class $x$2, sensitivity, specificity, balanced accuracy, and AUC.

Figure 3: Average ARI versus NS weight $x$3 for varying $x$4, comparing FSC-NS and SRS-based FSC. FSC-NS remains stable, SRS-based FSC rapidly degrades for larger $x$5.

Figure 4: Rare-class $x$6 score heatmaps for FSC-NS and SRS, as functions of $x$7. The misspecified SRS likelihood leads to catastrophic failures in the rare-small and moderately separated regime.

Figure 5: Average ARI versus $x$8 for different ranking qualities ($x$9), showing robustness of FSC-NS and rapid drop in SRS-based FSC with increased NS enrichment.

Figure 6: Average rare-class $g_k(x) = k\, f(x)\, F(x)^{k-1}$0 versus $g_k(x) = k\, f(x)\, F(x)^{k-1}$1: FSC-NS is invariant to ranking quality, whereas SRS degrades as $g_k(x) = k\, f(x)\, F(x)^{k-1}$2 increases.

Figure 7: Average RMSE of $g_k(x) = k\, f(x)\, F(x)^{k-1}$3 versus $g_k(x) = k\, f(x)\, F(x)^{k-1}$4. FSC-NS maintains low RMSE across all settings, while SRS inflates error as $g_k(x) = k\, f(x)\, F(x)^{k-1}$5 grows, especially for accurate ranking.

Key empirical findings:

• When the sampling design is ignored (i.e., using standard SRS FSC), adding weight to unlabeled nominated data actively damages parameter estimation and classification, resulting in degenerate solutions that may assign essentially all samples to the rare class (see Table 1 in the paper).
• Correct modeling (FSC-NS) yields well-calibrated estimates and stable, high ARI and $g_k(x) = k\, f(x)\, F(x)^{k-1}$6 scores across a wide range of settings, robust to both sample enrichment and imperfect ranking.
• The cost of ignoring the NS design increases with the efficacy of the ranker or set size: as the nomination mechanism more strongly enriches the rare class, bias and classification failure in the SRS approach intensify.

Real Data Illustration

A univariate FSC-NS analysis is performed on the Wisconsin Diagnostic Breast Cancer dataset, with artificial NS structure imposed using the largest nuclear radius as the ranking variable. The results mirror the simulation findings:

• FSC-NS improves both clustering (ARI) and classification (error, balanced accuracy, AUC) for moderate $g_k(x) = k\, f(x)\, F(x)^{k-1}$7 over the purely supervised baseline, especially with larger $g_k(x) = k\, f(x)\, F(x)^{k-1}$8.
• The estimated mixing proportions appropriately reflect enrichment; FSC-NS corrects for the over-representation of malignant cases in maxima-nominated samples whereas SRS-based FSC under- or overfits, depending on the weight.
• Excess weight on unlabeled data ($g_k(x) = k\, f(x)\, F(x)^{k-1}$9) in both methods leads to performance collapse, but the onset is much sharper for SRS-based FSC.

Theoretical and Practical Implications

This work demonstrates that integrating data selection design in statistical inference is mandatory for partially labeled learning when the unlabeled sample is nomination-enriched. The standard latent-indicator EM augmentation is structurally incompatible, and its use leads to systematic estimation bias, degenerate classifiers, and instability.

From a practical perspective:

• Properly leveraging NS data allows rare class detection with higher statistical efficiency and accuracy.
• Overweighting the unlabeled nominated sample without design correction should be explicitly avoided.
• The methodology is robust to reasonable ranking errors, allowing its deployment in field conditions where perfect ranking is infeasible.

From a theoretical standpoint:

• The FSC-NS likelihood and EM structure demonstrate a novel decomposability in mixture models with ranked set/enrichment designs.
• The latent pair $F(x)$0 paradigm generalizes readily to multi-component mixtures and variable set sizes.

Future research directions include asymptotic analysis of the weighted estimators, extension to high-dimensional and complex data types, rigorous model selection strategies for $F(x)$1, and adaptation to hierarchical and dependent sampling designs.

Conclusion

This paper provides a complete and technically sound foundation for FSC with maxima nominated samples, resolving longstanding methodological gaps for model-based classification in sample-enrichment regimes. By characterizing and curing the structural incompatibility of the standard EM augmentation under NS, it enables efficient and reliable inference for rare event and tail-focused applications and offers immediate practical guidelines for its adoption in applied statistics and biomedical diagnostics.