SCD-split: Smoothing in Conformal Prediction

• SCD-split is a smoothing-enhanced variant of split conformal prediction that integrates Fourier-based smoothing to merge fragmented prediction intervals while maintaining valid coverage.
• It applies a Gaussian low-pass filter to the conditional density estimator, reducing high-frequency noise and combining multiple subintervals into clearer, connected prediction bands.
• The method preserves rigorous statistical guarantees and offers user control over interval targeting, as demonstrated by both synthetic experiments and real-world applications.

SCD-split is a smoothing-enhanced variant of split conformal prediction methods that produces statistically valid, efficiently sized, and interpretable prediction sets in regression problems. Its key innovation is the integration of Fourier-based smoothing operations into the conformal prediction pipeline, which merges small disconnected intervals generated by CD-split, thereby yielding prediction bands that are easier to interpret without sacrificing coverage or interval efficiency (Zheng et al., 26 Sep 2025).

1. Motivation and Overview

Standard split conformal methods such as CD-split generate prediction sets that control marginal coverage under minimal assumptions but can result in multiple disconnected subintervals—particularly when using conditional density estimators in settings with multimodal or irregular conditional distributions. While such fragmented prediction sets offer statistical guarantees and can be more efficient than intervals, they may hinder interpretability in practical applications. SCD-split introduces a smoothing operation applied to the conditional density estimator, which reduces high-frequency structure and consequently merges fragmented intervals, aligning the prediction region closer to practitioners’ preference for fewer distinct intervals.

2. Smoothing Operations

SCD-split’s main technical step is spectral smoothing of the estimated conditional density. Given $\hat{f}(y \mid x)$, the method computes its Fourier transform: $$\mathcal{F}_y[\hat{f}](w\mid x) = \int_{-\infty}^{\infty} \hat{f}(y\mid x)\, e^{-2\pi i y w}\,dy$$ A Gaussian low-pass filter is then applied: $H_\sigma(w) = e^{-2\pi^2 \sigma^2 w^2}$ where $\sigma$ controls the degree of smoothing. The filtered density is recovered by inverse Fourier transform: $$\tilde{f}^{\mathrm{FS}}(y\mid x) = \int_{-\infty}^{\infty} \mathcal{F}_y[\hat{f}](w\mid x) H_\sigma(w) e^{2\pi i y w} dw$$ High-frequency peaks and oscillations causing disconnected intervals are attenuated, yielding a smoother density with broader modes and fewer spurious peaks.

3. Construction of Prediction Sets

SCD-split then follows split conformal prediction principles, applied to the smoothed density. On a partition cell $a$ of covariate space, the threshold is computed using the calibration set: $$t^S_\sigma(a) = \operatorname{Quantile}\left(\alpha;\left\{\tilde{f}^{\mathrm{FS}}(Y_i \mid X_i): X_i \in a \right\}\right)$$ For a query point $X_{n+1}$, the conformal prediction set is: $$\mathcal{C}^S_{1-\alpha}(X_{n+1}) = \left\{ y : \tilde{f}^{\mathrm{FS}}(y\mid X_{n+1}) \ge t^S \right\}$$ This procedure maintains exchangeability and the theoretical foundations of conformal prediction.

4. Theoretical Properties

SCD-split retains marginal coverage guarantees (with exactness under exchangeable calibration) as established in Theorem 4.1 (Zheng et al., 26 Sep 2025). Under minimal regularity, SCD-split provably does not increase the number of disconnected intervals compared to CD-split (Theorem 4.2), and strictly reduces them under specific shape conditions (“narrow valley” scenarios, Theorem 4.3). The length of the prediction region, a proxy for efficiency, is not substantially increased and is upper-bounded by a controlled function of the smoothing parameter (Theorem 4.4).

A user may set the target number of intervals $K_{\mathrm{target}}$, and tuning $\sigma$ adapts the tradeoff between sharpness/efficiency and interpretability (connectedness of the region). Thus, SCD-split provides direct control over a key aspect of practical use not addressed by CD-split.

5. Empirical Performance

Results on synthetic and real-world datasets verify the theoretical advantages:

• On multimodal synthetic regression problems, SCD-split reduces the average number of intervals markedly (e.g., from 2.85 with CD-split to 1.99 with SCD-split in a “complex” synthetic setting).
• The interval length of coverage sets is maintained at levels similar to those of CD-split, sometimes even lower due to the removal of noisy peaks from unsmoothed densities.
• On datasets such as Bike Sharing and Bio, the average number of intervals aligns closely to the user-specified target, with preserved coverage and competitive or shorter interval length.

6. Comparison with CD-split and Other Methods

CD-split produces regions that asymptotically approach the HPD sets, optimizing length but yielding potentially many subintervals. SCD-split achieves comparable efficiency and valid coverage, but its smoothing step merges intervals for enhanced interpretability. The smoothing is performed on the density estimator before computing conformity scores, preserving the symmetry needed for conformal validity.

Unlike alternative methods based solely on regression residuals or quantiles, SCD-split maintains the full nonparametric adaptability to complex conditional distributions while ensuring interpretable output. The user-controlled interval number feature distinguishes SCD-split from CD-split and quantile-based conformal approaches, especially for applications where concise region communication is critical.

7. Practical Implications and Applications

SCD-split is particularly advantageous where interpretability is essential. In clinical and financial domains, prediction sets consisting of a small number of intervals are easier to communicate and actionable. The smoothing parameter $\sigma$ provides a mechanism to tune the tradeoff between efficiency and interpretability based on application needs.

The Fourier smoothing embedding is general and can be integrated into any conformal prediction procedure relying on density estimation. As such, SCD-split increases the suite of distribution-free predictive tools for practitioners dealing with uncertainty quantification in complex regression settings.

Summary Table

Method	Coverage Guarantee	Avg. Number of Intervals	User Control	Interval Length
CD-split	Marginal	Higher	No	Efficient
SCD-split	Marginal	Lower (user-targeted)	Yes	Comparable

The SCD-split methodology exemplifies how spectral smoothing of nonparametric density estimators, combined with the rigorous framework of split conformal prediction, achieves a balance between statistical efficiency and interpretability in the construction of prediction sets (Zheng et al., 26 Sep 2025).