Report the Floor: A Training-Free Conformal Interval Is a Mandatory Baseline for Probabilistic Time-Series Forecasting

Abstract: Probabilistic forecasters are increasingly learned, yet the baselines they are compared against are often weak or omitted. We show that the simplest possible conformal interval - a last-value point forecast wrapped in a finite-sample split-conformal residual quantile, with no parameters and no training - is a far stronger baseline than its near-total absence from recent learned-forecasting and conformal-time-series comparisons would suggest. In one-step-ahead online forecasting across 2,217 real series from nine public sources (Monash, LOTSA, the LTSF traffic/electricity/weather suites, METR-LA, BOOM, nips/probts), this ConformalNaive interval decisively beats the naive value-quantile baselines, the entire NPTS family (NPTS 73%, SeasonalNPTS 64% of series), and the published Conformal Seasonal Pools (CSP) method (71% of series, bootstrap 95% CI [69,73], paired Wilcoxon p approx 7.6e-135); it is on par with the simpler learned conformal predictors (RCI, quantile regression; median relative Winkler within 2%) and is beaten only by the adaptive-online and ensemble methods (SPCI, ACI, AgACI), which track distribution shift and lead by 9-33% relative Winkler. It is also better calibrated than a trained neural forecaster: on the six datasets that introduced DeepNPTS, the trivial floors cover the truth 84-85% of the time at a nominal 95%, versus DeepNPTS's 66%. At multi-step seasonal horizons the picture inverts: the random-walk floor is the weakest method and the seasonal pool (CSP) wins - a boundary we map. Finally we give ConformalNaive+, a one-line, training-free, horizon-adaptive selector that attains the better of two complementary floors at every horizon with restored coverage. We argue the matching conformal naive floor must be a mandatory baseline whenever a learned probabilistic forecaster claims gains.

• The paper establishes that the ConformalNaive interval—a training-free method using last-value forecasts with split-conformal residual quantiles—serves as a mandatory baseline.
• It demonstrates that, in one-step forecasting, ConformalNaive outperforms 71% of competing methods while matching learned approaches without any training overhead.
• The evaluation reveals that while ConformalNaive excels in short-horizon settings, its performance drops in multi-step forecasts, highlighting the need for adaptive methods.

Mandatory Reporting of the ConformalNaive Interval as a Baseline in Probabilistic Time-Series Forecasting

Introduction and Motivation

The proliferation of learned probabilistic forecasters in time-series analysis has not been matched by rigorous benchmarking protocols, particularly concerning baseline methods. This paper systematically establishes that the most elementary conformal interval—comprised simply of a last-value point forecast augmented by a finite-sample split-conformal residual quantile, without any model parameters or training—serves as a much more stringent and appropriate baseline for probabilistic time-series forecasting than what is commonly reported. The analysis encompasses 2,217 real-world series from diverse public datasets and rigorously pits this "ConformalNaive" interval against a wide suite of classical, conformal, and neural probabilistic forecasters.

Methodological Framework

The core construction evaluated is the ConformalNaive interval: a prediction interval centered on the last observed value, with its width determined by the empirical quantile of past absolute one-step forecast errors. Formally:

• Point forecast: $\mu_h = y_t$
• Interval width: $Q_\alpha$ is the $(1-\alpha)$ split-conformal quantile of $\{|y_t - y_{t-1}|\}$
• No parameter tuning or model training involved

Variants include the ConformalSeasonalNaive (which uses the value from one seasonal lag as the point forecast) and ConformalNaive+, a horizon-adaptive hybrid that dynamically selects between the two, based on in-sample median error comparisons.

Protocols addressed both one-step-ahead online forecasting and multi-step seasonal batch forecasting, adhering to strict evaluation regimes.

Experimental Analysis and Results

One-Step-Ahead Online Forecasting

On the large-scale corpus, ConformalNaive demonstrates superior performance relative to all naive value-quantile baselines, all nonparametric probabilistic time-series (NPTS) variants, and the published Conformal Seasonal Pools (CSP) method. Specifically:

• ConformalNaive beats CSP on 71% of series and NPTS (strong) on 73%.
• Against 19 competitive baselines, only adaptive and ensemble conformal predictors (e.g., SPCI, AgACI, ACI) outperform ConformalNaive, by margins of 9–33% in median relative Winkler score.
• Crucially, it matches the best static learned conformal approaches and does so with no training overhead.
• Empirical coverage closely tracks the nominal 0.95 target (mean 0.91), exceeding that achieved by adaptive competitors (SPCI 0.89, CQR 0.86).

Multi-Step Seasonal Forecasting

Here the value proposition of ConformalNaive recedes:

• Its intervals remain fixed-width regardless of the forecast horizon, causing severe undercoverage and sharp loss in CRPS performance, particularly mid-season.
• In this setup, CSP and related seasonal methods (SeasonalNPTS) distinctly outperform the naive floor.
• Nevertheless, calibration of ConformalNaive and ConformalSeasonalNaive remains superior to neural forecasters, with DeepNPTS showing only 66% empirical coverage versus 84–85% for the trivial conformal floors.

Horizon-Adaptive Selection: ConformalNaive+

ConformalNaive+ adaptively routes each horizon to the better of the two primitive floors. This simple decision rule restores nominal coverage and tracks the stronger baseline as a function of horizon, while remaining parameter-free and training-free.

Implications and Theoretical Insights

The evidence clearly indicates that the ConformalNaive floor is not an artifact of benchmark composition. Within the canonical short-horizon regime, this method is decisively competitive with or surpasses nearly all static and many learned alternatives, diverging only when distribution shift is adaptively tracked or explicit sequence modeling confers an advantage. The coverage properties of this split-conformal scheme are robust, consistent with recent theoretical analyses establishing approximate split-conformal validity for time series under weak dependence (Barber et al., 2 Oct 2025, Rangapuram et al., 2023). The acute fall-off in multi-step accuracy and coverage illustrates a precise operational boundary for baseline sufficiency.

From a practical standpoint, this mandates that any claimed gains from a learned (especially a neural or adaptive) probabilistic forecaster must be validated against the ConformalNaive (and for multi-step regimes, the seasonal variant or a horizon-adaptive selection). Reporting a naive quantile or an NPTS method as the strong baseline is empirically indefensible; many published methods underperform relative to the true conformal floor and their alleged advantage may be an artifact of baseline omission.

Limitations

This study focuses on univariate, stationary or weakly non-stationary settings. It does not encompass exogenous/covariate forecasting, multivariate or hierarchical structures, cold-start inference, intermittent demand, or non-seasonal long-horizon protocols. Baseline superiority may not generalize to these regimes.

Future Directions

Several directions emerge. First, online or window-adaptive selectors using short-term performance metrics could narrow the gap to the oracle envelope in horizon adaptation. Second, variants handling skew/heteroskedastic error distributions, as well as extensions to distributional or Bayesian classical statistical baselines, are natural avenues. Finally, thorough calibration reporting at nominal coverage levels should become standard in the field, to guard against gains driven by overconfident (and under-covered) neural or combined methods.

Conclusion

A training-free, parameter-free last-value-plus-conformal residual interval is a stringent, replicable floor for probabilistic time-series forecasting benchmarks, outperforming widely reported baselines and matching most static conformal and learned approaches at short horizons. Only adaptively shift-tracking or ensemble conformal methods reliably exceed this floor, and then only at a quantifiable margin. Coverage of the ground truth is more reliable than for neural benchmarks even within their home regime. As such, any claim of improvement in probabilistic forecasting must be measured against this mandatory baseline, with empirical coverage reported as a core diagnostic (2606.09473).