ConformalNaive+: A Training-Free Forecasting Baseline

• The paper introduces ConformalNaive+ as a training-free baseline that adaptively switches between last-value and seasonal forecasts based on horizon-specific error comparisons.
• It computes symmetric split-conformal prediction intervals using order-statistic residual quantiles without any model training, ensuring robust coverage.
• ConformalNaive+ sets a benchmark for time-series forecasting by highlighting the strength of simple, static methods in both one-step and multi-step seasonal regimes.

Searching arXiv for the specified topic and closely related conformal forecasting papers. ConformalNaive+ is a training-free conformal forecasting baseline for probabilistic time-series prediction that augments the basic ConformalNaive interval with a horizon-dependent selector between two complementary naive floors: a last-value floor and a seasonal-last-value floor. It was introduced as a “one-line, training-free, horizon-adaptive selector that attains the better of two complementary floors at every horizon with restored coverage,” and was motivated by the observation that a trivial split-conformal interval around a last-value forecast is already a strong benchmark in one-step-ahead forecasting, while seasonal horizons require a different floor (Manokhin, 8 Jun 2026).

1. Definition and positioning

ConformalNaive+ is defined in a benchmarking context rather than as a learned forecasting model. Its purpose is to provide a minimal conformal floor against which learned probabilistic forecasters should be compared. The underlying paper argues that a last-value point forecast wrapped in a split-conformal residual quantile is substantially stronger than its omission from many forecasting comparisons would suggest, and that the multi-horizon case requires a selector that switches between random-walk and seasonal baselines by horizon (Manokhin, 8 Jun 2026).

The method inherits its two branches from ConformalNaive and ConformalSeasonalNaive. ConformalNaive uses the last observed value as forecast for every horizon, whereas ConformalSeasonalNaive uses the season-ago value. ConformalNaive+ does not train a model to choose between them; it applies a fixed error-comparison rule computed from the history. The selector is horizon-adaptive rather than per-window oracle-adaptive, which the paper identifies as a deliberate restriction: it remains training-free and simple, but does not attain the best-of-two oracle envelope on each test window (Manokhin, 8 Jun 2026).

A common point of confusion is terminological. ConformalNaive+ is a specific time-series forecasting baseline, not a generic name for any naive conformal method with an adjustment layer. Other papers discuss conceptually related “naive-plus” ideas—such as Jackknife+-rescaled local score adaptation (Deutschmann et al., 2023), trainable score rectification for better conditional coverage (Plassier et al., 22 Feb 2025), or density-based conformalisation of conditional normalising flows for joint regions (English et al., 2024)—but these are distinct constructions.

2. Base constructions and conformal intervals

The two floors used by ConformalNaive+ are fully specified by their point forecasts and residual definitions. ConformalNaive uses

$\mu_h = y_T,$

with absolute one-step differences

$s_t = |y_t - y_{t-1}|.$

ConformalSeasonalNaive uses

$\mu_h = y_{T+h-m},$

with seasonal residuals

$s_t = |y_t - y_{t-m}|.$

In both cases, the conformal interval is a symmetric split-conformal band around the chosen point forecast (Manokhin, 8 Jun 2026).

The split-conformal quantile is

$$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$

with $Q_\alpha = +\infty$ if $k>n$. The resulting prediction interval is

$[\,\mu_h - Q_\alpha,\ \mu_h + Q_\alpha\,].$

The paper emphasizes that this is a distribution-free split-conformal construction rather than the classical Gaussian naive interval. In particular, it does not use the parametric formula

$$\text{point} \pm z_{1-\alpha/2}\,\hat\sigma\sqrt{k+1},$$

and it does not widen with horizon via $\sqrt{k+1}$. The paper identifies this as a central reason the random-walk floor can become too narrow and undercovered at long horizons (Manokhin, 8 Jun 2026).

Component	Definition	Role
ConformalNaive	$s_t = |y_t - y_{t-1}|.$0	Last-value floor
ConformalSeasonalNaive	$s_t = |y_t - y_{t-1}|.$1	Seasonal floor
Split-conformal interval	$s_t = |y_t - y_{t-1}|.$2	Finite-sample conformal band under exchangeability

The significance of these definitions lies in their minimality. No model is trained, no probabilistic likelihood is fit, and no parametric forecast error law is assumed. The baseline is therefore intended as a floor: if a learned method cannot improve on it, the empirical value of that method is called into question (Manokhin, 8 Jun 2026).

3. Horizon-adaptive selector

ConformalNaive+ adds a horizon-dependent branch-selection rule over the two floors. The algorithm takes as inputs the history $s_t = |y_t - y_{t-1}|.$3, horizon $s_t = |y_t - y_{t-1}|.$4, seasonal period $s_t = |y_t - y_{t-1}|.$5, and miscoverage level $s_t = |y_t - y_{t-1}|.$6. It first computes a seasonal-lag error summary

$s_t = |y_t - y_{t-1}|.$7

which the paper describes as flat in $s_t = |y_t - y_{t-1}|.$8. It then loops over horizons $s_t = |y_t - y_{t-1}|.$9 and computes the horizon-$\mu_h = y_{T+h-m},$0 random-walk persistence error

$\mu_h = y_{T+h-m},$1

which is described as growing with $\mu_h = y_{T+h-m},$2 (Manokhin, 8 Jun 2026).

The branch rule is explicit. If $\mu_h = y_{T+h-m},$3, the method chooses ConformalNaive. Otherwise it chooses ConformalSeasonalNaive. For each horizon, it then emits the chosen branch’s point forecast $\mu_h = y_{T+h-m},$4, split-conformal interval $\mu_h = y_{T+h-m},$5, and samples, if needed. The output is therefore horizon-specific, but the selector is not learned and is computed from the training history only (Manokhin, 8 Jun 2026).

This horizon-adaptive construction is the defining “plus” modification. The paper motivates it by the complementarity of the two floors: the random-walk or last-value floor is best at short horizons, whereas the seasonal floor is better at longer seasonal horizons. A plausible implication is that ConformalNaive+ is best understood not as a new conformal scoring rule, but as a deterministic router over two already-conformalized forecast floors.

4. Statistical framing and coverage

The conformal machinery used by ConformalNaive+ is standard split conformal, but its deployment in time series requires qualification. The paper states that exact split-conformal finite-sample coverage is guaranteed only under exchangeability, which is violated in time series. It therefore does not claim a strict unconditional theorem for the reported time-series experiments, and instead treats coverage empirically. At the same time, it notes that near-nominal coverage is consistent with recent theory for weakly dependent, near-stationary series (Manokhin, 8 Jun 2026).

This framing distinguishes ConformalNaive+ from several neighboring lines of conformal research. In adaptive conformal regression with Jackknife+ rescaled scores, the core problem is local heteroscedasticity, and the method rescales a standard score by an estimate of local score distribution while preserving global validity through a Jackknife+ construction (Deutschmann et al., 2023). In rectified split conformal prediction, the method performs a trainable transformation of an arbitrary conformity score using an estimated conditional quantile to improve approximate conditional coverage while preserving exact marginal coverage (Plassier et al., 22 Feb 2025). In conformalised conditional normalising flows, the conformity score is the conditional density of the true future path under a conditional normalising flow, producing joint prediction regions that can be non-rectangular and disjoint in multi-step forecasting (English et al., 2024).

ConformalNaive+ does none of these things. It does not rectify or rescale a score using learned conditional structure, and it does not estimate a full conditional density. Its statistical role is instead austere: a distribution-free split-conformal band wrapped around two trivial point forecasts, plus a horizon rule that decides which trivial forecast to use.

5. Empirical regime dependence

In one-step-ahead online forecasting, the paper evaluates ConformalNaive across 2,217 real series from nine public sources, including Monash, LOTSA, the LTSF traffic/electricity/weather suites, METR-LA, BOOM, and nips/probts, with a 300-step test span and training length $\mu_h = y_{T+h-m},$6. Using Winkler score as the metric, ConformalNaive beats NaiveInterval on 90% of series, SeasonalNaiveInterval on 92%, NPTS_strong on 73%, SeasonalNPTS on 64%, and CSP on 71%, with bootstrap 95% CI $\mu_h = y_{T+h-m},$7 and paired Wilcoxon $\mu_h = y_{T+h-m},$8 (Manokhin, 8 Jun 2026).

Coverage in that one-step regime is also emphasized. ConformalNaive has mean empirical coverage $\mu_h = y_{T+h-m},$9 at nominal $s_t = |y_t - y_{t-m}|.$0, compared with SPCI $s_t = |y_t - y_{t-m}|.$1, ACI $s_t = |y_t - y_{t-m}|.$2, CQR $s_t = |y_t - y_{t-m}|.$3, and CSP $s_t = |y_t - y_{t-m}|.$4. The paper therefore characterizes it as not merely sharp but also well-calibrated, while noting that adaptive-online methods such as SPCI, ACI, and AgACI still beat it on Winkler and lead by 9–33% relative Winkler because they track distribution shift (Manokhin, 8 Jun 2026).

The multi-step seasonal regime reverses the picture. On the six GluonTS datasets on which DeepNPTS was introduced—electricity, exchange_rate, solar_energy, taxi, traffic, and wikipedia—using rolling-origin evaluation, $s_t = |y_t - y_{t-m}|.$5 for hourly series or $s_t = |y_t - y_{t-m}|.$6 for daily, $s_t = |y_t - y_{t-m}|.$7 samples, $s_t = |y_t - y_{t-m}|.$8, seed 0, and 380 forecast records per method, CSP wins, the random-walk floor ConformalNaive is last, and ConformalNaive+ together with ConformalSeasonalNaive beats NPTS and roughly ties the audited DeepNPTS rerun. DeepNPTS covers only $s_t = |y_t - y_{t-m}|.$9 at nominal $$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$0, whereas the trivial conformal floors cover $$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$1–$$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$2. For ConformalNaive+ specifically, it beats NPTS on 64% of windows with $$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$3, edges the matched DeepNPTS rerun on 51% with $$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$4, and still loses to CSP (Manokhin, 8 Jun 2026).

A horizon sweep on four hourly datasets supplies the direct empirical motivation for the “+” variant. Sweeping $$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$5, the two floors cross at about $$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$6–3; ConformalNaive is best at $$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$7, then deteriorates as the last value becomes stale, with coverage collapsing from about $$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$8 to $$Q_\alpha = s_{(k)}, \qquad k = \big\lceil (n+1)(1-\alpha) \big\rceil,$$9. ConformalSeasonalNaive remains near nominal across the horizon, and ConformalNaive+ tracks the better floor at each horizon, restoring coverage to about $Q_\alpha = +\infty$0–$Q_\alpha = +\infty$1 (Manokhin, 8 Jun 2026).

6. Relations, distinctions, and baseline status

ConformalNaive+ is explicitly distinguished from several other baselines and conformal methods. It is not a naive value-quantile baseline such as NaiveInterval, NaiveDiffInterval, or SeasonalNaiveInterval; those are classical intervals that rely on Gaussian or residual assumptions and often widen with horizon, whereas ConformalNaive uses an order-statistic residual quantile from absolute residuals (Manokhin, 8 Jun 2026).

It is also not NPTS, SeasonalNPTS, or CSP. NPTS and SeasonalNPTS are nonparametric predictive samplers based on empirical histories and recency or seasonal weighting. CSP mixes a seasonal empirical pool with seasonal residuals. ConformalNaive+ instead switches the point forecast by horizon between the random-walk and seasonal floors, and the paper explicitly states that it is not CSP. The empirical division of labor is correspondingly sharp: ConformalNaive dominates CSP at one step, CSP wins at seasonal multi-step horizons, and ConformalNaive+ functions as the horizon-adaptive bridge between those regimes (Manokhin, 8 Jun 2026).

The method is likewise distinct from learned conformal predictors such as RCI, CQR, EnbPI, EnsCQR, KOWCPI, QR, QEns, NexCP, ACI, AgACI, SPCI, and WeightedConformal. The paper’s conceptual distinction is that ConformalNaive is a training-free floor; learned conformal predictors typically rely on ridge, quantile-regression, ensemble, or online adaptation mechanisms; and ACI, AgACI, and SPCI explicitly update the quantile to handle distribution shift. The paper’s central interpretive claim is therefore about benchmark discipline: simple static conformal floors are already difficult to beat, and only methods that explicitly adapt to shift consistently do better (Manokhin, 8 Jun 2026).

This baseline status should not be conflated with more expressive conformal research. Conformalised conditional normalising flows replace residual scores with conditional density scores and target joint, potentially disjoint forecast regions in multivariate multi-step problems (English et al., 2024). Rectified conformity-score methods learn a transformation that improves approximate conditional coverage while retaining exact marginal coverage (Plassier et al., 22 Feb 2025). Jackknife+-rescaled adaptive conformal regression uses local score rescaling to address non-homogeneous coverage without sacrificing calibration set size (Deutschmann et al., 2023). ConformalNaive+ remains important precisely because it omits these modeling layers and still attains competitive empirical performance in the regimes identified by its defining paper.

The broader implication suggested by the available evidence is methodological rather than architectural. ConformalNaive+ serves as a lower bound on what can be achieved with no training, no probabilistic model fitting, and only a horizon-adaptive switch between two naive forecast anchors. In that sense, its significance lies in evaluation standards: at short horizons the relevant floor is ConformalNaive, at seasonal multi-step horizons the relevant floor is ConformalSeasonalNaive, and when the regime is mixed or unclear, ConformalNaive+ is the training-free selector that should also be reported (Manokhin, 8 Jun 2026).