Conformal Seasonal Pools (CSP) Forecasting

• CSP is a training-free probabilistic forecaster that combines same-season empirical draws with signed residuals to preserve realistic support and correct seasonal bias.
• It constructs an empirical predictive sample matrix by drawing from seasonal pools and a calibrated residual pool, achieving improved accuracy and coverage over learned methods.
• Empirical benchmarks show that both CSP-Adaptive and CSP-Fixed outperform DeepNPTS in CRPS, empirical coverage, and runtime efficiency in multi-step seasonal forecasting.

Searching arXiv for the cited CSP papers and closely related conformal forecasting work. Conformal Seasonal Pools (CSP) is a training-free probabilistic time-series forecaster that mixes same-season empirical draws with signed residual draws around a seasonal naive forecast, producing an empirical predictive sample matrix $S \in \mathbb{R}^{H \times B}$ without learned parameters or model training (Manokhin, 5 May 2026). In subsequent comparative work, CSP is treated as a published seasonal-pool conformal sampler that uses a seasonal empirical pool with seasonal residuals; that work identifies CSP as the stronger method in the multi-step seasonal regime, while also showing that it is not the appropriate default baseline for one-step-ahead online forecasting (Manokhin, 8 Jun 2026).

1. Construction and methodological rationale

CSP was proposed against the NPTS family, especially DeepNPTS, to test whether a training-free sampler built from empirical seasonal history and conformal-style residuals could outperform a learned non-parametric forecaster on both accuracy and calibration (Manokhin, 5 May 2026). The motivating critique is threefold: DeepNPTS requires training, is much slower when retrained across rolling-origin windows, and can be badly miscalibrated even when quantile-style losses appear competitive. The paper places this within a calibration-first view of probabilistic forecasting, arguing that low loss without reliable interval coverage is insufficient.

The CSP construction has two stochastic components for each forecast horizon $h$. The first is a same-season empirical pool, intended to preserve realistic support by sampling observations observed at the same seasonal phase as the target horizon. The second is a signed residual pool around a seasonal naive forecast, intended to recenter samples around the most recent seasonal level while preserving directional asymmetry in forecast error. This division of labor is explicit in the method description: seasonal empirical draws address support realism, whereas signed residual draws correct level and seasonal bias.

The seasonal naive anchor is

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

with a fallback to the nearest same-phase value if out of range. The seasonal empirical pool for horizon $h$ is

$$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$

with weights proportional to

$\propto \exp\bigl(-\lambda\,(T-t)\bigr).$

The signed residual calibration pool is

$$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$

where

$$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$

In the audited run, $\rho=0.5$. The predictive object is a sample matrix

$S \in \mathbb{R}^{H \times B},$

with $h$0 the forecast horizon and $h$1 the sample budget.

2. Mixture rule and CSP variants

For each horizon $h$2 and each sample $h$3, CSP draws a mixture indicator $h$4. If $h$5, the sample is drawn from the weighted seasonal pool $h$6. Otherwise CSP draws a residual $h$7 from $h$8 and forms

$h$9

The resulting predictive distribution is therefore a mixture between seasonal empirical sampling and residual-centered seasonal-naive sampling (Manokhin, 5 May 2026).

The paper defines two variants, CSP-Fixed and CSP-Adaptive. Their distinction lies entirely in the horizon-wise mixture weight $\mu_h = y_{T+h-m},$0.

Variant	Rule for $\mu_h = y_{T+h-m},$1	Interpretation
CSP-Fixed	$\mu_h = y_{T+h-m},$2 for all $\mu_h = y_{T+h-m},$3	Constant 50/50 mixture
CSP-Adaptive	$\mu_h = y_{T+h-m},$4 if $\mu_h = y_{T+h-m},$5	No seasonality available
CSP-Adaptive	$\mu_h = y_{T+h-m},$6 if $\mu_h = y_{T+h-m},$7	Seasonal pool too thin
CSP-Adaptive	$\mu_h = y_{T+h-m},$8 otherwise	Default seasonal mixing

At the values used in the audited benchmark, $\mu_h = y_{T+h-m},$9. The paper explicitly notes that at least half of every CSP draw comes from the conformally motivated residual pool except in the degenerate seasonal settings. This design makes CSP horizon-adaptive in a restricted sense—through the thinning-sensitive mixture weight—without introducing learned parameters or optimization.

3. Conformal interpretation, predictive outputs, and benchmark protocol

The “conformal” designation in CSP refers mainly to the residual construction and its split-conformal motivation rather than to a formal finite-sample validity theorem for the full mixture (Manokhin, 5 May 2026). The signed residual pool is described as inspired by inductive or split conformal prediction, but the paper is explicit that because time series are dependent, exchangeability is violated. Accordingly, the method does not claim a formal finite-sample coverage guarantee for CSP or for the residual-only sub-sampler. Coverage is instead treated as an empirical property to be measured.

CSP returns predictive samples rather than an analytically parameterized distribution. Point forecasts, quantiles, prediction intervals, and scoring-rule evaluations are derived directly from the empirical sample matrix $h$0. The interval construction used operationally is therefore the empirical interval induced by the sampled predictive distribution. This is a salient methodological distinction from residual-quantile conformal intervals that are written in closed form.

The principal benchmark is an audited rolling-origin evaluation on the six datasets originally used for DeepNPTS: electricity, exchange_rate, solar_energy, taxi, traffic, and wikipedia. The protocol uses 380 forecast records per method, the same windows and same seeds across methods, pairwise matched comparisons, and $h$1–$h$2 rolling-origin windows per series depending on dataset. The forecast horizon is $h$3 for hourly datasets and $h$4 for daily datasets. The compared methods are CSP-Adaptive, CSP-Fixed, AdaptiveWindowMCI, NPTS, SeasonalNPTS, and DeepNPTS.

4. Empirical performance against DeepNPTS and related baselines

On the six-dataset audited benchmark, CSP-Adaptive significantly outperforms DeepNPTS on every reported metric: CRPS, normalized mean quantile loss, and empirical $h$5 coverage (Manokhin, 5 May 2026). The reported paired Wilcoxon $h$6-values for CSP-Adaptive versus DeepNPTS are $h$7 for CRPS, $h$8 for normalized mean quantile loss, and $h$9 for coverage. CSP-Fixed is described as essentially the same.

The aggregate rank tables place CSP ahead of the learned non-parametric alternative. Across the six datasets, CSP-Adaptive attains mean CRPS rank $$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$0 and mean MQL rank $$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$1; CSP-Fixed attains CRPS rank $$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$2 and MQL rank $$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$3; DeepNPTS attains mean CRPS rank $$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$4 and mean MQL rank $$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$5. SeasonalNPTS is the strongest non-CSP comparator in the rank tables, with CRPS rank $$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$6.

Method	Mean CRPS rank	Mean empirical $$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$7 coverage
CSP-Adaptive	3.03	0.89
CSP-Fixed	3.09	0.89
DeepNPTS	3.60	0.66

The rank distribution is also asymmetric. CSP-Adaptive lands in ranks $$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$8–$$\mathcal S_h \leftarrow \bigl\{\, y_t \;:\; t \le T,\; t \equiv T+h \pmod m \,\bigr\},$$9 on about $\propto \exp\bigl(-\lambda\,(T-t)\bigr).$0 of CRPS windows, corresponding to $\propto \exp\bigl(-\lambda\,(T-t)\bigr).$1. DeepNPTS lands in the bottom band, ranks $\propto \exp\bigl(-\lambda\,(T-t)\bigr).$2–$\propto \exp\bigl(-\lambda\,(T-t)\bigr).$3, on $\propto \exp\bigl(-\lambda\,(T-t)\bigr).$4 of windows. In calibration, the contrast is sharper: mean empirical $\propto \exp\bigl(-\lambda\,(T-t)\bigr).$5 coverage is $\propto \exp\bigl(-\lambda\,(T-t)\bigr).$6 for CSP-Adaptive and CSP-Fixed, versus $\propto \exp\bigl(-\lambda\,(T-t)\bigr).$7 for DeepNPTS at a nominal target of $\propto \exp\bigl(-\lambda\,(T-t)\bigr).$8. The paper further reports DeepNPTS median per-window coverage $\propto \exp\bigl(-\lambda\,(T-t)\bigr).$9, lower decile coverage $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$0, and coverage standard deviation $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$1.

A particularly severe failure mode is reported for DeepNPTS. In the worst $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$2 of forecast windows, its $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$3 interval covers none of the forecast horizons $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$4 simultaneously. The paper treats this as more severe than aggregate undercoverage alone, because the entire multi-step trajectory misses the truth at every step. The associated discussion links this behavior to safety- and decision-critical applications such as healthcare, finance, energy operations, and autonomous systems.

Runtime is another major differentiator. CSP-Adaptive requires $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$5 minutes, whereas DeepNPTS requires $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$6 minutes, a $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$7 slowdown for DeepNPTS on CPU. The paper adds that the five non-DeepNPTS methods together take only $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$8 minutes, while DeepNPTS alone accounts for $$\mathcal R \leftarrow \bigl\{\, y_t - y_{t-m} \;:\; T - T_{\mathrm{cal}} < t \le T,\; t > m \,\bigr\},$$9 of total method-evaluation time. Against other baselines, CSP decisively beats NPTS on CRPS with $$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$0; against SeasonalNPTS, the CRPS difference is not significant at $$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$1 ($$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$2), but coverage remains significantly better ($$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$3).

5. Regime dependence and relation to conformal floors

A later comparison paper situates CSP within a broader regime map for training-free conformal forecasting (Manokhin, 8 Jun 2026). That paper contrasts two explicit “floors,” each based on a naive point forecast wrapped in a split-conformal residual-quantile interval: $$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$4 with $$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$5, and interval

$$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$6

For the seasonal floor,

$$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$7

Within that framing, ConformalNaive uses last-value or random-walk point forecasts and one-step absolute differences, while ConformalSeasonalNaive uses last-seasonal-value point forecasts and seasonal residuals.

CSP is explicitly distinguished from both. It is described as a published seasonal-pool conformal sampler that uses a seasonal empirical pool with seasonal residuals. It does not simply switch point forecasts by horizon in the manner of ConformalNaive+. Instead, it mixes seasonal information into the prediction construction itself. The paper therefore treats ConformalNaive+ as a selector between two explicit floors, whereas CSP is a seasonal pooling method with a different construction.

The comparative results are strongly regime-dependent. In one-step-ahead online forecasting across 2,217 real series from nine public sources, ConformalNaive beats CSP on $$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$8 of series, with bootstrap $$T_{\mathrm{cal}} \leftarrow \lfloor \rho\, T \rfloor.$$9 confidence interval $\rho=0.5$0 and paired Wilcoxon $\rho=0.5$1. The full-table value is a $\rho=0.5$2 win rate for ConformalNaive, with median relative Winkler improvement of ConformalNaive versus CSP equal to $\rho=0.5$3. In this one-step regime, CSP is therefore not the right default baseline.

At multi-step seasonal horizons, the picture reverses. On the six GluonTS datasets used in DeepNPTS, under a rolling-origin protocol with $\rho=0.5$4 hourly or $\rho=0.5$5 daily and CRPS as the main metric, the reported mean CRPS ranks are: CSP-Adaptive $\rho=0.5$6, CSP-Fixed $\rho=0.5$7, SeasonalNPTS $\rho=0.5$8, DeepNPTS $\rho=0.5$9, ConformalNaive+ $S \in \mathbb{R}^{H \times B},$0, ConformalSeasonalNaive $S \in \mathbb{R}^{H \times B},$1, NPTS $S \in \mathbb{R}^{H \times B},$2, and ConformalNaive $S \in \mathbb{R}^{H \times B},$3. Empirical $S \in \mathbb{R}^{H \times B},$4 coverage is $S \in \mathbb{R}^{H \times B},$5 for both CSP-Adaptive and CSP-Fixed, compared with $S \in \mathbb{R}^{H \times B},$6 for ConformalNaive+, $S \in \mathbb{R}^{H \times B},$7 for ConformalSeasonalNaive, $S \in \mathbb{R}^{H \times B},$8 for ConformalNaive, and $S \in \mathbb{R}^{H \times B},$9 for DeepNPTS. The paper’s horizon sweep places the crossover around $h$00: at short horizons ConformalNaive is best, while as the horizon grows and the last observation becomes stale, the seasonal method and CSP remain competitive. This suggests that CSP is properly understood as a specialized seasonal probabilistic forecaster rather than as a universal baseline.

6. Terminological scope and acronym ambiguity

The acronym “CSP” is heavily overloaded in the arXiv literature. In constraint programming, it denotes the Constraint Satisfaction Problem, with work on treewidth, backdoors, and fixed-parameter tractability using notions such as backdoor-treewidth (Ganian et al., 2016). In formal methods, it appears in tock-CSP and in process-algebraic work on encoding CSP into CCS, where “CSP” refers to Communicating Sequential Processes and its timed variants (Ribeiro et al., 2019, Hatzel et al., 2015). In energy-systems planning, “CSP” denotes Concentrating Solar Power, treated as a seasonal peak-regulation resource in high-renewable capacity expansion models (Li et al., 2023).

Within probabilistic time-series forecasting, however, CSP specifically denotes Conformal Seasonal Pools (Manokhin, 5 May 2026). This distinction matters because the forecasting usage is methodologically tied to training-free conformal-style sampling, seasonal empirical pooling, empirical coverage analysis, CRPS and normalized mean quantile loss evaluation, and rolling-origin benchmark design. Confusion with the unrelated expansions of “CSP” would obscure both the method’s construction and its regime-specific empirical role.