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Rollcast: Proper-Score Gated Rolling Anchors for Adaptive Probabilistic Time-Series Forecasting

Giancarlo Vercellino

arXiv:2609.05561v1stat.MLcs.LGstat.ME

TL;DR

Forecast combination is often built from separately estimated models, while Rollcast treats rolling summaries as local forecast hypotheses. It combines them using state-dependent proper-score gating and residual behavior retrieved from similar earlier states. In the synthetic benchmark, coverage falls below nominal levels on average, and local-trend and Markov-switching dynamics show the largest efficiency losses.

  • Problem

    Forecast combination is often organized around separately estimated forecasting models, whereas rolling summaries offer a different basis for local forecast hypotheses.

    Source: §1 · p. 2

  • Method

    Rollcast combines rolling statistical anchors using state-dependent proper-score gating, residual distributions from similar earlier states, adaptive persistence, and recursive resampling.

    Source: §3 · p. 3, §5 · p. 4, §14 · p. 11

  • Results

    Mean coverage is 0.862 for nominal 90% intervals and 0.915 for nominal 95% intervals; local trend and Markov-switching dynamics produce the largest efficiency losses.

    Source: §14 · p. 11

  • Takeaways & Limitations

    The conditional residual archive is closer to the oracle for stochastic volatility and heavy-tailed autoregression, while local trend and latent regimes are more challenging.

    Source: §13 · p. 10

    The eight fixed DGPs are descriptive stress scenarios, and the experiment does not establish superiority over fitted forecasting alternatives.

    Source: §13 · p. 10

Abstract

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Rollcast is a probabilistic forecasting method for univariate time series that combines a compact set of rolling statistical anchors rather than relying on a single global model. Rolling means, medians, extrema, regression endpoints, and quantiles define candidate forecast locations and a representation of the current state. A state-dependent softmax gate learns anchor probabilities by minimizing negative log predictive density, while residual distributions retrieved from similar historical states provide local uncertainty. Recursive simulation propagates the resulting mixture over multiple forecast horizons. The method is evaluated in a Monte Carlo study covering eight data-generating processes, including autoregressive, random-walk, local-trend, threshold, regime-switching, stochastic-volatility, heavy-tailed, and variance-break dynamics. Across 2,000 independent fitted series, Rollcast is compared with the true conditional predictive distribution generated by an oracle simulator. Overall empirical coverage is 86.2% for nominal 90% intervals and 91.5% for nominal 95% intervals. Predictive intervals are on average 13.6% wider than the oracle at the 90% level, while CRPS is 14.4% higher than oracle CRPS. Performance is closest to the oracle under autoregressive, threshold, stochastic-volatility, heavy-tailed, and variance-break dynamics, while local trends and regime switching are more challenging. The results indicate that Rollcast can construct competitive probabilistic forecasts from simple, interpretable local summaries, while also identifying limitations in calibration and recursive uncertainty propagation.

1 Introduction

Rollcast treats rolling summaries as interpretable local forecast hypotheses rather than separately estimated forecasting models. It combines those anchors using information from similar historical states and represents their configuration relative to the latest level and a robust local scale.

1 Introduction

  • Rolling means, medians, extrema, quantiles, and regression endpoints serve as interpretable forecast anchors whose relevance can vary with the observed state.The anchors are not expected to be individually correct; they provide a compact set of local hypotheses.Source: §1 · p. 2
  • Similar historical states provide anchor-specific residual distributions, which are combined with a state-dependent gate trained using a proper score.This differs from direct analog forecasting, which predicts from neighbors’ future trajectories.Source: §1 · p. 2

2 Forecast anchors and state representation

  • The state represents anchor configuration relative to the latest level and a robust local scale, with an optional standardized raw-anchor formulation.The reference scale uses median absolute deviation multiplied by 1.4826, with other scale measures and a numerical floor as fallbacks.Source: §1 · p. 2, §2 · p. 3

1 Introduction

  • Rollcast organizes proper-score pooling, nearest-state retrieval, residual smoothing, and dynamic mixture weights around a compact dictionary of rolling anchors.Anchor geometry supplies the gating state, and tuning uses common causal validation origins.Source: §1 · p. 2

2 Forecast anchors and state representation

  • The anchor dictionary includes rolling-window functionals, including a one-step regression endpoint; the median is not duplicated among user-specified quantiles.The endpoint is used instead of the mean fitted regression value, which equals the sample mean when the regression has an intercept.Source: §2 · p. 2
  • Soft responsibilities replace exclusive assignment to the ex post closest anchor, with τ controlling how sharply responsibility concentrates on the nearest anchor.The state representation also includes standardized anchor errors.Source: §2 · p. 3

3 Causal state-conditional component distributions

Rollcast turns deterministic anchors into predictive components by retrieving residual behavior from similar earlier states. The resulting component separates an anchor’s location from its local residual dispersion.

3 Causal state-conditional component distributions

  • For each target row, only earlier rows are eligible, and the K nearest standardized states provide historical residual behavior for anchor-specific predictive components.Neighbor states are ranked by root-mean-square Euclidean distance.Source: §3 · p. 3
  • Combined neighbor and anchor-responsibility weights determine which historical residual centers inform each component.Responsibilities are slightly smoothed to avoid zero support.Source: §3 · p. 3
  • A validation-only numerical floor keeps logarithmic scores finite but is not injected into simulations; with γ = 0, the forecast is a discrete mixture over anchors.The residual-density formulation is evaluated on the residual scale.Source: §3 · p. 3
  • Forecast simulation samples a historical residual center using the combined weights, then adds Gaussian kernel jitter around the selected anchor.This separates where an anchor places predictive mass from how much local residual dispersion is admitted.Source: §3 · p. 4

4 Proper-score gating

A multinomial logistic gate assigns state-dependent probabilities to forecast anchors. It is trained against realized mixture densities rather than ex post best-anchor labels.

4 Proper-score gating

  • The gate is a multinomial logistic model on standardized states, with one component fixed as the zero-reference coefficient vector for identifiability.Its softmax probabilities assign state-dependent weight to each anchor.Source: §4 · p. 4
  • The gate minimizes penalized negative log predictive density, rewarding probability assigned by the mixture density to realized observations rather than reproducing hard best-anchor labels.The objective aligns weight learning with the logarithmic proper score.Source: §4 · p. 4
  • The reference implementation optimizes the regularized objective numerically with BFGS.The practical distinction is that the gate is rewarded for improving realized-observation density, not for matching a hard label.Source: §4 · p. 4

5 State-adaptive persistence

Rollcast adapts gate persistence to standardized state movement, allowing the mixture to retain more weight on its previous value when the state changes little and respond faster when it moves more.

5 State-adaptive persistence

  • A raw gate may change too quickly for nearly identical states or too slowly under uniform persistence, motivating inertia based on standardized state movement.The persistence rule constrains the stabilized gate’s response to state changes.Source: §5 · p. 4
  • When the standardized state barely changes, persistence stays near ρ_max and gives the previous mixture more weight; greater movement shifts it toward ρ_min.The transition is smooth, and d_ρ controls how rapidly it occurs.Source: §5 · p. 4

6 Recursive multi-step predictive mixtures

Rollcast constructs multi-step forecasts by recursively combining anchor-based candidates across simulated paths. It aggregates their weights into horizon-specific predictive distributions and resamples candidates to propagate the paths.

6 Recursive multi-step predictive mixtures

  • Later-horizon particles carry their own recursively simulated windows, previous standardized states, and stabilized gates.Each path begins from the observed final rolling window at horizon one.Source: §6 · p. 5
  • At each horizon, Rollcast forms a weighted mixture of parent–anchor candidate levels to define the predictive distribution.The mixture contains BM candidates, with weights based on stabilized anchor probabilities.Source: §6 · p. 5
  • Predictive means, variances, medians, and quantiles are computed from mixture weights before resampling.Source: §6 · p. 5
  • Anchor-stratified systematic resampling allocates offspring approximately by anchor mass, reducing loss of low-probability components from Monte Carlo variation.The method first allocates anchor offspring counts, then selects particles within each anchor systematically.Source: §6 · p. 5

7 Algorithm

The algorithm builds rolling anchors and a causal archive of local residual information, fits a proper-score gate, then recursively aggregates and propagates anchor-based candidates to produce forecasts.

7 Algorithm

  • The procedure takes a univariate series, rolling-window and anchor settings, fitting and retrieval parameters, horizon, simulation count, and resampling rule as inputs.It returns horizon-specific predictive distributions and quantiles, simulated paths, anchor-probability summaries, and diagnostics.Source: §7 · p. 6
  • At each observable forecast origin, Rollcast computes rolling anchors, a robust scale, relative state, standardized anchor residuals, and soft responsibilities.Source: §7 · p. 6
  • The causal residual archive uses only earlier nearby states and combines state similarity with anchor responsibility to form conditional residual distributions.State standardization uses training-period information.Source: §7 · p. 6
  • A softmax gate is fit by minimizing penalized negative log predictive density for the complete mixture, with state-dependent inertia preserving more of the prior gate in stable states.Hyperparameter candidates are scored on common causal validation origins.Source: §7 · p. 6
  • For each particle and horizon, Rollcast draws conditional residuals for each anchor, aggregates weighted candidates into a predictive distribution, then resamples and updates particle histories.The updated windows and gate/state histories are carried forward through the forecast horizon.Source: §7 · p. 6

8 Practical statistical properties and validation design

Rollcast’s relative-state formulation preserves its behavior under shifts and positive rescaling of a series. Its fitting and tuning use causal information and common validation origins, after which the selected configuration is refit on available causal-density rows.

8 Practical statistical properties and validation design

  • Shifting and positively rescaling a series leaves relative anchor positions, standardized residuals, responsibilities, neighbor geometry, and gate inputs unchanged, while shifting and rescaling forecasts.Exactly constant windows use a small numerical scale floor.Source: §8 · p. 6
  • Causal fitting and tuning restrict residual information to earlier rows and compare rolling windows on common validation origins after the largest window has sufficient history.This keeps different window lengths from being evaluated on different realized target sets.Source: §8 · p. 6
  • Table 1 presents reference search spaces, with scalar values fixing parameters and vectors triggering staged search.Source: Table 1 · p. 7
  • After hyperparameter selection, the deployment gate is refit on all valid causal-density rows with the selected values held fixed.Source: §8 · p. 7

9 Hyperparameter selection

Hyperparameters can be fixed or searched through staged coordinate search on common validation origins. Rollcast distinguishes itself from related methods by combining rolling statistical anchors with state-dependent gating and locally retrieved residual distributions.

9 Hyperparameter selection

  • A scalar hyperparameter fixes its value, while a vector supplies candidate values for search across parameters including window width, gate, retrieval, residual, and persistence settings.Anchor quantiles are structural definition parameters rather than tuning candidates.Source: §9 · p. 7
  • Cached staged coordinate search tests candidates in a fixed order on common validation origins, replacing a parameter only when held-out negative log predictive density falls.This avoids a full Cartesian grid, whose evaluation count scales with the product of grid sizes rather than approximately the sum per pass; individual evaluations can remain costly.Source: §9 · p. 7

10 Relation to existing methods

  • Unlike combinations of separately fitted models, Rollcast combines statistical functionals of a single rolling empirical window as local forecast anchors.Source: §10 · p. 8
  • Rollcast conditions mixture weights on anchor-derived state variables and estimates component densities from local residual archives, rather than pooling fixed component densities.Source: §10 · p. 8
  • Nearest-state information estimates residual distributions around each rolling anchor rather than forecasting directly from neighbors’ future trajectories.The residual layer uses standardized anchor state, causal archives, responsibility-weighted neighbors, and kernel jitter.Source: §10 · p. 8
  • Rollcast’s gate is linear in standardized anchor geometry, and its persistence depends deterministically on observed state movement rather than a latent transition model.Source: §10 · p. 8

11 Synthetic calibration benchmark

The benchmark tests Rollcast against an oracle across eight fixed synthetic processes and finds undercoverage and higher probabilistic error despite wider intervals. Calibration also weakens with forecast horizon, while adaptive selections and persistence behavior vary across fits.

11.1 Design and benchmark

  • Eight fixed synthetic DGPs were tested over 2,000 fitted series against oracle distributions estimated from 10,000 conditional paths per replication.Each DGP had 250 replications, 300 training observations, and a six-step future; innovations were Gaussian unless otherwise stated, and evaluation was not against practical fitted alternatives.Source: §11.1 · pp. 8–9, Table 2 · p. 9

11.2 Results

  • Across DGP–horizon cells, mean coverage is 0.862 for nominal 90% intervals and 0.915 for 95%; intervals are 13.6% wider at 90%, and normalized CRPS is 14.4% above oracle.Oracle coverage is 0.901 and 0.952; 95% intervals are 15.9% wider, and wider intervals do not remove the coverage gap.Source: Table 3 · p. 9, §11.2 · p. 9
  • From h = 1 to 6, mean 90% coverage falls from 0.898 to 0.840 and 95% coverage from 0.940 to 0.901; local trend and Markov switching have the largest CRPS losses.The other six DGPs range from 1.081 to 1.196, with the smallest gap under stochastic volatility.Source: §11.2 · p. 9
  • γ = 1 is selected in 99.2% of fits, h_e = 0.55 in 95.2%, and W = 30 in 54.8%.All gate optimizations converge; persistence-stabilized training negative log score is lower than the raw-gate score in only 0.65% of fits, so persistence primarily acts as a smoothing regularizer in this experiment.Source: §11.2 · pp. 9–10

12 Computational considerations

Rollcast’s computational burden comes from repeated nearest-state residual retrieval during particle-based recursive forecasting. In the benchmark, calibration gaps and DGP-specific efficiency losses coexist with fixed-scenario and comparison limits.

12 Computational considerations

  • Without neighbor indexing, density construction is approximately quadratic in training states; particle-wise anchor and residual recomputation limits current implementation to moderate-dimensional univariate forecasting.Rcpp keeps only K nearest candidates in a heap rather than sorting every distance.Source: §12 · p. 10

13 Discussion

  • Unlike conventional pools, Rollcast combines rolling functionals rather than separately fitted models, using nearest states for anchor-specific residual densities and proper-score gating.Source: §13 · p. 10
  • Nominal 90% and 95% intervals cover 86.2% and 91.5% on average, with the calibration gap increasing by horizon despite wider-than-oracle intervals.The recursive distribution can still place probability mass in the wrong location or shape.Source: §13 · p. 10
  • Local trend produces intervals 65.9% wider than the oracle and a CRPS ratio of 1.461; Markov switching has a CRPS ratio of 1.287, while stochastic volatility and heavy-tailed autoregression reach 1.081 and 1.088.Source: §13 · p. 10
  • Eight fixed DGPs make cross-DGP averages descriptive; the true-model oracle is an efficiency reference, and no superiority over fitted alternatives is established.Source: §13 · p. 10
  • Horizon-aware residual calibration, richer state features, specialized anchors, and distributional-score tuning are suggested responses to the observed calibration and tracking weaknesses.Source: §13 · p. 11

14 Conclusion

Rollcast combines rolling statistical anchors through adaptive probabilistic forecasting components. Across 2,000 synthetic fits, it remains relatively close to the true-DGP oracle for several mechanisms, but calibration deteriorates with horizon and local-trend and Markov-switching dynamics incur the largest efficiency losses.

14 Conclusion

  • Rollcast treats rolling statistical summaries as forecast anchors, combining them through proper-score gating, causal residual conditioning, adaptive persistence, and recursive all-anchor resampling.Source: §14 · p. 11
  • Across 2,000 fits, coverage averages 0.862 for nominal 90% intervals and 0.915 for nominal 95% intervals; intervals are wider than the oracle on average.Source: §14 · p. 11
  • Multi-step conditional calibration is the main limitation; horizon-aware calibration, richer states, persistence ablation, and specialized trend or regime anchors are suggested extensions.Source: §14 · p. 11

Software and reproducibility

  • The reference implementation is the R package rollcast, and the reported benchmark protocol, raw forecast rows, diagnostics, metadata, and summaries are included in the reproducibility bundle.The protocol uses eight fixed DGPs, 250 replications per DGP, 300 training observations, horizons one through six, 1,000 Rollcast particles, and 10,000 oracle trajectories per replication.Source: Software and reproducibility · p. 11
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