Temporal Predictive Multiplicity: Equally Accurate Time Series Models Yield Different Forecast Trajectories

1 INTRODUCTION

• Because forecasters often face heterogeneous near-equivalent models but lack a principled way to characterize their temporal differences, deployment choices can be effectively arbitrary.No model class consistently dominates, and model rankings can shift with hyperparameters and evaluation protocols.Source: §1 · pp. 1–2
• Prior multiplicity frameworks focus on scalar predictions and treat outputs independently, so markedly different forecast trajectories can remain indistinguishable.Figure 1 illustrates trajectories with identical disagreement at every horizon but different temporal disagreement patterns.Source: §1 · p. 2, Figure 1 · p. 2
• The framework distinguishes pointwise, horizon-wise, and temporal multiplicity, showing that near-optimal performance and even tight horizon-wise agreement leave substantial trajectory differences.These differences can include predicted-change direction, event timing, and trajectory shape or frequency structure.Source: §1 · p. 2
• Experiments find that near-optimal models produce substantially different trajectories, with trajectory disagreement largely unrelated to horizon-wise disagreement and capable of affecting downstream decisions.The experiments cover 19 neural forecasting architectures across 11 public datasets.Source: §1 · p. 2

2 RELATED WORK

• Predictive multiplicity research examines conflicting predictions for individual instances, alongside differences in variable importance and feature attributions (Marx et al., 2020).Related work also distinguishes predictive multiplicity from procedural, explanatory, and dataset-induced forms (Ganesh et al., 2025).Source: §2 · p. 2
• Forecasting equivalence tests compare models through losses globally, per horizon, or jointly, but do not characterize how equivalent models’ forecasts differ.Source: §2 · p. 3
• Recent forecasting multiplicity studies, including horizon-constrained Rashomon sets, remain horizon-wise and therefore cannot fully characterize trajectory-level multiplicity (Kale et al., 2026).Other cited work studies recovery-rate forecasting, volatility forecasters, and feature reliance.Source: §2 · p. 3

3 A HIERARCHY OF PREDICTIVE MULTIPLICITY NOTIONS

• For models in a Rashomon set, forecast matrices pair origins with horizons; forecast, revision, and error trajectories extract rows, anti-diagonals, and error rows, respectively.The Rashomon set contains models whose loss is within tolerance ϵ of the best loss.Source: §3 · p. 3
• Forecast and revision trajectories use geometry also found in forecast-stability research, but here that geometry characterizes disagreement among equivalent-loss models rather than fluctuations within one model.Source: §3 · p. 3
• The hierarchy compares isolated predictions pointwise, predictions separately at each horizon, and complete forecast trajectories temporally.The first two levels compare scalar predictions by absolute difference; temporal multiplicity uses distances suited to temporal behavior.Source: §3 · p. 3
• Pointwise prediction range and ambiguity average per-instance maximal disagreement, while diameter and discrepancy measure the average drift of a single model pair.Pointwise ambiguity and discrepancy maximize over pairs of competing models rather than comparing against f⋆.Source: §3 · p. 4
• Horizon-wise multiplicity applies corresponding range, diameter, ambiguity, and discrepancy measures separately at each forecast horizon.Source: §3 · p. 4
• Temporal multiplicity compares complete trajectories using a distance function, whose choice determines which aspects of temporal disagreement are assessed.Possible aspects include magnitude, direction, volatility, shape, spectral structure, and timing; no single distance captures them all without conflation.Source: §3 · p. 4
• Only temporal statistics retain full trajectory structure; swapping origins at one horizon can change whether models agree on change direction without changing lower-level statistics.Pointwise statistics retain model alignment but disregard cell order, while horizon-wise statistics also distinguish horizons.Source: §3 · p. 5

4 HOW LOOSELY IS TEMPORAL MULTIPLICITY CONSTRAINED?

• Rashomon membership gives a single scalar loss budget over the error matrix, leaving error placement across instances and horizons unconstrained.The loss family includes MSE, MAE, and MASE-type losses.Source: §4 · p. 5
• For ℓ(e)=|e|^p, the pointwise bound grows as H^(1/p): H^1/2 under MSE and linearly under MAE.Longer horizons therefore weaken the control Rashomon membership provides over individual forecasts.Source: §4 · p. 5
• The bounds are attained in shell-rich model classes when the loss budget does not bind first.Shell-richness requires realizing error patterns whose loss lies between L⋆ and L⋆+ϵ.Source: §4 · pp. 5–6
• Horizon-wise tolerances τh bound absolute prediction differences at each horizon, yet leave direction, spectral behavior, and shape uncontrolled unless every τh=0.The all-zero case makes every model's forecast identical for every sample.Source: §4 · p. 6

5 INSTANTIATIONS OF TEMPORAL MULTIPLICITY

• The distance family is designed to distinguish level, local dynamics, and global trajectory structure, with complementary interpretations rather than one universally appropriate metric.The paper notes that no single trajectory distance is appropriate for every forecasting application.Source: §5 · p. 6
• Magnitude aggregates pointwise gaps and retains the least temporal structure, while increment-based distances compare patterns of change independently of absolute levels.The increment-based set includes increment, volatility, and direction distances; direction measures the fraction of increments with opposite signs.Source: §5 · p. 6
• Scale-free distances compare trajectory form rather than absolute values; timing measures the best-alignment lag as a fraction of the largest admissible lag.The scale-free family also includes shape and spectral distances.Source: §5 · p. 6
• Each distance applies to forecast trajectories, revision trajectories that track forecast updates, or error trajectories that capture temporal error structure.Forecast distances measure disagreement about the target's future path.Source: §5 · p. 6
• Substituting each distance into the framework yields metric-specific temporal-disagreement bounds, and Table 1 reports trajectory ranges and diameters across forecast, revision, and error trajectories.The bounds, proofs, and empirical tightness assessment are deferred to Appendix D.Source: §5 · p. 6, Table 1 · p. 7

• Across 11 datasets, 19 neural architectures, 10 random seeds, and multiple hyperparameter configurations, the empirical Rashomon set included models within 5% of best validation MASE.Multiplicity statistics were computed from rolling-origin test forecasts.Source: §6 · p. 7
• At a typical origin, some near-optimal model pair moved oppositely on 84% of trajectory increments (100% on Gasoline), while a fixed pair did so on 57%.Source: §6 · p. 7
• Temporal and pointwise measures ranked datasets differently: Weather had the fourth-largest pointwise range but the second-smallest directional disagreement.Source: §6 · p. 7
• For direction, shape, spectral behavior, and timing, average correlations on the least-correlated trajectory ranged from 0.04 to 0.16, and even the strongest correlations did not exceed 0.47.Source: §6 · p. 8
• Peak forecasts differed by 3.00 error budgets on average and by 57% of the horizon in timing; at least one model predicted the opposite direction on 80% of forecast cells.Source: §6 · pp. 8–9, Table 3 · p. 9
• Bitcoin forecasts split 75–98 on weekly direction, while Queensland peak timing split 165–167 between the same evening and following morning.Spanish electricity-price forecasts had the same intraday shape across models, but 98 predicted a higher price after 24 hours and 110 a lower one; New South Wales predicted peak increases ranged from 1.2 to 10.9 naive units.Source: §6 · p. 9, Figure 3 · p. 9

• The framework distinguishes pointwise, horizon-wise, and temporal disagreement to characterize trajectory features that conventional multiplicity metrics cannot capture.Source: §7 · p. 9
• Experiments across forecasting architectures and datasets found trajectory differences among near-optimal models that can affect downstream decisions.Source: §7 · p. 10
• Future work could extend the framework to temporal explanations, probabilistic forecasts, foundation-model conditioning choices, and uncertainty in model performance.Source: §7 · p. 10

DISCLAIMER

• The document is informational and is not investment research, investment advice, or a recommendation or offer to buy or sell financial products.Source: DISCLAIMER · p. 10
• Views or opinions expressed are solely those of the listed authors.Source: DISCLAIMER · p. 10
• The document states that it is © 2026 JP Morgan Chase & Co. and that all rights are reserved.Source: DISCLAIMER · p. 10

A EXPERIMENTAL SETUP

• The appendix provides details of the experimental setup.Source: A EXPERIMENTAL SETUP · p. 16

A.1 DATASETS

• The 11 publicly available datasets span finance, economics, energy, industry, and nature.Source: §A.1 · p. 16
• Strictly positive series spanning orders of magnitude are modeled in natural-log space, while nine non-positive FRED-MD series remain in raw levels.Source: §A.1 · p. 17
• Missing values are handled without future leakage: leading gaps are trimmed, interior gaps forward-filled, and an availability mask excludes imputed targets from losses and metrics.Source: §A.1 · p. 17
• Each input window is standardized by its own mean and standard deviation, and model outputs are transformed back to the original scale.Source: §A.1 · p. 17
• Historical non-target variables and four Fourier terms for declared seasonal periods are supplied as future-known features when architectures allow them.Source: §A.1 · p. 17
• Held-out portions use rolling-origin forecasts at stride one, producing an H-step trajectory at every origin.The validation portion immediately precedes the test tail; each holds out 15% of the calendar by default.Source: §A.1 · p. 17

A.2 MODEL TRAINING

• The experiments train 19 architectures across transformer, MLP/mixer, convolutional, recurrent, and near-linear families.The full model set is a Cartesian product of architectures, ten random seeds, batch sizes 32 and 64, and eligible input sizes four or eight times the forecast horizon.Source: §A.2 · pp. 17–18
• Training uses MSE and Adam, zero dropout, per-window input scaling, and fp32 evaluation to avoid precision confounds.Train-time operations use bfloat16 mixed precision where possible; parameter and optimizer state remain fp32.Source: §A.2 · p. 17
• Each run receives a throughput-based 15-minute GPU step budget, capped at 250 expected-coverage epochs and divided across per-target models.Dataset–architecture combinations exceeding available RAM or VRAM are excluded.Source: §A.2 · pp. 18–19
• Validation checkpoints are selected by lowest horizon- and origin-aggregated MASE; Table 8 describes minimum and mean performance across trained models.The reported metrics are MAE, RMSE, MASE, and RMSSE, averaged across the horizon and origins.Source: §A.2 · p. 19, Table 8 · p. 19

A.3 RASHOMON SET

• The empirical Rashomon set contains models within 5% of the best validation MASE, and its multiplicity statistics are computed on test forecasts.For datasets with multiple targets eligible for per-target modeling, a separate set is selected and measured per target.Source: §A.3 · p. 20

A.4 TEMPORAL MULTIPLICITY

• Trajectory magnitudes are scaled by seasonal-naive error for cross-series comparability, while direction, shape, spectral, and timing distances are scale-free.Source: §A.4 · pp. 20–21
• The experimental timing-distance implementation caps admissible lag at L, maximizes normalized correlation, and divides the selected lag by L; theoretical statements use a different definition.The implementation also requires an overlap of at least two points covering at least half the horizon.Source: §A.4 · pp. 21–22

B ADDITIONAL EXPERIMENTAL RESULTS

• The appendix breaks results out by target and reports target-level Spearman correlations to show variation across targets and trajectory aspects.Source: B ADDITIONAL EXPERIMENTAL RESULTS · p. 23

B.1 POINTWISE AND HORIZON-WISE PREDICTIVE MULTIPLICITY

• Pointwise ambiguity is almost one at both thresholds on most datasets, although small, accurate model sets may lack enough models to register high multiplicity.A single model pair disagrees beyond half the best MASE on close to half the cells on average.Source: §B.1 · p. 23
• Horizon-wise statistics grow with horizon on most datasets, but ATM 1W is an exception; the authors note that not all tasks necessarily exhibit chaotic behavior.They contrast these forecasting tasks with the chaotic systems studied by Kale et al. (2026).Source: §B.1 · p. 23

B.2 TEMPORAL PREDICTIVE MULTIPLICITY

• The per-target temporal results provide range and diameter in Table 11 and ambiguity and discrepancy in Table 12, using a threshold of 0.5 times the best validation MASE.Source: §B.2 · p. 24, Table 11 · p. 24, Table 12 · p. 25

B.3 TEMPORAL MULTIPLICITY IS UNCORRELATED WITH OTHER NOTIONS OF MULTIPLICITY

• Across individual targets, pointwise and horizon-wise correlations are close to one for magnitude, weaker for increment and volatility, and weak or absent for direction, shape, spectral behavior, and timing.Source: §B.3 · p. 24, Table 13 · p. 25

C PREDICTIVE MULTIPLICITY HIERARCHY CHARACTERIZATION

• Pointwise and horizon-wise statistics cannot detect trajectory changes caused by rearranging predictions across origins, while temporal statistics can.The appendix establishes this distinction through invariance results and examples where row permutations preserve the lower-level statistics but change temporal multiplicity.Source: C PREDICTIVE MULTIPLICITY HIERARCHY… · p. 26
• For two models, multiplicity reduces to expected prediction distance and the rate that distance exceeds τ; the inequalities ω ≤ ν and ς ≤ α can be strict only with at least three models.For a pair, range equals diameter and ambiguity equals discrepancy.Source: C PREDICTIVE MULTIPLICITY HIERARCHY… · pp. 26–27
• The seven trajectory distances yield trajectory-aware statistics, with ambiguity and discrepancy restricted to τ < 1 for direction, shape, spectral, and timing.For magnitude, increment, and volatility, ambiguity and discrepancy are trajectory-aware for any τ ≥ 0; for the other four distances, they are zero when τ ≥ 1.Source: C PREDICTIVE MULTIPLICITY HIERARCHY… · p. 27
• Row permutations can preserve loss and pointwise and horizon-wise multiplicity while changing temporal statistics for forecast, revision, and error trajectories.The examples establish this for the temporal statistics while retaining both models in the Rashomon set.Source: C PREDICTIVE MULTIPLICITY HIERARCHY… · pp. 26, 27, 29

D TEMPORAL MULTIPLICITY BOUNDS CHARACTERIZATION

• With Rashomon-set membership alone, magnitude, increment, and volatility disagreement are bounded, but direction, shape, spectral, and timing disagreement can remain maximal.The appendix translates the bound on the gap between forecasts at a single cell into bounds on trajectory distances.Source: D TEMPORAL MULTIPLICITY BOUNDS… · p. 30
• Additional pointwise or horizon-wise ambiguity control tightens bounds on magnitude, increment, and volatility, yet leaves direction, shape, spectral, and timing disagreement unconstrained under the stated assumptions.The corresponding per-cell gap is tightened to the chosen tolerances.Source: D TEMPORAL MULTIPLICITY BOUNDS… · p. 30
• The appendix evaluates each trajectory-distance bound by measuring the largest disagreement among trained models in the empirical Rashomon set and the fraction of its theoretical bound attained.Table 15 summarizes the bounds under Rashomon membership alone and additional pointwise or horizon-wise control; its first column specializes to MAE.Source: D TEMPORAL MULTIPLICITY BOUNDS… · p. 30, Table 15 · p. 30

D.1 THE PERFORMANCE CONSTRAINT OF THE RASHOMON SET MEMBERSHIP: EXACT CHARACTERIZATION AND BOUNDS FOR EACH TEMPORAL ASPECT

• Under Rashomon membership, forecasts can differ by as much as 2u at a single entry, and shell-richness allows this bound with identical aggregate loss.The attaining construction assigns opposite errors at that entry and matches the targets elsewhere.Source: §D.1 · pp. 30–31
• Magnitude and increment distances depend only on the forecast gap, but other distances can depend on the target; volatility, shape, and spectral results assume a constant target.Direction results allow a nonconstant target when its increments are small enough for the constructed errors to reverse predicted directions.Source: §D.1 · pp. 32, 34
• For magnitude disagreement, a concave ϕ(s) = ℓ(√s) favors concentrating the loss budget at one horizon, while a convex ϕ favors spreading it equally; power losses switch at p = 2.At p = 2, both allocations attain the same maximum; the maxima use opposite error rows with identical pointwise absolute errors.Source: §D.1 · p. 32
• For H ≥ 3, increment disagreement depends on error arrangement: interior spikes and alternating errors give lower bounds, while exact maxima are established for MAE and MSE.The displayed bounds and maxima are attained by opposite error rows with identical pointwise absolute errors.Source: §D.1 · pp. 33–34
• For a constant target and H ≥ 3, MAE volatility disagreement reaches 2u/(H −1) by pairing a flat forecast with one that fluctuates.The constructed pair uses the full loss budget and has identical aggregate loss.Source: §D.1 · pp. 34–35
• When every target increment has magnitude below 2ϱ, opposite alternating errors make the forecasts’ directions disagree at every step, attaining the maximum direction distance of 1.This condition holds automatically for a constant target when ϱ > 0.Source: §D.1 · p. 35
• For a constant target, opposite nonconstant error rows produce opposite centered forecasts and attain the largest possible shape disagreement.The construction preserves the models’ pointwise absolute errors.Source: §D.1 · pp. 35–36
• For a constant target and H ≥ 4, models with identical aggregate loss can attain maximum spectral disagreement using error rows with different single frequencies.The frequencies are chosen with disjoint spectral support.Source: §D.1 · p. 36
• Equal-loss forecasts formed from shifted samples of the same cosine wave achieve maximal timing disagreement, with best alignment at lag H −2 and timing distance 1.This construction applies when F is shell-rich, the target is constant, H ≥3, and B > 0.Source: §D.1 · pp. 36–37
• Identical aggregate loss can still permit disagreement in forecast magnitude, increments, volatility, direction, shape, frequency content, and timing.The section’s corollaries establish that performance constraints alone allow disagreement across these temporal aspects.Source: §D.1 · p. 37
• Forecast and error trajectories have matching cross-model gaps up to sign, so magnitude, increment, pointwise, and horizon-wise multiplicity statistics coincide.The stated bounds and constructions also apply to error trajectories without requiring a constant target or small target increments, subject to the other assumptions.Source: §D.1 · pp. 37–38

D.2 THE HORIZON-WISE MULTIPLICITY CONSTRAINT: EXACT CHARACTERIZATION AND BOUNDS FOR EACH TEMPORAL ASPECT

• Horizon-wise tolerances bound magnitude, increment, and volatility disagreement, while direction, shape, spectral, and timing disagreement can remain maximal under stated conditions.The latter distances can reach their maximum even when all tolerances are arbitrarily small but positive; the direction result also permits nonconstant targets with sufficiently small increments.Source: §D.2 · pp. 39, 41, 42
• Under horizon-wise control, direction, shape, spectral, and timing distances each attain 1 in the stated constructions.The constructions require shell-richness, positive tolerances, and a positive loss budget; forecast disagreement additionally requires a constant target for shape, spectral, and timing, while direction allows sufficiently small target increments.Source: §D.2 · pp. 41–42
• The horizon-wise results transfer to complete revision trajectories by reversing the tolerance order and placing the constructions along the relevant anti-diagonal.The same error patterns attain the stated horizon-wise bounds without assumptions on the target trajectory; forecast disagreement still requires the stated target conditions.Source: §D.2 · p. 42

Rashomon Membership Alone

• Direction, shape, and timing reach 100% of their Rashomon-membership bounds on every dataset, while spectral distance reaches 99.7%.These are scale-free distances; attaining their full range does not require large errors.Source: §D.3 · p. 43, Table 16 · p. 44
• Magnitude, increment, and volatility use at most 6.8% of their Rashomon-membership bounds, yet observed distances average 16.9 to 21.4 multiples of the naive error scale.Most datasets exceed 3 multiples of the naive error scale.Source: §D.3 · p. 43, Table 16 · p. 44

Vanishing Pointwise and Horizon-Wise Ambiguity

• Utilization compares each pair’s largest distance at any origin with its corresponding bound and tolerance, so it tests pair-specific tightness rather than attainment at one shared tolerance.These maxima indicate whether large disagreement occurs, not how typical it is across pairs or origins.Source: §D.3 · pp. 43–44
• For Exchange Rate, increment and volatility reach 99.8% and 95.5% of their horizon-wise bounds; remaining gaps may reflect the finite explored model set and training restrictions.The authors suggest limited predictability as a possible explanation for Exchange Rate’s especially small gap.Source: §D.3 · p. 45
• Increment and volatility remain below their bounds because equality requires maximal-amplitude horizon-to-horizon crossings and, for volatility, maximal total variation.Smooth, persistent forecast gaps can be large in magnitude while changing little across horizons.Source: §D.3 · p. 45
• Horizon-wise control yields higher utilization because separate horizon tolerances avoid applying the largest gap everywhere.Source: §D.3 · p. 45
• Under vanishing ambiguity, magnitude reaches 99.8% pointwise and 100% horizon-wise; increment and volatility improve from 65.4% and 53.5% pointwise to 82.5% and 67.9% horizon-wise.Direction, shape, and timing reach 100%, and spectral distance reaches 99.7%.Source: §D.3 · p. 44, Table 17 · p. 45