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

Emanuele Albini; Francesca Toni; Saumitra Mishra; Francesco Leofante

cs.LGOct 7, 2026

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

Authors: Emanuele Albini, Francesca Toni, Saumitra Mishra, Francesco Leofante

Organizations: J.P. Morgan AI Research Imperial College London London, UK · Imperial College London London, UK · J.P. Morgan AI Research London, UK

Abstract

Models with near-identical predictive performance can yield substantially different predictions, a phenomenon known as predictive multiplicity. Prior work has mostly studied this at the level of individual scalar outputs. In time-series forecasting, however, predictions across horizons jointly define a trajectory, and horizon-wise comparisons can hide important differences in predictive behavior. To address this problem, we introduce temporal predictive multiplicity, a framework that characterizes disagreement over complete forecast trajectories among models with near-identical predictive performance. We show that constraining predictive performance alone can still admit a broad range of different trajectories. We further show that constraining multiplicity at individual horizons partially reduces, but does not eliminate, trajectory-level multiplicity. Experiments with 19 neural forecasting architectures on 11 datasets confirm that near-optimal models can exhibit substantial variability in the forecast trajectories they produce, and trajectory-level disagreement is largely unrelated to horizon-wise disagreement. Our framework, therefore, exposes a gap in existing multiplicity studies: models with indistinguishable predictive performance imply fundamentally different temporal trajectories, with consequential downstream effects.

Figures & tables

Panel of Figure 1 — Figure 1: Examples of forecast trajectories that are indistinguishable under current multiplicity frameworks. At every horizon, the disagreement among forecasts is identical across the three panels. Yet, the panels exhibit markedly different patterns of temporal disagreement at the trajectory level.

Panel of Figure 2 — Figure 2: Pictorial representation of the structure that each of the loss and the three levels of the multiplicity hierarchy is aware of on a running example.

Dataset	${d}_{\mathrm{magnitude}}$	${d}_{\mathrm{increment}}$	${d}_{\mathrm{volatility}}$	${d}_{\mathrm{direction}}$	${d}_{\mathrm{shape}}$	${d}_{\mathrm{spectral}}$	${d}_{\mathrm{timing}}$	$\nu^{(1)}$ / $\omega^{(1)}$	$\nu^{(H)}$ / $\omega^{(H)}$	$\nu$ / $\omega$	MASE ∗
ATM 1D	1.38 / 0.66	1.55 / 0.81	0.87 / 0.35	100% / 59%	96% / 54%	85% / 36%	100% / 54%	1.79 / 0.59	1.94 / 0.62	1.85 / 0.55	1.05
ATM 1W	1.46 / 0.67	1.38 / 0.65	0.94 / 0.49	100% / 55%	99% / 56%	85% / 32%	100% / 58%	1.75 / 0.71	1.67 / 0.61	1.68 / 0.59	0.93
Bitcoin	7.00 / 2.99	3.66 / 2.08	2.01 / 0.88	100% / 62%	99% / 61%	82% / 42%	100% / 69%	5.16 / 2.35	9.41 / 3.34	7.10 / 2.63	2.67
Electricity D. A	1.56 / 0.62	0.47 / 0.33	0.21 / 0.15	79% / 49%	83% / 39%	82% / 43%	98% / 33%	1.59 / 0.46	2.19 / 0.58	1.93 / 0.53	0.46
Electricity D. Z	0.80 / 0.30	0.20 / 0.15	0.07 / 0.05	69% / 50%	79% / 44%	77% / 49%	99% / 40%	0.62 / 0.22	1.17 / 0.32	0.97 / 0.24	0.24
Electricity Price	0.95 / 0.45	0.40 / 0.29	0.18 / 0.11	81% / 46%	72% / 35%	76% / 38%	96% / 30%	0.98 / 0.34	1.29 / 0.48	1.24 / 0.37	0.82

Table 1: Pointwise, horizon-wise, and temporal predictive multiplicity in empirical Rashomon sets. Each cell reports the trajectory range / diameter (

\nu_{{d}}/\omega_{{d}}

), maximized across forecast, revision, and error trajectories. For multi-target datasets, multiplicity statistics are maximized and MASE is averaged across targets. Magnitude, increment, and volatility are reported in seasonal-naive units; direction, timing, shape, and spectral distances are scale-free and reported as percentages.

Dataset

Pointwise

Horizon-Wise

{d}_{\mathrm{magnitude}}

{d}_{\mathrm{increment}}

{d}_{\mathrm{volatility}}

{d}_{\mathrm{direction}}

{d}_{\mathrm{shape}}

{d}_{\mathrm{spectral}}

{d}_{\mathrm{timing}}

{d}_{\mathrm{magnitude}}

{d}_{\mathrm{increment}}

{d}_{\mathrm{volatility}}

{d}_{\mathrm{direction}}

{d}_{\mathrm{shape}}

{d}_{\mathrm{spectral}}

{d}_{\mathrm{timing}}

ATM 1D

0.98 / 0.98

0.58 / 0.75

0.35 / 0.37

0.15 / 0.28

0.24 / 0.61

0.05 / 0.42

0.07 / 0.24

0.98 / 0.98

0.58 / 0.75

0.35 / 0.37

0.15 / 0.28

0.24 / 0.61

0.05 / 0.42

0.07 / 0.24

ATM 1W

0.99 / 0.99

0.56 / 0.70

0.39 / 0.47

0.13 / 0.18

0.17 / 0.40

0.04 / 0.24

0.14 / 0.20

0.99 / 0.99

0.56 / 0.70

0.39 / 0.47

0.13 / 0.18

0.17 / 0.40

0.04 / 0.24

0.14 / 0.20

Bitcoin

0.99 / 1.00

0.61 / 0.69

0.43 / 0.49

-0.02 / 0.19

-0.01 / 0.32

-0.05 / 0.26

0.02 / 0.14

0.96 / 0.98

0.60 / 0.67

0.41 / 0.48

-0.03 / 0.17

-0.03 / 0.30

-0.05 / 0.25

-0.01 / 0.13

Electricity D. A

0.99 / 0.99

0.46 / 0.66

0.16 / 0.36

-0.03 / 0.26

-0.00 / 0.64

-0.08 / 0.49

0.05 / 0.36

0.96 / 0.97

0.47 / 0.65

0.16 / 0.36

-0.03 / 0.26

-0.03 / 0.63

-0.10 / 0.48

0.04 / 0.35

Electricity D. Z

0.99 / 0.99

0.23 / 0.52

0.09 / 0.28

-0.08 / 0.23

0.07 / 0.52

-0.05 / 0.38

0.10 / 0.37

0.98 / 0.99

0.24 / 0.52

0.09 / 0.28

-0.07 / 0.23

0.07 / 0.51

-0.06 / 0.38

0.10 / 0.37

Table 2: Spearman rank correlation, over all (model pair, origin) samples of the empirical Rashomon set, between the pair’s mean pointwise gap

\bar{\delta}^{fg}_{t}

and its trajectory distance

{d}({\bm{v}}^{f}_{t},{\bm{v}}^{g}_{t})

on the same forecast origin, read as in Equation 3 with

\mathcal{L}^{\star}_{h}\equiv 1

(Pointwise) or as is (Horizon-Wise). Each row reports the minimum / maximum over the three trajectories and, for multi-target datasets, its targets.

Dataset	Extremes Value		Extremes Location		Increment Direction
	$\mathrm{range}({\bm{v}}^{M})$	$\mathrm{range}({\bm{v}}^{m})$	$\mathrm{range}({\bm{\ell}}^{M})$	$\mathrm{range}({\bm{\ell}}^{m})$	Contested Cells	Minority Share	$\mathrm{range}({\bm{\theta}})$
ATM 1D	1.91 (6.22)	1.43 (5.40)	18.7% (85.7%)	48.3% (85.7%)	51.0%	9.2%	55.8 (159)
ATM 1W	1.83 (6.82)	1.56 (8.17)	68.3% (75.0%)	70.7% (75.0%)	89.8%	21.8%	69.2 (153)
Bitcoin	2.83 (7.35)	2.48 (9.21)	84.7% (85.7%)	85.2% (85.7%)	100%	28.7%	101 (164)
Electricity D. A	5.44 (39.4)	3.13 (17.1)	43.9% (97.9%)	28.2% (97.9%)	60.9%	10.3%	34.6 (143)
Electricity D. Z	4.18 (20.9)	2.19 (7.52)	30.3% (99.0%)	17.7% (99.0%)	70.1%	13.3%	17.2 (89.3)

Table 3: Disagreement across empirical Rashomon-set members about the value and location of forecast minima and maxima, and about the direction of forecast increments. Value ranges are expressed in multiples of the best validation MASE; locations are fractions of the forecast horizon; and increment-angle ranges are in degrees (up to

180^{\circ}

, opposite steep moves). Ranges are reported as mean (maximum) across origins, or across forecast cells for increment angles; directional shares are averaged across cells. We report the maximum for multi-target datasets.

Figure 3: Examples of temporal multiplicity within empirical Rashomon sets. Each panel shows member forecast trajectories at one origin, as deviations from the realized level at that origin, together with the realized trajectory (black). Left: Bitcoin and Spanish electricity-price forecasts, colored by net direction over the horizon (blue: up; orange: down). Right: Australian electricity-demand forecasts, colored by the timing of their maximum (dots); annotations report disagreement in peak value and timing.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Domain	Source	License	Distribution Channel
UK ATM Withdrawals - Daily	finance	Crone (2008) ; Godahewa et al. (2021)	CC BY 4.0	GluonTS ( Alexandrov et al., 2020 )
UK ATM Withdrawals - Weekly	finance	Crone (2008) ; Godahewa et al. (2021)	CC BY 4.0	GluonTS ( Alexandrov et al., 2020 )
Bitcoin Price	finance	Godahewa et al. (2021)	CC BY 4.0	GIFT-Eval ( Aksu et al., 2024 )
Australian Electricity Demand	energy	AEMO () ; Godahewa et al. (2021)	CC BY 4.0	GluonTS ( Alexandrov et al., 2020 )
Zurich Electricity Consumption	energy	Stadt-Zürich ()	CC0 Public Domain	Darts ( Herzen et al., 2022 )
Hourly Energy Pricing - Spain	energy	Jhana (2019)	CC0 Public Domain	Darts ( Herzen et al., 2022 )

Appendix

Table 4: Provenance of the datasets used in our experiments.

Dataset	Frequency	Variables	Targets	Timestamps ( $T$ )	Input Size †	Horizon	Seasonality	Poolable ∗
UK ATM Withdrawals - Daily	D	108	108	791	28 (4W) / 56 (8W)	7 (1W)	7 (1W)	Yes
UK ATM Withdrawals - Weekly	W	108	108	113	16 (16W) / 26 (26W)	4 (4W)	52 (52W)	Only
Bitcoin Price	D	18	1	2629	28 (4W) / 56 (8W)	7 (1W)	–	No
Australian Electricity Demand	30min	5	5	230736	336 (1W)	48 (1D)	48 (1D), 336 (1W), 17,520 (52W)	Yes
Zurich Electricity Consumption	15min	10	2	268705	672 (1W)	96 (1D)	96 (1D), 672 (1W), 35,040 (52W)	No
Hourly Energy Pricing - Spain	h	20	1	35064	168 (1W)	24 (1D)	24 (1D), 168 (1W), 8,760 (52W)	No

Appendix

Table 5: Properties of the datasets used in our experiments.

Dataset	Length		Samples			Samples ∗			Forecasts
	Train	Val & Test	Train	Train †	Val & Test	Train	Train †	Val & Test	Train	Val & Test
UK ATM Withdrawals - Daily	553	119	526 / 498	531 / 503	113	56808 / 53784	57348 / 54324	12204	56808 / 53784	12204
UK ATM Withdrawals - Weekly	79	17	–	–	–	6912 / 5832	7128 / 6048	1512	6912 / 5832	1512
Bitcoin Price	1841	394	1814 / 1786	1819 / 1791	388	–	–	–	1814 / 1786	388
Australian Electricity Demand	161516	34610	161181	161227	34563	805905	806135	172815	805905	172815
Zurich Electricity Consumption	188093	40306	187422	187516	40211	–	–	–	374844	80422

Appendix

Table 6: Train/validation/test splits of the datasets used in our experiments.

Family	Model	Reference
Transformer	PatchTST	Nie et al. (2022)
	iTransformer	Liu et al. (2024)
	TimeXer	Wang et al. (2024b)
	TFT	Lim et al. (2021)
MLP / Mixer	NHITS	Challu et al. (2023)
	NBEATSx	Olivares et al. (2023)

Appendix

Table 7: Neural forecasting architectures used to generate the empirical Rashomon set, grouped by model family.

Dataset		Val								Test
		MAE		RMSE		MASE		RMSSE		MAE		RMSE		MASE		RMSSE
	Models	Min	Mean	Min	Mean	Min	Mean	Min	Mean	Min	Mean	Min	Mean	Min	Mean	Min	Mean
ATM 1D	2080	4.23	4.68	6.88	7.56	1.05	1.16	1.09	1.20	3.16	3.45	4.95	5.23	0.73	0.80	0.71	0.76
ATM 1W	1520	15.4	18.0	22.3	25.9	0.93	1.11	1.05	1.26	12.9	13.9	18.2	19.5	0.75	0.81	0.80	0.86
Bitcoin	2800	0.06	0.07	0.08	0.10	2.67	3.26	2.58	3.03	0.05	0.06	0.07	0.09	2.35	3.01	2.16	2.76
Electricity D. A (NSW)	2160	171	206	274	323	0.38	0.45	0.41	0.48	171	206	274	320	0.38	0.46	0.41	0.48

Appendix

Table 8: Validation and test performance of the full set of trained models of each dataset: minimum and mean across all models of the horizon-averaged MAE, RMSE, MASE, and RMSSE, on the validation portion (left) and on the test portion (right).

Dataset					MASE		MAE		RMSE		RMSSE
	$H$	$\|\hat{{\mathcal{F}}}\|$	$\|{\mathcal{R}}_{\epsilon}\|$	Ratio	Best	Worst	Best	Worst	Best	Worst	Best	Worst
ATM 1D	7	2080	88	0.04	1.05	1.10	4.23	4.44	6.88	7.51	1.09	1.19
ATM 1W	4	1520	81	0.05	0.93	0.98	15.4	16.2	22.3	24.5	1.05	1.16
Bitcoin	7	2800	173	0.06	2.67	2.81	0.06	0.06	0.08	0.09	2.58	2.81
Electricity D. A (NSW)	48	2160	135	0.06	0.38	0.40	171	179	274	298	0.41	0.44
Electricity D. A (QLD)	48	2160	332	0.15	0.41	0.43	100	105	153	168	0.40	0.44

Appendix

Table 9: Empirical Rashomon sets: forecast horizon

H

, the size of the full set of trained models

|\widehat{{\mathcal{F}}}|

, Rashomon set size

|\widehat{{\mathcal{R}}}_{{\epsilon}}|

, the Rashomon ratio, and the best and worst validation score of the set’s members on the selection metric (MASE, in bold) and on the other reported metrics.

Dataset	Pointwise						Horizon-Wise
	$\nu$	$\omega$	$\alpha(\tau_{0})$	$\varsigma(\tau_{0})$	$\alpha(\tau_{1})$	$\varsigma(\tau_{1})$	$\nu^{(1)}$	$\nu^{(H)}$	$\omega^{(1)}$	$\omega^{(H)}$	$\alpha^{(1)}(\tau_{0})$	$\alpha^{(H)}(\tau_{0})$	$\varsigma^{(1)}(\tau_{0})$	$\varsigma^{(H)}(\tau_{0})$	$\alpha^{(1)}(\tau_{1})$	$\alpha^{(H)}(\tau_{1})$	$\varsigma^{(1)}(\tau_{1})$	$\varsigma^{(H)}(\tau_{1})$
ATM 1D	1.85	0.55	1.00	0.67	1.00	0.41	1.79	1.94	0.59	0.62	1.00	1.00	0.70	0.76	1.00	1.00	0.44	0.49
ATM 1W	1.68	0.59	0.99	0.72	0.98	0.48	1.75	1.67	0.71	0.61	0.99	0.99	0.76	0.75	0.98	0.98	0.55	0.51
Bitcoin	7.10	2.63	1.00	0.79	0.98	0.62	5.16	9.41	2.35	3.34	1.00	1.00	0.77	0.85	0.93	1.00	0.59	0.69
Electricity D. A (NSW)	1.38	0.35	1.00	0.78	1.00	0.58	0.76	1.64	0.24	0.42	1.00	1.00	0.76	0.83	1.00	1.00	0.53	0.67
Electricity D. A (QLD)	1.93	0.48	1.00	0.85	1.00	0.70	1.37	2.19	0.45	0.56	1.00	1.00	0.85	0.87	1.00	1.00	0.70	0.75

Appendix

Table 10: Pointwise and horizon-wise predictive multiplicity statistics. Ambiguity

\alpha

and discrepancy

\varsigma

are measured at thresholds

\tau_{0}=0.25\cdot\mathcal{L}(f^{\star})

and

\tau_{1}=0.5\cdot\mathcal{L}(f^{\star})

. Horizon-wise statistics (range, diameter, ambiguity, and discrepancy) are reported at the first and the last horizon only.

Dataset	${d}_{\mathrm{magnitude}}$	${d}_{\mathrm{increment}}$	${d}_{\mathrm{volatility}}$	${d}_{\mathrm{direction}}$	${d}_{\mathrm{shape}}$	${d}_{\mathrm{spectral}}$	${d}_{\mathrm{timing}}$
ATM 1D	1.38 / 0.66	1.55 / 0.81	0.87 / 0.35	99.7% / 58.9%	95.9% / 54.1%	85.1% / 36.0%	99.9% / 54.1%
ATM 1W	1.46 / 0.67	1.38 / 0.65	0.94 / 0.49	99.9% / 55.3%	99.4% / 56.0%	85.4% / 31.6%	100% / 57.6%
Bitcoin	7.00 / 2.99	3.66 / 2.08	2.01 / 0.88	99.9% / 61.7%	99.4% / 61.1%	81.5% / 41.6%	100% / 68.9%
Electricity D. A (NSW)	1.17 / 0.42	0.32 / 0.23	0.14 / 0.09	75.4% / 48.5%	81.1% / 39.2%	78.9% / 42.9%	97.9% / 32.7%
Electricity D. A (QLD)	1.55 / 0.57	0.41 / 0.28	0.19 / 0.13	76.6% / 48.9%	83.3% / 38.6%	81.7% / 43.3%	98.3% / 33.1%
Electricity D. A (SA)	1.56 / 0.62	0.47 / 0.33	0.21 / 0.15	73.4% / 47.4%	75.1% / 33.9%	76.8% / 41.8%	94.9% / 29.0%

Appendix

Table 11: Temporal multiplicity of the empirical Rashomon Sets. Each cell reports the range and diameter (

\nu_{{d}}/\omega_{{d}}

), maximized over the three trajectories (forecast, revision, and error).

{d}_{\mathrm{magnitude}}

{d}_{\mathrm{increment}}

and

{d}_{\mathrm{volatility}}

are in seasonal-naive units while

{d}_{\mathrm{direction}}

{d}_{\mathrm{timing}}

{d}_{\mathrm{shape}}

, and

{d}_{\mathrm{spectral}}

are scale-free on

[0,1]

and reported as percentages to allow comparisons across different datasets.

Dataset	${d}_{\mathrm{magnitude}}$	${d}_{\mathrm{increment}}$	${d}_{\mathrm{volatility}}$	${d}_{\mathrm{direction}}$	${d}_{\mathrm{shape}}$	${d}_{\mathrm{spectral}}$	${d}_{\mathrm{timing}}$
ATM 1D	100% / 62%	100% / 76%	90% / 16%	100% / 53%	100% / 58%	100% / 23%	100% / 56%
ATM 1W	98% / 63%	98% / 61%	90% / 44%	100% / 58%	100% / 58%	100% / 23%	100% / 41%
Bitcoin	99% / 75%	96% / 70%	73% / 24%	100% / 62%	100% / 65%	100% / 34%	100% / 79%
Electricity D. A (NSW)	100% / 95%	98% / 67%	13% / 2%	100% / 42%	95% / 27%	99% / 30%	98% / 29%
Electricity D. A (QLD)	100% / 99%	100% / 77%	31% / 10%	100% / 44%	96% / 29%	99% / 32%	98% / 30%
Electricity D. A (SA)	100% / 94%	99% / 71%	20% / 7%	100% / 37%	88% / 22%	97% / 28%	95% / 25%

Appendix

Table 12: Temporal multiplicity of the empirical Rashomon Sets. Each cell reports the ambiguity and discrepancy (

\alpha_{{d}}/\varsigma_{{d}}

) with threshold

\tau_{1}=0.5\cdot\mathcal{L}(f^{\star})

, maximized over the three trajectories (forecast, revision, and error).

Dataset

Pointwise

Horizon-Wise

{d}_{\mathrm{magnitude}}

{d}_{\mathrm{increment}}

{d}_{\mathrm{volatility}}

{d}_{\mathrm{direction}}

{d}_{\mathrm{shape}}

{d}_{\mathrm{spectral}}

{d}_{\mathrm{timing}}

{d}_{\mathrm{magnitude}}

{d}_{\mathrm{increment}}

{d}_{\mathrm{volatility}}

{d}_{\mathrm{direction}}

{d}_{\mathrm{shape}}

{d}_{\mathrm{spectral}}

{d}_{\mathrm{timing}}

ATM 1D

0.98 / 0.98

0.58 / 0.75

0.35 / 0.37

0.15 / 0.28

0.24 / 0.61

0.05 / 0.42

0.07 / 0.24

0.98 / 0.98

0.58 / 0.75

0.35 / 0.37

0.15 / 0.28

0.24 / 0.61

0.05 / 0.42

0.07 / 0.24

ATM 1W

0.99 / 0.99

0.56 / 0.70

0.39 / 0.47

0.13 / 0.18

0.17 / 0.40

0.04 / 0.24

0.14 / 0.20

0.99 / 0.99

0.56 / 0.70

0.39 / 0.47

0.13 / 0.18

0.17 / 0.40

0.04 / 0.24

0.14 / 0.20

Bitcoin

0.99 / 1.00

0.61 / 0.69

0.43 / 0.49

-0.02 / 0.19

-0.01 / 0.32

-0.05 / 0.26

0.02 / 0.14

0.96 / 0.98

0.60 / 0.67

0.41 / 0.48

-0.03 / 0.17

-0.03 / 0.30

-0.05 / 0.25

-0.01 / 0.13

Electricity D. A (NSW)

0.99 / 0.99

0.51 / 0.66

0.16 / 0.36

-0.01 / 0.16

0.04 / 0.63

-0.08 / 0.46

0.08 / 0.33

0.97 / 0.97

0.52 / 0.65

0.16 / 0.36

-0.01 / 0.16

0.03 / 0.61

-0.09 / 0.45

0.07 / 0.32

Electricity D. A (QLD)

0.99 / 0.99

0.53 / 0.60

0.19 / 0.33

-0.01 / 0.23

0.04 / 0.63

-0.05 / 0.44

0.08 / 0.36

0.97 / 0.97

0.54 / 0.59

0.19 / 0.33

-0.01 / 0.23

0.03 / 0.63

-0.06 / 0.44

0.07 / 0.35

Appendix

Table 13: Spearman rank correlation, over all (pair, origin) samples of the empirical Rashomon set, between the pair’s mean pointwise gap and its trajectory distance

{d}

on the same forecast origin, read as in Equation 3 with

\mathcal{L}^{\star}_{h}\equiv 1

(Pointwise) or as is (Horizon-Wise). Each row reports the minimum / maximum over the three trajectories.

Statistic	Multiplicity-Aware not invariant to per-model cell permutations	Horizon-Aware not invariant to cell permutations	Trajectory-Aware not invariant to row permutations
Loss $\mathcal{L}$	$\times$	$\times$	$\times$
Pointwise Multiplicity	✓	$\times$	$\times$
Horizon-wise Multiplicity	✓	✓	$\times$
Temporal Multiplicity	✓	✓	✓

Appendix

Table 14: Classification of the loss and of the multiplicity statistics of the hierarchy.

Temporal Statistic	$\bm{{\mathcal{R}}_{\epsilon}}$ only	$\bm{+}$ pointwise $\bm{\alpha_{\tau}=0}$	$\bm{+}$ horizon-wise $\bm{\alpha^{(h)}_{\tau_{h}}=0}$
Pointwise $\|\hat{y}^{f}-\hat{y}^{g}\|$	$2TH(\mathcal{L}^{\star}+{\epsilon})$	$\tau$	$\tau_{h}$
Magnitude ${d}_{\mathrm{magnitude}}$	$2\sqrt{H}\,T(\mathcal{L}^{\star}+{\epsilon})$	$\tau$	$\sqrt{\frac{1}{H}\sum_{h}\tau_{h}^{2}}$
Increment ${d}_{\mathrm{increment}}^{\lx@sectionsign}$	$\frac{2\sqrt{2}}{\sqrt{H-1}}\,TH(\mathcal{L}^{\star}+{\epsilon})$	$2\tau$	$\sqrt{\frac{1}{H-1}\sum_{h<H}(\tau_{h}+\tau_{h+1})^{2}}$
Volatility ${d}_{\mathrm{volatility}}^{\dagger\lx@sectionsign}$	$\frac{2}{H-1}\,TH(\mathcal{L}^{\star}+{\epsilon})$	$2\tau$	$\frac{1}{H-1}\sum_{h<H}(\tau_{h}+\tau_{h+1})$
Direction ${d}_{\mathrm{direction}}^{\dagger}$	free	free	free
Shape ${d}_{\mathrm{shape}}^{\dagger}$	free	free	free

Appendix

Table 15: Summary of the bounds proved in Sections D.1 and D.2 : how Rashomon set membership and additional pointwise or horizon-wise multiplicity constraints bound disagreement between forecast, revision, and error trajectories at each origin. The first column is reported for the MAE. † For forecast trajectories, the bounds are attained under conditions that rule out degenerate cases. § For revision trajectories, the bounds hold when the trajectories are traversed in reverse order.

Dataset	${\mathcal{R}}_{\epsilon}$ only
	${d}_{\mathrm{magnitude}}$	${d}_{\mathrm{increment}}$	${d}_{\mathrm{volatility}}$	${d}_{\mathrm{direction}}$	${d}_{\mathrm{shape}}$	${d}_{\mathrm{spectral}}$	${d}_{\mathrm{timing}}$
ATM 1D	6.54 (0.0%)	6.20 (0.0%)	3.52 (0.0%)	100%	100%	97.6%	100%
ATM 1W	6.66 (0.1%)	5.47 (0.1%)	3.62 (0.1%)	100%	100%	99.4%	100%
Bitcoin	19.7 (0.3%)	11.1 (0.1%)	7.39 (0.3%)	100%	100%	95.4%	100%
Electricity D. A	7.86 (0.0%)	1.66 (0.0%)	0.75 (0.0%)	90.2%	98.8%	95.0%	100%
Electricity D. Z	3.73 (0.0%)	0.72 (0.0%)	0.31 (0.0%)	83.2%	98.4%	93.2%	100%

Appendix

Table 16: Worst-origin range and utilization of the Rashomon-membership bound (Equation 6 ). Magnitude, increment, and volatility are in naive units, with utilization in parentheses; scale-free distances are percentages of their bound

1

. Values are maxima over trajectories, averaged over targets where applicable; the last row averages over datasets.

Dataset	${d}_{\mathrm{magnitude}}$	${d}_{\mathrm{increment}}$	${d}_{\mathrm{volatility}}$	${d}_{\mathrm{direction}}$	${d}_{\mathrm{shape}}$	${d}_{\mathrm{spectral}}$	${d}_{\mathrm{timing}}$
ATM 1D	90.4% / 100%	49.8% / 69.2%	41.5% / 47.7%	100% / 100%	100% / 100%	97.6% / 97.6%	100% / 100%
ATM 1W	99.6% / 100%	62.7% / 74.8%	54.4% / 65.1%	100% / 100%	100% / 100%	99.4% / 99.4%	100% / 100%
Bitcoin	99.3% / 100%	53.4% / 71.0%	32.8% / 44.8%	100% / 100%	100% / 100%	95.4% / 95.4%	100% / 100%
Electricity D. A	85.5% / 99.1%	18.1% / 23.5%	7.7% / 9.9%	90.2% / 90.2%	98.8% / 98.8%	95.0% / 95.0%	100% / 100%
Electricity D. Z	81.5% / 95.4%	11.5% / 14.0%	4.8% / 6.5%	83.2% / 83.2%	98.4% / 98.4%	93.2% / 93.2%	100% / 100%
Electricity Price	84.7% / 99.1%	26.3% / 31.9%	11.8% / 14.5%	95.7% / 95.7%	96.5% / 96.5%	92.4% / 92.4%	100% / 100%

Appendix

Table 17: Maximum pointwise / horizon-wise utilization over pairs (Equation 8 ), each evaluated at its smallest admissible tolerances (Equation 7 ). Scale-free distances use only origins satisfying the target-flatness filter. Aggregation over trajectories, targets, and datasets follows Table 16 .

Figure 4: Pairwise bound utilization against normalized tolerance: $\tau^{fg}/\mathcal{L}^{\star}$ for pointwise control (top), and $\max_{h}(\tau^{fg}_{h}/\mathcal{L}^{\star}_{h})$ for horizon-wise control (bottom). Each marker represents one pair; lines show each dataset’s running maximum.

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Chaos Divergence

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Predictive multiplicity and chaotic dynamics represent two fundamental challenges in machine learning that have evolved independently despite their conceptual connections. We bridge this gap by introducing horizon-constrained Rashomon sets, a theoretical framework that characterizes how model multiplicity evolves with prediction horizon in chaotic systems. Unlike static prediction tasks where the Rashomon set remains fixed, chaos induces exponential divergence among initially similar models, fundamentally transforming the nature of predictive equivalence. We prove that the effective Rashomon set contracts exponentially with lead time at a rate determined by the maximum Lyapunov exponent and introduce Lyapunov-weighted metrics that provide tighter bounds on predictive disagreement. Leveraging these insights, we develop decision-aligned selection algorithms that choose among near-optimal models based on downstream utility rather than forecast accuracy alone. Extensive experiments on synthetic chaotic systems (Lorenz-96, Kuramoto-Sivashinsky) and real-world applications (wind power, traffic, weather) demonstrate that our framework improves decision quality by 18-34% while maintaining competitive predictive performance. This work establishes the first rigorous connection between chaos theory and predictive multiplicity, providing principled guidance for deploying machine learning in safety-critical chaotic domains.

Abstract

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