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Interweaving Marginals into Multivariate Sample Paths: Training-Free Dependence Construction for Probabilistic Time Series Foundation Models

Jinmyeong Choi, Jinkwan Jang, Seul Lee, Taesup Kim

arXiv:2609.25980v1cs.LGcs.AI

TL;DR

Probabilistic TSFM marginals do not determine a joint distribution over multivariate future trajectories. The paper couples frozen marginals without task-specific retraining and evaluates fixed-marginal, direct-sampling, and native multivariate settings. Historical temporal and channel couplings improve targeted diagnostics, supporting dependence reconstruction as a distinct post-processing problem.

  • Problem

    Probabilistic TSFM marginal forecasts do not specify the joint distribution over future trajectories, including whether values across channels or neighboring horizons occur together.

    Source: §1 · p. 1

  • Method

    The paper compares training-free temporal, channel, channel–time, and historical rank-based couplings of frozen TSFM marginals, including fixed-marginal and direct-sampling evaluations.

    Source: §1 · p. 2, §2 · p. 3

  • Results

    Historical temporal and channel couplings improve their targeted diagnostics with fixed empirical marginals; the pattern persists under direct sampling and Channel–Time remains beneficial under native multivariate inference.

    Source: §3 · pp. 3–4, §4 · p. 4

  • Takeaways & Limitations

    The findings support treating dependence reconstruction as a distinct post-processing problem for frozen probabilistic TSFMs.

    Source: §4 · p. 4

    Findings are limited to evaluated tasks and backbones; historical coupling depends on informative pre-forecast dependence, and the separable Gaussian copula limits channel-pair-specific lead–lag patterns.

    Source: §4 · p. 4

Abstract

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Probabilistic time series foundation models (TSFMs) provide coordinate-wise predictive distributions, but these marginals do not determine a joint distribution over multivariate future trajectories. We study training-free coupling of frozen TSFM marginals into multivariate forecast sample paths. Our primary evaluation fixes the empirical marginal sample multiset at every channel--horizon coordinate across methods, isolating the effect of coupling alone. Historical temporal and channel relations substantially improve their corresponding dependence diagnostics. The same pattern persists when the fixed-marginal constraint is removed and paths are sampled directly, and remains present under native multivariate backbone inference. These results support treating dependence reconstruction as a distinct post-processing problem for probabilistic TSFMs.

1 Introduction

Probabilistic TSFMs can forecast marginal distributions across channels and horizons, but those marginals do not specify a joint distribution over future trajectories. This paper asks how channel–horizon coupling contributes when frozen TSFM marginals are held fixed.

1 Introduction

  • Marginal forecasts do not determine whether high values across channels or neighboring horizons occur together, so independent sampling imposes a coupling without identifying dependence from the marginals.The gap concerns the joint distribution over future trajectories, not the coordinate-wise forecasts themselves.Source: §1 · p. 1
  • Copulas separate dependence modeling from marginal distributions, and a Gaussian copula models dependence through correlated latent Gaussian variables transformed to marginal probability levels and predictive quantiles.The Gaussian assumption applies to latent dependence variables, not to the TSFM's predictive marginal distributions.Source: §1 · p. 1
  • Prior work includes Schaake Shuffle and ensemble copula coupling [Clark et al., 2004, Schefzik et al., 2013], temporal Gaussian copulas for multi-step TSFMs [2025], and approaches for pretrained forecasting models [Redhead et al., 2026, Benechehab et al., 2025].The paper asks what channel–horizon coupling contributes when frozen TSFM marginals are held fixed.Source: §1 · p. 1
  • The study compares temporal, channel, channel–time, and historical rank-based couplings with independent assembly while holding every finite marginal sample set fixed.Complementary experiments remove the fixed-marginal constraint and evaluate univariate and native multivariate backbone inference.Source: §1 · p. 2

2 Controlled Coupling of Forecast Marginals

The method couples frozen TSFM marginal forecasts into joint paths by changing coordinate-wise assignments while preserving the finite empirical marginal samples. It compares Gaussian and historical-rank constructions, including a direct-sampling evaluation that removes the fixed-sample constraint.

2 Controlled Coupling of Forecast Marginals

  • At each forecast origin, the frozen TSFM supplies predictive marginal CDFs and quantile functions for each channel and horizon; a copula couples those marginals into joint forecasts.The notation suppresses conditioning on observed history.Source: §2 · p. 2
  • In the controlled experiment, every method receives the same coordinate-wise sample sets and changes only their permutations, preserving each empirical marginal while varying how values co-occur.Gaussian coupling uses latent-draw ranks to assign the fixed values; all procedures leave the backbone frozen and use pre-forecast information.Source: §2 · p. 2
  • Factorial Gaussian coupling combines channel and temporal relations: channel structure comes from same-time rank-Gaussianized history correlations, while temporal structure uses an AR(1)-style correlation with median lag-one correlation.The four constructions include IID, Temporal, Channel, and Channel–Time; the separable construction is a controlled decomposition, not a general separability assumption about true dependence.Source: §2 · p. 2
  • Historical rank templates reorder the same fixed samples, while equicorrelation and randomly relabeled historical relations test correctly aligned heterogeneous channel structure against controls.The Schaake-style construction uses historical D × H rank templates.Source: §2 · p. 3
  • Direct copula sampling removes the fixed-marginal constraint by mapping copula probabilities through predictive quantiles, making it an end-to-end rather than coupling-only comparison.Finite marginal sample sets may differ across methods in this evaluation.Source: §2 · p. 3, Table 1 · p. 3

3 Experiments and Results

Across the evaluated methods, historical temporal and channel couplings improve their targeted joint-forecast diagnostics, with Channel–Time retaining both improvements. The targeted pattern also appears in direct sampling and under native multivariate inference.

3 Experiments and Results

  • 19.61% Time-VS skill for Temporal and 4.23% Ch.-VS skill for Channel are achieved with empirical marginals held fixed; Channel–Time retains both targeted improvements, as does Schaake.Temporal has near-zero Ch.-VS effects and Channel has near-zero Time-VS effects, as expected by construction; Energy skill for Channel is 0.02%.Source: §3 · p. 3
  • Historical Channel–Time has lower aggregate Ch.-VS than equicorrelation and randomly relabeled controls, with mean paired raw-score differences of −6.78 and −7.73, respectively.The 95% hierarchical bootstrap confidence intervals are [−12.77, −1.71] and [−14.10, −2.25], supporting correctly aligned heterogeneous structure beyond these tested controls.Source: §3 · pp. 3–4
  • Figure 1 illustrates direct sample paths: Temporal adds within-channel temporal dependence, while Channel–Time additionally incorporates the estimated historical cross-channel relation.The illustration uses frozen Chronos-2 marginal forecasts on a fixed synthetic multivariate history.Source: Figure 1 · p. 4, §3 · p. 4
  • At N = 100, Channel–Time achieves 4.94% Ch.-VS skill and 21.86% Time-VS skill in direct sampling, with lower scores than IID on both diagnostics in all 30 seeds.The pattern persists across sample sizes N ∈ {10, 50, 100}.Source: §3 · p. 4
  • Channel–Time versus IID yields positive joint-score skill under both univariate and native multivariate inference for all three backbones, so post-hoc coupling remains useful when marginals condition on multivariate history.The benchmark evaluates 14 multivariate forecasting tasks with three frozen backbones and N = 10 sample paths.Source: §3 · pp. 3–4

4 Conclusion and Limitations

Historical temporal and channel couplings improve targeted joint-forecast diagnostics while preserving empirical marginals, and the pattern persists in direct sampling and native multivariate inference. The findings are limited by evaluated scope and by the dependence structures and historical information available to the methods.

4 Conclusion and Limitations

  • The findings support treating dependence reconstruction as a distinct post-processing problem for frozen probabilistic TSFMs, while motivating study of richer post-hoc dependence models.The conclusion reports targeted improvements with fixed empirical marginals, persistence under direct sampling, and benefit under native multivariate inference.Source: §4 · p. 4
  • Evidence is limited to the evaluated tasks and backbones; historical couplings require pre-forecast dependence to remain informative, and the separable Gaussian copula cannot represent channel-pair-specific lead–lag patterns.Schaake-style coupling also depends on sufficient, relevant historical rank templates.Source: §4 · p. 4

A.1 Benchmark population and backbones

The benchmark evaluates three frozen probabilistic TSFMs on multivariate forecasting tasks, with metrics and aggregation designed for comparisons across datasets. It also specifies how marginal quantiles and historical channel relations are constructed.

A.1 Benchmark population and backbones

  • The benchmark uses three frozen TSFMs with native multivariate inference across 14 tasks; one declared Air Quality task is excluded for incomplete future ground truth.Table 2 reports task dimensionality, forecast horizon, context length, and evaluated item–origin cells per backbone.Source: §A.1 · p. 6, Table 2 · p. 6

A.2 Metrics and history standardization

  • Joint scores use channel-wise standardization based only on pre-forecast context, while variogram diagnostics measure same-horizon channel structure and within-channel temporal structure.Global-mean and global-max CRPS evaluate functionals of the standardized future field; Energy Score is also reported.Source: §A.2 · pp. 6–7

A.3 Dataset-equal aggregation and relative skill

  • IID-relative skill compares each method with IID on shared support, averaging backbones equally within datasets and then weighting datasets equally.Positive skill indicates a lower score than IID.Source: §A.3 · p. 7

A.4 Hierarchical uncertainty estimation

  • Controlled-benchmark uncertainty uses 10,000 paired hierarchy-aware bootstrap replicates, resampling datasets, then items and forecast origins.Sampling seeds are averaged within each evaluation cell before resampling.Source: §A.4 · p. 7

B.1 Marginal quantile construction

  • Marginal quantile knots are monotonically rearranged, interpolated between interior levels, and logarithmically extrapolated in the tails.The tails join continuously at u = 0.1 and u = 0.9; probabilities are clipped before evaluating the quantile function.Source: §B.1 · pp. 7–8

B.2 Historical channel relation

  • Historical channel dependence is estimated as the Pearson correlation matrix of independently rank-Gaussianized channels over timestamps with complete observations.The implementation falls back to independence in specified sparse-history or single-channel cases and stabilizes the matrix before factorization.Source: §B.2 · p. 8

B.3 Historical temporal relation

The paper estimates temporal persistence from historical rank-Gaussianized observations and uses it in Gaussian and historical-rank coupling procedures. In the controlled benchmark, the coupling methods preserve finite marginal samples while varying their co-occurrence.

B.3 Historical temporal relation

  • Across 219 Core cells, temporal coefficients had mean 0.921 and median 0.955; 151 exceeded 0.9 and 64 were clipped at 0.99.The authors interpret the estimator mainly as strong historical persistence, rather than evidence that precise cell-specific estimation of ρ is necessary.Source: §B.3 · p. 8

B.4 Gaussian rank coupling

  • The Gaussian variants share base-normal draws, combine temporal and channel factors, and reassign each coordinate’s fixed marginal values by latent ranks.This controlled rank assignment permutes the common finite marginal samples without introducing new forecast values.Source: §B.4 · p. 8

B.5 Historical empirical rank coupling

  • Schaake-style coupling uses historical D × H rank templates to reorder the fixed samples, preferring recent non-overlapping blocks before using overlapping blocks when needed.At N = 10, spacing-one fallback was used in 342 of 654 scored cells; no scored cell required deterministic template reuse.Source: §B.5 · p. 9

B.6 Fixed-marginal audit

  • The audit found identical sorted coordinate-wise forecast samples across coupling methods in every controlled Core cell; this exact identity does not extend to direct sampling.Gaussian coupling only permutes the common fixed samples.Source: §B.6 · p. 9

C Full Controlled Benchmark Results

  • The controlled benchmark compares five methods on 654 shared cells under native multivariate inference, with identical empirical marginals at every coordinate.Methods differ only in their coupling within each cell.Source: C Full Controlled Benchmark Results · p. 9

C.1 Absolute scores

  • Table 3 reports dataset-equal absolute scores for the fixed-marginal N = 10 benchmark; Mean CRPS and Max CRPS score global field functionals, not coordinate-wise marginal CRPS.All listed metrics are lower-is-better.Source: Table 3 · p. 9, §C.1 · p. 9

C.2 IID-relative skill

  • Table 4 reports IID-relative skill on the same 654-cell support, with ratios aggregated across backbones and datasets after within-pair normalization.These skill values cannot be recovered by taking ratios of Table 3 aggregate absolute scores.Source: §C.2 · p. 9, Table 4 · p. 10

C.3 Paired uncertainty for the controlled factorial

  • Table 5 reports Gaussian-factorial IID-relative skill alongside paired raw-score differences and hierarchy-aware 95% bootstrap intervals for those differences.Negative differences favor the coupling; the intervals apply to differences, not skill percentages.Source: Table 5 · p. 10, §C.3 · p. 10

C.4 Factorial interpretation

Historical temporal and channel couplings improve their respective dependence diagnostics, while untargeted effects are consistent with construction-implied invariances. Energy Score is less responsive to the channel-only intervention, and its reported values do not establish a direct ranking between methods.

C.4 Factorial interpretation

  • Temporal leaves same-horizon cross-channel dependence unchanged, while Channel leaves within-channel cross-horizon dependence unchanged; small untargeted differences are sampling variation.The corresponding paired intervals include zero, making these design checks rather than independent evidence about forecasting ability.Source: §C.4 · p. 10
  • Temporal lowers Time-VS by 18.72 raw-score units and Channel lowers Ch.-VS by 8.263 relative to IID, with both paired intervals excluding zero.Channel–Time also improves both diagnostics; Schaake has positive skill on both axes, though its paired uncertainty is not reported here.Source: §C.4 · p. 10
  • Channel achieves 4.23% Ch.-VS skill but 0.02% Energy skill, indicating Energy Score is less responsive to this channel-only intervention under the reported relative-skill measure.Its small paired Energy interval below zero does not imply a large effect.Source: §C.4 · p. 10
  • Energy skills of 2.83% for Temporal and 2.89% for Channel–Time compare each method with IID and do not establish that Channel–Time outperforms Temporal.That ranking would require a direct paired comparison between the methods.Source: §C.4 · p. 11

D Jointization skill by inference mode

Channel–Time coupling retains positive IID-relative skill across the reported inference modes, including native multivariate inference. Negative controls test whether historical channel relations add information beyond average correlation or randomly relabeled structure while preserving temporal coupling and fixed marginals.

D Jointization skill by inference mode

  • Under native multivariate inference, Channel–Time Ch.-VS skill ranges from 3.15% to 4.42% and Time-VS skill from 17.14% to 22.18%.The coupling yields positive IID-relative skill for every backbone and inference mode shown; the comparison holds mode-specific empirical marginals fixed.Source: D Jointization skill by inference mode · p. 11, Table 6 · p. 11
  • Uni–Multi row differences are not interpreted causally because changing inference mode also changes the frozen marginal forecasts, and jointization gains are not uniformly larger under Multi.The supported conclusion is that IID-to-Channel–Time gains remain within each inference mode.Source: D Jointization skill by inference mode · p. 11

E Negative Controls for Cross-Channel Structure

  • The negative-control question is whether historical channel structure adds information beyond average signed correlation or randomly relabeled correlations, with temporal relation and fixed empirical marginals retained.Both controls use the same temporal relation and the controlled fixed-marginal protocol.Source: E Negative Controls for Cross-Channel… · pp. 11–12, §E.1 · p. 12

E.1 Control definitions

  • Equicorrelation preserves average signed correlation while removing pair-specific heterogeneity; the relabeled relation preserves the correlation-entry multiset and spectrum but changes alignment with channel labels.The relabeling control averages five permutations, and individual permutations may leave some pairs unchanged, especially for small D.Source: §E.1 · p. 12

E.2 Paired evaluation and results

Historical Channel–Time coupling outperforms two cross-channel controls on Ch.-VS in the paired evaluation. Direct-sampling evaluation removes the fixed empirical-marginal constraint, while Schaake’s finite template pool constrains interpretation of scenario-budget sensitivity.

E.2 Paired evaluation and results

  • The paired evaluation uses exact Gaussian support for each historical-versus-control comparison, and reports paired raw-score differences with hierarchy-aware bootstrap intervals and dataset win/tie/loss counts.Negative differences favor the historical relation; all comparisons retain the same temporal relation and fixed empirical marginals.Source: Table 7 · p. 12, §E.2 · p. 12

E.3 Interpretation

  • Historical Channel–Time yields lower Ch.-VS than each control on 12 of 14 datasets; mean paired differences are −6.78 versus equicorrelation and −7.73 versus random relabeling.The respective 95% intervals are [−12.77, −1.71] and [−14.10, −2.25], supporting correctly aligned heterogeneous channel structure beyond these two alternatives.Source: §E.3 · p. 12, Table 7 · p. 12
  • Energy contrasts also favor the historical relation, but raw Energy and Ch.-VS changes are not compared because the metrics have different scales; supplementary functional CRPS intervals include zero.Time-VS differences are near zero with intervals including zero, as all conditions share Rtime and differ only in channel relation.Source: §E.3 · p. 13

F End-to-End Direct Sampling

  • Direct copula sampling is an end-to-end evaluation rather than a coupling-only comparison because realized coordinate-wise sample multisets are not forced to coincide.The stated scenario budgets are N ∈ {10, 50, 100}, using 30 seeds on the 654-cell common support.Source: F End-to-End Direct Sampling · p. 13

F.1 Direct Gaussian scenario generation

  • Direct Gaussian scenarios use the same historical factors by coupling variant and map latent Gaussian values through the normal CDF and frozen marginal quantile functions.Schaake draws historical rank templates with replacement; at N = 100, some cells may reuse a small pool, and increasing N does not enlarge that pool.Source: §F.1 · p. 13, §F.2 · p. 13

F.3 Sampling-budget sensitivity

The direct-sampling sensitivity study finds that targeted dependence gains persist across scenario budgets, although their magnitudes vary. Additional analyses examine marginal scores, seed stability, absolute scores, a synthetic illustration, and the deterministic Median reference.

F.3 Sampling-budget sensitivity

  • Across N = 10, 50, and 100, Temporal improves Time-VS, Channel improves Ch.-VS, and Channel–Time improves both; untargeted contrasts remain near zero.Table 8 reports IID-relative skill means over 30 sampling seeds on the 654-cell common support.Source: §F.3 · p. 13, Table 8 · p. 14
  • Channel–Time Ch.-VS skill falls from 6.94% at N = 10 to 4.94% at N = 100, while its Time-VS skill remains above 21%.Temporal and Channel–Time retain approximately 2.5–2.7% Energy skill across budgets; Channel-only Energy skill remains near zero.Source: §F.3 · p. 14

F.4 Coordinate-wise marginal CRPS

  • At N = 100, aggregate marginal-CRPS skill magnitude is below 0.02% for every method, alongside substantial targeted joint-score gains.This does not establish equality of realized empirical marginals or rule out differences at individual coordinates or evaluation cells.Source: §F.4 · p. 14, Table 9 · p. 14

F.5 Seed stability

  • For Temporal, paired Energy-difference standard deviation across seeds decreases from approximately 0.036 at N = 10 to 0.013 at N = 100; Channel–Time decreases from 0.063 to 0.021.At N = 100, seed-level comparisons favor Temporal and Channel–Time over IID on Energy in all 30 seeds, but do not establish improvement in every evaluation cell.Source: §F.5 · p. 15

F.6 Absolute scores

  • Table 10 reports dataset-equal absolute scores at each budget, but its cross-dataset ratios cannot reconstruct Table 8’s within-dataset–backbone IID-relative skills.The absolute-score table covers direct stochastic scenario generation on the 654-cell common support; all metrics are lower-is-better.Source: §F.6 · p. 15, Table 10 · p. 15

F.7 Synthetic illustration

  • The synthetic illustration uses history-estimated channel correlation Rch[1, 2] = 0.8875 and temporal correlation ρ = 0.9571 to show coupling, not benchmark accuracy.It uses one fixed realization and common marginal quantile functions and base-normal tensor for the three coupling methods.Source: §F.7 · pp. 15–16

G.1 Construction

  • The Median reference is the coordinate-wise Qd,h(0.5) trajectory from rearranged marginals and has no predictive spread.The primary construction reuses the same rearranged marginal distributions as the probabilistic scenarios.Source: §G.1 · p. 16

G.2 Comparison on the primary common support

  • Probabilistic scenarios score lower than Median, while IID already improves Energy, Ch.-VS, Time-VS, and global-mean CRPS on average; structured couplings add targeted gains beyond IID.Median is a point-forecast reference, not a coupling ablation; the Gaussian comparison on all 657 cells retains the same qualitative comparison as the 654-cell support.Source: Table 11 · p. 17, §G.2 · p. 17, §G.3 · p. 17

G.5 Implementation checks

Implementation checks verify the Median reference and the numerical construction, while documenting the backbones, assets, randomization, evaluation support, and audit procedures used in the experiments.

G.5 Implementation checks

  • The primary Median trajectory is bitwise identical to canonical Q(0.5) in all 657 cells; raw cached median knots can differ when quantiles cross, with negligible aggregate-score changes.Deterministic-score identities hold numerically, and the support contains 654 cells when Schaake availability is required versus 657 for Gaussian-only comparisons.Source: §G.5 · p. 18

H.1 Backbones and cached marginals

  • The experiments use frozen Chronos-2, TimesFM-3, and TiRex-2; the primary controlled benchmark uses native multivariate inference for all three.Direct sampling reuses cached marginal forecasts without new backbone forward calls.Source: §H.1 · p. 19

H.2 Existing assets, licenses, and terms of use

  • The FEV collection has no single permissive license for all source datasets; use is limited to the research-only evaluation setting described by its dataset card.The paper does not claim that all benchmark datasets share the fev software package’s Apache 2.0 license.Source: §H.2 · p. 19, Table 14 · p. 19

H.3 Randomness and common random numbers

  • Gaussian methods share a deterministically derived base-normal stream across methods within a seed; controlled stochastic methods use 20 seeds per cell, while direct sampling uses 30.Direct-sampling budgets are N ∈ {10, 50, 100}.Source: §H.3 · p. 19

H.4 Common support

  • The primary all-method comparison uses 654 cells because one UCI Air Quality 1D origin is unavailable to Schaake; 657 cells are declared across the three backbones.The direct-sampling experiment uses the same 654-cell support at every budget.Source: §H.4 · p. 20

H.5 Numerical audit

  • The fixed-marginal audit found identical sorted coordinate-wise samples across couplings, controls, and Schaake, with no target values used in dependence estimation.A synthetic correlation check matched Rch ⊗ Rtime to numerical Monte Carlo precision, with no channel–time indexing swap found.Source: §H.5 · p. 20

H.6 Scope of sensitivity results

The sensitivity results have a specific scope: the fixed-marginal all-multivariate benchmark is reported at N = 10, while the multi-budget sensitivity study is a separate direct-sampling experiment.

H.6 Scope of sensitivity results

  • The fixed-marginal all-multivariate benchmark uses N = 10; N ∈ {10, 50, 100} sensitivity results come from a separate direct-sampling experiment, not a fixed-marginal budget sweep.Older sensitivity artifacts mixing univariate TimesFM-3 paths with multivariate Chronos-2 and TiRex-2 are excluded from the final main-paper conclusions.Source: §H.6 · p. 20
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