Efficiently Generating Correlated Sample Paths from Multi-step Time Series Foundation Models

TL;DR

The paper tackles the problem of generating correlated multi-step forecast sample paths from time series foundation models, which typically provide only marginal predictions. It introduces a copula-based, one-pass method that decouples marginals from the correlation structure, using a Gaussian copula with AR(1)-type Toeplitz covariance and IQF-based marginal reconstruction. The approach yields sample paths with realistic temporal correlations and competitive or improved marginal accuracy (CRPS) while delivering orders-of-magnitude speedups over autoregressive sampling. This enables scalable, joint forecasting in practical settings and opens avenues for learned copula parameterizations and richer dependence structures.

Abstract

Many time series applications require access to multi-step forecast trajectories in the form of sample paths. Recently, time series foundation models have leveraged multi-step lookahead predictions to improve the quality and efficiency of multi-step forecasts. However, these models only predict independent marginal distributions for each time step, rather than a full joint predictive distribution. To generate forecast sample paths with realistic correlation structures, one typically resorts to autoregressive sampling, which can be extremely expensive. In this paper, we present a copula-based approach to efficiently generate accurate, correlated sample paths from existing multi-step time series foundation models in one forward pass. Our copula-based approach generates correlated sample paths orders of magnitude faster than autoregressive sampling, and it yields improved sample path quality by mitigating the snowballing error phenomenon.

Figures (8)

• Figure 1: Mechanisms for generating sample paths from multi-step time series models. 100 sample paths are shown for each method, with three randomly-selected sample paths highlighted. The naive approach generates sample paths in $\mathcal{O}(1)$ time but produces jagged trajectories with unrealistic correlation structures. Autoregressive sampling produces smooth correlated trajectories, but takes $\mathcal{O}(N \cdot H)$ forward passes, which can be prohibitive, and can result in snowballing errors. Our copula-based approach provides realistic correlated sample paths in $\mathcal{O}(1)$ time.
• Figure 2: Median percent improvement from copula-based method by horizon for M4 Daily dataset. Switching from autoregressive sampling to our copula-based approach yields high-quality sample paths with realistic correlation structures, and significant improvements in CRPS for longer forecast horizons. Here, we plot the CRPS term at each horizon separately, rather than the cumulative sum.
• Figure 3: Copula-based sampling mitigates snowballing errors in autoregressive sampling. For each method, we plot 3 random sample paths. The true future trajectory of the time series is shown in gray.
• Figure A1: Sample path quality for M4 Daily dataset. Switching from autoregressive sampling to our copula-based approach yields improved sample paths with realistic correlation structures, at a fraction of the time. Points show the median performance across series after dividing by the corresponding score of the SeasonalNaive baseline. The $x$-axis indicates the time taken to generate 10 sample paths per time series on an A100 GPU.
• Figure A2: Median CRPS results for other datasets. The copula approach yields sample paths with comparable or higher quality than autoregressive sampling but is significantly faster.