SGA: Uncertainty Quantification for Multi-Step Forecasting in Time Series Foundation Models

# SGA: Uncertainty Quantification for Multi-Step Forecasting in Time Series Foundation Models
Source: [https://arxiv.org/html/2609.28582](https://arxiv.org/html/2609.28582)
Xin\-Yu Hu1,2Shuang Liang1,2Cheng Feng3,4Shao\-Qun Zhang1,2,4,🖂 1National Key Laboratory for Novel Software TechnologyChina\. 2School of Intelligent Science and TechnologyNanjing UniversityChina\. 3Siemens Data and AI ResearchBeijingChina\. 4Nanjing University – Siemens Joint Research Center on Industrial AISuzhouChina\. zhangsq@lamda\.nju\.edu\.cn

###### Abstract

The recent emergence of Time Series Foundation Models \(TSFMs\) has significantly advanced multi\-step forecasting performance, enabling accurate predictions over extended future horizons\. However, existing TSFMs often suffer from significantly inherent uncertainty, which typically manifests as derived forecast branches emerging at each time step and spreading to subsequent steps; different forecast branches often exhibit varying forecasting performance, thereby undermining the credibility of TSFM forecasts\. In this paper, we propose the Slicing\-Graphing\-Alignment \(SGA\) method to quantify the uncertainty of multi\-step TSFM forecasts\. The proposed SGA first characterizes the topology of all potential forecast branches using a directed acyclic graph, such that the graph complexity bounds the uncertainty of multi\-step forecasts, and then precisely measures the graph complexity by integrating both topological information and TSFM\-inherent stochasticity\. Experimental results conducted on 11 TSFMs and 27 datasets demonstrate that \(i\) SGA achieves the best performance when ranking predictive errors with uncertainty estimates; \(ii\) SGA works with a more extensive and more precise sampling coverage than those of existing UQ methods, deriving a quantification mechanism fundamentally different from those of established ones; and \(iii\) larger model scales of TSFMs correlate with lower uncertainty estimates of multi\-step forecasts, suggesting another empirical scaling law for uncertainty quantification of multi\-step TSFM forecasts\.

Key words:Time Series Foundation Model, Multi\-Step Forecasting, Uncertainty Quantification, Slicing\-Graphing\-Alignment

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/overview.png)Figure 1:Workflow of time series multi\-step forecasting and horizon uncertainty quantification\.## 1Introduction

Many real\-world applications, such as quantitative trading[Sezer et al\. \(2020\)](https://arxiv.org/html/2609.28582#bib.bib1), weather forecasting[Shi et al\. \(2015\)](https://arxiv.org/html/2609.28582#bib.bib2), and industrial power\-load forecasting[Hong and Fan \(2016\)](https://arxiv.org/html/2609.28582#bib.bib3), can be modeled as multi\-step forecasting tasks, which have been studied for decades and remain a hot topic[Ben Taieb et al\. \(2012\)](https://arxiv.org/html/2609.28582#bib.bib5);[Ansari et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib10)\. Recently developed Time Series Foundation Models \(TSFMs\) have demonstrated superior predictive accuracy in multi\-step forecasting compared with many previous approaches[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9);[Ansari et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib10);[Liu et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib13), marking a significant milestone in the field\.

Apart from accurate multi\-step forecasts, their credibility is often required by the developers for risk\-benefit trade\-offs, especially when they apply TSFMs in high\-stakes scenarios\. For example, traders frequently encounter a critical dilemma when TSFMs forecast sustained price growth for the next week without providing any credibility measures such as uncertainty; ignoring this forecast means missing out on potential profits, whereas trusting it to trigger an excessively large trading position risks substantial financial losses[Fu et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib4)\. Such credibility concerns primarily stem from model uncertainty arising from the training process and aleatoric uncertainty inherent in the data[Chinta et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib6)\. Therefore, it is necessary and important to quantify the uncertainty of the multi\-step forecasting process performed by TSFMs to assess the credibility of their forecasts\. However, little attention has been paid to this topic\.

There are two intuitive perspectives on extending established Uncertainty Quantification \(UQ\) methods to the UQ of multi\-step forecasting\. From the perspective of the processes concerned, one can leverage the UQ methods developed for the one\-step forecasting process\. These methods typically quantify the uncertainty of a single\-step forecast using the width of a quantile\-based interval[Adler et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib15), which is either directly produced by the model[Ansari et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib10)or empirically estimated from multiple forecast samples[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9)\. Hence, it is intuitive to extend such approaches by averaging the uncertainty estimates across individual time steps\. From the perspective of the models concerned, one can adapt the UQ methods developed for Large Language Models \(LLMs\)\. These methods typically model generated outputs as sequences of tokens and exploit information\-theoretic metrics[Fomicheva et al\. \(2020\)](https://arxiv.org/html/2609.28582#bib.bib16)of a single sequence or diversity\-based measures of multiple sampled sequences[Farquhar et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib17)to quantify uncertainty\. Thus, an intuitive approach is to model a multi\-step forecast as a sequence of forecast slices, each comprising one or more consecutive time steps\. Nevertheless, these two intuitive extensions struggle to precisely describe the potential branches of multi\-step forecasts\. Such branches emerge at each time step due to the intrinsic uncertainty of TSFMs[Chinta et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib6)and spread to the subsequent steps, forming a complex topology we term the forecasting space, which encompasses all possible multi\-step forecasts\. Therefore, achieving a precise characterization of potential branches during multi\-step forecasting is fundamental to the UQ of this process, yet remains challenging\.

Built upon this recognition, we propose the Slicing\-Graphing\-Alignment \(SGA\) method to quantify the uncertainty of the multi\-step forecasting process\. The slicing stage samples multiple forecasts and segments each into slices spanning one or more consecutive time steps, where slices covering the same temporal span represent potential forecast branches\. The graphing stage constructs a Directed Acyclic Graph \(DAG\), where nodes and edges separately indicate slices and temporal dependency between consecutive slices, thereby roughly describing the topology of the forecasting space\. The alignment stage aligns slices using the Dynamic Time Warping \(DTW\) method[Sakoe and Chiba \(1978\)](https://arxiv.org/html/2609.28582#bib.bib29), measures their similarity based on the resulting DTW distance, and merges similar slices to reveal the intrinsic topology of the forecasting space\. The uncertainty of multi\-step forecasting is then quantified by the forecasting space complexity, which can be approximated by graph complexity\. We construct graph complexity by integrating both topological information and TSFM\-inherent stochasticity\. Experimental results across 11 TSFMs and 27 datasets demonstrate that \(i\) SGA significantly outperforms its contenders in ranking predictive errors with uncertainty estimates; \(ii\) SGA covers the forecasting space more extensively and precisely than existing methods in sampling, leading to a quantification mechanism inherently distinct from those of established ones; and \(iii\) models with larger scales tend to exhibit lower uncertainty estimates of multi\-step forecasts, indicating another empirical scaling law for UQ of multi\-step TSFM forecasts\.

The rest of this paper is organized as follows\. Section[2](https://arxiv.org/html/2609.28582#S2)reviews related work\. Section[3](https://arxiv.org/html/2609.28582#S3)formally introduces the SGA method\. Section[4](https://arxiv.org/html/2609.28582#S4)conducts experiments to validate the effectiveness of SGA and investigate the relation between model scales and uncertainty estimates in multi\-step TSFM forecasting\. Section[5](https://arxiv.org/html/2609.28582#S5)concludes this work\.

## 2Related Work

Current TSFMs can be roughly divided into three types according to their output formats, involving vocabulary\-based, quantile\-based, and trajectory\-based models\. Vocabulary\-based models like Chronos[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9)produce a probability distribution over a frozen vocabulary, where each token in the vocabulary represents a forecast value at one time step\. A multi\-step forecast is generated by autoregressively sampling from the predicted distributions, and multiple such forecasts are averaged to obtain the point forecast\. Quantile\-based models, including TimesFM\-2\.5[Das et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib11), Chronos\-2[Ansari et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib10), and Timer\-S1[Liu et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib12), output a matrix of forecasts, the row and column of which separately indicate the quantile level and future time step, and the median \(50%50\\%quantile\) is typically used as the multi\-step point forecast\. Trajectory\-based models such as Sundial[Liu et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib13)and Aurora[Wu et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib14)map a sample from a standard multivariate Gaussian distribution to a single multi\-step forecast, termed a trajectory, and obtain the point forecast by averaging multiple trajectories\.

Existing UQ methods of LLMs can be broadly divided into three categories, involving reflexive\-based, information\-based, and diversity\-based methods[Vashurin et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib27)\. Reflexive\-based methods\([Kadavath et al\., 2022](https://arxiv.org/html/2609.28582#bib.bib19);[Xiong et al\., 2024](https://arxiv.org/html/2609.28582#bib.bib20)\)prompt the LLM to judge the uncertainty of its own generation, making them unsuitable for adaptation to the UQ of multi\-step forecasting, since TSFMs cannot be prompted in the same manner\. Information\-based and diversity\-based UQ methods model LLM generations as sequences, then quantify uncertainty via information\-theoretic metrics[Fomicheva et al\. \(2020\)](https://arxiv.org/html/2609.28582#bib.bib16);[Malinin and Gales \(2021\)](https://arxiv.org/html/2609.28582#bib.bib23)and heuristic diversity metrics[Lin et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib21);[Farquhar et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib17);[Duan et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib18), respectively\. Thus, an intuitive adaptation is to model a multi\-step forecast as a sequence of forecast slices, each comprising one or more consecutive time steps\.

Existing studies primarily focus on UQ for one\-step forecasting, where uncertainty is typically quantified by quantile\-based interval widths[Adler et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib15)either directly produced by the model[Ansari et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib10)or empirically estimated from multiple forecast samples[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9)\. Thus, UQ for multi\-step forecasting remains underexplored\.

## 3Our Methods

In this section, we formulate the task of quantifying the uncertainty during multi\-step TSFM forecasting processes in Subsection[3\.1](https://arxiv.org/html/2609.28582#S3.SS1)and then formally propose the Slicing\-Graphing\-Alignment \(SGA\) method in Subsection[3\.2](https://arxiv.org/html/2609.28582#S3.SS2)\. We start with some useful notations\. Let\[N\]=\{1,2,…,N\}\[N\]=\\\{1,2,\.\.\.,N\\\}be an integer set forN∈ℕ\+N\\in\\mathbb\{N\}^\{\+\}, and\|⋅\|\|\\cdot\|denotes the number of elements in a collection, e\.g\.,\|\[N\]\|=N\|\[N\]\|=N, or the width of an interval, e\.g\.,\|\[0,1\]\|=1\|\[0,1\]\|=1\.

### 3\.1Horizon Uncertainty Quantification

Here, we formulate the task of quantifying the uncertainty during multi\-step TSFM forecasting processes, named Horizon Uncertainty Quantification \(HUQ\)\. Provided a historical time series𝒙1:t=\(x1,…,xt\)∈ℝt\\boldsymbol\{x\}\_\{1:t\}=\(x\_\{1\},\\dots,x\_\{t\}\)\\in\\mathbb\{R\}^\{t\}of lengtht∈ℕ∗t\\in\\mathbb\{N\}^\{\*\}and a forecast horizonh∈ℕ∗h\\in\\mathbb\{N\}^\{\*\}, a TSFM produces thehh\-step\-ahead forecasts, denoted by𝒙^t\+1:t\+h=\(x^t\+1,…,x^t\+h\)∈ℝh\\hat\{\\boldsymbol\{x\}\}\_\{t\+1:t\+h\}=\(\\hat\{x\}\_\{t\+1\},\\dots,\\hat\{x\}\_\{t\+h\}\)\\in\\mathbb\{R\}^\{h\}\. Following seminal work[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9), we refer to𝒙^t\+1:t\+h\\hat\{\\boldsymbol\{x\}\}\_\{t\+1:t\+h\}as a forecast trajectory indicated by the nexthhtimestamps, wherex^t\+s\\hat\{x\}\_\{t\+s\}is a forecast value at time stepssfors∈\[h\]s\\in\[h\]\. The goal of HUQ is to assign an uncertainty estimateu∈ℝu\\in\\mathbb\{R\}to the entire forecast trajectory𝒙^t\+1:t\+h\\hat\{\\boldsymbol\{x\}\}\_\{t\+1:t\+h\}for quantifying the credibility of multi\-step TSFM forecasts\.

Compared with the traditional UQ task for one\-step forecasting[Adler et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib15), the investigated HUQ task is not only more general since it takes the traditional one as a particular case once developers seth=1h=1, but also more challenging because the inevitable step\-wise error accumulation[Ben Taieb et al\. \(2012\)](https://arxiv.org/html/2609.28582#bib.bib5)in multi\-step forecasting often amplifies uncertainty simultaneously[Girard et al\. \(2002\)](https://arxiv.org/html/2609.28582#bib.bib8)\.

### 3\.2Slicing\-Graphing\-Alignment

This subsection formally proposes the Slicing\-Graphing\-Alignment \(SGA\) method for HUQ, comprising stages of slicing, graphing, and alignment\. We recognize that potential forecast branches would emerge at each time step due to the inherent uncertainty of TSFMs[Chinta et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib6);[Dey et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib7), with such branching spreading to subsequent steps to form a complex topology that covers all possible multi\-step forecasts, which we refer to as the forecasting space\. Built upon this recognition, the slicing stage first samples multiple forecasts and then segments each into several slices, each containing one or more consecutive time steps, where slices spanning the same time steps represent potential forecast branches\. The graphing stage further models these slice\-based forecast branches and their temporal dependency as nodes and edges in a DAG, respectively, thereby roughly describing the topology of the forecasting space\. Finally, the alignment stage aligns similar nodes if their DTW distance does not exceed the pre\-specified thresholdτ∈ℝ∗\\tau\\in\\mathbb\{R\}^\{\*\}, and thus reveals the intrinsic topology of the forecasting space\. The above three stages are illustrated in Figure[1](https://arxiv.org/html/2609.28582#S0.F1)and further detailed in the following paragraphs\.

#### Slicing

This stage aims to represent potential forecast branches with sliced forecasts\. First, we aim to obtainK∈ℕ∗K\\in\\mathbb\{N\}^\{\*\}forecast trajectory samples\. The vocabulary\-based and trajectory\-based TSFMs produce a point multi\-step forecast by averagingKKforecast trajectory samples, which can be directly retained without incurring any additional computational cost\. The quantile\-based TSFMs output a matrix𝐀=\(am​n\)g×h∈ℝg×h\\mathbf\{A\}=\(a\_\{mn\}\)\_\{g\\times h\}\\in\\mathbb\{R\}^\{g\\times h\}, wheream​na\_\{mn\}indicates the forecast value with themm\-th smallest quantile levelqmq\_\{m\}form∈\[g\]m\\in\[g\]at time stepn∈\[h\]n\\in\[h\]\. The largest and the smallest quantile levels are denoted asqmaxq\_\{\\textrm\{max\}\}andqminq\_\{\\textrm\{min\}\}, respectively\. The key idea is first to construct a horizon\-level joint inverse Cumulative Distribution Function \(CDF\)F:ℝ→ℝhF:\\mathbb\{R\}\\to\\mathbb\{R\}^\{h\}by linearly interpolating between the forecast valuesam​na\_\{mn\}andam\+1​na\_\{m\+1\\ n\}form∈\[g−1\]m\\in\[g\-1\]and forn∈\[h\]n\\in\[h\]\. Specifically, thess\-th component of the mappingFFis defined as

Ft\+s​\(x\)=am⁡\(x\)​s\+\(x−qm⁡\(x\)\)​\(am⁡\(x\)\+1​s−am⁡\(x\)​s\)qm⁡\(x\)\+1−qm⁡\(x\),F\_\{t\+s\}\\left\(x\\right\)=a\_\{m\(x\)s\}\+\\frac\{\\left\(x\-q\_\{m\(x\)\}\\right\)\\left\(a\_\{m\(x\)\+1\\ s\}\-a\_\{m\(x\)s\}\\right\)\}\{q\_\{m\(x\)\+1\}\-q\_\{m\(x\)\}\}\\ ,wherem⁡\(x\)=max⁡\{j∈\[g−1\]∣qj≤x\}m\(x\)=\\max\\\{j\\in\[g\-1\]\\mid q\_\{j\}\\leq x\\\}identifies the interval\[qm⁡\(x\),qm⁡\(x\)\+1\]\[q\_\{m\(x\)\},q\_\{m\(x\)\+1\}\]containingxx\. Based on the constructed inverse CDF, we can perform inverse transform sampling\([Devroye, 1986](https://arxiv.org/html/2609.28582#bib.bib30)\)by drawing a quantile levelq∈ℝq\\in\\mathbb\{R\}from the uniform distribution𝒰⁡\(qmin,qmax\)\\mathcal\{U\}\(q\_\{\\textrm\{min\}\},q\_\{\\textrm\{max\}\}\)and obtaining the forecast trajectory sampleF⁡\(q\)∈ℝhF\(q\)\\in\\mathbb\{R\}^\{h\}, with such sampling procedure repeatedKKtimes to obtainKKsamples\. The sampling processes for three types of TSFMs are illustrated in Figure[1](https://arxiv.org/html/2609.28582#S0.F1)\. Next, we segment thekk\-th sampled forecast trajectory𝒙^t\+1:t\+hk\\hat\{\\boldsymbol\{x\}\}\_\{t\+1:t\+h\}^\{k\}into a sequence of slices𝒃1:nk=\(b1k,…,bnk\)\\boldsymbol\{b\}\_\{1:n\}^\{k\}=\(b\_\{1\}^\{k\},\\ldots,b\_\{n\}^\{k\}\), where each slicebjk=\(x^t\+ls​\(j−1\)\+1k,…,x^t\+ls​jk\)b\_\{j\}^\{k\}=\(\\hat\{x\}\_\{t\+l\_\{s\}\(j\-1\)\+1\}^\{k\},\\dots,\\hat\{x\}\_\{t\+l\_\{s\}j\}^\{k\}\)containsls∈ℕ∗l\_\{s\}\\in\\mathbb\{N\}^\{\*\}consecutive forecast values fork∈\[K\]k\\in\[K\],n∈ℕ∗n\\in\\mathbb\{N\}^\{\*\}, andj∈\[n\]j\\in\[n\]\. Slices with the same indexjjrepresent potential forecast branches during the corresponding temporal span\.

#### Graphing

Built upon the slice\-based representation of potential branches during a certain temporal span, we can exploit a DAG to characterize the forecasting space\. Specifically, we model the historical time series𝒙1:t\\boldsymbol\{x\}\_\{1:t\}as the root, and each slicebjkb\_\{j\}^\{k\}as the subsequent node\. Next, directed edges indicating the temporal dependency are added from the root to the initial sliceb1kb\_\{1\}^\{k\}, and from the slicebjkb\_\{j\}^\{k\}to the consecutive onebj\+1kb\_\{j\+1\}^\{k\}fork∈\[K\]k\\in\[K\]andj∈\[n−1\]j\\in\[n\-1\]\.

#### Alignment

This stage aligns similar nodes in the constructed DAG for revealing the intrinsic topology of the forecasting space\. Two nodes are considered similar if their corresponding slices share the same indexjj, and the DTW distance of these two slices is below a thresholdτ∈ℝ∗\\tau\\in\\mathbb\{R\}^\{\*\}\. We setτ\\tauby leveraging the seasonal\-scaled error[Hyndman and Athanasopoulos \(2018\)](https://arxiv.org/html/2609.28582#bib.bib36)as

τ=λt−S​∑i=1t−S\|xi−xi\+S\|,\\tau=\\frac\{\\lambda\}\{t\-S\}\\sum\_\{i=1\}^\{t\-S\}\{\|\}x\_\{i\}\-x\_\{i\+S\}\|\\ ,whereλ∈ℝ∗\\lambda\\in\\mathbb\{R\}^\{\*\}is the threshold coefficient andS∈\[t−1\]S\\in\[t\-1\]is seasonality parameter[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9)inherent in most datasets for TSFMs\. Such a setting roughly matches the similarity threshold scale to that of MASE, which varies significantly across different time series or slices, thereby greatly simplifying the tuning ofτ\\tau\. After merging similar nodes, one can obtain an aligned DAGG=\(V,E\)G=\(V,E\), whereVVandEEdenote the collections of vertices and edges, respectively\.

#### Graph Complexity Estimation

Given the aligned DAGGG, we construct graph complexity by integrating both TSFM\-inherent stochasticity and topological information\. The key idea of leveraging TSFM\-inherent stochasticity is to estimate the time\-step\-level entropy as

U⁡\(x^t\+sk\)=−𝔼X∼Pt\+sk​log⁡Pt\+sk​\(X\),U\(\\hat\{x\}\_\{t\+s\}^\{k\}\)=\-\\mathbb\{E\}\_\{X\\sim P\_\{t\+s\}^\{k\}\}\\log P\_\{t\+s\}^\{k\}\(X\)\\ ,wherePt\+sk​\(⋅\)P^\{k\}\_\{t\+s\}\(\\cdot\)denotes the time\-step\-level distribution at time stepssfor thekk\-th forecast trajectory sample\. Hence, it suffices to derive this time\-step\-level distribution\. Since vocabulary\-based TSFMs directly output such a distribution, we can obtain the distribution concerned without incurring any additional computations\. For quantile\-based TSFMs, we use thess\-th component of the constructed inverse CDF, i\.e\.,Ft\+sF\_\{t\+s\}, to obtain time\-step\-level distribution asPt\+sk​\(x^\)=dd​x​Ft\+s−1​\(x^\)P^\{k\}\_\{t\+s\}\(\\hat\{x\}\)=\\tfrac\{\\mathrm\{d\}\}\{\\mathrm\{d\}x\}F\_\{t\+s\}^\{\-1\}\(\\hat\{x\}\)\. For trajectory\-based TSFMs, we leverage kernel density estimation\([Rosenblatt, 1956](https://arxiv.org/html/2609.28582#bib.bib33)\)to approximate the time\-step\-level distribution as

Pt\+sk​\(x^\)=1K​ηt\+s​∑i=1K𝒦⁡\(x^−x^t\+siηt\+s\),P^\{k\}\_\{t\+s\}\(\\hat\{x\}\)=\\frac\{1\}\{K\\eta\_\{t\+s\}\}\\sum\_\{i=1\}^\{K\}\\mathcal\{K\}\\left\(\\frac\{\\hat\{x\}\-\\hat\{x\}\_\{t\+s\}^\{i\}\}\{\\eta\_\{t\+s\}\}\\right\)\\ ,where𝒦\(x\)=\(2π\)−1/2exp\(−x2/2\)\\mathcal\{K\}\(x\)=\(2\\pi\)^\{\-1/2\}\\exp\(\-x^\{2\}/2\)is the Gaussian kernel andηt\+s∈ℝ∗\\eta\_\{t\+s\}\\in\\mathbb\{R\}^\{\*\}denotes the bandwidth\. Specifically, we follow Scott’s rule[Scott \(1992\)](https://arxiv.org/html/2609.28582#bib.bib34)and setηt\+s=K−1/5σt\+s\\eta\_\{t\+s\}=K^\{\-1/5\}\\sigma\_\{t\+s\}, whereσt\+s\\sigma\_\{t\+s\}denotes the standard deviation of\{x^t\+sk\}k=1K\\\{\\hat\{x\}\_\{t\+s\}^\{k\}\\\}\_\{k=1\}^\{K\}\. Next, we quantify the slice\-level uncertainty as

U⁡\(bjk\)=1ls​∑s=ls​\(j−1\)\+1ls​jU⁡\(x^t\+sk\),U\\left\(b^\{k\}\_\{j\}\\right\)=\\frac\{1\}\{l\_\{s\}\}\\\!\\sum\_\{s=l\_\{s\}\(j\-1\)\+1\}^\{l\_\{s\}j\}\\\!\{U\\left\(\\hat\{x\}\_\{t\+s\}^\{k\}\\right\)\}\\ ,where a largerU⁡\(bjk\)U\(b^\{k\}\_\{j\}\)indicates higher TSFM\-inherent stochasticity over this slice’s temporal span\. This value is then treated as the uncertainty of the slice’s corresponding node\. Now, we integrate both topological information and TSFM\-inherent stochasticity by exploiting the Bonacich–Lloyd alpha\-centrality[Bonacich and Lloyd \(2001\)](https://arxiv.org/html/2609.28582#bib.bib32)as

B⁡\(v\)=U⁡\(v\)\+α​∑\(w,v\)∈EB⁡\(w\),B\(v\)=U\(v\)\+\\alpha\\\!\\sum\_\{\(w,v\)\\in E\}\\\!B\(w\)\\ ,whereB⁡\(v\)B\(v\)is the centrality score of nodevv,wwdenotes any predecessor ofvv,U⁡\(v\)U\(v\)indicates the node uncertainty calculated by the aforementioned process, andα\\alphais the attenuation factor that is typically set to0\.10\.1[Bucur and Holme \(2020\)](https://arxiv.org/html/2609.28582#bib.bib31)\. Thus, a large value ofB⁡\(v\)B\(v\)reflects not only high slice\-level TSFM\-inherent stochasticity but also high local topological complexity, since nodevvreceives connections from numerous predecessors and accumulates their centrality\. The graph complexity is derived by summing all uncertainty\-aware node centrality over the DAGGGas

GC​\(G\)=∑v∈VB​\(v\)\.\\textrm\{GC\}\(G\)=\\sum\_\{v\\in V\}B\(v\)\\ \.Therefore, a large value ofGC​\(G\)\\textrm\{GC\}\(G\)reflects both high TSFM\-inherent stochasticity during multi\-step forecasting and a complex topology of the whole forecasting space\. Algorithms[1](https://arxiv.org/html/2609.28582#alg1)and[2](https://arxiv.org/html/2609.28582#alg2)summarize the aforementioned SGA and graph complexity estimation procedures, respectively, wheredDTWd\_\{\\text\{DTW\}\}denotes the computation of DTW distance\.

Algorithm 1The SGA AlgorithmInput:historical time series𝒙1:t\\boldsymbol\{x\}\_\{1:t\}, forecast horizonhh, sampling timesKK, distance thresholdτ\\tau Output:GraphG=\(V,E\)G=\(V,E\) Procedures:

1:Initialize

G=\(V,E\)G=\(V,E\)with root node

𝒙1:t\\boldsymbol\{x\}\_\{1:t\}
2:Initialize

U\(𝒙1:t\)←0U\(\\boldsymbol\{x\}\_\{1:t\}\)\\leftarrow 0
3:for

k∈\[K\]k\\in\[K\]do

4:Sample a forecast trajectory

𝒙^t\+1:t\+hk\\hat\{\\boldsymbol\{x\}\}\_\{t\+1:t\+h\}^\{k\}
5:Segment it into slices

𝒃1:nk=\(b1k,…,bnk\)\\boldsymbol\{b\}\_\{1:n\}^\{k\}=\(b\_\{1\}^\{k\},\\ldots,b\_\{n\}^\{k\}\)
6:Calculate slice\-level uncertainty

U⁡\(bjk\)U\(b\_\{j\}^\{k\}\)
7:Initialize

U​C​\(bjk\)←\{U⁡\(bjk\)\}UC\(b\_\{j\}^\{k\}\)\\leftarrow\\\{U\(b\_\{j\}^\{k\}\)\\\}for all

j∈\[n\]j\\in\[n\]
8:

V←V∪\{bjk\}j∈\[n\]V\\leftarrow V\\cup\\\{b\_\{j\}^\{k\}\\\}\_\{j\\in\[n\]\}and

E←E∪\{\(𝒙1:t,b1k\)\}E\\leftarrow E\\cup\\\{\(\\boldsymbol\{x\}\_\{1:t\},b\_\{1\}^\{k\}\)\\\}
9:

E←E∪\{\(bjk,bj\+1k\)\}j∈\[n−1\]E\\leftarrow E\\cup\\\{\(b\_\{j\}^\{k\},b\_\{j\+1\}^\{k\}\)\\\}\_\{j\\in\[n\-1\]\}
10:endfor

11:Initialize processed node set

Vp←∅V\_\{p\}\\leftarrow\\emptyset⊳\\trianglerightStart alignment

12:for

j∈\[n\]j\\in\[n\]do

13:

Vp←Vp∪\{bjk\}k∈\[K\]V\_\{p\}\\leftarrow V\_\{p\}\\cup\\\{b\_\{j\}^\{k\}\\\}\_\{k\\in\[K\]\}
14:Initialize node set

Va←\{bjk\}k∈\[K\]V\_\{a\}\\leftarrow\\\{b\_\{j\}^\{k\}\\\}\_\{k\\in\[K\]\}for alignment

15:for

v∈Vav\\in V\_\{a\}do

16:

Va←Va∖\{v\}V\_\{a\}\\leftarrow V\_\{a\}\\setminus\\\{v\\\}
17:for

w∈Vaw\\in V\_\{a\}do

18:if

dDTW​\(v,w\)≤τd\_\{\\text\{DTW\}\}\(v,w\)\\leq\\tauthen

19:

U​C​\(v\)←U​C​\(v\)∪U​C​\(w\)UC\(v\)\\leftarrow UC\(v\)\\cup UC\(w\)
20:Get the parent

ppof

wwwhere

\(p,w\)∈E\(p,w\)\\in E
21:Get the child

ccof

wwwhere

\(w,c\)∈E\(w,c\)\\in E
22:

E←E∪\{\(p,v\),\(v,c\)\}E\\leftarrow E\\cup\\\{\(p,v\),\(v,c\)\\\}⊳\\trianglerightMerge nodes

23:

Vp←Vp∖\{w\}V\_\{p\}\\leftarrow V\_\{p\}\\setminus\\\{w\\\}and

Va←Va∖\{w\}V\_\{a\}\\leftarrow V\_\{a\}\\setminus\\\{w\\\}
24:endif

25:endfor

26:endfor

27:endfor

28:

U​\(v\)←Average​\(U​C​\(v\)\)U\(v\)\\leftarrow\\text\{Average\}\(UC\(v\)\)for all

v∈Vpv\\in V\_\{p\}
29:Update

GGto retain only nodes in

Vp∪\{𝒙1:t\}V\_\{p\}\\cup\\\{\\boldsymbol\{x\}\_\{1:t\}\\\}

Algorithm 2Graph Complexity EstimationInput:GraphG=\(V,E\)G=\(V,E\), node uncertainty\{U⁡\(v\)\}v∈V\\\{U\(v\)\\\}\_\{v\\in V\}, attenuation factorα\\alpha Output:Graph complexityGC​\(G\)\\text\{GC\}\(G\) Procedures:

1:

B⁡\[v\]←U⁡\(v\)B\[v\]\\leftarrow U\(v\)for all

v∈Vv\\in V
2:

L←TopologicalSort​\(G\)L\\leftarrow\\text\{TopologicalSort\}\(G\)⊳\\trianglerightGet traversal order

3:for

v∈Lv\\in Ldo

4:

B⁡\[v\]←B⁡\[v\]\+α​∑\(w,v\)∈EB⁡\[w\]B\[v\]\\leftarrow B\[v\]\+\\alpha\\sum\_\{\(w,v\)\\in E\}B\[w\]
5:endfor

6:

GC​\(G\)←∑v∈VB​\[v\]\\text\{GC\}\(G\)\\leftarrow\\sum\_\{v\\in V\}B\[v\]

## 4Experiments

This section conducts experiments to answer three questions of whether and to what extent \(Q1\) the proposed SGA outperforms existing UQ methods in ranking predictive errors of multi\-step forecasts, \(Q2\) SGA covers a broader forecasting space than existing UQ methods in sampling, and \(Q3\) model complexity of TSFMs affects their uncertainty estimates\.

#### Configurations

The evaluated TSFMs can be categorized into three types according to the output formats, including \(i\) vocabulary\-based models such as Chronos\-T5 \(abbreviated as C\-T5\) family[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9)that are denoted by \-Tiny, \-Mini, \-Small, \-Base, and \-Large over increasing parameter sizes, \(ii\) quantile\-based models such as Chronos\-2 \(denoted as C\-2\)[Ansari et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib10), TimesFM\-2\.5[Das et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib11)and Timer\-S1[Liu et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib12), and \(iii\) trajectory\-based models like Sundial[Liu et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib13)and Aurora[Wu et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib14)\. All TSFMs with various UQ methods were evaluated on 27 datasets, which come from Chronos Benchmark II[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9)\. Experiments were conducted on NVIDIA RTX 5090 32GB GPUs×\\times4\.

We extended two types of UQ methods for HUQ as contenders\. First, UQ methods designed for one\-step forecasting, such as Native Calibration \(NC\)[Adler et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib15), are extended in HUQ by averaging their step\-wise uncertainties to obtain an overall uncertainty estimate of a multi\-step forecast\. Second, we selected 8 representative sequence\-modeling\-based UQ methods developed for LLMs, including information\-based methods such as Perplexity \(Ppl\)\([Fomicheva et al\., 2020](https://arxiv.org/html/2609.28582#bib.bib16)\)and Predictive Entropy \(PE\)\([Malinin and Gales, 2021](https://arxiv.org/html/2609.28582#bib.bib23)\), and the diversity\-based methods such as Eccentricity \(Ecc\), Sum of Eigenvalues \(Eig\), Degree matrix \(Deg\) of Graph Laplacian\([Lin et al\., 2024](https://arxiv.org/html/2609.28582#bib.bib21)\), Semantic Density \(SD\)\([Qiu and Miikkulainen, 2024](https://arxiv.org/html/2609.28582#bib.bib22)\), Shifting Attention to Relevance \(SAR\)\([Duan et al\., 2024](https://arxiv.org/html/2609.28582#bib.bib18)\), and Semantic Entropy \(SE\)\([Farquhar et al\., 2024](https://arxiv.org/html/2609.28582#bib.bib17)\)\. We adapt these methods by first treating each forecast trajectory as a sequence of forecast values, and then replacing the standard similarity measure between token sequences with a DTW\-based similarity between forecast trajectories𝒙^\\hat\{\\boldsymbol\{x\}\}and𝒙^′\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}, computed ass⁡\(𝒙^,𝒙^′\)=exp⁡\(−dDTW​\(𝒙^,𝒙^′\)\)s\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)=\\exp\(\-d\_\{\\text\{DTW\}\}\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)\)\. Moreover, we consider the UQ baseline that randomly samples uncertainty estimates uniformly from the interval\[0,1\]\[0,1\], denoted as “Random \(Rnd\)”\. Details regarding contenders are provided in Appendix[B\.3](https://arxiv.org/html/2609.28582#A2.SS3)\.

For evaluations of the performance of the point forecast trajectory, we follow[Ansari et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib10)and adopt the Mean Absolute Scaled Error \(MASE\) metric\. Following seminal studies\([Farquhar et al\., 2024](https://arxiv.org/html/2609.28582#bib.bib17);[Lin et al\., 2024](https://arxiv.org/html/2609.28582#bib.bib21)\), we evaluate a UQ method via the downstream task of selective prediction\([Geifman and El\-Yaniv, 2017](https://arxiv.org/html/2609.28582#bib.bib24)\), which reflects its ability to rank predictions by uncertainty such that highly erroneous predictions are assigned high uncertainty estimates\. To quantify such ability, we employ Normalized Excess Area Under the Risk\-Coverage Curve \(NEAURC\), which is a normalized variant of Excess Area Under the Risk\-Coverage Curve \(EAURC\)\([Geifman et al\., 2019](https://arxiv.org/html/2609.28582#bib.bib26)\)\. The lower the NEAURC, the better the ranking performance\. We bootstrap datasets 1000 times and report the mean and standard deviation \(std\) of NEAURC\. Notably, the NEAURC of Rnd is theoretically constant at100%100\\%\. Appendix[B\.1](https://arxiv.org/html/2609.28582#A2.SS1)provides details regarding evaluations\.

UQVocabulary\-basedQuantile\-basedTrajectory\-basedC\-T5\-TinyC\-T5\-MiniC\-T5\-SmallC\-T5\-BaseC\-T5\-LargeC\-2\-SmallC\-2TimesFM\-2\.5Timer\-S1SundialAuroraRnd100\.000\.00100\.000\.00100\.000\.00100\.000\.00100\.000\.00100\.000\.00100\.000\.00100\.000\.00100\.000\.00100\.000\.00100\.000\.00NC93\.7211\.01100\.319\.7494\.9310\.7595\.2211\.0997\.9911\.8188\.2310\.7989\.8111\.5993\.9412\.0096\.9910\.7093\.9311\.37100\.3110\.35Ppl89\.3311\.5287\.8110\.2492\.2112\.8288\.8812\.1693\.4311\.4286\.9110\.5584\.7410\.4187\.0311\.8193\.119\.9693\.4611\.14100\.1410\.89PE84\.2011\.6284\.619\.1490\.8912\.6689\.4710\.4491\.7711\.8691\.2110\.5388\.8011\.2188\.1011\.6493\.9110\.1496\.0711\.91101\.8310\.77Eig96\.7311\.6599\.7010\.5495\.0110\.7394\.5510\.7697\.6211\.3590\.9310\.7791\.3711\.2799\.2111\.8896\.2310\.3398\.1011\.94102\.2011\.10Ecc95\.5010\.8494\.4110\.4694\.4610\.6996\.2910\.0898\.8611\.1289\.4111\.0893\.9710\.4696\.6511\.5997\.6610\.3097\.3912\.41101\.3010\.93Deg96\.2811\.7399\.4910\.4194\.3010\.7294\.1210\.8097\.4511\.4891\.1310\.7091\.5211\.3499\.5012\.0896\.2810\.4397\.8511\.91101\.9911\.02SD89\.9311\.8588\.5211\.7892\.6613\.2689\.4912\.6494\.7612\.6689\.659\.7390\.3610\.3194\.7911\.8398\.579\.9599\.5611\.55105\.5110\.56SAR86\.2811\.6092\.6910\.3591\.1711\.5290\.7310\.8890\.2511\.8390\.1610\.3388\.7810\.9388\.2211\.3493\.309\.9494\.5111\.34101\.8411\.30SE84\.8111\.2784\.669\.3791\.9212\.9790\.9410\.2092\.6111\.3090\.4610\.2287\.1410\.7787\.3411\.4793\.4710\.1995\.0511\.47102\.1311\.10SGA58\.898\.5260\.307\.9361\.237\.2361\.338\.1262\.209\.5349\.818\.2244\.876\.3149\.757\.4947\.766\.2051\.017\.1257\.747\.01

Table 1:Comparisons of the overall NEAURC of SGA and its contenders, averaged across 27 datasets and 11 TSFMs of 3 types, where bold and underlined values denote the best and second\-best results, respectively\.
#### Verifications on SGA

Table[1](https://arxiv.org/html/2609.28582#S4.T1)presents the comparisons of the overall NEAURC of SGA and its contenders, averaged across 27 datasets and 11 TSFMs of 3 types, where bold and underlined values denote the best and second\-best results, respectively\. There are two key observations\. First, all contenders yield comparable overall performance to that of the Rnd baseline\. Specifically, according to the3​σ3\\sigmarule in statistics[Moore et al\. \(2017\)](https://arxiv.org/html/2609.28582#bib.bib28), a UQ method is considered to significantly outperform the Rnd baseline if its performance exceeds the baseline by three standard deviations, a criterion that all contenders fail to meet\. Therefore, we can conclude that all contenders typically exhibit near\-random ranking ability\. Second, it is obvious that SGA outperforms the strongest contenders by34\.49%34\.49\\%in NEAURC and meets the3​σ3\\sigmarule\. This observation indicates that our proposed SGA can effectively assign high uncertainty to a forecast trajectory with high MASE, achieving a favorable ranking ability, and thus answers Q1\.

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Overlap/main.png)Figure 2:Visualized comparisons of the sampling coverage between SGA and NC across 11 TSFMs of 3 types\.TypeTSFMASGA∩ANCANC\\tfrac\{A\_\{\\text\{SGA\}\}\\cap A\_\{\\text\{NC\}\}\}\{A\_\{\\text\{NC\}\}\}ASGA∩ANCASGA\\tfrac\{A\_\{\\text\{SGA\}\}\\cap A\_\{\\text\{NC\}\}\}\{A\_\{\\text\{SGA\}\}\}Vocabulary\-basedC\-T5\-Tiny100\.0050\.24C\-T5\-Mini100\.0049\.47C\-T5\-Small100\.0048\.68C\-T5\-Base100\.0048\.75C\-T5\-Large100\.0048\.77Quantile\-basedC\-2\-Small91\.4093\.51C\-291\.0293\.86TimesFM\-2\.592\.2893\.31Timer\-S191\.1893\.26Trajectory\-basedSundial100\.0046\.81Aurora100\.0034\.64Table 2:Comparisons of the overall overlap area ratios for 11 TSFMs of 3 types, averaged across 27 datasets\.
#### Comparisons of Sampling Coverage

The sampling coverageCCof a UQ methodaais a set of step\-wise coverage intervalsI⁡\(a\)I\(a\)defined as

C⁡\(a\)=\{It\+s​\(a\)\}s∈\[h\],C\(a\)=\\left\\\{I\_\{t\+s\}\\left\(a\\right\)\\right\\\}\_\{s\\in\[h\]\}\\ ,where

It\+s​\(a\)≜\[mink∈\[K\]⁡\{x^t\+sk​\(a\)\},maxk∈\[K\]⁡\{x^t\+sk​\(a\)\}\],I\_\{t\+s\}\\left\(a\\right\)\\triangleq\\left\[\\min\_\{k\\in\[K\]\}\\left\\\{\\hat\{x\}\_\{t\+s\}^\{k\}\(a\)\\right\\\},\\max\_\{k\\in\[K\]\}\\left\\\{\\hat\{x\}\_\{t\+s\}^\{k\}\(a\)\\right\\\}\\right\]\\ ,andx^t\+sk​\(a\)\\hat\{x\}\_\{t\+s\}^\{k\}\(a\)denotes thekk\-th forecast sample value at time stepss, drawn with UQ methodaa\. A broader sampling coverage indicates that a method explores a more extensive region of the forecasting space\. Therefore, such a method can contribute to describing the forecasting space more precisely\. Figure[2](https://arxiv.org/html/2609.28582#S4.F2)visualizes the comparisons of the sampling coverage between SGA and its contender NC across 11 TSFMs of 3 types, with the former shown in red and the latter in blue\. It is obvious that the red regions typically cover their blue counterparts, which suggests that SGA achieves a broader sampling coverage than NC\. To further quantify such comparisons, we define the areaA∈ℝA\\in\\mathbb\{R\}of the sampling coverage as the sum of widths of the coverage intervals over horizonhh, that is,

A⁡\(C⁡\(a\)\)=∑s=1h\(maxk∈\[K\]⁡\{x^t\+sk​\(a\)\}−mink∈\[K\]⁡\{x^t\+sk​\(a\)\}\),A\(C\(a\)\)=\\sum\_\{s=1\}^\{h\}\{\\left\(\\max\_\{k\\in\[K\]\}\\left\\\{\\hat\{x\}\_\{t\+s\}^\{k\}\(a\)\\right\\\}\-\\min\_\{k\\in\[K\]\}\\left\\\{\\hat\{x\}\_\{t\+s\}^\{k\}\(a\)\\right\\\}\\right\)\}\\ ,where a larger area indicates a broader sampling coverage\. LetASGAA\_\{\\text\{SGA\}\}andANCA\_\{\\text\{NC\}\}denote the sampling coverage areas of the sampling mechanisms inherent in SGA and its contender NC, respectively\. We define the overlap in sampling coverage between SGA and NC as the sum of the intersection widths of their coverage intervals across horizonhh, that is,

ASGA∩ANC≜∑s=1h\|It\+s​\(SGA\)∩It\+s​\(NC\)\|\.A\_\{\\text\{SGA\}\}\\cap A\_\{\\textrm\{NC\}\}\\triangleq\\sum\_\{s=1\}^\{h\}\{\|I\_\{t\+s\}\\left\(\\text\{SGA\}\\right\)\\cap I\_\{t\+s\}\\left\(\\textrm\{NC\}\\right\)\|\}\\ \.Thus, a higher ratio\(ASGA∩ANC\)/ANC\(A\_\{\\text\{SGA\}\}\\cap A\_\{\\text\{NC\}\}\)/A\_\{\\text\{NC\}\}indicates that SGA envelops a larger portion of NC, while a lower ratio\(ASGA∩ANC\)/ASGA\(A\_\{\\text\{SGA\}\}\\cap A\_\{\\text\{NC\}\}\)/A\_\{\\text\{SGA\}\}implies that SGA can explore a more extensive area beyond the enclosed NC\. Table[2](https://arxiv.org/html/2609.28582#S4.T2)shows the comparisons of the overall overlap area ratios for 11 TSFMs of 3 types, averaged across 27 datasets\. There are two key observations\. First, all ratios in the first column exceed 90%, indicating that SGA encompasses the vast majority of NC sampling coverage\. Second, the ratios in the second column are typically low, especially for vocabulary\-based and trajectory\-based TSFMs, suggesting that SGA covers a much broader region beyond NC’s existing sampling coverage\. Overall, the aforementioned observations answer Q2\.

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Scaling/main.png)Figure 3:Plots of averaged uncertainty versus ARMASE of TSFMs over diverse scales, averaged across 27 datasets\.
#### Relation between Uncertainty and Scale

Figure[3](https://arxiv.org/html/2609.28582#S4.F3)shows the plots of averaged uncertainty versus Aggregated Relative Mean Absolute Scaled Error \(ARMASE\)[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9)of TSFMs over diverse scales, averaged across 27 datasets, where computations of these metrics and results for each TSFM\-dataset pair are detailed in Appendices[B\.1](https://arxiv.org/html/2609.28582#A2.SS1)and[B\.5](https://arxiv.org/html/2609.28582#A2.SS5), respectively\. It is observed that larger circles tend to appear in the bottom\-left region, whereas smaller circles are generally located in the top\-right, indicating that TSFMs with larger scales usually achieve lower predictive errors and lower uncertainty estimates\. This observation reveals two conclusions\. First, TSFMs with larger scales usually achieve higher forecasting accuracy, which is consistent with the scaling law of model performance\([Kaplan et al\., 2020](https://arxiv.org/html/2609.28582#bib.bib35)\)\. Second, larger\-scale TSFMs tend to produce multi\-step forecasts with lower uncertainty estimates\. This insight reveals an empirical scaling law of TSFM uncertainty, thus answering Q3\.

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Ablation/main.png)Figure 4:Ablation comparison of the overall performance of SGA on 11 TSFMs of 3 types, averaged across 27 datasets\.![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Sensitivity/main.png)Figure 5:Impact of the number of samplesKK\(left\), the slicing lengthlsl\_\{s\}\(middle\), and the threshold coefficientλ\\lambda\(right\) on the overall performance of SGA for 11 TSFMs of 3 types, averaged across 27 datasets\.
#### Ablation Analyses

Figure[4](https://arxiv.org/html/2609.28582#S4.F4)shows the ablation comparisons of the overall performance of SGA on 11 TSFMs of 3 types, averaged across 27 datasets\. There are two key observations\. First, the green bars are significantly taller than the blue bars, demonstrating that node alignment is critical to the performance of SGA\. Second, the red bars are slightly taller than their blue counterparts, which suggests that when applying SGA to selective prediction, topological information contributes more significantly to the construction of graph complexity than TSFM\-inherent stochasticity\. Results for each TSFM\-dataset pair are provided in Appendix[B\.6](https://arxiv.org/html/2609.28582#A2.SS6)\.

#### Sensitivity Analyses

Our proposed SGA involves three key hyperparameters, that is, the number of samplesKK, the slicing lengthlsl\_\{s\}, and the threshold coefficientλ\\lambda\. The left, middle, and right panels of Figure[5](https://arxiv.org/html/2609.28582#S4.F5)separately show the impact ofKK,lsl\_\{s\}, andλ\\lambdaon the overall performance of SGA for 11 TSFMs of 3 types, averaged across 27 datasets\. Since a largerKKleads to higher computational costs, we recommendK=20K=20for the C\-2 family, TimesFM\-2\.5, Sundial, and Aurora, andK=30K=30for the C\-T5 family and Timer\-S1 to achieve a favorable trade\-off between efficiency and performance\. Moreover, we recommendls=4l\_\{s\}=4andλ=0\.25\\lambda=0\.25for all 11 TSFMs according to the overall performance\. Appendix[B\.7](https://arxiv.org/html/2609.28582#A2.SS7)provides individual results on each dataset\.

## 5Conclusions

In this paper, we investigated the task of quantifying the uncertainty of multi\-step forecasts in TSFMs\. This work resorts to characterizing the topology of all potential forecast branches of TSFMs using a DAG, such that the graph complexity bounds the uncertainty of multi\-step forecasts, and then precisely measures the graph complexity by integrating both topological information and TSFM\-inherent stochasticity\. Empirical results across 11 TSFMs and 27 datasets not only demonstrated the effectiveness of our method, but also revealed an empirical scaling law for multi\-step forecasting uncertainty, where a larger parameter scale of TSFMs typically leads to lower uncertainty estimates\.

## Acknowledgments

This research was supported by the National Science Foundation of China \(62406138, 62632005\) and Nanjing University – Siemens Joint Research Center on Industrial AI\.

## Appendix

This appendix provides the supplementary materials for our work “SGA: Uncertainty Quantification for Multi\-Step Forecasting in Time Series Foundation Models”, constructed according to the corresponding sections therein\.

## Appendix AAdditional Implementation Details of SGA

This section provides additional implementation details of the proposed SGA method, introducing the configurations of hyperparameters in this work\.

#### Configuration of Forecasting and Sampling Parameters

For each time series, we use the most recent history as the model input\. The length of the input is set tot=min⁡\(lh,512\)t=\\min\(l\_\{h\},512\), wherelhl\_\{h\}denotes the length of the historical time series available before the forecast horizon\.

- •For Evaluating Predictive Error\.The point forecast used to compute MASE is obtained according to the native output form of each model\. For vocabulary\-based and trajectory\-based models, we use 20 forecast samples to obtain the point forecast\. For quantile\-based models, we use the median forecast \(50%50\\%quantile\)\.
- •For Evaluating Uncertainty\.Following existing TSFM studies\([Ansari et al\., 2024](https://arxiv.org/html/2609.28582#bib.bib9);[Liu et al\., 2025](https://arxiv.org/html/2609.28582#bib.bib13)\), which generate 20 forecast trajectory samples for probabilistic forecasting, we drawK=20K=20samples from each model for quantifying horizon uncertainty\. In the experiments of comparisons of sampling coverage, we sampleK=50K=50forecasts to showcase the difference in sampling coverage both for SGA and for NC when NC is applied to vocabulary\-based and trajectory\-based models\. Sampling parameters vary for three types of TSFMs\. For the vocabulary\-output models considered in our experiments, namely the Chronos\-T5 family, autoregressive sampling is performed with a temperature ofT=1\.0T=1\.0and top\-k=50k=50\. For quantile\-output models, namely the Chronos\-2 family, TimesFM\-2\.5, and Timer\-S1, we setqmin=0\.1q\_\{\\textrm\{min\}\}=0\.1andqmax=0\.9q\_\{\\textrm\{max\}\}=0\.9to draw a quantile levelq∈ℝq\\in\\mathbb\{R\}from the uniform distribution𝒰⁡\(0\.1,0\.9\)\\mathcal\{U\}\(0\.1,0\.9\), since\[0\.1,0\.9\]\[0\.1,0\.9\]is the widest quantile\-level range shared by all four quantile\-based models\. To balance efficiency and performance, we construct an 8\-step joint inverse CDFF:ℝ→ℝ8F:\\mathbb\{R\}\\to\\mathbb\{R\}^\{8\}and sample the next 8 consecutive forecast valuesF⁡\(q\)∈ℝ8F\(q\)\\in\\mathbb\{R\}^\{8\}\. This procedure is repeated autoregressively until the complete forecast trajectory is generated\. For trajectory\-output models, namely Sundial and Aurora, we retain their default sampling configurations and directly generateK=20K=20forecast trajectory samples\.

#### Configuration of SGA

The hyperparameters in SGA include the slicing lengthlsl\_\{s\}and the DTW threshold coefficientλ\\lambda\. For our implementation, we setls=4l\_\{s\}=4andλ=0\.25\\lambda=0\.25\. The middle and right panels of Figure[5](https://arxiv.org/html/2609.28582#S4.F5)in Section[4](https://arxiv.org/html/2609.28582#S4)show the impact oflsl\_\{s\}andλ\\lambdaon the overall performance of SGA for 11 TSFMs of 3 types, respectively\. It can be observed that SGA is relatively insensitive to small variations inlsl\_\{s\}andλ\\lambdaaround our default configuration and maintains stable performance across a broad range of reasonable settings, which further demonstrates the robustness of SGA\. Whenlsl\_\{s\}does not divide the forecast horizonhh, we set the number of slices asn=⌈h/ls⌉n=\\lceil h/l\_\{s\}\\rceil, where the symbol⌈z⌉\\lceil z\\rceilindicates the smallest integer not less thanz∈ℝz\\in\\mathbb\{R\}\. The last slice is then defined asbnk=\(x^t\+ls​\(n−1\)\+1k,…,x^t\+hk\)b\_\{n\}^\{k\}=\(\\hat\{x\}\_\{t\+l\_\{s\}\(n\-1\)\+1\}^\{k\},\\ldots,\\hat\{x\}\_\{t\+h\}^\{k\}\), which contains the remaining forecast values\.

## Appendix BAdditional Experimental Details on SGA

This section provides additional experimental details of the proposed SGA method\.

### B\.1Evaluations

#### NEAURC

Following seminal studies[Farquhar et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib17), we evaluate a UQ method via the downstream task of selective prediction\([Geifman and El\-Yaniv, 2019](https://arxiv.org/html/2609.28582#bib.bib25)\), where a time series prediction with higher uncertainty is expected to incur a higher predictive error\. We adopt the Normalized Excess Area Under the Risk\-Coverage Curve \(NEAURC\), a normalized variant of Excess Area Under the Risk\-Coverage Curve \(EAURC\)[Geifman et al\. \(2019\)](https://arxiv.org/html/2609.28582#bib.bib26), as a metric to evaluate the performance of a UQ method on such a downstream task\. The lower the NEAURC, the better the performance\. Specifically, NEAURC normalizes the traditional EAURC by calculating the ratio of the difference between the evaluated estimator and the oracle estimator to the difference between the random baseline and the oracle estimator, where the random baseline samples uncertainty estimates from\[0,1\]\[0,1\]uniformly, and the oracle estimator always assigns higher uncertainty to predictions with higher predictive error\. Therefore, NEAURC measures the extent to which the evaluated UQ estimator matches the ideal oracle ranking\.

Specifically, leteke\_\{k\}denote the error of thekk\-th forecast withinnnmulti\-step forecasts, which is measured by MASE in our experiments\. Letπ\\pibe the ordering induced by increasing uncertainty\. At the coverage levelmn\\frac\{m\}\{n\}forn∈ℕ\+n\\in\\mathbb\{N\}^\{\+\}andm∈\[n\]m\\in\[n\], the selective risk isRisk​\(mn\)=1m​∑j=1meπj\\text\{Risk\}\(\\frac\{m\}\{n\}\)=\\frac\{1\}\{m\}\\sum\_\{j=1\}^\{m\}e\_\{\\pi\_\{j\}\}\. The AURC is defined as the average selective risk over all coverage levels

AURC=1n​∑m=1nRisk​\(mn\)=1n​∑m=1n1m​∑j=1meπj\.\\text\{AURC\}=\\frac\{1\}\{n\}\\sum\_\{m=1\}^\{n\}\\text\{Risk\}\\left\(\\frac\{m\}\{n\}\\right\)=\\frac\{1\}\{n\}\\sum\_\{m=1\}^\{n\}\\frac\{1\}\{m\}\\sum\_\{j=1\}^\{m\}e\_\{\\pi\_\{j\}\}\\ \.Therefore, the calculation of NEAURC follows

NEAURC=AURC−AURCOracleAURCRandom−AURCOracle,\\text\{NEAURC\}=\\frac\{\\text\{AURC\}\-\\text\{AURC\}\_\{\\text\{Oracle\}\}\}\{\\text\{AURC\}\_\{\\text\{Random\}\}\-\\text\{AURC\}\_\{\\text\{Oracle\}\}\}\\ ,whereAURCRandom\\text\{AURC\}\_\{\\text\{Random\}\}andAURCOracle\\text\{AURC\}\_\{\\text\{Oracle\}\}denote the AURC of the random baseline and the oracle estimator, respectively\.

#### ARMASE

Following[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9), we adopt the aggregated relative MASE for evaluating the overall performance of point forecasts across multiple datasets\. We first compute the relative MASE of each model on each dataset by dividing its MASE by that of a baseline model\. Chronos\-T5\-Tiny and Chronos\-2\-Small serve as the baseline models for the Chronos\-T5 and Chronos\-2 families, respectively\. Next, we aggregate relative MASE across all datasets using geometric mean, deriving the aggregated relative MASE\. Specifically, given the historical time series𝒙1:t\\boldsymbol\{x\}\_\{1:t\}, the ground\-truth future time series𝒙t\+1:t\+h\\boldsymbol\{x\}\_\{t\+1:t\+h\}, and the point forecast𝒙^t\+1:t\+h\\hat\{\\boldsymbol\{x\}\}\_\{t\+1:t\+h\}, MASE is defined as

MASE=1h​∑s=1h\|x^t\+s−xt\+s\|1t−S​∑i=1t−S\|xi−xi\+S\|,\\text\{MASE\}=\\frac\{\\frac\{1\}\{h\}\\sum\_\{s=1\}^\{h\}\\left\|\\hat\{x\}\_\{t\+s\}\-x\_\{t\+s\}\\right\|\}\{\\frac\{1\}\{t\-S\}\\sum\_\{i=1\}^\{t\-S\}\\left\|x\_\{i\}\-x\_\{i\+S\}\\right\|\}\\ ,whereS∈\[t−1\]S\\in\[t\-1\]denotes the seasonality parameter\. Therefore, the calculation of ARMASE onnndatasets follows

ARMASE=\(∏i=1nMASE\(i\)MASEBaseline\(i\)\)1n,\\text\{ARMASE\}=\\left\(\\prod\_\{i=1\}^\{n\}\\frac\{\\text\{MASE\}^\{\(i\)\}\}\{\\text\{MASE\}\_\{\\text\{Baseline\}\}^\{\(i\)\}\}\\right\)^\{\\frac\{1\}\{n\}\}\\ ,whereMASE\(i\)\\text\{MASE\}^\{\(i\)\}denotes the MASE of the evaluated model on theii\-th dataset andMASEBaseline\(i\)\\text\{MASE\}\_\{\\text\{Baseline\}\}^\{\(i\)\}denotes the MASE of the baseline model on theii\-th dataset fori∈\[n\]i\\in\[n\]\.

### B\.2Details on Datasets

In this work, we conducted experiments on Chronos Benchmark II[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9), containing 27 datasets spanning several application domains including energy, finance and economics, healthcare, nature, retail, and mobility and transport\. These datasets cover a wide range of sampling frequencies, including 15\-minute, 30\-minute, hourly, daily, weekly, monthly, quarterly, and yearly frequencies\. Such diversity provides a comprehensive testbed for evaluating our proposed SGA in HUQ tasks across heterogeneous forecasting regimes\. For each dataset, we follow the Chronos Benchmark II evaluation protocol, where the finalhhobservations are held out as the forecast horizon and the preceding observations are used as historical time series\.

### B\.3Details on UQ Contenders

This subsection introduces the details of the investigated contenders\.

#### UQ Methods of One\-Step Forecasting

To adapt the native calibration method[Adler et al\. \(2026\)](https://arxiv.org/html/2609.28582#bib.bib15), we calculate the average of widths of central80%80\\%intervals with the10%10\\%and90%90\\%quantile forecasts from TSFMs\. For quantile\-based models, the quantiles used in NC are obtained directly from model outputs\. For vocabulary\-based and trajectory\-based models, we follow[Ansari et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib9);[Liu et al\. \(2025\)](https://arxiv.org/html/2609.28582#bib.bib13)and the official implementation of Chronos111https://github\.com/amazon\-science/chronos\-forecasting/blob/main/src/chronos/chronos\.py\#L513\-L533, and obtain each required quantile at each time step by linearly interpolating between adjacent sampled values after sorting these values\.

#### UQ Methods of LLMs

Table[3](https://arxiv.org/html/2609.28582#A2.T3)lists 8 representative sequence\-modeling\-based UQ methods for LLMs that we used as contenders in our experiments, where multiple sampling means the method needs to sample multiple sequences for UQ\. All multi\-sampling contenders use the same sampling procedure as SGA and obtainK=20K=20sampled forecast trajectories, ensuring a consistent sampling protocol\. Our implementations of these 8 methods are based on LM\-Polygraph222https://github\.com/IINemo/lm\-polygraph\([Vashurin et al\., 2025](https://arxiv.org/html/2609.28582#bib.bib27)\)\.

CategoryAbbr\.Full NameMultipleExploiting InherentLiteratureSampling?Stochasticity?Information\-basedPplPerplexity×\\times✓\\checkmark[Fomicheva et al\. \(2020\)](https://arxiv.org/html/2609.28582#bib.bib16)PEPredictive Entropy✓\\checkmark✓\\checkmark[Malinin and Gales \(2021\)](https://arxiv.org/html/2609.28582#bib.bib23)Diversity\-basedSESemantic Entropy✓\\checkmark✓\\checkmark[Farquhar et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib17)SARShifting Attention to Relevance✓\\checkmark✓\\checkmark[Duan et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib18)SDSemantic Density✓\\checkmark✓\\checkmark[Qiu and Miikkulainen \(2024\)](https://arxiv.org/html/2609.28582#bib.bib22)EigSum of Eigenvalues✓\\checkmark×\\times[Lin et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib21)DegDegree Matrix✓\\checkmark×\\times[Lin et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib21)EccEccentricity✓\\checkmark×\\times[Lin et al\. \(2024\)](https://arxiv.org/html/2609.28582#bib.bib21)Table 3:Overview of the UQ contenders for LLMs\.To adapt these methods to multi\-step forecasting, we treat each forecast trajectory as the input of the method, while replacing the token\-level distribution with the time\-step\-level distribution constructed in Subsection[3\.2](https://arxiv.org/html/2609.28582#S3.SS2)\. Table[4](https://arxiv.org/html/2609.28582#A2.T4)shows the correspondence between the key elements used in UQ for LLMs and HUQ for TSFMs\.

UQ for LLMsHUQ for TSFMsToken sequenceForecast trajectoryTokenForecast valueToken\-level distributionTime\-step\-level distributionSemantic equivalenceDTW\-based equivalenceSemantic similarityDTW\-based similarityTable 4:Correspondence between the key elements used in UQ for LLMs and HUQ for TSFMs\.For information\-based methods, we calculate the information\-theoretic measures directly after modeling forecast trajectories as sequences\. For diversity\-based methods, we additionally replace their text\-based measures of semantic equivalence or similarity with a DTW\-based criterion when measuring the diversity\. Specifically, let𝒙^\\hat\{\\boldsymbol\{x\}\}and𝒙^′\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}be two forecast trajectories, anddDTW​\(𝒙^,𝒙^′\)d\_\{\\text\{DTW\}\}\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)denotes the DTW distance between them\. Their DTW\-based equivalenceℰ⁡\(𝒙^,𝒙^′\)\\mathcal\{E\}\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)holds when their DTW distance does not exceed the thresholdτ\\tau, i\.e\.,ℰ\(𝒙^,𝒙^′\)=𝕀\{dDTW\(𝒙^,𝒙^′\)≤τ\}\\mathcal\{E\}\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)=\\mathbb\{I\}\\\{d\_\{\\text\{DTW\}\}\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)\\leq\\tau\\\}\. Their DTW\-based similaritys⁡\(𝒙^,𝒙^′\)s\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)is defined ass⁡\(𝒙^,𝒙^′\)=exp⁡\(−dDTW​\(𝒙^,𝒙^′\)\)s\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)=\\exp\(\-d\_\{\\text\{DTW\}\}\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)\)\. The further adaptations are listed as follows\.

- •SE\.We replace semantic equivalence between sentences with DTW\-based equivalenceℰ⁡\(𝒙^,𝒙^′\)\\mathcal\{E\}\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)between forecast trajectories\.
- •SAR\.We replace token\-level and sentence\-level relevance with DTW\-based forecast\-value\-level and forecast\-trajectory\-level relevance, respectively\. The forecast\-value\-level relevance of the valuex^′\\hat\{x\}^\{\\prime\}in theii\-th trajectory𝒙^i\\hat\{\\boldsymbol\{x\}\}^\{i\}is defined asri​\(x^′\)=1−s⁡\(𝒙^i,𝒙^i∖x^′\)r^\{i\}\(\\hat\{x\}^\{\\prime\}\)=1\-s\(\\hat\{\\boldsymbol\{x\}\}^\{i\},\\hat\{\\boldsymbol\{x\}\}^\{i\}\\setminus\\hat\{x\}^\{\\prime\}\), where𝒙^i∖x^′\\hat\{\\boldsymbol\{x\}\}^\{i\}\\setminus\\hat\{x\}^\{\\prime\}denotes the trajectory with the forecast valuex^′\\hat\{x\}^\{\\prime\}removed\. Then we compute the forecast\-trajectory\-level relevance with DTW\-based similarity∑j=1,j≠iKs⁡\(𝒙^i,𝒙^j\)​p​\(𝒙^j\)\\sum\_\{j=1,j\\neq i\}^\{K\}s\(\\hat\{\\boldsymbol\{x\}\}^\{i\},\\hat\{\\boldsymbol\{x\}\}^\{j\}\)p\(\\hat\{\\boldsymbol\{x\}\}^\{j\}\), wherep⁡\(𝒙^j\)p\(\\hat\{\\boldsymbol\{x\}\}^\{j\}\)is the forecast\-value\-level SAR probability of𝒙^j\\hat\{\\boldsymbol\{x\}\}^\{j\}and is derived from forecast\-value\-level relevance\.
- •SD\.We replace the semantic similarity kernel between sentences with a DTW\-based similarity kernel between forecast trajectories, which is defined as𝒦DTW​\(𝒙^,𝒙^′\)=s⁡\(𝒙^,𝒙^′\)\\mathcal\{K\}\_\{\\mathrm\{DTW\}\}\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)=s\(\\hat\{\\boldsymbol\{x\}\},\\hat\{\\boldsymbol\{x\}\}^\{\\prime\}\)\.
- •Eig/Deg/Ecc\.We replace the similarity matrix over text sequences with the DTW\-based similarity matrix over forecast trajectories\. Specifically, the\(i,j\)\(i,j\)\-th entrywi​jw\_\{ij\}of the similarity matrix𝐖\\mathbf\{W\}is defined ass⁡\(𝒙^i,𝒙^j\)s\(\\hat\{\\boldsymbol\{x\}\}^\{i\},\\hat\{\\boldsymbol\{x\}\}^\{j\}\), representing the DTW\-based similarity between two sampled forecast trajectories𝒙^i\\hat\{\\boldsymbol\{x\}\}^\{i\}and𝒙^j\\hat\{\\boldsymbol\{x\}\}^\{j\}\.

### B\.4UQ Performance Evaluations

This subsection provides additional details on the evaluation of UQ performance\.

Figure[6](https://arxiv.org/html/2609.28582#A2.F6)visualizes the overall evaluation ranks of the UQ methods, where the ranking scores are calculated from the average performance across 27 datasets and 11 models\. A lower rank indicates better overall performance of a UQ method\. It can be observed that SGA achieves the best rank among all methods, further demonstrating the strong advantage of SGA\.

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/UQ_performence/ranking.png)Figure 6:Overall evaluation ranking of 10 UQ methods across 27 datasets and 11 TSFMs\.Tables[5](https://arxiv.org/html/2609.28582#A2.T5)–[15](https://arxiv.org/html/2609.28582#A2.T15)list the comparisons of NEAURC of SGA and its contenders across 27 datasets for each individual TSFM, where bold and underlined values denote the best and second\-best results, respectively\. It is observed that SGA exceeds all contenders in 260 dataset\-model pairs among a total of11×27=29711\\times 27=297pairs, thus validating the effectiveness and generality of SGA in various kinds of datasets\.

DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity88\.7052\.2091\.4062\.0077\.4051\.60121\.7051\.00107\.0041\.90121\.7050\.5091\.4061\.50105\.1052\.5077\.4049\.9026\.8043\.30CIF 2016138\.3021\.80176\.4038\.10136\.5030\.10132\.4017\.50126\.7020\.70132\.5017\.80169\.8037\.10128\.9019\.20135\.3030\.2050\.1011\.40Car Parts118\.004\.40101\.803\.70121\.204\.60137\.606\.00142\.206\.30137\.705\.70129\.505\.80123\.704\.50121\.204\.2057\.101\.70Covid Deaths76\.2013\.3059\.1010\.7079\.8017\.9089\.7015\.7095\.6015\.6087\.6016\.3072\.208\.0093\.0017\.0083\.7017\.0056\.7012\.70Dominick32\.900\.9065\.200\.9033\.900\.8035\.901\.0035\.901\.0034\.500\.9035\.401\.0032\.700\.8033\.800\.8029\.800\.90ERCOT Load128\.4032\.30117\.5032\.5074\.1030\.5091\.8046\.9046\.9034\.3091\.8047\.60117\.5034\.1073\.1041\.0074\.1030\.7075\.6028\.90ETT \(15 Min\.\)95\.3024\.8060\.5022\.7019\.5011\.30110\.7021\.10115\.8019\.10110\.2021\.9058\.3023\.0026\.5014\.2019\.309\.8017\.5010\.80ETT \(Hourly\)96\.0021\.10127\.4024\.6072\.7039\.20112\.1020\.70106\.9023\.90112\.1021\.40130\.4026\.2060\.6035\.4088\.3041\.0040\.4024\.80Exchange Rate84\.6032\.9045\.7029\.9085\.7049\.3093\.8035\.2082\.3031\.9093\.8035\.8031\.1036\.3062\.0038\.7062\.1037\.4052\.5019\.10FRED\-MD80\.208\.8058\.909\.5047\.905\.0079\.209\.6075\.909\.8076\.8010\.0059\.109\.7066\.608\.4052\.405\.9046\.106\.30Hospital104\.703\.80106\.403\.4097\.803\.50102\.603\.7096\.603\.50102\.403\.60107\.803\.5098\.103\.2097\.803\.4044\.002\.90M1 \(Monthly\)89\.204\.6081\.403\.8082\.603\.8087\.804\.7087\.405\.1087\.804\.6081\.303\.9082\.703\.9079\.403\.7057\.503\.90M1 \(Quarterly\)88\.607\.5065\.006\.0068\.105\.3085\.107\.6090\.606\.3084\.407\.2066\.805\.7079\.105\.8069\.705\.2075\.206\.70M1 \(Yearly\)80\.108\.1072\.306\.7071\.807\.6079\.209\.1089\.407\.6078\.609\.1073\.106\.8076\.309\.5072\.407\.6072\.209\.30M3 \(Monthly\)68\.502\.2070\.102\.1072\.002\.0072\.502\.1091\.202\.9071\.102\.2070\.702\.2071\.902\.2072\.102\.1053\.101\.90M3 \(Quarterly\)74\.803\.3071\.103\.2062\.102\.6068\.302\.9088\.704\.0068\.403\.1075\.503\.4062\.002\.7062\.002\.7056\.602\.60M3 \(Yearly\)115\.905\.6095\.203\.7088\.703\.70116\.705\.1097\.804\.60114\.905\.0094\.503\.6089\.803\.6088\.703\.7064\.703\.50M4 \(Quarterly\)76\.500\.7070\.900\.8064\.000\.7074\.500\.7093\.600\.9074\.700\.8072\.700\.8066\.600\.6064\.000\.6062\.300\.70M4 \(Yearly\)87\.801\.0082\.500\.9075\.500\.8089\.201\.00100\.901\.0089\.201\.0085\.700\.9077\.800\.9075\.600\.9072\.901\.00M5111\.705\.3090\.101\.90112\.604\.40115\.306\.00114\.106\.20116\.205\.8092\.703\.60129\.504\.90128\.104\.9071\.902\.10NN5 \(Daily\)84\.309\.4088\.909\.5086\.008\.00119\.9011\.30110\.1011\.10119\.9011\.3093\.0010\.4094\.107\.7094\.409\.3043\.207\.60NN5 \(Weekly\)106\.909\.5099\.708\.50104\.508\.40105\.409\.1092\.3012\.20105\.408\.9099\.108\.30106\.809\.20104\.408\.6091\.806\.10Tourism \(Monthly\)115\.705\.5076\.004\.4069\.905\.10127\.605\.40111\.005\.50127\.105\.4076\.404\.9091\.406\.1069\.905\.1073\.605\.70Tourism \(Quarterly\)115\.104\.6093\.805\.0067\.804\.60108\.104\.70120\.306\.30108\.304\.5095\.704\.9078\.505\.1067\.804\.7073\.305\.00Tourism \(Yearly\)68\.903\.4090\.704\.80143\.604\.5080\.003\.8088\.604\.0080\.303\.7086\.204\.6099\.604\.30134\.405\.10125\.705\.10Traffic161\.706\.90196\.105\.20200\.805\.20108\.406\.5082\.004\.60108\.506\.70199\.605\.20185\.206\.40200\.805\.4065\.604\.10Weather41\.403\.4057\.806\.5057\.003\.3066\.106\.1088\.602\.3063\.705\.9062\.504\.5067\.905\.3060\.704\.3033\.801\.90

Table 5:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Chronos\-T5\-Tiny, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity182\.7038\.60106\.5049\.409\.1011\.70187\.4027\.3076\.7043\.90187\.4020\.80106\.5069\.00187\.4025\.309\.1011\.00118\.0059\.10CIF 2016131\.8016\.30111\.4015\.00128\.2026\.50119\.6015\.40100\.8010\.20119\.7016\.00111\.2013\.90121\.4016\.40129\.9027\.8037\.706\.60Car Parts120\.104\.30114\.403\.90130\.205\.00146\.706\.30139\.406\.50146\.406\.30137\.406\.00131\.905\.20127\.404\.2057\.301\.40Covid Deaths87\.3012\.9059\.7011\.5069\.3013\.7081\.7012\.8081\.6012\.8081\.0012\.9060\.2036\.8077\.6013\.2075\.6014\.3049\.708\.40Dominick32\.700\.9063\.501\.0037\.701\.2038\.201\.0037\.801\.0037\.401\.0034\.701\.0037\.001\.2037\.901\.2031\.001\.00ERCOT Load100\.0038\.5087\.1038\.7091\.8042\.00117\.9040\.5058\.1033\.30117\.9038\.8074\.7035\.1093\.6043\.6091\.8042\.5025\.2011\.80ETT \(15 Min\.\)113\.8019\.6098\.2022\.0029\.2012\.8097\.3022\.50110\.9021\.2097\.3024\.00114\.3021\.0051\.2021\.4033\.8013\.2024\.208\.70ETT \(Hourly\)123\.1019\.20104\.6019\.4075\.9034\.30123\.1019\.50100\.8020\.30125\.9020\.40104\.5019\.7070\.5031\.8087\.1031\.8024\.8012\.00Exchange Rate137\.9019\.2059\.7029\.70154\.0017\.10103\.2039\.2097\.2036\.30103\.2039\.0066\.6029\.70131\.9031\.50130\.3024\.00111\.7028\.60FRED\-MD84\.108\.3059\.707\.6060\.006\.2079\.808\.3078\.907\.9078\.408\.5061\.407\.6066\.007\.6058\.806\.9038\.004\.40Hospital100\.203\.30104\.903\.90101\.603\.50100\.903\.4097\.103\.60100\.903\.40104\.504\.10100\.303\.40102\.203\.4046\.403\.10M1 \(Monthly\)83\.804\.6085\.304\.3083\.904\.2084\.604\.5083\.604\.5084\.104\.4083\.504\.3079\.304\.0080\.104\.2055\.203\.50M1 \(Quarterly\)88\.507\.2078\.106\.4068\.605\.1085\.407\.5083\.207\.7085\.207\.5079\.606\.2081\.306\.4068\.505\.2073\.606\.90M1 \(Yearly\)75\.608\.1063\.506\.2060\.706\.8076\.208\.9080\.707\.7076\.308\.8064\.406\.3070\.208\.7059\.506\.3068\.0011\.00M3 \(Monthly\)72\.802\.2076\.002\.4077\.402\.2073\.902\.2085\.802\.7073\.902\.3076\.002\.4076\.802\.4077\.302\.3055\.002\.30M3 \(Quarterly\)75\.703\.4068\.303\.4059\.302\.8065\.802\.9090\.904\.3065\.203\.2071\.303\.6060\.002\.9059\.402\.8054\.903\.00M3 \(Yearly\)106\.604\.4094\.604\.4081\.304\.10112\.205\.50102\.404\.70108\.305\.7096\.004\.3081\.903\.8081\.303\.9059\.903\.40M4 \(Quarterly\)78\.300\.7071\.900\.7067\.300\.6076\.300\.7091\.200\.8076\.500\.7073\.500\.7069\.500\.6067\.400\.7064\.900\.70M4 \(Yearly\)84\.700\.9079\.100\.9071\.500\.8086\.701\.00100\.101\.0086\.701\.0082\.500\.9073\.100\.8071\.600\.9070\.500\.90M5112\.305\.0091\.402\.00113\.204\.40117\.105\.80116\.905\.90116\.905\.9094\.004\.70130\.104\.60127\.404\.5070\.402\.30NN5 \(Daily\)98\.9013\.5074\.807\.8076\.208\.30121\.7013\.40131\.4013\.20121\.6013\.9079\.108\.6083\.808\.8077\.908\.1028\.104\.00NN5 \(Weekly\)101\.808\.7091\.908\.4092\.808\.60102\.109\.10117\.208\.50102\.209\.3092\.908\.30102\.309\.7091\.708\.5082\.009\.30Tourism \(Monthly\)118\.305\.9089\.305\.2076\.605\.10122\.005\.20121\.505\.80122\.305\.2090\.404\.8092\.605\.9076\.605\.2086\.106\.00Tourism \(Quarterly\)115\.504\.60100\.105\.4076\.605\.20109\.004\.9094\.204\.90109\.504\.90101\.705\.4081\.505\.2076\.605\.0084\.905\.30Tourism \(Yearly\)67\.503\.3086\.104\.60134\.604\.7076\.403\.5084\.003\.6076\.903\.5082\.904\.60101\.104\.20126\.105\.30111\.304\.50Traffic178\.306\.20195\.705\.40201\.705\.60131\.207\.20107\.507\.60131\.207\.60192\.805\.90187\.606\.70202\.305\.3070\.604\.50Weather36\.003\.2055\.006\.8055\.704\.4055\.506\.1079\.102\.5053\.806\.0053\.503\.2062\.804\.1058\.304\.4028\.701\.50

Table 6:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Chronos\-T5\-Mini, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity69\.0041\.2071\.7048\.6085\.4043\.4052\.3035\.9095\.6046\.6052\.3036\.6071\.7047\.2052\.3036\.7085\.4042\.3012\.4018\.80CIF 2016114\.5016\.40156\.9035\.10118\.7028\.70121\.0015\.90108\.9014\.50121\.0016\.40159\.2036\.90115\.3016\.20117\.8029\.1026\.805\.30Car Parts125\.205\.60118\.003\.90128\.505\.40139\.706\.30132\.506\.20139\.506\.30134\.806\.30130\.305\.60126\.304\.6060\.201\.60Covid Deaths83\.2012\.7073\.3016\.3065\.7011\.6082\.108\.9080\.808\.9081\.909\.3076\.209\.3082\.5010\.8073\.3011\.3058\.3015\.50Dominick33\.600\.9062\.701\.0037\.901\.1039\.101\.0036\.901\.0036\.701\.0035\.301\.0037\.401\.2038\.301\.1031\.201\.10ERCOT Load119\.6039\.6090\.8039\.00101\.6036\.10132\.8040\.40121\.6030\.20132\.8039\.4096\.3043\.10124\.2039\.10101\.6036\.0021\.3015\.80ETT \(15 Min\.\)109\.5020\.50126\.2059\.90104\.9049\.5073\.4017\.4080\.7017\.7072\.2017\.30132\.0058\.30105\.0050\.00127\.3060\.3034\.1015\.50ETT \(Hourly\)151\.1020\.70148\.7031\.40131\.0037\.40136\.9027\.80118\.4030\.60134\.7028\.50154\.9028\.00112\.8036\.80125\.1036\.2039\.1016\.30Exchange Rate71\.2031\.9046\.7025\.5072\.1045\.30118\.3039\.90117\.6040\.40118\.3038\.9052\.7035\.1032\.9022\.9072\.1044\.90153\.0021\.60FRED\-MD95\.508\.4075\.9010\.7056\.906\.5081\.907\.2079\.207\.8079\.807\.3069\.708\.7072\.007\.8055\.605\.8038\.905\.40Hospital100\.203\.20100\.203\.30103\.503\.80103\.503\.70102\.803\.80103\.403\.60100\.703\.30104\.003\.60102\.603\.8047\.003\.10M1 \(Monthly\)89\.904\.5081\.603\.9081\.004\.0091\.604\.7089\.604\.9091\.304\.5081\.204\.1084\.503\.8079\.203\.8059\.704\.10M1 \(Quarterly\)91\.207\.8077\.207\.1071\.806\.3086\.807\.3083\.306\.7086\.106\.8078\.107\.0082\.006\.3071\.106\.3081\.8011\.10M1 \(Yearly\)75\.509\.1064\.506\.6064\.206\.3080\.608\.2074\.406\.6079\.908\.1064\.606\.5072\.008\.8065\.505\.4082\.5012\.80M3 \(Monthly\)70\.902\.3075\.002\.3072\.802\.3072\.802\.2089\.003\.0072\.002\.2076\.202\.4072\.902\.3072\.802\.3057\.402\.10M3 \(Quarterly\)73\.803\.6069\.203\.4062\.303\.0070\.603\.4093\.903\.7069\.603\.6071\.803\.5063\.103\.0062\.303\.1057\.703\.00M3 \(Yearly\)109\.904\.70102\.304\.2088\.103\.80116\.606\.00104\.005\.20115\.205\.80103\.704\.5090\.803\.9088\.203\.9061\.403\.00M4 \(Quarterly\)77\.600\.8071\.900\.7065\.300\.6076\.900\.7093\.600\.8077\.100\.8073\.700\.7067\.700\.7065\.300\.7064\.500\.70M4 \(Yearly\)85\.300\.9079\.400\.8075\.400\.8089\.000\.9099\.201\.0089\.001\.0082\.000\.8077\.200\.8075\.400\.9071\.300\.90M5112\.205\.6093\.502\.00112\.705\.30117\.706\.10118\.906\.00118\.205\.9094\.202\.10127\.805\.00124\.304\.9065\.600\.90NN5 \(Daily\)106\.7016\.2064\.207\.5079\.507\.40120\.5015\.00101\.1011\.10120\.4015\.5071\.109\.9096\.1010\.6086\.9010\.8039\.206\.90NN5 \(Weekly\)101\.609\.6088\.609\.2097\.808\.40101\.708\.8092\.708\.80100\.808\.7088\.309\.3099\.209\.0097\.808\.3091\.207\.80Tourism \(Monthly\)124\.005\.80101\.705\.1084\.105\.40130\.105\.00107\.706\.10129\.805\.00101\.404\.70107\.805\.4084\.105\.5085\.005\.80Tourism \(Quarterly\)118\.304\.70110\.205\.2089\.405\.80113\.004\.70112\.605\.70109\.704\.90110\.605\.0095\.505\.4089\.405\.20100\.305\.80Tourism \(Yearly\)66\.803\.3091\.704\.80145\.904\.9082\.503\.9075\.504\.2082\.204\.0087\.704\.70102\.204\.50134\.105\.30119\.504\.70Traffic147\.207\.60195\.305\.40202\.305\.4082\.706\.0063\.004\.7082\.705\.90176\.406\.70180\.306\.40201\.704\.8066\.704\.20Weather39\.602\.6052\.403\.2055\.103\.2051\.302\.3076\.902\.4049\.402\.2057\.208\.9073\.704\.5058\.403\.5027\.001\.50

Table 7:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Chronos\-T5\-Small, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity89\.6054\.50118\.4054\.30119\.7042\.1068\.1046\.70148\.5046\.0068\.1046\.30118\.4054\.2083\.2053\.70119\.7042\.40111\.9044\.10CIF 2016117\.7016\.30109\.8026\.20101\.0021\.20109\.0015\.6093\.0013\.50109\.1016\.00111\.1027\.30103\.8016\.90101\.5021\.6033\.706\.30Car Parts132\.805\.10119\.303\.70130\.205\.00141\.306\.70133\.306\.80141\.706\.70140\.906\.40132\.805\.50132\.905\.1067\.301\.90Covid Deaths91\.9010\.0064\.0010\.6075\.1010\.5097\.7011\.0096\.4012\.2095\.1010\.8069\.3025\.0086\.4011\.2079\.7010\.1053\.208\.50Dominick33\.200\.9059\.301\.0041\.001\.3040\.001\.1037\.801\.0038\.101\.0035\.401\.0039\.901\.3041\.401\.3031\.601\.00ERCOT Load110\.5040\.4098\.7041\.8064\.2022\.70109\.5039\.3064\.4031\.20109\.5038\.2098\.7042\.70100\.3040\.9064\.2022\.9026\.8013\.50ETT \(15 Min\.\)90\.5021\.0064\.2023\.6024\.907\.7066\.6020\.60105\.2021\.0066\.2021\.3063\.1023\.9027\.5010\.8021\.907\.4028\.909\.30ETT \(Hourly\)109\.6022\.5062\.3027\.00115\.9036\.30110\.9020\.10115\.5020\.10108\.5020\.6061\.4027\.7058\.1024\.80126\.9033\.2036\.7014\.20Exchange Rate113\.3026\.10110\.7047\.50121\.2040\.30150\.0024\.40154\.7025\.00150\.0024\.30120\.4032\.90157\.4027\.20147\.8036\.30109\.0043\.70FRED\-MD87\.209\.5070\.7012\.1048\.806\.7080\.908\.5081\.408\.8080\.308\.9070\.5012\.3073\.209\.2050\.107\.4041\.806\.40Hospital103\.903\.2099\.703\.00103\.903\.40103\.703\.20101\.003\.40103\.803\.3099\.803\.20104\.303\.20103\.503\.5046\.403\.00M1 \(Monthly\)88\.604\.4079\.303\.7082\.304\.4090\.804\.8090\.205\.3090\.404\.5078\.903\.7086\.704\.2079\.704\.1058\.904\.60M1 \(Quarterly\)88\.207\.2079\.907\.0069\.205\.6089\.307\.4083\.506\.9087\.707\.4081\.507\.0082\.906\.2069\.905\.8067\.906\.00M1 \(Yearly\)71\.908\.7061\.306\.4060\.706\.2074\.908\.0071\.806\.5074\.508\.0062\.506\.4064\.209\.0058\.305\.9086\.7011\.10M3 \(Monthly\)72\.602\.3074\.002\.5075\.302\.4076\.402\.6095\.703\.1076\.302\.6074\.302\.5075\.602\.4075\.302\.4056\.902\.00M3 \(Quarterly\)79\.303\.6076\.503\.6065\.003\.1073\.603\.0094\.404\.2073\.003\.2080\.403\.8065\.403\.0065\.103\.1057\.603\.00M3 \(Yearly\)105\.505\.5085\.004\.3082\.304\.20112\.006\.00111\.206\.50111\.006\.2085\.604\.0087\.105\.0082\.404\.2061\.803\.60M4 \(Quarterly\)76\.500\.7069\.800\.7064\.600\.6075\.700\.7093\.300\.9075\.900\.7071\.500\.7066\.900\.7064\.700\.7063\.900\.70M4 \(Yearly\)85\.400\.9078\.600\.8074\.100\.8089\.200\.9098\.001\.0089\.301\.0081\.000\.8075\.500\.8074\.100\.8071\.200\.90M5112\.405\.3088\.502\.10110\.604\.20115\.605\.80114\.505\.80115\.905\.7089\.002\.60128\.404\.50123\.104\.2063\.101\.00NN5 \(Daily\)93\.8015\.9081\.2010\.30111\.7019\.00105\.2016\.1078\.909\.60105\.1016\.8086\.5012\.10100\.5014\.20104\.3018\.4027\.905\.10NN5 \(Weekly\)93\.9011\.0091\.2010\.1079\.908\.6095\.909\.2088\.408\.3096\.409\.3091\.4010\.0091\.909\.6079\.908\.3071\.907\.30Tourism \(Monthly\)139\.306\.50103\.405\.2095\.505\.70136\.806\.10111\.907\.00136\.806\.30104\.004\.80111\.907\.3095\.505\.8078\.406\.40Tourism \(Quarterly\)114\.004\.60101\.005\.00101\.306\.30106\.804\.90106\.405\.60106\.405\.00100\.804\.70100\.406\.00101\.306\.0086\.105\.30Tourism \(Yearly\)69\.103\.70109\.105\.50146\.304\.4081\.804\.0075\.704\.1082\.404\.10104\.605\.70102\.104\.40136\.805\.20118\.504\.90Traffic165\.306\.90191\.506\.40197\.605\.5098\.607\.4076\.906\.6098\.707\.10177\.806\.70177\.106\.60199\.005\.2072\.904\.40Weather34\.902\.6052\.403\.8053\.303\.8052\.606\.3077\.901\.8051\.106\.2057\.309\.3066\.305\.2056\.404\.1024\.901\.10

Table 8:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Chronos\-T5\-Base, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity100\.5045\.70142\.8044\.10116\.9056\.1085\.1047\.2089\.6048\.5085\.1047\.20142\.8042\.7085\.1046\.00116\.9055\.9044\.9051\.60CIF 2016121\.3017\.10176\.1036\.40115\.3024\.70107\.6015\.00136\.3021\.10107\.6015\.40176\.4037\.90103\.9015\.90114\.9026\.2036\.906\.60Car Parts126\.005\.30113\.503\.60130\.404\.70142\.406\.40140\.107\.00142\.606\.50136\.505\.90132\.004\.70126\.904\.1060\.901\.50Covid Deaths102\.8012\.7081\.1016\.6093\.5017\.40102\.3014\.30100\.6013\.20102\.9014\.9081\.4019\.5099\.3015\.3087\.1013\.7068\.2010\.70Dominick32\.200\.9059\.901\.0039\.001\.3037\.901\.0036\.601\.0036\.301\.0033\.901\.0037\.901\.3039\.501\.4029\.901\.00ERCOT Load143\.5051\.3041\.8022\.70100\.5030\.00101\.1044\.10125\.3033\.00101\.1044\.2053\.0027\.50101\.1044\.80100\.5031\.40120\.2044\.90ETT \(15 Min\.\)74\.6025\.0060\.4017\.8037\.2011\.2085\.2025\.3090\.7024\.5085\.2026\.5067\.3020\.9026\.408\.9041\.309\.1024\.507\.80ETT \(Hourly\)126\.0018\.9080\.0024\.60121\.2039\.10123\.5019\.90116\.3020\.50123\.5019\.1093\.4022\.2093\.1032\.90120\.0039\.2041\.4015\.20Exchange Rate112\.1038\.30118\.7031\.30139\.9036\.00156\.0022\.80143\.2022\.80156\.0023\.50122\.2031\.2099\.7041\.90155\.6026\.7080\.1033\.10FRED\-MD79\.609\.2073\.4012\.0050\.507\.1086\.709\.5089\.3010\.5086\.1010\.0077\.9013\.4071\.0010\.5052\.908\.5038\.605\.70Hospital98\.503\.1099\.203\.00100\.203\.10101\.603\.1099\.003\.20101\.503\.1099\.403\.00101\.202\.9099\.903\.1042\.402\.70M1 \(Monthly\)90\.204\.9078\.703\.5080\.004\.2088\.804\.8088\.005\.6088\.704\.6078\.503\.8083\.404\.3078\.504\.3064\.504\.90M1 \(Quarterly\)89\.107\.3078\.607\.0073\.206\.5082\.407\.0087\.107\.0081\.706\.7078\.607\.2081\.806\.1072\.906\.3078\.809\.90M1 \(Yearly\)74\.608\.6083\.8018\.7061\.907\.5069\.108\.9072\.106\.0068\.809\.0084\.9018\.0063\.809\.4057\.906\.9080\.3012\.00M3 \(Monthly\)69\.402\.4069\.502\.2071\.902\.4074\.102\.5084\.902\.6073\.702\.4069\.702\.2071\.702\.3071\.902\.5059\.302\.40M3 \(Quarterly\)77\.103\.4072\.503\.1066\.603\.2072\.403\.50102\.904\.7072\.803\.5075\.603\.4067\.103\.1066\.703\.2058\.903\.20M3 \(Yearly\)106\.405\.3080\.103\.8082\.605\.30119\.307\.90100\.308\.70117\.708\.4079\.903\.6087\.607\.4082\.805\.4067\.904\.10M4 \(Quarterly\)77\.200\.7070\.800\.8065\.000\.7076\.700\.8095\.501\.0077\.200\.9072\.700\.8067\.400\.7065\.100\.7064\.800\.80M4 \(Yearly\)86\.800\.9080\.700\.9074\.100\.8089\.101\.0099\.601\.1089\.201\.0083\.300\.9075\.300\.9074\.200\.9072\.500\.90M5113\.605\.7091\.102\.10114\.504\.80119\.406\.40118\.906\.20119\.306\.1091\.602\.30131\.404\.80128\.604\.9063\.301\.10NN5 \(Daily\)112\.6016\.1095\.6012\.7081\.6012\.90125\.9015\.50104\.6013\.90125\.9016\.10101\.5012\.60110\.8014\.8079\.809\.7031\.305\.00NN5 \(Weekly\)93\.6011\.00103\.708\.7094\.1012\.3097\.509\.30112\.8014\.0097\.809\.50103\.709\.3092\.409\.6093\.8011\.7086\.809\.10Tourism \(Monthly\)139\.706\.00111\.505\.1085\.305\.10141\.705\.70104\.405\.60142\.006\.00111\.904\.90106\.606\.7085\.304\.9085\.506\.40Tourism \(Quarterly\)118\.204\.80109\.805\.8087\.005\.40105\.905\.10102\.906\.00105\.505\.00110\.105\.8094\.605\.4087\.005\.3088\.905\.80Tourism \(Yearly\)71\.003\.60101\.809\.80143\.308\.0087\.805\.4079\.705\.5088\.405\.6098\.409\.40106\.305\.40134\.908\.4094\.504\.80Traffic172\.106\.50195\.006\.50197\.105\.50104\.006\.5070\.805\.30104\.006\.70177\.807\.10178\.206\.70205\.505\.0067\.904\.50Weather36\.904\.3052\.604\.5054\.904\.9052\.207\.5077\.801\.8050\.507\.0056\.1025\.4067\.706\.8060\.005\.8026\.201\.60

Table 9:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Chronos\-T5\-Large, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity63\.4042\.3063\.4044\.60120\.4043\.20120\.4042\.40129\.8043\.60120\.4041\.3052\.0038\.30120\.4042\.80120\.4043\.5021\.2036\.90CIF 2016118\.4016\.50112\.6015\.50118\.4016\.50112\.6015\.20122\.5018\.30114\.7015\.80108\.6014\.60113\.7015\.60116\.8015\.9027\.907\.40Car Parts129\.906\.90127\.205\.60136\.105\.40131\.206\.20121\.805\.40131\.806\.40136\.506\.40123\.804\.60122\.404\.4054\.501\.70Covid Deaths69\.309\.1056\.007\.3071\.609\.9063\.707\.5063\.707\.3063\.907\.3062\.707\.2067\.909\.3070\.309\.9019\.403\.00Dominick48\.502\.5036\.100\.7050\.801\.3048\.102\.4046\.402\.4047\.302\.4039\.401\.2051\.600\.7046\.400\.8029\.801\.00ERCOT Load140\.7040\.50142\.0041\.90144\.6042\.20118\.2040\.709\.6040\.00118\.2038\.90126\.1029\.50144\.6041\.40144\.6042\.8066\.7040\.30ETT \(15 Min\.\)63\.6019\.2052\.6019\.8053\.1019\.0060\.5019\.3069\.8020\.9060\.5019\.1060\.2021\.0052\.4019\.6047\.5018\.8014\.4011\.00ETT \(Hourly\)131\.7023\.70140\.1026\.80133\.3029\.00121\.1021\.10122\.0021\.40120\.8020\.20130\.5023\.90112\.6022\.20133\.3028\.6025\.5010\.70Exchange Rate49\.1031\.8049\.1033\.4029\.7020\.8048\.6032\.2051\.4034\.8048\.6032\.1048\.6030\.8046\.6031\.4029\.7021\.4085\.6039\.10FRED\-MD86\.6010\.1067\.808\.0079\.109\.1086\.609\.3084\.509\.2086\.709\.9070\.508\.6076\.208\.4076\.308\.7037\.606\.40Hospital107\.703\.40104\.203\.20101\.203\.20103\.003\.50103\.004\.00102\.303\.60101\.303\.10101\.003\.10100\.703\.2054\.202\.50M1 \(Monthly\)89\.304\.2087\.204\.4084\.904\.1088\.104\.5089\.104\.9087\.904\.3086\.804\.4084\.704\.0085\.304\.2057\.403\.20M1 \(Quarterly\)88\.307\.3085\.707\.2085\.907\.3089\.107\.3088\.007\.5090\.307\.4084\.007\.0086\.506\.8085\.206\.9056\.505\.60M1 \(Yearly\)72\.109\.7072\.4010\.3071\.2010\.2071\.9011\.5075\.9010\.6071\.0010\.9077\.108\.7071\.7010\.4071\.9010\.2050\.007\.40M3 \(Monthly\)73\.702\.4072\.302\.3071\.002\.2070\.802\.4088\.402\.9070\.802\.4085\.702\.9071\.602\.4071\.102\.3052\.302\.20M3 \(Quarterly\)76\.504\.0073\.403\.5072\.603\.5085\.204\.80100\.004\.6085\.904\.7086\.603\.2076\.904\.4072\.803\.7064\.003\.80M3 \(Yearly\)62\.004\.6064\.905\.1062\.704\.9081\.806\.20106\.508\.1082\.606\.4084\.805\.5066\.104\.8062\.604\.6039\.003\.60M4 \(Quarterly\)75\.500\.7073\.600\.8074\.100\.7078\.700\.7091\.900\.7079\.200\.7082\.900\.7075\.300\.7074\.100\.7053\.700\.60M4 \(Yearly\)76\.700\.8076\.800\.8077\.800\.9082\.300\.8094\.900\.9082\.800\.9088\.600\.8078\.200\.9077\.900\.9062\.500\.80M5108\.707\.6085\.201\.60107\.006\.90115\.509\.40111\.909\.50115\.709\.7091\.602\.7090\.701\.9085\.701\.4065\.801\.90NN5 \(Daily\)102\.9010\.90102\.6010\.20110\.2011\.50110\.2011\.9090\.6010\.50109\.2011\.90108\.0011\.30109\.1010\.90114\.4010\.6035\.005\.70NN5 \(Weekly\)93\.009\.6097\.9010\.0096\.409\.4091\.307\.3092\.407\.4094\.008\.10103\.708\.3096\.108\.8096\.409\.3085\.107\.40Tourism \(Monthly\)112\.405\.20111\.105\.20113\.705\.50110\.805\.30110\.705\.60111\.105\.70102\.205\.60113\.805\.40113\.705\.6060\.605\.20Tourism \(Quarterly\)99\.604\.6099\.204\.8099\.504\.8096\.704\.6098\.304\.5096\.004\.4095\.104\.50100\.404\.7099\.404\.7065\.804\.70Tourism \(Yearly\)59\.603\.3059\.503\.4060\.103\.4072\.603\.5078\.403\.5072\.903\.6072\.804\.0060\.703\.4060\.203\.5054\.703\.90Traffic139\.707\.00191\.705\.20188\.605\.60152\.007\.00125\.307\.70152\.007\.10188\.005\.60189\.305\.80202\.504\.8089\.604\.70Weather43\.303\.5042\.103\.3048\.803\.9044\.103\.7047\.202\.9043\.803\.6046\.203\.0052\.504\.5060\.904\.6016\.001\.30

Table 10:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Chronos\-2\-Small, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity66\.0054\.9043\.4036\.1065\.5052\.3089\.3058\.10138\.9047\.8089\.3057\.8043\.4034\.4089\.3058\.4065\.5051\.7012\.2028\.80CIF 2016111\.5015\.70100\.6015\.90105\.4014\.50106\.6014\.80113\.4016\.00106\.7015\.0094\.2013\.80106\.6015\.20105\.2014\.7024\.905\.40Car Parts117\.905\.60109\.904\.00130\.905\.30123\.205\.50114\.204\.60123\.705\.60124\.504\.30116\.504\.40120\.704\.2054\.601\.70Covid Deaths70\.008\.5054\.209\.4055\.908\.6063\.707\.0062\.807\.0063\.806\.8063\.608\.0054\.506\.8051\.506\.0021\.703\.40Dominick47\.302\.5033\.000\.9038\.401\.0047\.202\.4045\.602\.5046\.602\.4039\.301\.2042\.100\.9037\.500\.8026\.501\.00ERCOT Load125\.1050\.40119\.3050\.10119\.3049\.70108\.0044\.10105\.6029\.10108\.0043\.90124\.1048\.50128\.6049\.60119\.3049\.806\.604\.40ETT \(15 Min\.\)61\.8017\.4047\.7017\.9045\.3017\.2059\.4018\.1063\.4017\.8059\.4018\.6059\.9020\.3033\.4016\.8020\.9013\.007\.104\.70ETT \(Hourly\)154\.4025\.00154\.4025\.50147\.8028\.40140\.9022\.70134\.7021\.40140\.9022\.40153\.1026\.30140\.8023\.10142\.6029\.6027\.3012\.90Exchange Rate97\.2030\.2095\.3032\.5095\.3030\.50104\.2029\.70114\.4031\.20104\.2029\.70112\.1030\.0097\.2030\.5098\.2029\.2029\.9036\.90FRED\-MD95\.709\.8060\.707\.6063\.507\.5095\.009\.6094\.709\.3094\.809\.9087\.209\.2067\.907\.6069\.608\.5039\.307\.10Hospital108\.703\.50105\.703\.40104\.503\.40104\.303\.50102\.403\.90103\.903\.70105\.103\.30103\.303\.20104\.403\.6057\.103\.00M1 \(Monthly\)90\.504\.4084\.004\.4085\.604\.3088\.804\.3088\.504\.9088\.704\.2086\.504\.5084\.304\.3087\.304\.8055\.603\.30M1 \(Quarterly\)88\.806\.7086\.306\.7086\.506\.8089\.106\.8084\.107\.7090\.407\.0086\.106\.0087\.206\.4086\.306\.6057\.506\.00M1 \(Yearly\)67\.0010\.1066\.5010\.1067\.1010\.9066\.9012\.1068\.2011\.3066\.8011\.8069\.209\.0066\.9010\.6067\.1010\.8052\.608\.30M3 \(Monthly\)72\.102\.4068\.902\.3070\.502\.1069\.402\.2095\.303\.1069\.302\.2089\.902\.7068\.702\.1070\.502\.3050\.402\.10M3 \(Quarterly\)74\.603\.4068\.603\.2069\.103\.3082\.004\.1098\.704\.6082\.904\.3083\.503\.3073\.003\.6069\.203\.3060\.002\.80M3 \(Yearly\)65\.504\.8062\.304\.9066\.105\.1079\.305\.2088\.506\.4079\.805\.1064\.904\.8068\.004\.9066\.104\.8040\.203\.30M4 \(Quarterly\)74\.600\.7068\.300\.7070\.500\.7078\.000\.7091\.300\.7078\.500\.7076\.700\.7072\.400\.7070\.400\.7052\.600\.60M4 \(Yearly\)78\.900\.8074\.400\.9077\.300\.9084\.401\.0095\.501\.0084\.801\.0080\.900\.9079\.000\.9077\.500\.9064\.200\.80M5107\.307\.7082\.001\.40103\.404\.70115\.008\.80110\.008\.10115\.109\.1099\.004\.4089\.901\.5088\.401\.0066\.501\.80NN5 \(Daily\)109\.2014\.00103\.1012\.70110\.6012\.90117\.5011\.80104\.7012\.60117\.2012\.20112\.1012\.60109\.2012\.20108\.5012\.3044\.605\.60NN5 \(Weekly\)91\.5010\.4092\.909\.0092\.809\.4085\.607\.3083\.807\.0088\.808\.2097\.707\.7091\.308\.8091\.609\.3078\.907\.10Tourism \(Monthly\)113\.205\.30109\.305\.10115\.005\.40112\.105\.30105\.506\.10111\.405\.50100\.805\.00114\.705\.30115\.005\.4055\.305\.00Tourism \(Quarterly\)105\.404\.80104\.205\.30104\.905\.10102\.104\.90101\.604\.80101\.004\.8091\.005\.60105\.804\.90104\.805\.0062\.604\.60Tourism \(Yearly\)60\.103\.5059\.903\.4060\.803\.5074\.303\.5078\.203\.7074\.503\.6064\.803\.9061\.303\.5061\.003\.5059\.504\.00Traffic131\.207\.20191\.205\.20198\.805\.30140\.106\.80108\.906\.80140\.107\.00186\.804\.70196\.205\.40198\.805\.2089\.804\.60Weather39\.303\.3041\.802\.6046\.704\.0040\.703\.9044\.303\.0040\.403\.7043\.203\.3049\.003\.5054\.803\.8014\.001\.20

Table 11:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Chronos\-2, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity115\.5052\.8067\.5063\.1091\.2055\.70169\.2039\.9084\.6055\.60169\.2039\.0087\.0057\.00115\.5053\.7091\.2055\.3018\.3022\.30CIF 2016122\.9018\.80101\.0015\.7099\.5016\.20123\.1017\.50143\.3023\.50125\.4019\.20101\.6016\.80108\.6018\.80105\.8020\.1029\.405\.10Car Parts142\.507\.5097\.903\.1098\.103\.90136\.706\.20133\.106\.60136\.706\.40111\.404\.2087\.902\.9086\.802\.8054\.801\.60Covid Deaths71\.708\.1037\.804\.7053\.707\.0077\.009\.1081\.709\.7077\.809\.2059\.705\.8049\.805\.0059\.407\.9016\.202\.40Dominick44\.702\.0031\.100\.7036\.001\.0042\.401\.8042\.601\.7041\.901\.7033\.000\.8037\.301\.1035\.201\.1024\.000\.80ERCOT Load81\.3051\.5098\.2058\.1098\.2056\.2098\.2057\.1089\.4030\.1098\.2057\.9087\.9051\.7098\.2057\.6098\.2057\.2064\.8043\.40ETT \(15 Min\.\)84\.5022\.5079\.5022\.0057\.0021\.0092\.9022\.0092\.4022\.5092\.9022\.6090\.6022\.5050\.4022\.3036\.8022\.2018\.0012\.40ETT \(Hourly\)150\.2023\.50148\.6024\.90150\.3026\.90152\.2023\.30145\.2021\.40152\.0023\.60132\.4024\.80153\.3023\.00156\.6029\.6028\.9015\.80Exchange Rate75\.6035\.1077\.6033\.4075\.5033\.7094\.8038\.7096\.8039\.0094\.8040\.1094\.8039\.5088\.1030\.3076\.5025\.9034\.5024\.60FRED\-MD83\.1010\.8059\.907\.4047\.806\.7082\.3010\.3093\.1010\.4082\.3010\.8086\.409\.8043\.006\.1045\.206\.2050\.409\.10Hospital109\.603\.10108\.803\.10109\.103\.00108\.503\.30109\.103\.50108\.603\.30105\.003\.30108\.503\.10108\.003\.0047\.902\.70M1 \(Monthly\)84\.503\.9083\.204\.7084\.905\.5083\.704\.2081\.804\.4083\.504\.1084\.104\.3087\.605\.5086\.105\.9060\.203\.80M1 \(Quarterly\)88\.307\.4071\.806\.1072\.506\.0090\.507\.3087\.907\.9091\.507\.7086\.406\.3079\.606\.4072\.606\.2070\.707\.40M1 \(Yearly\)82\.409\.7072\.507\.5075\.908\.5082\.209\.9086\.508\.5081\.209\.8081\.506\.9077\.608\.4076\.208\.3055\.005\.80M3 \(Monthly\)72\.002\.2074\.502\.9071\.302\.6075\.402\.3097\.403\.1075\.302\.3097\.402\.6069\.402\.5071\.202\.5046\.201\.80M3 \(Quarterly\)88\.904\.2074\.603\.1074\.003\.3097\.305\.80105\.904\.8097\.505\.3091\.803\.3075\.103\.8074\.103\.6070\.903\.50M3 \(Yearly\)85\.603\.9083\.803\.5082\.003\.7095\.004\.40100\.805\.6096\.004\.7092\.503\.7085\.803\.4082\.203\.5057\.603\.60M4 \(Quarterly\)80\.600\.8070\.900\.7069\.500\.6083\.700\.8094\.500\.8083\.900\.8091\.500\.8069\.900\.6069\.500\.6054\.300\.60M4 \(Yearly\)83\.600\.9077\.400\.7075\.700\.7087\.100\.9093\.100\.9087\.200\.9091\.100\.8076\.800\.7075\.700\.7069\.500\.70M5116\.008\.0087\.304\.20100\.805\.70119\.608\.90116\.108\.70119\.608\.9085\.304\.2090\.501\.5086\.201\.8069\.501\.60NN5 \(Daily\)118\.4012\.80113\.4011\.80122\.1013\.20124\.1013\.90119\.1013\.80124\.3014\.20120\.2013\.20123\.1013\.50121\.0012\.9039\.506\.00NN5 \(Weekly\)99\.309\.1097\.608\.60103\.809\.1095\.107\.30106\.107\.2099\.408\.00108\.007\.80101\.708\.70104\.208\.8075\.207\.40Tourism \(Monthly\)110\.605\.20119\.006\.10105\.405\.20110\.305\.5095\.904\.90110\.505\.70124\.606\.40109\.905\.40105\.505\.3047\.604\.40Tourism \(Quarterly\)105\.204\.80131\.008\.90108\.905\.30104\.604\.5096\.305\.30103\.904\.30117\.508\.90109\.504\.60108\.905\.2072\.605\.50Tourism \(Yearly\)70\.104\.1064\.403\.8067\.503\.8077\.003\.9078\.804\.1077\.204\.0075\.403\.8066\.603\.9067\.403\.8057\.203\.60Traffic124\.007\.40183\.406\.50194\.305\.90132\.808\.0092\.405\.70132\.907\.70176\.406\.50162\.807\.20196\.905\.9091\.504\.90Weather45\.304\.0037\.203\.5053\.603\.9043\.104\.0045\.603\.2042\.904\.0045\.703\.7055\.406\.3060\.703\.5018\.501\.30

Table 12:Comparisons of NEAURC of SGA and its contenders across 27 datasets for TimesFM\-2\.5, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity90\.0057\.4081\.0053\.9092\.4057\.6092\.4058\.20133\.8057\.3092\.4058\.00107\.6046\.3092\.4058\.8092\.4057\.708\.9014\.50CIF 2016114\.6018\.7099\.7016\.60102\.4014\.10105\.1015\.10111\.8016\.30105\.6015\.10111\.4015\.90104\.3014\.90102\.0014\.0032\.906\.00Car Parts158\.407\.20112\.703\.80127\.505\.50160\.207\.00154\.307\.10160\.307\.10123\.903\.70106\.703\.40115\.504\.4059\.301\.50Covid Deaths67\.007\.1025\.203\.4025\.203\.7068\.807\.1068\.007\.1068\.206\.8059\.205\.6034\.704\.4031\.903\.8012\.802\.70Dominick47\.102\.4032\.201\.0035\.201\.0048\.102\.4045\.702\.5047\.302\.3037\.801\.3037\.401\.0036\.501\.2027\.001\.10ERCOT Load90\.8039\.30125\.6037\.2082\.2040\.3066\.9034\.3074\.0034\.6066\.9034\.80117\.4048\.3085\.6041\.7082\.2039\.0044\.0026\.10ETT \(15 Min\.\)75\.5020\.2058\.5020\.2048\.0017\.2073\.6018\.5074\.9018\.5073\.6019\.5081\.9018\.5034\.9014\.7037\.5014\.3014\.907\.70ETT \(Hourly\)144\.6022\.60149\.5024\.70143\.8026\.90128\.2021\.20129\.1021\.60128\.2020\.60133\.2020\.30140\.8022\.40149\.3029\.3015\.609\.00Exchange Rate182\.6015\.20181\.5015\.10181\.5014\.20188\.4015\.00179\.8017\.20188\.4015\.40183\.6015\.50182\.8015\.20171\.7021\.0039\.5025\.60FRED\-MD74\.007\.30105\.4011\.8081\.909\.8073\.707\.5082\.407\.7073\.507\.80106\.0011\.4088\.209\.7086\.9010\.8030\.504\.40Hospital107\.603\.30105\.303\.4098\.803\.00105\.603\.30106\.503\.50104\.803\.20100\.203\.0098\.603\.2097\.803\.2057\.103\.10M1 \(Monthly\)90\.804\.3086\.304\.4084\.804\.1090\.704\.6089\.004\.8090\.404\.3090\.504\.5085\.804\.1083\.204\.4066\.503\.70M1 \(Quarterly\)87\.706\.6077\.106\.9083\.607\.0090\.107\.1088\.807\.7091\.607\.5080\.906\.8086\.306\.6082\.906\.7071\.208\.10M1 \(Yearly\)81\.509\.8080\.109\.1081\.009\.3079\.7010\.4086\.209\.2079\.6010\.6080\.708\.9080\.809\.3081\.109\.3056\.209\.30M3 \(Monthly\)85\.902\.4082\.302\.3082\.102\.3084\.602\.4088\.102\.6084\.702\.5088\.302\.6081\.702\.5082\.202\.5060\.902\.10M3 \(Quarterly\)91\.304\.0086\.603\.9085\.903\.8094\.603\.90104\.203\.7094\.804\.2090\.603\.5089\.304\.2086\.004\.0069\.204\.40M3 \(Yearly\)102\.903\.90101\.504\.10102\.204\.00104\.104\.6099\.804\.70104\.604\.7094\.404\.00104\.504\.00102\.303\.7046\.003\.00M4 \(Quarterly\)80\.600\.7075\.000\.7075\.500\.7084\.200\.8094\.400\.8084\.500\.8083\.800\.7077\.600\.7075\.600\.7052\.600\.60M4 \(Yearly\)80\.600\.8078\.200\.7078\.900\.8084\.800\.8096\.000\.9085\.100\.9088\.800\.7080\.000\.8078\.800\.8064\.300\.70M5124\.108\.6094\.601\.90101\.402\.70124\.708\.50120\.308\.80124\.908\.80104\.704\.3090\.401\.5088\.601\.2072\.201\.60NN5 \(Daily\)106\.0011\.20104\.8010\.80117\.0011\.70116\.2012\.1095\.1010\.50116\.6012\.20114\.5011\.90118\.5012\.30117\.9010\.4034\.704\.70NN5 \(Weekly\)103\.6010\.00104\.509\.50108\.0010\.2094\.308\.90100\.408\.7094\.009\.00108\.409\.40103\.409\.70104\.3010\.1076\.408\.00Tourism \(Monthly\)100\.805\.0099\.804\.90104\.205\.20102\.205\.1099\.405\.50102\.205\.4084\.204\.90103\.505\.10104\.205\.3055\.804\.40Tourism \(Quarterly\)93\.304\.3092\.904\.3093\.404\.4092\.804\.1091\.103\.9092\.603\.9088\.704\.2093\.704\.2093\.404\.3061\.204\.50Tourism \(Yearly\)69\.004\.0068\.604\.1069\.004\.1078\.303\.8086\.804\.2079\.004\.1085\.704\.3068\.904\.1069\.104\.0070\.404\.90Traffic122\.908\.00164\.607\.00188\.405\.90120\.608\.2088\.305\.80120\.608\.10170\.306\.40192\.105\.80197\.605\.5074\.704\.30Weather45\.504\.7040\.603\.1061\.204\.2045\.404\.1048\.602\.8045\.104\.1044\.701\.8056\.304\.0072\.803\.4014\.701\.40

Table 13:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Timer\-S1, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity122\.0053\.60125\.0051\.60125\.0053\.00111\.5052\.30132\.9070\.40111\.5053\.00149\.4047\.10125\.0053\.10125\.0051\.8072\.4043\.30CIF 2016111\.1016\.00111\.5014\.60110\.7014\.10105\.7014\.4085\.5011\.10104\.3014\.0099\.6012\.60107\.5014\.10108\.4013\.7022\.904\.80Car Parts145\.506\.40146\.206\.00150\.605\.90140\.605\.40127\.205\.60140\.705\.50143\.605\.50144\.205\.80151\.005\.7055\.901\.60Covid Deaths69\.907\.0069\.007\.4071\.608\.2077\.508\.7078\.3010\.9076\.008\.4069\.607\.3071\.608\.0071\.408\.8012\.101\.80Dominick51\.902\.4051\.502\.6052\.602\.4056\.002\.4052\.602\.2053\.902\.3052\.602\.5054\.702\.5053\.002\.5025\.100\.90ERCOT Load126\.3043\.50153\.3048\.3093\.3040\.10119\.0046\.20109\.0035\.20119\.0046\.30114\.5045\.30121\.1045\.5093\.3040\.4098\.9032\.40ETT \(15 Min\.\)96\.0024\.6087\.8023\.1093\.4022\.7093\.3023\.30111\.1022\.6093\.3023\.5085\.4024\.1096\.5022\.6090\.7021\.7033\.7014\.80ETT \(Hourly\)146\.8020\.90137\.2024\.10134\.6025\.70126\.8021\.50118\.8019\.20126\.8021\.20122\.4022\.50125\.3023\.20134\.6025\.2015\.7010\.40Exchange Rate11\.4034\.3010\.4027\.10117\.9051\.80121\.3049\.10120\.6056\.50121\.3047\.9086\.9039\.5057\.2032\.9083\.4041\.103\.0015\.00FRED\-MD77\.007\.2081\.907\.6079\.107\.4080\.907\.6085\.808\.0080\.207\.8081\.007\.6079\.708\.0078\.807\.7027\.003\.50Hospital104\.603\.50103\.103\.20105\.503\.40105\.303\.60102\.003\.40105\.103\.70105\.903\.90106\.003\.70105\.903\.6057\.603\.30M1 \(Monthly\)90\.604\.1090\.004\.4090\.304\.2090\.604\.4089\.104\.2090\.304\.2087\.204\.2089\.704\.2089\.904\.4079\.604\.30M1 \(Quarterly\)87\.707\.5088\.107\.6088\.207\.8088\.107\.9083\.508\.3088\.207\.8089\.107\.7089\.307\.5088\.707\.9065\.506\.20M1 \(Yearly\)78\.208\.7077\.209\.0077\.909\.1078\.509\.5078\.209\.2078\.909\.7083\.108\.4079\.109\.1077\.909\.0049\.707\.80M3 \(Monthly\)90\.102\.5087\.302\.4088\.902\.4089\.502\.2093\.303\.0089\.402\.3094\.303\.3086\.202\.4088\.902\.6068\.002\.70M3 \(Quarterly\)95\.404\.1091\.903\.8094\.003\.9075\.503\.40101\.104\.5075\.703\.2091\.104\.2084\.403\.5094\.004\.1055\.303\.90M3 \(Yearly\)89\.803\.7090\.603\.7094\.603\.70108\.203\.40104\.405\.10108\.003\.7089\.104\.3094\.703\.6094\.603\.4039\.802\.20M4 \(Quarterly\)85\.400\.7084\.600\.7085\.000\.7085\.000\.7095\.700\.7084\.900\.7092\.300\.7084\.400\.7085\.000\.7054\.800\.60M4 \(Yearly\)70\.500\.7070\.500\.7071\.800\.7077\.800\.7097\.700\.8077\.800\.7093\.100\.8072\.400\.7071\.800\.7046\.700\.50M5140\.2010\.10137\.909\.70140\.0010\.20135\.3010\.70117\.7011\.00135\.4010\.80142\.9010\.10142\.1010\.10141\.2010\.2061\.901\.00NN5 \(Daily\)112\.3012\.80109\.9011\.40110\.4011\.80120\.2012\.5097\.7010\.30120\.3012\.30117\.0013\.20111\.6011\.60111\.9011\.6045\.005\.00NN5 \(Weekly\)87\.408\.9082\.107\.8083\.407\.9087\.307\.9084\.207\.8087\.508\.1090\.908\.2086\.207\.9085\.508\.1073\.107\.00Tourism \(Monthly\)84\.404\.5085\.704\.6086\.004\.6095\.404\.60101\.806\.1095\.704\.70104\.407\.2087\.504\.8086\.004\.7067\.305\.10Tourism \(Quarterly\)92\.104\.3092\.804\.6092\.904\.4096\.704\.40104\.505\.8095\.304\.4092\.104\.7092\.804\.5092\.904\.4072\.204\.40Tourism \(Yearly\)69\.303\.9069\.604\.1070\.804\.2079\.104\.2077\.204\.2079\.104\.1089\.704\.3071\.004\.2070\.804\.1059\.803\.30Traffic148\.807\.70139\.508\.00137\.008\.20137\.407\.90106\.406\.50137\.408\.00153\.507\.80133\.308\.70140\.608\.0085\.804\.90Weather51\.403\.5048\.702\.7048\.503\.0066\.303\.4073\.102\.5066\.003\.3067\.504\.9058\.403\.3051\.203\.5028\.601\.50

Table 14:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Sundial, where bold and underlined values denote the best and second\-best results, respectively\.DatasetNCPplPEEigEccDegSDSARSESGAAustralian Electricity173\.4026\.00152\.4043\.60154\.6040\.40152\.4043\.10152\.4043\.00152\.4042\.60152\.4042\.30152\.4047\.70154\.6041\.5026\.6025\.30CIF 2016130\.7016\.50124\.9012\.50129\.8013\.90119\.6016\.60103\.9018\.80119\.7017\.30113\.5017\.60122\.2015\.90131\.7015\.9069\.109\.80Car Parts185\.309\.40178\.908\.00174\.407\.70165\.607\.40143\.507\.20165\.807\.60170\.306\.90168\.108\.10175\.107\.8061\.002\.10Covid Deaths106\.0015\.30106\.8015\.00117\.8015\.60120\.8016\.70119\.0015\.50120\.9016\.70117\.2016\.40121\.6017\.00119\.0016\.8080\.609\.80Dominick76\.001\.7076\.501\.6077\.101\.6076\.701\.4076\.101\.4075\.501\.4073\.001\.5076\.101\.6076\.901\.6042\.900\.30ERCOT Load86\.8043\.1091\.4044\.6091\.4044\.4095\.8046\.9059\.6036\.4095\.8046\.7084\.7032\.4095\.8046\.4091\.4042\.9053\.7023\.40ETT \(15 Min\.\)110\.7022\.20116\.0019\.90110\.3019\.20111\.9018\.20115\.5018\.40111\.9018\.50121\.7018\.00117\.2019\.00115\.3019\.8069\.0019\.90ETT \(Hourly\)128\.5022\.50127\.5020\.90129\.9020\.30118\.1020\.00113\.8020\.00118\.1020\.40126\.2020\.70128\.9020\.70133\.8021\.8027\.208\.80Exchange Rate73\.8038\.8086\.8043\.8098\.5038\.7092\.6043\.2087\.2042\.7092\.6040\.5089\.9040\.3092\.7042\.8092\.7042\.3028\.5019\.90FRED\-MD58\.006\.8062\.407\.1063\.207\.5063\.707\.4067\.107\.7063\.407\.4063\.107\.1063\.807\.5063\.407\.7047\.504\.60Hospital94\.602\.8092\.402\.8096\.903\.3094\.003\.6095\.103\.5093\.603\.5090\.503\.5094\.503\.2096\.603\.4057\.003\.10M1 \(Monthly\)82\.104\.2083\.804\.5082\.304\.0081\.203\.9080\.404\.1081\.003\.8080\.403\.8081\.303\.8082\.104\.2066\.403\.80M1 \(Quarterly\)91\.606\.6089\.506\.5090\.006\.5090\.006\.5079\.706\.9088\.906\.7087\.006\.7090\.106\.2089\.806\.7078\.5010\.90M1 \(Yearly\)72\.006\.8073\.406\.9072\.807\.6075\.507\.4080\.107\.7075\.607\.5076\.806\.5073\.207\.2072\.907\.5034\.903\.60M3 \(Monthly\)79\.502\.6081\.802\.5080\.602\.4086\.902\.60100\.003\.4086\.902\.60104\.103\.4081\.202\.6080\.702\.5051\.301\.80M3 \(Quarterly\)124\.003\.60126\.603\.70126\.003\.80111\.403\.5099\.503\.70111\.303\.4093\.103\.30121\.103\.40126\.003\.8063\.203\.60M3 \(Yearly\)80\.203\.7081\.804\.2083\.304\.1091\.604\.40102\.804\.2090\.404\.0098\.804\.5083\.704\.1083\.304\.1029\.901\.70M4 \(Quarterly\)94\.000\.7093\.900\.7094\.100\.7095\.000\.7097\.900\.7095\.000\.7098\.000\.7094\.400\.7094\.100\.7063\.500\.50M4 \(Yearly\)69\.200\.6069\.700\.5070\.200\.6069\.000\.50100\.600\.7068\.700\.6091\.200\.7070\.200\.6070\.200\.6034\.700\.30M5139\.809\.40143\.509\.90143\.2010\.00148\.409\.40144\.709\.10148\.409\.40150\.609\.00147\.309\.50142\.9010\.0063\.001\.10NN5 \(Daily\)103\.807\.60103\.807\.70113\.109\.10126\.607\.70120\.907\.70127\.207\.20122\.908\.50117\.108\.10113\.509\.5076\.908\.50NN5 \(Weekly\)79\.408\.9080\.607\.7082\.009\.1080\.509\.2081\.808\.4080\.409\.1079\.508\.6081\.109\.0081\.808\.8069\.707\.00Tourism \(Monthly\)70\.404\.2070\.104\.0070\.604\.2079\.104\.0096\.705\.1079\.004\.2095\.805\.1071\.204\.2070\.604\.2052\.503\.80Tourism \(Quarterly\)72\.103\.6071\.303\.6072\.403\.9073\.603\.7089\.804\.5073\.603\.7094\.305\.5073\.103\.8072\.403\.7059\.103\.80Tourism \(Yearly\)61\.903\.6062\.903\.7062\.303\.5067\.703\.5080\.603\.9066\.203\.5091\.304\.0062\.503\.4062\.303\.6052\.503\.30Traffic143\.506\.00133\.405\.90137\.405\.90140\.705\.70113\.705\.70140\.706\.00153\.505\.90138\.505\.70140\.105\.70106\.905\.00Weather121\.202\.30121\.602\.20125\.102\.90130\.902\.50132\.804\.80130\.802\.50129\.002\.20130\.402\.90124\.202\.5092\.803\.50

Table 15:Comparisons of NEAURC of SGA and its contenders across 27 datasets for Aurora, where bold and underlined values denote the best and second\-best results, respectively\.
### B\.5Relation between Uncertainty and Scale

Figure[7](https://arxiv.org/html/2609.28582#A2.F7)shows the plots of averaged uncertainty versus MASE of TSFMs over diverse scales on 27 datasets\. It is observed that across some individual datasets, larger circles do not appear in the bottom\-left direction relative to smaller circles, indicating that the scaling law of TSFM performance or uncertainty is not consistently observable with every single dataset\.

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Scaling/all.png)Figure 7:Plots of averaged uncertainty versus MASE of TSFMs over diverse scales across 27 datasets\.
### B\.6Additional Ablation Analyses

This subsection further analyzes how node alignment and node uncertainty calculation affect the performance of SGA on diverse datasets\. Figures[8](https://arxiv.org/html/2609.28582#A2.F8),[9](https://arxiv.org/html/2609.28582#A2.F9),[10](https://arxiv.org/html/2609.28582#A2.F10)show the ablation comparison of UQ performance of SGA on 11 TSFMs of 3 types over 27 datasets\. There are two key observations\. First, the green bars are significantly taller than the blue bars, demonstrating that node alignment is critical to SGA’s performance\. Second, the red bars are typically slightly taller than their blue counterparts, which suggests that when applying SGA to selective prediction, topological information contributes more significantly to the construction of graph complexity than TSFM\-inherent stochasticity\.

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Ablation/all_1.png)Figure 8:Ablation comparison of UQ performance of SGA on 11 TSFMs of 3 types over the first 9 datasets\.![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Ablation/all_2.png)Figure 9:Ablation comparison of UQ performance of SGA on 11 TSFMs of 3 types over the middle 9 datasets\.![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Ablation/all_3.png)Figure 10:Ablation comparison of UQ performance of SGA on 11 TSFMs of 3 types over the last 9 datasets\.
### B\.7Additional Sensitivity Analyses

This subsection further investigates how three key hyperparameters in SGA, namely the number of samplesKK, the slicing lengthlsl\_\{s\}, and the threshold coefficientλ\\lambda, affect the HUQ performance of SGA on 27 datasets\. Figure[11](https://arxiv.org/html/2609.28582#A2.F11)shows the impact of the number of samplesKKon the performance of SGA across 27 datasets and 11 TSFMs of 3 types\. Since a largerKKleads to higher computational costs, we recommendK=20K=20for the C\-2 family, TimesFM\-2\.5, Sundial, and Aurora, andK=30K=30for the C\-T5 family and Timer\-S1 to achieve a favorable trade\-off between efficiency and performance\.

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Sensitivity/all_k.png)Figure 11:Impact of the number of samplesKKon the performance of SGA across 27 datasets and 11 TSFMs of 3 types\.Figure[12](https://arxiv.org/html/2609.28582#A2.F12)shows the impact of the slicing lengthlsl\_\{s\}on the performance of SGA across 27 datasets and 11 TSFMs of 3 types\. According to the overall performance, we recommendls=4l\_\{s\}=4\.

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Sensitivity/all_l_s.png)Figure 12:Impact of the slicing lengthlsl\_\{s\}on the performance of SGA across 27 datasets and 11 TSFMs of 3 types\.Figure[13](https://arxiv.org/html/2609.28582#A2.F13)shows the impact of the threshold coefficientλ\\lambdaon the performance of SGA across 27 datasets and 11 TSFMs of 3 types\. According to the overall performance, we recommendλ=0\.25\\lambda=0\.25\.

![Refer to caption](https://arxiv.org/html/2609.28582v1/Figs/Sensitivity/all_lambda.png)Figure 13:Impact of the threshold coefficientλ\\lambdaon the performance of SGA across 27 datasets and 11 TSFMs of 3 types\.

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