Investigating Hallucinations of Time Series Foundation Models through Signal Subspace Analysis
Authors: Yufeng Zou
Zijian Wang
Diego Klabjan
Han Liu
Venue: NeurIPS 2025 poster ()
Keywords: foundation models
hallucinations
time series forecasting
Primary area: deep_learning

Abstract: Times series foundation models (TSFMs) have emerged as a promising paradigm for time series analyses and forecasting, showing remarkable generalization performance across different domains. Despite the efforts made on hallucinations of foundation models, hallucinations of TSFMs have been underexplored in existing literature. In this paper, we formally define TSFM hallucinations in the zero-shot forecasting setting by examining whether a generated forecast exhibits different dynamics from those of the context. Our study reveals that TSFM hallucinations are associated with the loss of context information in hidden states during forward propagation. As such, we propose a methodology to identify signal subspaces of TSFMs and magnify the information through intervention. Experiments demonstrate that our proposed intervention approach effectively mitigates hallucinations and improves forecasting performance. The signal strength measure computed from signal subspaces shows strong predictive power of hallucinations and forecasting performance of the model. Our work contributes to deeper understanding of TSFM trustworthiness that could foster future research in this direction.

Investigating Hallucinations of Time Series Foundation Models through Signal Subspace Analysis
Yufeng Zou Zijian Wang Diego Klabjan' Han Liu
DDepartment of Computer Science, 'Department of Industrial Engineering
and Management Sciences, DDpartmeet of Statistics and Data Science, Northwestern University Sccool of Computer Science, The University of Sydney sureng.zou@u yufeng zou@u wwwtwwester..nneeemmmooo. edu
zwan09986uni, sydno/gyeeem...nn........m.. sydney.edu. edu.au {d- {d-klabjan, fanlin/@northvesterr.cmm edu
Abstract
Times series foundation models (TSFMs) have emerged as promising paradigm for time series analyses and forecasting, showing remarkable generalization per- formance across different domains. Despite the efforts made on hallucinations of foundation models, hallucinations of TSFMs have been underexplored existing literature. In this paper, we formally define TSFM hallucinations in the zero-shot forecasting setting by examining whether a generated forecast exhibits different dynamics from those of the context. Our study reveals that TSFM hallucinations are associated with the loss of context information in hidden states during forward propagation. As such, TSFMs and magnify the information through intervention. Experiments demon- strate that proposed intervention approach effectively mitigates hallucinations and improves forecasting performance. The signal strength measure computed from signal subspaces shows strong predictive power of hallucinations and fore- casting performance the model. Our work contributes to deeper understanding of TSFM trustworthiness that could foster future research in this direction.
propose methodology identify signal subspaces of
Introduction
Times series analysis is major research field that facilitates decision making and scientific inference across broad range of domains, from energy and weather to economy, transport, and system management. As key task, time series forecasting has motivated the development of distinct approaches including statistical 6 37] and deep learning [31 45 models. Despite competitive performance for specific tasks, these models are typically trained on single domain, without sufficient capability to generalize to different domains. Inspired by the success of foundation models in fields like natural language processing 717, time series foundation models (TSFMss have recently emerged as a new paradigm towards universal forecasters 44 g
26
44 Pretrained on
large-scale time series data, TSFMs have shown remarkable few-shot and even zero-shot forecasting performance across numerous domains [3] [1), substantially reducing the need for downstream data. The hidden representations of TSFMs are also useful for downstream tasks through the extraction of
context time series information 355
Yet, the performance of TSFMs is often plagued by hallucinations, as with other foundation models. Broadly referring to the generation of unsupported statements nonsensical content, hallucinations are attributed to incorrect knowledge or insufficient inference capability of model [20]. Among various hallucination detection and mitigation approaches proposed, mechanistic interpretability and
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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test-time intervention require no additional training and have demonstrated effectiveness for large
language models (LLMs) 23 27
36
48 and large vision-language models (LVLMs) 25
In zero-shot forecasting, where a TSFM is tasked with generating extrapolations based on the extracted information of context time series such trends, periodicity, and patterns [9]. accurate processing of the context information is essential for generating high-quality forecasts. As such, we study TSFM hallucinations by examining whether forecast exhibits drastically different dynamics from those of the context, e.g., Figure D
(a) versus (b), We investigate the underlying mechanisms of TSFM hallucinations through the lens of hidden representations and develop a novel intervention approach to address the identified causes. As far as we know, limited effort has been made on similar research problems in the existing literature. We strive to address these knowledge gaps and contribute to deeper understanding of TSFM trustworthiness, fostering future research in this direction.
We formally define TSFM hallucinations in the zero-shot forecasting setting in $ and outline the knowledge rules for checking hallucinations in practice. In $4.1] we gain insights on TSFM hallucinations through experimental analyses, where we find that hallucinations are associated with a lack of context information in hidden states during forward propagation. We then propose a method to identify the signal spaces along with measure (SSAS) to quantify the signal strength of hidden states in $4.2 Built upon these results, we propose a novel intervention approach (SSIM) that mitigates hallunittioons by magnifying the signal information of hidden states in $4.3 Extensive experiments in $5.2 demonstrate that while the forecasting performance of TSFMs suffers from hallucinations, our test-time intervention effectively mitigates hallucinations and improves the quality of forecasts, yielding up to 6.62% reduction in the hallucination rate, 93.83% gain in R², and 13.52% gain in correlation. Moreover, the signal strength measure we propose has strong predictive power of both hallucinations and forecasting performance of TSFMs.
Our main contributions in this work are as follows. (1) We formally define the problem of TSFM hallucinations and outline a set of procedures to check hallucinations. We are the first to systematically study this problem our best knowledge. (2) We propose method identify the signal subspaces of TSFMs along with a measure to quantify the signal strength in TSFM hidden states. (3) We propose a test-time intervention approach to mitigating hallucinations by magnifying the signal information in TSFM hidden states. (4) We conduct extensive experiments on synthetic and real- world datasets across various domains to examine the impact of TSFM hallucinations and demonstrate the effectiveness of our proposed signal strength measure and test-time intervention.
2 Related Work
Times series foundation models. TSFMs represent a promising paradigm towards generalization across different time series domains and tasks leveraging the knowledge from large-scale pretraining data [4] 8 17 26 34 [4]. TSFMs not only substantially reduce the need for downstream data but have also shown capabilities of producing accurate forecasts even in zero-shot scenarios when forecasting on inputs from unseen domains 19I. While most TSFMs are Transformer based (38] and open sourced, they are diverse in architectural design, tokenization strategies, and pretraining objectives. For instance, Chronos 4 and Chronos-Bolt adopt an encoder-decoder architecture, while TimesFM 11 is decoder-only. Chronos Poronos-Boottee and TimesFM truncate the normalized time series inputs into patches, while Chronos discretely quantizes the scaled inputs into fixed vocabulary. Yet, the forecasting performance of TSFMs suffers from hallucinations when they fail to adequately capture the signal information from inputs, an issue which we study on models from both families.
Hallucinations. Hallucination, defined as the generation of unfaithful or nonsensical content, is a fundamental challenge in large foundation models due to their black-box nature (20). Recent research has examined model hidden representations for hallucination detection and mitigation, based on the
hypothesis that factual knowledge is encoded in these states [10] 12 16]. Studies have identified diagnostic signals in hidden states, showing that outlier or inconsistent activation patterns during generation can indicate potential hallucinations 2 9 13 36 41). Complementary approaches focus on hidden state manipulation, demonstrating that truthfulness can be elicited through targeted neuron activation interventions, offering promising directions for reducing hallucinations [3 25
36 46 47
We present the first attempt to formally define and systematically study hallucinations of time series foundation models, developing a methodology to detect and mitigate TSFM hallucinations.

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Intervention. Hidden state intervention has emerged as powerful technique for controlling neural models' behaviors, these internal representations serve as causal factors influencing model outputs. Disentangled time series representations have been learned using variational autoencoders [24 32
demonstrates effective control over model outputs through activation steering, which identifies iinear-nnterpretable directions in the representation space and guides hidden states along these pathways. Some achieve model output modification by selectively masking specific neuron activations, preventing corresponding generations from occurring [T0] 27 33 40). The approach [43] alters TSFM outputs but does not address specific challenges of time series forecasting. Differently, propose novel intervention approach TSFM hallucination mitigation, which is context adaptive and selectively intervenes model layers.
Research in 22
25 48
Definitions and Preliminaries
Formally, we first describe the forecasts of a time series foundation model and the problem of hallucinations.
Definition 1 (TSFM forecasts). A pretrained time series foundation model, denoted as Ma, takes a time series Frontezt = [z),...,,p xpl I of context length p as the input and generates a forecast [p+t1...... pp+g] of horizon y 2. We call i in o; or Tpti the i-th time time series foundation model encoder or decoder,
= -Ma (eeontezz))
series step. For an L--laer transformer-base denote the hidden states different positions of layer I (the outputs of the layer) as matrix R= where d is the dimension of hidden states.
H = ()) ,..., 11 ....,,..]
Definition 2 (Time series forecasting hallucinations). Suppose for a time series Full [f1...., ) a knowledge set K can be inferred from a partial time series Fcontest [x,, ......... ........ 2.l. I 1 < j T for any i and j. The knowledge set K comprises time-dependent knowledge rules that hold true for 2 full, i.c. r(x,, zero-shot time series forecasting, we consider a hallucination to be forecast that does not conform to the knowledge rules inferred from the context time series. We define the set of hallucinations as Hallu( (ccoteezt)
for all S T, or r(zful)
in short. In
{#: ArEK r r()) 0}.
The goal of hallucination detection is to define a score function f that discriminates hallucinated forecasts of the model, such that for any a Mo((Eontezr), we have [( & 0) > (allu(@contenn)
if an only if & on hidden states so that with the intervention operation I. we obtain non con-baalucinated forecasts Ma. (Pconteet)))) & H Hallu((contezt).
We mitigate hallucinations through test-time intervention
In practice, we sequentially extract a set of knowledge rules IK ( I from the context time series to check whether a forecast is hallucinated.
Trend. The trend rule checks whether the trend of the forecast conforms to those of the context. We perform ordinary least-square (OLS) regression on & and take the first-degree coefficient è as the trend if it is significant with the p-value < 0.01. We then perform OLS on rolling windows of Econtent and take significant trends (c.,....cu c,]. With the relative difference between trends computed as f(c é) = 1/. the trend rule is satisfied iff the minimum relative difference min; diff é) < 8, or neither the forecast nor the context exhibits a significant trend.
Frequency. The frequency rule checks whether the spectral density of the forecast conforms to those of the context. After removing the trend, compute the spectral densities [fi..... ... fn] of rolling windows on Fcontezt using short-time Fourier transform (STFT) 18] and also the spectral density f of é. With the Jaccard distance between spectral densities computed D(J, f) Σ. min the frequency rule is satisfied iff the minimum Jaccard distance min, D(f;, f) <
Σ, max fu.f()
Pattern.
The pattern rule checks whether the pattern of the forecast is close those of the context.
After removing the trend, we compute the relative absolute errors between the forecast and rolling
windows [Wi...., wn]] on Feontent- With the relative absolute error computed as RAE( a)
Ei the pattern rule is satisfied iff the minimum relative error miny R.A( (w x) < 8.
3

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17945
1.00
2.0
Hallacinated
Nor-ballachsted
0.75
13
0.5
0.0
0.5
-100
2.0
100
200
300
400
5.0
100
200
200
400
500
(a) Hallucinated
(b) Non-hallucinated
(c) UMAP visualization
Figure 1: (a) (b) Examples of hallucinated and non-hallucinated forecasts by Chronos (c) UMAP visualization of hidden states at the final model layer for both examples.
ARMA. The ARMA rule checks whether the ARMA dynamics of the forecast conform to those of the context, which complements the pattern rule since time series that exhibits strong ARMA dynamics may not have distinct patterns. After removing the trend, we fit first-order Autoregressive moving average (ARMA) model [6] on Ceontest and take the AR and MA coefficients & and & both are significant with p-values less than 0.01. Let á and be the first-order ARMA coefficients on é the ARMA rule is satisfied iff the relative differences
and 1
TSFM forecast that violates either the trend, frequency. or both the pattern and ARMA rules is considered be hallucinated, since it is ungrounded in the context time series information. Further details are provided in Appendices A and D.3.
Methodology
To understand the cause of TSFM hallucinations, we first gain insights from observations, provide intuitive explanations, and then perform experimental analyses to justify our claims. Afterwards, we propose a signal strength measure to help detect hallucinations. Finally, we develop a novel test-time intervention approach that mitigates hallucinations by addressing the identified causes
4.1
Observations and Findings
We begin with a brief case analysis. Figure (a) presents hallucination example where TSFM fails to generate a forecast consistent with the context time series. The 2-dimensional UMAP [28 visualization of hidden states the final layer in Figure (c) shows irregular patterns, with a high mean pairwise cosine similarity at 0.8763 and a low mean activation std across positions at 11.0570
(we observe two clusters since the context time series binary). We speculate that the forecast failure is caused by the loss of context information in hidden states during forward propagation. We find that injecting a small amount of random perturbation the context time series with Gaussian noise alleviates such information loss, as shown in Figure (b). We observe that the hidden states become more evenly distributed in each cluster Figure (c), with the mean pairwise cosine similarity substantially reduced to 0.6338 and the mean activation std raised to 14.5772.
To analyze the effects of context signal and noise on TSFM when hallucination occurs, collect the activations over 10 random perturbations to context signal with Gaussian noise of varying magnitudes. From Figures 2 (b), we observe that hidden state activations exhibit the greatest variance across the positions of layer with clean signal. As the signal mixes with greater noise, despite the increase of input variance, hidden state activations become less variant. The decline in mean activation variance with noise is more salient at higher layers, suggesting that a TSFM incrementally extracts signals and reduces noises from the input at each layer.
'This not based rolling windows for computational efficiency.
We check hallucinations based on the rules defined in 3 with parameters specified in observations not sensitive these parameters.
5.1 The results and

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0m
448
(a) Chronos
(h) TimesFM
(c) Chronos
(d) TimesFM
Figure 2: (a) (b) The mean standard deviations of hidden state activations across the positions of layer under varying noise magnitudes. (c) (d) The mean pairwise cosine similarities of hidden states across the positions of layer. The z-axes represent the standard deviation of Gaussian noise.
Based on this, we posit that the hidden state space H()) at each TSFM layer can be decomposed into signal and noise subspaces (H))) g(1) @ NO) C Id, respectively handling the signals and noises of the input. Decomposition based null spaces and similar concepts have been considered in the past 14 21 30). but herein we introduce the idea of signal and noise. In forward model propagation (1)
HDL,, the signal components of a hidden state Nga h are further processed by subsequent layers, while the noise components yg ( are repressed and eventually removed. Since the signal components more variant and dissimilar across hidden state positions than the noise components, the hidden states would exhibit greater distinctiveness across positions with strong signal presence at layer (Figures ( (c)
Reporting Figures D (a) of TSFM hallucinations, the inactivity signal subspaces of model leads to homogeneous hidden states across positions. In such case, a proper amount of input random perturbation injects variances that help restore the activity level of signal subspaces and facilitate the propagation of context signal information. Nonetheless, it is hard determine the optimal amount of perturbation, since too much perturbation obscures the input signal and degrades forecast quality. Moreover, single perturbation is not robust (25]. while performing multiple perturbations hampers efficiency. Our goal to magnify the signal information of hidden states through test-time intervention, which would enable mitigate hallucinations in controllable and efficient manner.
4.2
Signal Subspace Identification
Next, we develop a novel method to identify the signal subspaces TSFM layers and provide empirical analyses. We aim identify set of hidden state neurons that most active to context signals by examining the variance activations across hidden state positions, enlightened by the associations between the activation variance and signal strength observed in the previous section. Let H (ar) be the activation neuron j at position of layer given input x: and (1) (a:) be the mean activation of neuron with respect to i. We compute the activity score of neuron j
A x)
Σ
h
(1)
The neuron activity measure we propose is more nuanced compared with the magnitude of neuron activations used in prior works [39 40) as not only reflects the overall magnitude but also measures the deviation of neuron activations across time series steps.
To measure neuron activity in the presence of signals, we collect the activity scores on a synthetic dataset comprising common waveforms that is described in $5.1 We also vary the magnitude of noises injected to the context signals and initialize them with different random seeds for robustness. With D Diignal denoting the set of synthetic time series inputs where hallucination occurs, consider neurons with the activity score consistently top ranked across the samples as candidate signal neurons, i.e., Cand(() = NNeE sggnat (j A x) qe}. where qe is the e quantile of the
underlying sequence (in this case ( x)). We compute the signal activity score of each neuron (!)
as the sample mean signal (j) D Pananatl ExeP A x)).
5

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e.g., signal processing and noise removal in concurrence, we aim to identify neurons that are primarily responsible for signal processing. To this end, we collect the activity scores A using Gaussian noises as the input in the same way and compute the contrastive neuron activity score between signal and noise inputs:
Since hidden state neurons may fulfill multiple roles [5 47
A contrastive (j) = Asgmal (j) - A ())
(2)
For each model layer / E (1,...,b) L), we pick candidate neurons with top-ranked contrastive activity scores the signal neurons, i.e., Sig(() Cand(l) { | A Acontrastion (5) q.). Figure displays the distributions of contrastive neuron activity scores across model layers. We note that at each layer only a small proportion of neurons are exclusively sensitive signal noise. Moreover, greater contrastive scores are observed at higher layers, where neurons are more specialized.
Ranking signal neurons by contrastive tivity score, we select the top signal neu- ron's activity score at the final layer strength measure of signal information the model has processed, i.e., () x) with contrastine (i'). j arg y A (LL
Loyer
Laver
Lyyer
Layer
15 32
Contertee www.urt
Loyer 12
Layer 53
4.
The final layer is selected as it shows the greatest contrastive scores between signal and noise inputs (Figure 3). We call the proposed measure Signal Subspace Activ- ity Score (SSAS) and will verifyy its efficacy of hallucination detection and forecasting performance prediction in (5.2) With this, claim that TSFM implicitly expresses in the hidden state subspaces how much signal information is able to capture from the context.
200
400 Neurtre
000
160
230
8do
1003
Neuten
(a) Chronos
(b) TimesFM
Figure 3: Ranked contrastive neuron activity scores
across model layers.
4.3 Signal Subspace Intervention
Built upon previous results, we propose a Center-Project-Seee (CPS) intervention operation to mitigate hallucinations by magnifying hidden state signal information. Setting ((O) to Sig((), CPS works for the hidden states H ((0)))) E Raxd of a TSFM layer during forward propagation as follows:
1. Centering H (x) by subtracting the mean across hidden state positions to obtain H (x)= H (z) (x):
2. Computing the projections on signal subspaces Tgay H (x)) at all positions;
3. Scaling the signal components by factor X so that (!! (x) H! + +2
1) ggT (x);
4. Adding back the mean to obtain the intervened hidden states H (x)
H (2) (x).
The intervened hidden states inputted to the next layer. We center the hidden states in Step to emphasize the activation differences across positions. Arranging the basis of g(() into an orthogonal matrix P = (e.,..., ex] E Rdd where e; is the indicator vector a signal neuron, the projection in Step 2 can be computed by matrix product. The CPS operation can be formulated simply as (x) (1) (x) + (A H! (z)PTT P. which can be efficiently computed each layer in ((ddk) cost, with k & d. The cost can be further reduced to (Okk) leveraging sparsity of P.
The CPS operation has desirable properties. First, the mean of hidden state neuron activations is unaltered by the operation, and the standard deviation scales proportionally with X, which makes the operation easy control and causes no distribution drift neuron activations. Moreover, different
from prior approaches that add a static steering vector to hidden representations [2 25 43) our
approach adaptively alters neuron activations based on their distributions, improving the contrast of hidden states and clustering effects. We mathematically show that in many cases the CPS operation can reduce the cosine similarity between hidden states (see proofs in Appendix B).
We further propose an adaptive scaling approach that helps identify the cases necessary to apply the intervention. Since the signal activity scores A sigmat measure neuron activity in the presence of

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strong signals, we use them as a reference. At each layer, we compute the mean activity scores of signal neurons A (x) = EjEsin()) (j x). We then compute the adaptive scaling factor as ratio ( ( signat/ (x)). with A Aignal x (t A signat (i), and apply the intervention only when \ > 1. In this way, we selectively intervene layers with weak signal information and scale the activations of signal neurons match those of the reference. We call the complete test-time intervention approach Signal Subspace Intervention through Magnification (SSIM).
Experiments
In this section, we conduct experiments to address the following questions: (1) How do hallucinations affect the performance of each type of TSFM? (2) What is the effect of proposed test-time intervention on hallucination mitigation? (3) What is the performance of our proposed signal strength measures? (4) How does each our designed component impact the intervention performance?
5.1 Experimental Settings
Datasets. We curate a synthetic dataset comprising of common waveforms of sine, square, sawtooth, triangle, and pulse waves with varying slopes from {-0.01, O, 0.013. We vary the number of periods in the context from {8, 10, 12, 14, 16, 18, 20} and the standard deviation of Gaussian noise added the context signal from {0, 0.1, 0.2, 0.4,
addition, we adopt read-world datasets from the GIFT-Exal 3 benchmark covering various domains. We take fixed number of final observations from each time series, dividing them into context and ground truth of fixed lengths. We discard time series instances with over 10% missing values and impute missing values with the segment mean. As defined in ]3] we retain time series instances whose ground truth satisfies the knowledge rules extracted from the context such that the context contains sufficient information for forecasting, Each dataset randomly split into validation (20%) and test (80%) sets. Further details are provided in Appendix D.2.
Baselines. For hallucination mitigation, we compare SSIM with input denoising by smoothing as well as input perturbation with output averaging (25). Additional comparison study with finetuning presented in Appendix D.4. For hallucination detection, we compare SSAS with other statistics discussed in $4.1, including the mean pairwise cosine similarity of hidden states and the mean standard deviation of neuron activations.
Evaluation metrics. We evaluate forecast quality with R² and the Pearson correlation coefficient, which are scale invariant. Metric R² measures the goodness of fit of the forecast the ground truth, ranging in (-o0, 1]: the Pearson correlation coefficient measures the strength and direction of the linear relationship between the forecast and ground truth, ranging in [-1, 1] (invalid values are imputed with 0). Whether forecast is hallucinated is determined according to the knowledge rules defined in $3 We evaluate the effect of hallucination mitigation with hallucination rate reduction and forecast quality improvement. We assess the accuracy of hallucination detection with AUROC and forecasting performance prediction with the Spearman rank correlation coefficient.
Implementation details. We evaluate on three mainstream TSFMs: Chronos 4 Chronos- and Times/M [TI. We set the context length to 500 and the forecasting horizon 64 for zero-shot time series forecasting in our main experiments, using the base versions of Chronos and Chronos-Bolt together with TimesFM-2 O. As Chronos produces probabilistic forecasts, we set the number of decoding samples to 1 and fix the random seed to ensure reproducibility. We set the frequency configuration of TimesFM frequency, pattern, and ARMA rules to 0.25, 0.5, 0.5, and 0.25, respectively, based on validation. For SSIM, we perform grid search for the proportion of selected top neurons E {0.1, 0.2, 0.3. 0.4, 0.5) and set it to 0.1 for Chronos and TimesFM and 0.2 for Chronos- Chronos-Bolt based on validation. For baselines methods, we denoise input time series using the mean of sliding windows of size 5. We perturb input time series by Gaussian noise with a standard deviation of 0.05 times that of the input for 10 runs with different random seeds. We leverage signal strength measures for forecasting performance prediction and their negations for hallucination detection.
For hallucination check, we set the tolerance thresholds å of the trend,
7

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Table 1: Comparison of forecasting performance across domains, with the best results boldfaced.
Model
Domain
Hal
Original R² 1
Corz [
Hol 4
Denvising R2º
Corr
Periursation Averaging
R²-
Hol
SSIM (ours)
Corr
Syntheric
0.4534
-B.1625
0.6265
0.4420
-0.6334
0.5714
0.433
-0.1055
0.6992
0.4145
0.1854
0.7150
Bcoe/Pin
0.4115
-3.3554
0.4751
0.5007
4.5011
0.3413
0.4689
-3.5727
0.435
0.4061
-3,2837
0.5146
Chronos
Energy
0.1389
-0:4830
0.7180
0.2504
-3.0764
0.5315
0.1212
-0.2073
0.7241
0.1191
0.0268
0.7707
Nature
-10.7363
0.0457
0.9514
8. 3558
0.0575
0.8436
-7.2975
0.0553
407577
0,1082
Transport
-1.6244
0.5127
0.7565
4804
0.5295
0.4461
-1,4582
0.3938
0.6081
Woh0ps
0.5801
414,8937
0.2762
0.8833
139 8035
0.1559
06115
-79,3298
0.2822
0.6052
-2.8389
0,3369
Aggrgaued
04451
-82.3762
0.4458
0.5991
-28 6399
0.3336
0.4759
-17.2700
044599
0.4231
-5.0845
0.5061
Chronos Hols
Foon/Fin
Syathetic
0,8866
0.5381
-1,3759
0.0152
0.5589
0.5727
0.4770
0.5810
-1.2344
0.0625
0.5302
0.4243
0.5500
0.4911
-1,4191
0.0099
0.5586
0.5929
0.4787
0.5231
-1.2891
0.0238
0.5672
0.5811
Energy
0.1499
0.7094
0,1712
000411
0.6291
00002
0,1035
0.7671
0,0843
6.1508
0.7766
Nature
0.9426
-0.7744
0.1400
0.95%
0.6290
0.1057
09404
0.0875
0.1462
0,9316
4.0657
0.1472
Transpont
0.6684
0.2039
0.6501
0.8446
0.2129
0.4072
0.6632
0.2015
0.6188
0.6522
0.3124
0.6563
Wehlpps
0.6777
-0.6529
0.3591
0.90124
-I 2424
0.1777
0.6707
-0.4822
0.5625
0.6632
410680
0.3646
Apgregated
-0,4260
0.5099
0.6084
0.6062
0.3848
0.5321
403766
0.5171
Syathetic
0.5661
0.1452
0.7568
0.1190
0,5688
0.9044
0.5699
0.9194
Ecce/Fin
0.3868
-187115
0.7793
0.4672
5.1532
0.3722
0.4005
1.8277
0.7657
0.3771
4,3161
0.7847
TixedFM
Energy
0.1357
0,2745
0.8065
0.1987
0.0881
0,6150
0.1341
0.2916
0.7911
0,1222
0.1301
0.8133
Nature
0.9558
-0.1678
0.1620
0.9691
0.2561
0.1302
0,9492
-0.1715
0.1451
0.9536
-00902
0.1559
Transport
0.5751
0.4245
0.7027
0.632]
0.4210
0.3540
0.5648
0.4236
0.7028
0.5733
0.4201
0.7076
WebOps
0.8676
20.0077
0.2324
0.6202
-7,5448
0.1216
4.7480
0.4291
Aggregated
0.4441
-4,5461
0.6368
0.5251
-3.3271
0,4119
0.4418
2,0435
0.6275
0.4348
-1,2529
0.6410
5.2 Main Experimental Results
TSFM hallucinations (RQI). Table summarizes the performance of original TSFMs. We note that the hallucination rate varies drastically across domains. On domain Energy the time series have more periodic patterns, resulting in low hallucination rates; while on domain Nature the time series contain more abrupt changes, making it harder for TSFMs to process the context information. The forecasts appear to have stronger correlations with the ground truths on domains where the hallucination rate is lower. Table 2 compares the performance of hallucinated forecasts versus non-hallucinated forecasts by TSFMs. We observe that hallucinated forecasts are consistently outperformed by non- non-hallucinated forecasts. For Chronos-Bolt and TimesFM, the mean 122 of non-hallucinated forecasts stays positive on each domain. Hallucinated forecasts have substantially weaker correlations with ground truths than non-hallucinated forecasts for all models, indicating weaker signal information captured from the context. Unpaired t-tests yield p < 10 against the null hypothesis that the performance of hallucinated and oonhhalluedetee forecasts is the same. These results underscore the significance of TSFM forecasting performance being affected by hallucinations.
Figure 4 (a) compares the distributions of TSFM hallucinations, with Type referring to the violation of the trend rule, Type 2 the frequency rule, and Type 3 to both the pattern and ARMA rules. We observe that Type 3 hallucinations are the most com- mon ones, since the pattern and ARMA rules demand more sophisticated reasoning of context information. Chronos exhibits fewer Type I and Type 2 hallucinations than the other models as it does not apply input patching, which enables more accurate processing of trend and frequency information.
Type
3pp
Type
Imporial (H)
Wopotton
(a) Original
(b) SSIM
Figure 4: Distributions of hallucination types.
Hallucination mitigation (RQ2). Table compares the forecasting performance with SSIM and the baseline methods. SSIM attains the best performance overall, yielding up to 6.62% reduction on hallucination rate, 93.83% gain on Ht² and 13.52% gain on correlation over the original models. Although denoising improves R²2 in some cases by reducing the impact of outliers, it causes higher hallucination rates and lower correlation in general due to the loss of context information. Pertur- bation averaging improves the forecast quality to some extent, but it does not sufficiently mitigate hallucinations and demands considerably more computation. In comparison, SSIM pre-compuees the signal neurons only once for each TSFM and incurs minor additional overheads during test time. The performance margin between SSIM and baselines is statistically significant at p < 0.01 by the Friedman-Nemeny test [15] 29). hallucinations. From Figure
We further analyze the effect of SSIM on the distributions of (b), we note that SSIM has the greatest effect on Type 3 hallucinations,
8

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Table 2: Performance comparison of hallucinated and non-hallucinated forecasts by TSFMs.
Chronos-Boll Now--uu
TimesFM Non-hul
Metric
Domain
Hul
Non-hal
Dij
Hul
Dijf
Hul
Dij
Synthetic
-1.1206
0.6290
1.7496
-0.4450
0.5512
0.9962
-1.5404
0.8379
2.3783
Econ/Fin
-6.7964
-0,9492
5.8473
-3.3098
0.4497
3.7595
-5.7181
0.5553
6.2734
Energy
-2,8794
-0,0974
2.7820
-1.1625
0.2934
1,4599
-0.8971
0.4585
1.3556
R²1
Nature
-12.8573
-2.0209
10.8364
-0.0846
0.0933
0.1779
-0.1923
0.3626
0.5549
Transport
-3,9208
0.0018
3,9226
0.0548
0.7253
0.7800
0.2046
0.7223
0.5178
WebOps
633,8971
-109900
523.9916
-8,1848
0.4657
1.6505
-34,4655
0.4127
34,8783
Aggregated
161.6823
16.6599
145.0224
-.1647
0.4096
1.5743
10.9572
0,5757
11,5329
Syuthetie
0,3478
0.8568
0.5090
0.3596
0.7910
0.4314
0.7436
0,9364
0.1927
Econ/Fin
0,0853
0.7476
0.6624
0.2558
0.8719
0.6162
0.5927
0.8969
0.3042
Energy
0.4738
0.7574
0.2836
0.5716
0.7910
0.2194
0.6300
0,8342
0.2043
Corv 1
Nature
0.0226
0.1401
0.1176
0.0991
0.8118
0.7127
0.1312
0.8289
0.6977
Transport
0.2776
0.6827
0.4051
0.5407
0.8708
0.3301
0.5744
0.8765
0.3021
WebOps
0,1036
0,5148
0,4113
0. 1760
0.7441
0. 5686
0.2135
0,7984
0,5849
Aggregated
0,J459
0.6943
0.5484
0.2441
0.8105
0.5664
0.3429
0.8716
0.5286
Table 3: The results of TSFM hallucination detection and forecasting performance prediction, with the best results boldfaced. For each method, the first column shows AUROC of hallucination detection and the latter two columns show rank correlations with the performance metrics. The statistical significance of positive rank correlations is indicated with for p 0.05 and ** for p < 0.01.
Cosine Similarity At
Activation Variance
SSAS (Ours) R²" 1
Model
Domain
Hal
Corr
f
It²
Corr
Hal
Corr
Syathetic
0.7847
0.3834
0.3262
0.6786
0,3734
0.4052** **
0.8316
0.4299**
0,5111
Feon/Fin
0.6501
0.6208
0.6927
0.5096 **
0,5507
0.7833
0.5034
0.5258 **
Chronos
Eneray
0.7124
0.5088
0.3528
0.8093
10.1
0.0373
0.8096
0.1166**
0.0363
Nature
0.4978
0.3382
0.1524
0.5384
0.3490
0,1886
0.5925
0,3430
0.1507**
Transport
0.6601
0.4706*
0.5550
0.7466
0.5351 *
0.6083 *
0.6767
0.415%
0.5234
WebCpt
0.2328
0.3710
0.5060
0,2720
0.5740
0.1693**
0.2526 **
Aggregated
0.7903
0.5866
0,5804
0.7226
0:4197
0.5277
0,8086
0.5082*
0.5758
Synthetic
0.4416
-0.1767
-00331
0.4011
-0.3806
-033569
0.5282
0.2363*
0.2142
Chronaz Rolt
inon/Tin
0.2247
-0.5253
-0.5530
0.3851
-0.3600
00.4283
0.8528
0.6289
0.5979**
Energy
0.5360
0.2052
0.1488
0.4739
0.4158
0.3741
0.7451
-0.0603
-0.8665
Nature
0.9343
-0.1155
0.3819
0.9395
-0.1003
0.3842 *
0.6340
0,0895
0.1690
Transport
0.7183
0.2601 *
0.2394
0.6656
0,2730
0,2064
0.6416
0.2850
0,2858*
WebOps
0.6656
0,1315
0.4091
0.5224
-0.0236
0.1767
0.6518
0.2188
0.3423
Agpregated
0.6279
0.0462
0.2607
0.6131
0.0789
0.1840"
0.7991
0.3860
0,5037
Synthetic
0.4892
-0.2210
-0.1684
0.3821
-0.3302
-0,2738
0.4902
-0.0081
-0,0359
icon/Tin
-0.6896
-0.6568
0.1930
-0,5539
0.4625
-0.0932
TineaFW
Energy
0.2898
-0.2597
-0.1570
0.2777
-0.1531
-0.1911
0.7469
0.1407
0.1042
Nature
0.4042
-0.0565
-0.9041
0.7912
0.0587
0,3480
0.8934
0.1744
0.3870
Transport
0.4881
-0.0669
-0.0174
0.6121
0.1892 **
0 **
0.6529
0.3483*
0.3622
WebOps
-0.1001
0.1046 *
0.6365
0.0637
0,2418
Agpregated
0.3963
-0.2608
-0.1767
0.5211
-0.0517
0.0634
0.6890
0.2636
0.3552
yielding up to 5% reduction. The facilitation of signal information propagation helps TSFM better capture patterns from the context.
Hallucination detection and performance prediction (RQ3). We report the performance of different measures for hallucination detection and forecasting performance prediction in Table 3 SSAS has consistently strong predictive power across domains for different TSFMs, attaining high AUROC for hallucination detection and significantly positive rank correlations with forecasting performance, demonstrating the effectiveness of our proposed signal strength measure and the critical role of signal neurons in generating high-quality forecasts. Simply using the mean neuron activation variance measure yields inferior and less consistent results overall, as it is obscured by the activity of irrelevant neurons. The mean pairwise cosine similarity of hidden states exhibits relatively strong predictive power for Chronos and Chronos- Boolt but fails to generalize to TimesFM.
Ablation study (RQ4). We compare the performance of SSIM intervention with the following reduced variants: (1) w/o adaptive scaling: using constant scaling factor À for each layer; (2) w/o centering: scaling neuron activations without subtracting the mean across the positions of
layer [10 40). From Figure 5] SSIM consistently outperforms the variants. The performance

---

54
adspdive
centieting
0.62
SGIM
% Hallciration
Correlation
0.60
48
0.58
0.52
0.50
1330
2001
40
Chronos
Chrones-Bolt
TimesFM
Chrones
Chronos-Bolt
TimesFM
Figure 5: The aggregated mean performance of SSIM and the variants for TSFMs.
differences are significant at < 0.01 by paired t-tests, highlighting the effectiveness of our design. The adaptive scaling enables more refined control of the intervention, providing greater magnification when weak signal information is detected at layer. The centering operation places an emphasis on activation differences that helps reduce the homogeneity of hidden states without causing distribution drifts to the activations of intervened neurons.
6 Conclusion
Time series foundation models represent a promising paradigm for time series analyses and fore- casting, yet the issue of hallucinations has been underexplored in existing literature. In this paper, we have formally defined TSFM hallucinations in the zero-shot forecasting setting and outlined of knowledge rules for checking hallucinations in practice. We have found that hallucinations are associated with a lack of context information in hidden states through experimental analyses. We have proposed a method to identify the signal spaces along with measure to quantify the signal strength of hidden states. We have further developed an intervention approach that mitigates hallucinations by magnifying the signal information of hidden states. Extensive experiments across various domains have demonstrated that our test-time intervention effectively mitigates hallucinations and improves the quality of TSFM forecasts. The signal strength measure we proposed has shown strong predictive power of both hallucinations and forecasting performance. Our work contributes to deeper understanding of TSFM trustworthiness that could foster future research.
Acknowledgments
Yufeng Zou is partially supported by the Walter Murphy Fellowship. The content solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies.
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Appendices
Knowledge Rules
21
B Properties of CPS Operation
21
SSIM Algorithm
23
Experimental Details
23
D.I
Experimental Platform
23
D.2
Dataset Details
23
D.3
Rule Ablations
24
D.4
Additional Baseline
24
D.5
Parameter Study
24
Additional Experimental Results
25
E.1 Model Size
25
E.2 No Data Filtering
26
Case Study
27
Limitations
27
H Impact Statement
27
Ethics Statement
28
Reproducibility Statement
28
A
Knowledge Rules
The rolling windows we apply on the context always have the same size as the forecasting horizon. 30 We adopt the OLS and ARMA (6) implementations in the statsmodels package We adopt the STFT 18 implementation in the SciPy package with unsymmetrical Parzen windows and hop=1.
B Properties of CPS Operation
Center-Project-Sean (CPS) is our proposed intervention operation that magnifies the signal informa- tion in hidden states with a scaling factor A. We use S to denote the identified signal subspaces. CPS has several mathematical properties that help with the effectiveness of hidden state intervention:
Theorem 1 (Mean Invariance). The mean of hidden state neuron activations across the positions of layer is unaltered by the CPS operation, i.e., E(h) = h.
Proof. Let h denote the mean of hidden states across the positions of layer. For a hidden state h, the hidden state after CPS becomes h h + (A [ (h h). The mean of hidden states after CPS is
E(h) 1 Σn h Σn 1 (A-1))
1 n Ei (A 1)]1s(h= h). h) = (A 1 D (h, h) = 1h h) = 0, we
h)] 1 En h
Since 1 Liml
1g h4
have E (h) ΣR h, +0=h. h.
Theorem 2 (Standard Deviation Scaling). In CPS, the standard deviation of signal neuron activa-
tions across the positions of a layer scales with X, i.e., A(j) = AAA()) any j E S.
OLS. html
ARIMA.
html
http:///eeeerrrrrccccc////////ed Shon
htmla #ww.pysc.ssgg..... www.ssigaaa..rrTinigoteee
21

---

Proof. For any j E 8,
A()- =
H - F((h))
1=1
His + (A (H --)- h5g]
(From Theorem
His + (A ()(His h₂) - =n,]²
(AHij- - Xhz)² 2
Hi.j -hj)
A(A
Theorem 3 (Cosine Similarity Reduction). For two hidden states h2 and h2 that have differ- ent projection directions in the signal subspace after subtracting the mean, the CPS operation reduces the cosine similarity of these two hidden states with sufficiently large scaling factor X, i.e.
cos(h1, h22) < cos(h1, h22 when X > max (81+82) 1,
28 11 1.
2 a 1,1}.
Proof. The original hidden states can be decomposed as h1 = + ni + h and h2 = 82 + n2 + h. where 81 11s h- h), nı 11g(h, h), 82 llg h). and n2
(h2 h). We have h h 81 82 + (8) + 82) + residual terms. The hidden states after CPS can be written as h \sı + + and h2 = As2 + + h. We have h1 h2 = X281 82 + +A(11
(((s++20 + +2) h residull terms. The residual terms are n, n2 (n1 n22) h + |||| in both cases. Then,
h1 h2
h1 h2
X 82 + X(s) + 82) h < 81 (12 1)81
82 + (8) + 82) h
82 + () + 82) h <0
(A 1)81 82 (8) + 82) h <0
(Assume À >>> 1)
(81 82) h
A
1.
(Assume 81 882 < <)
81 82
Also,
||h,|| > hhil|
2²1 81 + + h + ||]]]
81 + 281 h + ||11
(X² 1)s1 81 +2( - 1)81 h> 0
(A+1)81 + 23 h>0 0
(Assume À > 1)
28 h || 2
1
-1
Similarly, have ||aa|| >
||2||| when A
2
and X 1.
|
Hence, when À > max(- (#1+e2)
1,
2 =]ll2
1,
2 82
1, and 81 82 < 0, we have
81.82
h h
h h2
poshhi h22) =
cos
h hall
|/
h2ll
22

---

Since we select signal neurons with top activity scores as the bases of signal subspaces, we often have 81 < O when the time series information encoded in h1 and h2 is distinct. The CPS operation reduces hidden state homogeneity by increasing the contrast time series signal information across the positions of a layer.
Corollary 4. In CPS. the cosine similarity of two hidden states tends the cosine similarity of their projections the signal subspace as the scaling factor tends to infinity, i.e., limx-+o cos(h, h22) cos($1, $2).
The proof' above explain how the CPS operation improves the clustering effects of hidden states by magnifying the time series signal information.
C SSIM Algorithm
Algorithm T details the full procedures of SSIM. The additional computation overhead each layer is O(k)), with & being the number of signal neurons at the layer.
Algorithm 1: SSIM: Signal Subspace Intervention through Magnification
Input :TFFM Mg with layers, context time series Econteat: signal neurons Sig, reference neuron activity scores Asignal
Output:Forecastt Forecasts
( <- Preprocess( (atprooeess((cmmteeeel)
2 for f 1..... ......L L do
H t MM (1-))
(
Σ (ESig(() (j x))
5
signal < E (t)ig()) signat( (J)
1
A
Aignal (x)
7
A > 1 then
H (1) H (0)
Center
H t H + (A - H) ( I, Sig(
Project and scale
10
end
11 end
12 2 < Predict( H (1)
return t
D Experimental Details
D.1 Experimental Platform
All experiments are conducted on the Ubuntu 22.04.4 LTS operating system, 16 Intel(R) Core(TM) 17-7820X CPUs, and 4 NVIDIA GeForce RTX 2080 Ti GPUs, with the framework of Python 3.11.9 and PyTorch 1.12.1.
D.2 Dataset Details
Synthetic dataset. The times series we generate takes the form of xxtt) = signal(t) + frend(t) moise(t). For signal component, we adopt common waveforms of sine, square, sawtooth, triangle, and pulse waves as implemented in the SciPy packagd We vary the number of signal periods in the context input from {8, 10, 12, 14, 16, 18, 20). We vary the slope of the trend component from {-0.01, O, 0.01). The noise component is Gaussian noise with mean of 0, and we vary the standard deviation from (0, 0.1, 0.2, 0.3, 0.4). In this way, we generate 525 time series instances in total.
23

---

Real-world dataset. We adopt the GIFT-Eval benchmark [3]. We discard time series instances with over 10% missing values and impute missing values with the mean of the segment. As explained in $3 of the main text, we retain time series instances whose ground truth satisfies the knowledge rules extracted from the context such that the context contains sufficient information for forecasting. The datasets are categorized into five different domains, with the bracket showing the number of time series instances after preprocessing:
Econ/Fin: m4 daily (294), m4 m4_ hourly (391), m4 monthly (168), m4 qua_ttrryyly (13), m4 fe__weekly (42);
Energy: electricity/IST (235), electricity/D (43), electricity/H (318), solar/10T (43), solar/H (132):
Nature: kdd_ car__uu___0000___ pep_2018_wwar 2018_wwth_aameeeeeeeeeeeeeeellae with_missing/H (66), temperature_rant painn with_missing (2358);
Transport: LOOP SESTTLEETT (37), LOOP SEATTLE/H (129), M_DENSE/D (9),
M. DENSE/H (25), TAXIIIST (26), SZ TAXI/H (12);
WebOps: bitbrains_ Settranns_fat_sste fast__torageti storage/ (261), citrrais_fasttstooree (203), bitbrains_mn//55 (124), bitbrains_md/I (127).
To preserve data balance, randomly sample 500 time series instances from oversized datasets.
D.3 Rule Ablations
we report the ablation results on the effect of each knowledge rule that constitutes our definition of time series forecasting hallucinations in $3 of the main text. We note that all three rules contribute to the differentiation of the forecast quality, with the aggregated Pearson correlation of non-hallucinated forecasts substantially higher than that of hallucinated forecasts in all cases. The patternAARM, rule effectively differentiates both R² and correlations in all cases. The frequency rule differentiates correlations in all cases but not R², affected by the negative R² values from the misalignment between forecasts and ground truths on some outlier test instances. The synergy of these rules gives the best performance differentiation overall.
In Table 4
We further examine the effect of pattern and ARMA rules separately. From Table 4 both pattern and ARMA rules contribute to the differentiation of TSFM forecast quality on their own, with positive differences of both R²2 and correlations in all cases. The pattem+ARMA rule narrows down the scope of hallucinations by considering forecasts that violate both the pattern and ARMA rules as hallucinations. captures a smaller of forecasting hallucinations with poor quality and thus provides superior performance differentiation overall.
D.4 Additional Baseline
As stronger hallucination mitigation baseline, we full-parameter finetune TSFMs using the validation data of each domain with 1,000 training steps. 5 compares the forecasting performance across domains. We observe that the performance of SSIM for Chronos is par with full-parameter finetuning on most domains and better on some domains such as Energy, while finetuning suffers poor performance even more severe overfitting, leading to inferior performance on all domains except Energy. Energy domain with frequent and regular patterns. SSIM saves the overheads of finetuning and better preserves the generalization performance of TSFMs.
some test instances due to overfitting. For TimesFM, finetuning suffers from
D.5 Parameter Study
Figure 6 reports the validation performance selecting varying proportions of top neurons for SSIM. We observe that while intervening a reasonable number of signal neurons boosts the forecasting performance of TSFMs, selecting too many neurons degrades the performance because the activations of noise neurons may also get scaled. Based on the results, we set this parameter to 0.1 for Chronos and TimesFM and 2 for Chronos-Boll
24

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Table 4: Comparison of the aggregated performance between hallucinated and non- con-hallucinated TSFM forecasts by checking with different knowledge rules.
R² f
Corr 1 Non-hal
Models
Rules
Hal
Non-hal
Diff
Hul
Diff
All
-161.6823
-16,6599
145.0224
0.1459
0.6943
0,5484
Chronos
Trend
-7.4013
86.5622
-79.1609
0.1512
0,4623
0.3111
Frequency
-0.6223
-84.2801
-83.6578
0.2505
0.4504
0.1999
Pattern
-168.4090
-1622990
152.1100
0.1343
0.6851
0.5507
Pattern
-143.4747
-18.1746
125,3001
0.1536
0.7529
0,5994
ARMA
1008.4763
-10564
107. 419.4199
0,3672
0,6909
0.3237
Chronos -Bolt
All
-1.1647
0.4096
1.5743
0.2441
0.8105
0,5664
Trend
-1.3373
-0.2686
1.0686
0.1313
0.5753
0.4440
Frequency
-0.0674
-0.5082
-0.4408
0.1278
0.5974
0.4697
Pattern
-1.1767
0.3555
1.5322
0,2370
0.7940
0.5570
Pattern
-1.778
0.4547
1.6355
0.2455
0.8192
0.5737
ARMA
-0.5322
0.3277
0.8599
0.4677
0.8096
0.3420
All
-1099572
0,5757
11,5329
0.3429
0.8716
0,5286
TimeaFM
Trend
-24.6562
0.2459
24.9021
0.1132
0.7616
0.6483
Frequency
-0.2205
-5.3556
-5.1351
0.1027
0.7367
0.6340
PatternAARMA
-11.0627
0.4657
11.5284
0.3416
0.8639
0.5223
Pattern
-10.5886
0.5633
11.1519
0.3410
0.8869
0.5459
ARMA
-5.3132
0.3740
5.6872
0.6040
0.8471
0.2431
Table 5: Performance comparison between tull-parameter finetuning and SSIM.
Model
Chronoa
TineaFX
Finetuning R² f
SSIM 1
Finetuming R2"
SSIM
Domain
Hol 4
Corv
Hol 4
Corr
Fa |
Corr
Hal
1
Core
0.90
Ecun/Fin
0.40
-00.11
0.54
0.41
-3.20
0.51
0.81
-63.93
0.35
0.38
40,32
0.78
Energy
0.79
0.77
Nature
0.56
-5.46
0.14
0.67
-0.76
0.11
0.92
-1.22
0.17
0.95
-0.09
0.16
Transport
0.36
-0.04
0.64
0.39
-0.22
0.61
0.85
-0.59
0.57
0.42
0.71
WebOps
0.44
-7906.68
0.35
0.61
-2.84
0.34
0.87
-334666.9
0.17
0.64
6.75
0.43
Aggregated
0.33
-1524.60
0.55
0.42
-5.08
0.51
0.65
-83519.74
0.48
0.43
-1.25
0.64
E
Additional Experimental Results
E.1
Model Size
We examine the effectiveness of our proposed approaches on models of different sizes from the Chronos family [ From Table 6 SSIM effectively reduces the hallucination rate and improves the forecasting quality of different models. From Table 7 the efficacy SSAS for hallucination detection and performance prediction is better for larger models. This is because larger models have
52-
0.60
$6 HHallcihation
50
Correlation
drioos
Chrones Chronee-Bo Bot TimeedN
88
chronos-ut But
46
44
0.45
42
0.0
0.3
0.4
Figure 6: The aggregated mean validation performance under varying proportions of top neurons.
25

---

Table 6: Comparison of forecasting performance across domains, with the improvements boldfaced.
Method
Domain
Fal l
Chronos-Mini
Coer
Find
Chronon-Sna Rf
Small
Corz
Hol
Chromos-lose -lase R6'
Core
Chrsnos-Largo
Corr
Synthatic
0.6024
-085781
0.5426
0.5714
-0.7335
0.5633
0.6665
0.3048
0.1046
0.7153
EaondFia
0.4307
-2.5519
0.4227
0.4184
-2.8915
014766
0.415
-3,3554
0.4511
0.4321
-4.0115
0.4874
Onginal
Energy
0.1567
-13.0010
0.6869
0.1309
-9.1815
0.6807
0.1389
-04839
0.7180
0.1486
-0.2673
0.7065
Nature
0,7792
-14.2961
0.0613
0.7748
-14.3360
0.0074
0.0035
100.7283
0.0457
0.7815
-10.5791
0,4155
Transport
0.4560
-1.0088
0.5013
0.3938
-1.5306
0.5415
0.4197
-1 116444
0.5127
0.5130
-1.3101
0.5397
We
0.6760
-183017650
-13.1433
0.5801
414,8953
0.2762
0.5470
-56443522
0.2791
Approganed
0.4531
-8.3760
Synthotic
0.5095
-0.1507
0.6166
0.4500
-0.0955
0.6451
0.4145
0,1854
0.7150
0.2452
0.4994
0.7658
Econt-in
0.43561
-2.0534
0.5522
0.4387
-2.2133
0.5336
0.3061
-3.2037
0,5146
0.4319
-1.3995
0.5769
SSIM
Energy
0.1373
-0.6871
0.7427
0.1228
-1.3961
0.7258
0.1191
0.0268
0.7707
0.1292
-0.5916
0.7543
Nature
0.71.9
-1.2148
0.0980
0.6894
-0.3257
0.1050
0.6715
-0.7575
0.1082
0.6960
-0.5360
0.6860
Tranppori
0.4197
-0.1980
0.5880
0,3893
-0.2187
0.6032
0.3938
-0.2221
0.6081
0.5078
-0.25%3
0,5954
0.6400
-2.2954
0.3413
0,5625
-1,9302
0.2899
21,8389
0.3369
0.5261
-1.4402
0.3685
Aggregased
0.4621
-1.3024
0.4745
-1,2709
-5,0845
0,5061
0.4051
-0.7534
0.5370
Table 7: The results of hallucination detection and forecasting performance prediction for TSFMs of different sizes, with the best results boldfaced. For each method, the first column shows AUROC of hallucination detection and the latter two columns show rank correlations with the performance metrics. The statistical significance of positive rank correlations is indicated with * for p < 0.05 and for < 0.01.
Cosine Similarity
Activation Variance It²
SSAS (Ours) Ita 1
Model
Domain
Hal 1
AP, 1
Carr
Hal 1
Carr
Hal
Corr 1
Chronos Large
Syuthetic
0.7878
0.3553
0.2976*
0.5719
0.1594**
0.1459** *
0.7725
0.4818*
0.4905**
Rcon/Fin
0.8619
0.6584**
0.6523**
0.7922
0.6079
0.6474
0.8780
0.6966**
0.7055
Energy
0.6952
0.1309
0.1190**
0.6883
-0.1911
-0.0404
0.8190
-0.0030
0.1407 **
Nature
0,5008
0.3057
0.1232
0.5020
0.2979 **
0.1239 **
0.5479
0.4150
0.2080 *+
Transport
0.6228
0
0.5423**
0.6994
0,5187**
0.5728
0.7044
0.5523
0.6872
WehOps
0.6121
0.2366
0.3800
0.5222
0,2237
0,2785
0.5477
0,2532
0.3525**
Aggrogated
0.7800
0.5168
0.5388*
0.7240
0,4455
0.5033
0,8035
0.5787
0.6492**
Syothetic
0,7847
0.3834
0.3262
0.6786
0.3734
0.4052
0.8316
0.4299**
0.5111
Chronos: -aaa
Reon/Tin
0.8495
0.6501
0.6208
0.6927
0.5096 **
0.5507 **
0.7833
0.5034
0.5258 **
Energy
0.7124
0.5088 *
0.3528**
0.8093
-0.1116
0.8096
0.1166
0.0363
Nature
0.4978
0.3382
0.1524
0.5384
0.3490
0.1886"
0.5925
0,3430*
0,1507**
Transport
0.6601
0.4706*
0.5550
0.7466
0.5351
0.6083*
0.6767
0.4158
0.5234**
WebOps
0.5542
0.2328
0.3710
0.5060
0.1363 *
0,2720 *
0.5740
0.1693*
0.2526 **
Aggregated
0.7903
0.5866
0.5804**
0.7226
0,4197 **
0,5277 **
0.8086
0.5082
0.5758 **
Chronos Small
Synthetic
0.7748
0.4697**
0.3234**
0.5204
0.4696 **
0.5616 *
0.7367
0.4725**
0.4265**
Econ/Tin
0.8686
0.6334
0.6167*
0.6337
0,3839 *
0.4081 *
0.6405
0.4002
0.4556 **
Energy
0.6909
0.6957
0.6036
0.6000
0.0611
0.1903
0.7932
0,4817*
0,5093**
Nature
0.4989
0.3520
0.2850
0.5236
0,358y
0.3672
0.5595
0.3803
0.3253
Transport
0.7160
0.6267 **
0.6033 **
0,7683
0,4515 **
0.6326 **
0.7822
0,5387**
0.6956"* **
WobOps
0.5695
0.168%*
0.2756" **
0,4830
-0,0445
0.1239** ** 0.5524
0.0906*
0.2306**
Agpregated
0.7733
0.6013
0.5874
0.6255
0.3299
0.4396
0.6876
0.4356*
0.5408
Synthetic
0.4669* *
0.2308** *
0.3921
-0.1445
0.5656
0.1675**
0.0889*
Chronoa Vini
Econ/Fin
0,8479
0.6233*
0.5881
0.2835
-0.1790
-0.1390
0,5310
0.2172
0.2582**
Energy
0.7092
0.6510
0.5645
0.6565
-0.0651
0.1352 **
0.7568
0.0121
0.1435**
Nature
0,5288
0.3352
0.2117*
0.5769
0,4002 **
0.3066 **
0.6528
0,5848
0.4637**
Transport
0.6610
0
0.3709
0.6627
0.4066 ++
0.4655
0.6817
0.5747
0.6822**
WebOps
0.5263
0.1289**
0.1983
0.5149
0.1870
0.1203
0.5349
0.1148"
0.1532
Aggregated
0.7580
0.5664
0.5317**
0.5855
0.1707
0,2853
0.6147
0.3091 **
0.3975
stronger capability of distinguishing context signals from noises through the processing of larger and more layers, leading to more distinct activation behaviors of signal neurons with respect different types of inputs.
E.2 No Data Filtering
We study the impact of TSFM hallucinations without filtering the time series instances using the knowledge rules in data preprocessing. We randomly sample 500 time series instances from oversized datasets to preserve data balance. From Tables 8 and we note that although there is a small decline in the overall performance compared with the results with data filtering in Tables and 2 of the main text, the quality of non non-hallucinated forecasts is still substantially better than that of hallucinated
26

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Table 8: Comparison of forecasting performance across domains without data filtering.
Chronos R²
Chronos-Bolz R² f
TimesFM R² 1
Domain
Hal
Corr
Hal 4
Corr
Hol 4
Corr
Synthetic
0.4610
-0.1701
0.6292
0.5448
0.0013
0.5344
0.1105
0.5983
0.9141
Econ/Fin
0.6526
-5.5863
0.3556
0.6825
-2.0782
0,4702
0.6937
-1.0520
0.6423
Energy
0.4078
-2.2455
0.5606
0.3643
-0.4111
0.6521
0.3938
-0.5814
0.6625
Nature
0.8736
-2,8898
0,0875
0,9347
-0,6486
0.1628
0.9432
-0.2972
0.1918
Transport
0.7358
-7.6617
0.2747
0,8752
-0,8877
0.3657
0.8507
-0,3036
0.4527
WcbOps
0.8172
-433.0063
0.1532
0.8616
-1.8096
0.2428
0.8407
-6,8100
0.2511
Aggregated Mean
0.6696
-117.9305
0.3228
0.7120
-1.2357
0.4055
0.6804
-2.1914
0,4915
Table 9: Performance comparison of hallucinated and non-hallucinated forecasts by TSFMs without data filtering.
Metric
Domain
Chronos Non
Chronos-Boli
TimeaFM
Hol
Dif
Hal
Diff
Dij
Synthetic
-1.1127
0.6360
1.7487
-0.4383
0.5273
0.9656
-1.3704
0.8428
2.2132
Fenn/Pin
-7.766
-1,8883
5.7280
-2,0005
40,3106
2.5899
-1.5933
0.1737
1.7670
Energy
-3.5878
-1.3212
2.2667
-1.1147
-0.0079
1.1099
-1.5737
0.0632
1.6369
R² f
Nature
-2.9187
-2,6900
0.2286
-0.6573
-0,5235
0.1338
+0.3369
0.3629
0.6999
Transport
-8.4450
-5.4807
2.9642
-0,9532
40,4285
0.5247
+0.4530
0.5478
1,0008
WebOps
-524.3276
24.6696
499.6580
-1.3156
4.8857
-3,5701
-7.1521
-5.0045
2.1476
Aggregated
-173,5672
-5.1648
168. 4024
-1.4589
-0.6837
0.7753
-3.0391
-0.3866
2.6524
Synthetic
0.3617
0.8581
0.4964
0.3568
0.7909
0.4341
0.7481
0.9347
0.1866
Econ/Fin
0.1642
0.7151
0.2854
Energy
0,3303
0.7191
0.3888
0.4583
0.7631
0.3048
0.4491
0.8012
0.3521
Corv
Nature
0.0619
0.2643
0.2024
0.1192
0,7860
0.6668
0.1561
0.7855
0. 0.6295
Transport
0.4539
0.3858
0.8336
WehOps
0.1037
0.3744
0.2706
0.1778
0.6474
0.4696
0.1750
0.6530
0.4780
Aggregated
0,1629
0.6467
0.4838
0,2603
0,7643
0.5040
0.3375
0.8193
0.4818
forecasts at 10-5 by unpaired (-tests. This demonstrates the general impact of the hallucination problem we have formulated on the forecasting performance TSFMs.
F
Case Study
Figures 7 illlssrrte examples of hallucinated and non-hallucinated forecasts by different TSFMs across world domains. The inferior performance of hallucinated forecasts due to the TSFM failing to adequately capture the context information and generating hallucinations in the forecast that do not exist in the context, such as peaks, slopes, or patterns. The hallucinations generally deviate from the ground truths, resulting in low performance evaluation scores.
G Limitations
While our formulation of time series forecasting hallucinations is broadly applicable to all types of TSFMs, the proposed hallucination detection and intervention methods are applicable to white-box models only. A future direction is to include further TSFMs our study.
H Impact Statement
We are the first to formulate and systematically study TSFM hallucinations to our best knowledge. We have formally defined the problem of TSFM hallucinations and outlined a set of procedures to check ballucinations in practice. We have proposed method to identify the signal subspaces TSFMs along with measure to quantify the signal strength in TSFM hidden states. We have also proposed simple and efficient intervention approach to mitigate hallucinations by magnifying the signal information in hidden states. Our work contributes deeper understanding of TSFM trustworthiness that could foster future research in this direction.
27

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I Ethics Statement
No human subjects were involved in this research, and all experiments were conducted using publicly available models and datasets, adhering to their respective licenses and use policies.
J Reproducibility Statement
We have provided comprehensive details our proposed methods in $4 the main text and Appen- dices A and C. The experimental setups, including model configurations, datasets, and evaluation metrics, have been thoroughly described $5,1 of the main text and Appendix D. We utilize publicly available TSFMs and detail any modifications or specific settings used during experimentation All datasets employed in our evaluation are standard benchmarks.
28

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03c 12770 9012 0036 N2. 34266 .347
HCC 17A5.ITH RE-S TEE, Can 5.0011
Non Par 7/1711 11
Cyntoa
Unid
Conuat Presicies
(a) Econ/Fin, hallucinated
(b) Econ/Fin, non-hallucinated
Target
Prediccion
Predictiee
(c) Energy, hallucinated
(d) Energy, non con-hallueinated
2257085L -07244 d:30
N0W 9372 E2006 Can 0.59
(e) Nature, hallucinated
(1 Nature, non con-hallueidteed
1635, 20026
Prediction
gg) Transport, hallucinated
(h) Transport, non non- ha-llllucintedd
409
Treeet
Trrgrt
WehOps, hallucinated
WebOps, non-hallucinated
Figure 7: Examples of hallucinated and non con-hallucinated forecasts by Chronos across domains.
29

---

TAZAOATIA
1000
Co:1001
Predicion
Frediccion
(a) Econ/Fin, hallucinated
(b) Econ/Fin, non hallucinated
Twret
11
Taree Crea
Presec
(c) Energy, hallucinated
(d) Energy, non-ballucinated
1S0 8186622, 92 11565, Curn 03332
We0 9:09 9938 #2: BIAIOO Even E.990
Cannost
Contoot gen
(e) Nature, hallucinated
(1 Nature. non hallucinaledd
650L Com 2A26
Prestiction
g Transport, hallucinated
(h) Transport, non-hallucinated
155 2072009113555 20---4266..0 -25765
21902. 6:4099
000
3007
Target Pisdiction
( WebOps, hallucinated
i WebOps, non-hallucinated
Figure 8: Examples of ballucinated and non non-hallucinated forecasts by Chronos Chronos-Bolt across do- mains.
30

---

P0e: 1332214 DOOK, RO 2.56 Can 2,566
ICC SXPAZER RE-S 20TS, Can 5.556
2007
380
p
082
Contost
Conuat
(a) Econ/Fin, hallucinated
(b) Econ/Fin, non- aa-hhaluucinttdd
Torget Proticiinn
Prodiccion
(c) Energy, hallucinated
(d) Energy, non non-hallueinated
MCC 0.2342
N0C edG 4335 E.856, Can
20-
M
Taignt
(e) Nature, hallucinated
( Nature, non hallunanleed
1758
Predittion
gg) Transport, hallucinated
(h) Transport, non ha-lalluuinaedd
300
609
Svpet
WehOps, hallucinated
WebOps, non-hallucinated
Figure 9: Examples of hallucinated and non non-hallucinated forecasts by TimesFM across domains.
31


Reviews:
[{"review_id": "a186AnTgRF", "reviewer": "Reviewer_pd2E", "summary": "This paper explores the phenomenon of hallucinations in Time Series Foundation Models (TSFMs), introduces a formal definition in the zero-shot forecasting setting, and proposes a test-time intervention method based on identifying and amplifying signal subspaces in model hidden states.", "questions": "Kindly see the questions outlined in the weaknesses section.", "limitations": "This work has some limitations. It focuses on zero-shot forecasting, leaving other probing and finetuning unexplored. Identifying signal subspaces may not generalize to all TSFMs, and the signal strength measure needs further interpretation. The intervention adds test-time overhead, and the evaluation criteria might not cover all types of forecast errors in practice.", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n- Timely Topic: Hallucination analysis in TSFMs is underexplored. Drawing parallels with hallucinations in NLP and vision models is meaningful and relevant.\n- Insightful Hypothesis: The idea that hallucinations stem from diminished signal representation in hidden states is plausible and interesting.\n- Signal Subspace Analysis and Intervention Strategy:  Proposing a Signal Subspace Activity Score (SSAS) and identifying \"signal neurons\" using contrastive analysis adds a fresh angle to understanding hidden states in TSFMs. The proposed test-time intervention (via signal subspace magnification) is simple and avoids retraining, which is attractive for practical deployments.\n\n**Weaknesses**\n\n1. Literature Positioning & Missing Related Work\n - There's insufficient discussion on similar latent space analyses in time series models or comparable intervention techniques in other domains like disentanglement for time series\n\n2. Framing and Definitions\n - The framing of hallucination is imprecise and overly broad in places. For example, defining hallucinations strictly based on deviations from handcrafted rules (trend, frequency, ARMA) is fragile and may not generalize across domains. \n- Definitions (esp. Definition 2 and 3) are complicated and sometimes tautological,  hallucinations are defined via violations of rules, but those rules are subjective and can depend on thresholds chosen arbitrarily.\n\n3. Unclear Methodology\n - The contrastive neuron activity score is underdeveloped theoretically. Why this specific measure? Why should variance across TS steps signify \"signal\" rather than something else (e.g., noise sensitivity or instability)?\n - The intervention mechanism remains vague: What exact operation is applied to hidden states during test-time intervention?\n\n4. Experimental Validation\n - a) The experiments lack rigorous baselines. There's no comparison to other hallucination mitigation strategies or any ablation on how each rule contributes to hallucination detection.\n - b) The evaluation of the proposed SSAS metric as a predictor for hallucinations and forecast quality is unconvincing. Could the authors clarify under which conditions SSIM might fail or be less effective, especially given the varying results of baseline methods? Additionally, how does SSIM’s performance scale with model complexity and computational overhead compared to perturbation averaging? \n - c) There is a lack of variation analysis across different seeds for each method, such as Denoising and Perturbation+Averaging. These methods may be sensitive to hyperparameter settings.\n\n5. Clarity & Presentation\n - a) Clarity and Organization: The paper is cluttered and difficult to follow in several sections. The overall presentation would benefit from clearer structure and more focused explanations.\n\n - b) Figures Not Well Integrated: Many figures, such as Figure 1, are not clearly explained or referenced in the main text. For example, it’s unclear what exactly is being visualized on the right-hand side of Figure 1, and how it supports the authors' claims.\n\n - c) Cosine Similarity Ambiguity: It is not clear what the cosine similarity measures in this context. Is it computed between embeddings, hidden states, or the projection layer weights? This distinction is important for interpreting the results.\n\n - d) Gap in Activation Statistics: The difference in standard deviation between hallucinated and non-hallucinated forecasts appears small. Why is such a narrow statistical gap sufficient to explain the shift from hallucination to accurate forecasting?\n\n - e) Interpretation of Representation Alignment: The authors seem to suggest that when representations are aligned in a certain direction, hallucinations are less likely. If this is the case, it could be an interesting and valuable insight for the community. However, this idea is currently underexplained and would benefit from a more rigorous and intuitive justification.\n\n - f) UMAP Projection: The direction in the UMAP projection (Figure 1, right) corresponding to models that do not hallucinate is not clearly defined. Without a clear definition, it is difficult to assess the significance of the observed alignment and determine whether it meaningfully correlates with hallucination behavior.", "was_revised": true, "final_justification": "The authors’ rebuttal addresses my comments only partially, and several critical concerns remain unresolved. The explanation for the HAL performance gap lacks quantitative detail on the frequency and impact of outliers; claims of statistical significance are not supported by p-values, effect sizes, or per-domain analysis; the “pattern frequency” concept remains undefined in operational terms and is unaccompanied by quantitative evidence; and the metrics section, while clearer, still omits important specifics and methodological justifications. These shortcomings reduce confidence in the validity and robustness of the reported results. Given the absence of sufficient new evidence to address these core methodological issues, **I maintain my score at a borderline reject.**", "rebuttal": "Dear Reviewer pd2E,\n\nThank you for acknowledging the importance of our research topic, novelty of our methodology, and potential impact. \nWe want to address the posed questions to improve our paper's quality and clarity. Please find the below replies to your comments.\n\nW1 We have discussed on [38] in the paper.\nWe add discussions on [1,2,3]. If there are additional works, we would appreciate pointing them out.\n\nW2 The knowledge rules we propose are basic rules that capture fundamental characteristics of general time series (trend, frequency, and patterns). \nThese rules are scale invariant, robust to noise, and generalizable across different domains.\nSimilar TS pattern matching approaches with sliding windows have been adopted in [4].\nExperiments on real-world datasets across five different domains show that our rules effectively differentiate TSFM forecast performance (Table 2 and Appendix Table 4).\nThe thresholds of rules allow for a reasonable degree of generalization of context signals in TSFM forecasts. \nAs mentioned in $\\S5.1$, the thresholds are tuned based on validation performance such that TSFM performance differentiation is maximized.\n\nW3 The neuron activity score (Equation 1) stems from our observations that hallucinated forecasts associate with \nlow activation variance and high cosine similarity of hidden states across TS steps (Figure 1),\nwhile non-hallucinated forecasts with clean context signal associate with high activation variance and low cosine similarity on average (Figure 2). \nThe intuition and motivation are elaborated in $\\S4.1$. Essentially, activation variance reflects the sensitivity of a neuron to the context TS.\nAs such, we propose to leverage the standard deviation of activations across TS steps as a measure of neuron activity. \nTo distinguish neurons that are sensitive to context signal from those sensitive to context noise, \nwe compute the contrastive neuron activity score (Equation 2) between data inputs without causing hallucinations and Gaussian noises.\nWe select the activity score of the signal neuron with top contrastive score from the last hidden layer as the hallucination detector and forecast performance predictor, i.e., SSAS.\nExperiments on real-world datasets show that while cosine similarity and mean activation variance alone have hallucination detection and performance prediction power to some extent, \nSSAS yields superior performance by considering the activity of identified signal neurons (Table 3 and Appendix Table 2).\n\nIn SSIM, we scale the activations of candidate signal neurons via our proposed Center-Project-Scale (CPS) operation described in $\\S4.3$. \nThe properties of CPS operation are mathematically proved in Appendix B. The algorithm of SSIM is outlined in Appendix C. \n\nW4 a) We add hallucination mitigation baselines. \nWe add an ablation study on how each rule contributes to hallucination detection. \nWe are working on these experiments.\n\nW4 b) SSIM could fail when the number of pattern occurrences in the context is low.\nIt is easier for TSFMs to capture frequent patterns than rare patterns from the context TS. \n\nThe comparison of SSIM performance for models of different complexity is presented in Appendix E.1. \nThe computational overhead of SSIM at each layer is $O(nk)$, where $n$ is the number of TS steps and $k$ is the number of signal neurons.\nThe computational cost of perturbation averaging is the number of perturbations times the original computational cost of TSFM inference.\n\nW4 c) Denoising with sliding windows does not involve randomization. We set the window size to 5 as it works the best overall.\nIn Perturbation+Averaging, we perturb the input TS with Gaussian noise for 10 runs with different random seeds and average the outputs.\n\nW5 b) Figure 1 (c) and (d) compare the distributions of last layer hidden states \ncorresponding to the hallucinated forecast in Figure 1 (a) and non-hallucinated forecast in Figure 1 (b).\nWe observe the hidden states in (d) are more evenly distributed within each cluster than those in (c).\nFigure 1 has been referenced in $\\S1$ and $\\S4.1$ main text. \nWe note that the subfigures should instead be listed as (a) (b) ... (c) (d) ... in Figure 1 caption. \n\nW5 c) We compute the mean pairwise cosine similarity of hidden states across TS steps,\nwhich has also been used as a measure in [5].\n\nW5 d) We assume the activation statistics in Figure 1 (c) (d) are being referred to in the comment. \nThese show the difference of the mean activation standard deviation of all neurons. \nBy identifying the signal neurons, SSAS yields greater gaps\nbetween hallucinated and non-hallucinated forecasts with substantially higher hallucination detection and \nperformance prediction accuracies than mean activation standard deviation (Table 3 and Appendix Table 2) \n\n\nW5 e) Our observations suggest that hallucinated forecasts associate with \nlow activation variance and high cosine similarity of hidden states across TS steps (Figure 1),\nwhile non-hallucinated forecasts with clean context signal associate with high activation variance and low cosine similarity on average (Figure 2).\nTSFM hallucinations are partly caused by the inactivity of signal neurons, leading to more similar hidden states.\nExperiments on real-world datasets show that cosine similarity and mean activation variance of hidden states have hallucination detection and performance prediction power to some extent (Table 3 and Appendix Table 2).\nWe mathematically show in Appendix Theorem 3 that our CPS intervention operation \neffectively reduces the cosine similarity of hidden states that have different projection directions in the signal subspace.\n\nW5 f) Through UMAP projection, we aim to show in Figure 1 (c) (d) the difference in the distributional characteristics of last layer hidden states corresponding to the hallucinated forecast in Figure 1 (a) and non-hallucinated forecast in Figure 1 (b)\nrather than alignment to certain directions.\nWe observe the hidden states in (d) are more evenly distributed within each cluster than those in (c).\n\n\n[1] Santamaria-Valenzuela, Inmaculada, et al. \"Decoding Latent Spaces: Assessing the Interpretability of Time Series Foundation Models for Visual Analytics.\" arXiv preprint arXiv:2504.20099 (2025).\n\n[2] Li, Yuening, et al. \"Towards learning disentangled representations for time series.\" Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2022.\n\n[3] Oublal, Khalid, et al. \"Disentangling time series representations via contrastive independence-of-support on l-variational inference.\" The Twelfth International Conference on Learning Representations. 2024.\n\n[4] Shokoohi-Yekta, Mohammad, et al. \"Discovery of meaningful rules in time series.\" Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining. 2015.\n\n[5] Nguyen, Tam, Tan Nguyen, and Richard Baraniuk. \"Mitigating over-smoothing in transformers via regularized nonlocal functionals.\" Advances in Neural Information Processing Systems 36 (2023): 80233-80256."}, {"review_id": "odxdVgdCjN", "reviewer": "Reviewer_43oT", "summary": "Disclaimer: While I have experience in time series forecasting and deep learning, I do not have sufficient expertise in the specific techniques or theoretical frameworks employed in this paper (e.g., signal subspace analysis, internal representation intervention). My review may not fully reflect the technical depth of this work, and I encourage the AC to consider assigning an additional reviewer with relevant background.\n\nRecommendation to AC: Given the specific technical depth of this work, particularly in representation manipulation and signal processing, I believe this paper would benefit from an additional review by someone with a strong background in neural representation interpretability or signal-space analysis in deep networks.\n\nThis paper investigates hallucination phenomena in Time Series Foundation Models (TSFMs). It proposes a formal definition of hallucination, introduces a method for identifying signal subspaces, and applies test-time interventions to reduce hallucination. The methods are validated on both synthetic and real-world datasets.", "questions": "While the motivation is clear and the empirical results appear thorough, I find it difficult to evaluate the novelty and technical correctness of the proposed intervention method and subspace decomposition approach. These appear to rely on internal mechanisms of TSFMs and assumptions that I am not qualified to fully assess.", "limitations": "N/A", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "While the motivation is clear and the empirical results appear thorough, I find it difficult to evaluate the novelty and technical correctness of the proposed intervention method and subspace decomposition approach. These appear to rely on internal mechanisms of TSFMs and assumptions that I am not qualified to fully assess.", "was_revised": true, "final_justification": "N/A", "rebuttal": "Dear Reviewer 43oT,\n\nThank you for acknowledging the clarity of motivation and the completeness of experimental results. We appreciate your time and efforts."}, {"review_id": "Sz6oyd3pZa", "reviewer": "Reviewer_Q2NU", "summary": "This paper defines the hallucination of time series foundation models.  The underlying mechanisms are investigated and the signal strength of hidden states are used for checking of hallucination.  Activity score is proposed to  rank the signal neurons. Then the center-project-scale (CPS) intervention approach is proposed for mitigation of the hallucination is proposed.   It magnifies the signal information in hidden states.  Experimental results have shown the effects of the hallucination detection and mitigation.", "questions": "Section 4.1 describes a hallucination example and illustrates that the hidden states gives us clues about the presence of hallucinated forecasts.  It is not clear if this is just an example or if this can be generalized to most cases.   Also, are all TSFMs showing similar phenomena?  The authors should elaborate this point in order to make the proposed work convincing.\n\nIt is said that type 3 hallucinations is the most common one which include violation of the pattern and ARMA rules. It is also claimed that SSIM has the greatest impact on Type 3 hallucinations.  It is puzzling that the authors did not separate these two types of rules and analyze them individually.  As in terms of prediction, any small adjustments of ARMA coefficients may affect the prediction results.  The adjustment could be caused by the training data in the learning process.    Thus it may be fruitful to study them separately.\n\nThe legend of figure 1 may have the sub-figures (b) and (c) mixed up.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "The paper is clear written and well organized.  The definition of hallucination in time series based on the knowledge set is reasonable.   The paper presents the observed the patterns of the hidden states and hence form the central idea of this paper.   The only question is whether this signal and noise space of the hidden states can be generalized to all kinds of hallucination and TSFMs.  The proposed hallucination detection and hallucination mitigation methods are clearly presented.  The works are original and it may have substantial impact to future research work in this area.   The use of the knowledge set is important in this work.  Currently, trend, frequency, pattern and ARMA are considered.  It is not clear why pattern and ARMA rules are combined in this study.   Also, is it possible to include other domain knowledge in the knowledge set?", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer Q2NU,\n\nThank you for acknowledging the novelty of our work and the clarity of our paper. \nWe want to address the posed questions to improve our paper's quality and clarity. Please find the below replies to your comments.\n\nW The knowledge rules we propose are basic rules that capture fundamental characteristics of general time series (trend, frequency, and patterns). \nThese rules are scale invariant, robust to noise, and generalizable across different domains.\nExperiments on real-world datasets across five different domains show that our rules effectively differentiate TSFM forecast performance (Table 2 and Appendix Table 4).\nIt is possible to further include domain knowledge as rules, but this may affect \nthe generalizability of the knowledge set to other domains.\n\nQ1 Experiments on real-world datasets show that cosine similarity and mean activation variance alone have hallucination detection and performance prediction power to some extent (Table 3 and Appendix Table 2).\nAs discussed in $\\S5.2$ RQ3 and Appendix E.1, the phenomena of cosine similarity are applicable to the\nChronos family and Chronos-Bolt. The phenomena of mean activation variance are applicable to\nall three types of models but are less consistent across domains. \n\nQ2 As explained in $\\S3$, the ARMA rule is purposed to complement the pattern rule\nwhen the TS exhibits strong ARMA dynamics.\nSince only a small proportion of TS exhibit strong ARMA dynamics, we have not reported \nthe experimental results of the standalone ARMA rule.\nWe choose to use the first-order ARMA model to capture the fundamental characteristics of \ngeneral time series and avoid overfitting.\nWe are working on separate results for the pattern and ARMA rules. \n\n\nQ3 Thank you for pointing this out. We fix the typo in the revision."}, {"review_id": "ZcfCC6RPe1", "reviewer": "Reviewer_A1xq", "summary": "The paper addresses hallucinations in Time Series Foundation Models (TSFMs), particularly in zero-shot forecasting settings, by defining what constitutes a hallucination based on violations of time-series dynamics inferred from the context. It proposes a novel intervention approach (SSIM) that operates on signal subspaces of hidden representations to mitigate hallucinations, and introduces a signal strength score (SSAS) to detect them. Experiments are conducted across multiple TSFMs and datasets.", "questions": "- How sensitive are the hallucination rules (especially ARMA/pattern-based) to scale normalization and window size?\n- Have you compared SSIM against any stronger TSFM-specific debiasing or attention control methods?\n- Can the proposed signal subspace activity score (SSAS) generalize to classification or anomaly detection tasks?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n- Problem Definition: The paper makes an effort to formalize the notion of hallucination in TSFMs, filling a relatively unexplored gap in the literature.\n\n- Methodological Clarity: The intervention mechanism (CPS and SSIM) is well-described, mathematically clean, and efficient to compute at inference time.\n\n- Empirical Evaluation: Experiments are conducted on multiple models and both synthetic and real-world datasets, showing moderate improvements over baselines in both hallucination reduction and predictive performance.\n\n- Potential Impact: If substantiated, the idea that hidden-state subspace structure directly correlates with hallucination likelihood could provide useful insights into TSFM interpretability.\n\n**Weakness**\n\n- The hallucination definition relies on handcrafted rules (trend, frequency, ARMA, etc.) that are domain- and scale-sensitive, and often ill-defined or unstable in noisy, non-stationary time series.\n\n- It is unclear whether failure on these rules always corresponds to a hallucination, or if these mismatches are natural consequences of generalization to difficult signals.\n\n- While the CPS operation is clean, the intervention idea (projecting onto a signal subspace and magnifying) is conceptually similar to many LLM steering methods (e.g., activation steering, neuron editing).\n\n- The main novelty seems incremental in the context of adapting these ideas to TSFMs, and prior work (e.g., [38]) has already discussed intervention in TSFM hidden spaces.\n\n- The baseline comparisons (e.g., denoising, perturbation + averaging) are relatively weak. Stronger baselines, such as diffusion-based or fine-tuned TSFMs with dynamic attention masking, are missing.\n\n- The correlation between SSAS and hallucination is heavily empirical; no theoretical insights are provided about why neuron variance should reliably reflect semantic grounding.\n\n- Many insights are drawn from synthetic datasets (e.g., sinusoidal signals with additive noise), which may not reflect the complex patterns of real-world time series.\n\n- The lack of domain-specific case studies reduces confidence in the generalizability of the findings.", "was_revised": true, "final_justification": "I would like to raise my score to 4 since the authors make hard works in responding most of my questions and most of their answers are satisfying. However, frankly speaking, my main concern to this paper is still not adequately addressed, which is how to correctly define \"Hallucination\" in the context of time series forecasting. This is because of the core difference between language and time series. Language is an abstract description to the real world, while time series are direct signals from our real world. In my understanding, the knowledge rules that the authors define are more like a set of \"common errors\" of \"common failures\". If other reviewers or the Area Chair have any concerns and would like to reject this paper, I will not oppose.", "rebuttal": "Dear Reviewer A1xq,\n\nThank you for acknowledging the novelty of our problem definition, methodological clarity, and potential impact. \nWe want to address the posed questions to improve our paper's quality and clarity. Please find the below replies to your comments.\n\nW1 The knowledge rules we propose are basic rules that capture fundamental characteristics of general time series (trend, frequency, and patterns). \nThese rules are scale invariant, robust to noise, and generalizable across different domains.\nSimilar TS pattern matching approaches with sliding windows have been adopted in [1].\nExperiments on real-world datasets across five different domains show that our rules effectively differentiate TSFM forecast performance (Table 2 and Appendix Table 4).\n\n\nW2 The thresholds of knowledge rules allow for a reasonable degree of generalization of context signals in TSFM forecasts. \nAs mentioned in $\\S5.1$, we tune the thresholds based on validation performance such that TSFM performance differentiation is maximized.\n        \nW3 While our approach is intervention-based, \nthe center-project-scale (CPS) operation is not used in existing works to our best knowledge.\nOur approach adaptively determines the scaling factor.\nMoreover, few existing works investigate the use of activation variance \nas the hallucination detector and forecast performance predictor. \n\nW4 As discussed in $\\S2$, the work in [38] does not address specific challenges, \nsuch as hallucination detection and mitigation and performance prediction in our work.\n\nW5 Thanks for the suggestions. We add hallucination mitigation baselines. \nWe are working on the experiments.\n\nW6 The neuron activity score (Equation 1) stems from our observations that hallucinated forecasts associate with \nlow activation variance and high cosine similarity of hidden states across TS steps (Figure 1),\nwhile non-hallucinated forecasts with clean context signal associate with high activation variance and low cosine similarity on average (Figure 2). \nThe intuition and motivation are elaborated in $\\S4.1$. Essentially, activation variance reflects the sensitivity of a neuron to the context TS.\nAs such, we propose to leverage the standard deviation of activations across TS steps as a measure of neuron activity. \nTo distinguish neurons that are sensitive to context signal from those sensitive to context noise, \nwe compute the contrastive neuron activity score (Equation 2) between data inputs without causing hallucinations and Gaussian noises.\nWe select the activity score of the signal neuron with top contrastive score from the last hidden layer as the hallucination detector and forecast performance predictor, i.e., SSAS.\n\nW7 While our observations in $\\S4.1$ are drawn from synthetic data, \nexperiments on real-world datasets show that cosine similarity and mean activation variance alone have hallucination detection and performance prediction power to some extent (Table 3 and Appendix Table 2),\nvalidating the generality of observations. \nOur approach only utilizes synthetic datasets without requiring any domain-specific real-world data.\n\n\nW8 Our experiments have been conducted on real-world datasets from five different domains. We add case studies in the revision.\n\nQ1 The rules are invariant to scale as they consider the relative differences. \nCentering the context TS may have an impact on the results, \nbut we do not normalize the context TS in the experiments.\nAs mentioned in Appendix A, the rolling window size is always the same as the forecast horizon,\nwhich is necessary for computing the relative differences between the forecast and context windows.\n\nQ2 We add hallucination mitigation baselines to our evaluations. \nWe are working on the experiments.\n\nQ3 We think SSAS could generalize to classification and anomaly detection tasks. \nSSAS detects hallucinations by characterizing the abnormal behaviors in hidden states\nwhen hallucinated forecasts are generated. Similarly, it could be useful for detecting \nwhether anomalies exist in the context TS. \n\n[1] Shokoohi-Yekta, Mohammad, et al. \"Discovery of meaningful rules in time series.\" Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining. 2015."}]

Author rebuttal:

Meta-review: The paper studies hallucinations in time series foundation models, especially in the zero-shot setting. It suggests a method to detect hallucination (SSAS) and another method to mitigate hallucination (SSIM).

The hallucination is defined as a change in certain statistics of the data between the context given to the foundation model and the predicted output. These statistics, or knowledge set as the paper calls them, are based on the Trends, Frequency, a certain definition of pattern, and the fit of an ARMA model.


We have a wide range of opinions about this paper. There is one accept (5), two borderline accepts (4), and one borderline reject (3). One of the borderline accepts is of low confidence, so we effectively have scores of 3, 4, 5 with high confidence.


The positive aspects of the paper is the novelty and originality of the method, which may in the future lead to better and more effective methods. 

There are some negative aspects too:

The concern of Reviewer A1xq (Score: 4) is how to correctly define Hallucination in the context of time series, and whether the proposed criteria is the correct on. On the other hand, Reviewer Q2NU (score: 5) believes that the definition is reasonable.

In my own reading of the paper, I tend to agree that the definition is a bit arbitrary, as it depends on how the "knowledge set" is defined. Someone else could choose a different knowledge sets with different predictions potentially being detected as hallucination. This does not, however, invalidate the whole framework, as the proposed methods would still be applicable, though with different results.

There was a concern (Reviewers pd2E and A1xq) about missing baselines. The authors have added new results, based on model fine-tuning, during the rebuttal period, and at least partially addressed it.

Another concern is related to the Statistical Significance of the results, which is raised by Reviewer pd2E. While the authors perform significance tests for some of the results, some others lack any statistical significance. For example, in Figure 4, the proportion of hallucinations - type 3 for Chronos-Bolt is reduced from 51.00 to 49.87, and it is reduced from 43.44 to 41.30 in Chronos. These numbers are very close to each other and may not even be statistically different.

Another issue is the low R2 score for almost all methods, baselines and the introduced SSIM, as they often range in the negative values – this is worse than a very simple estimator that predicts the average value. Yet, the proposed method is better than the alternative baselines. This is perhaps more an issue with the foundation models for time-series prediction than the proposed method. During the rebuttal, the authors attributes very negative R2 scores to a few outlier test instances. Reviewer pd2E is not fully satisfied.

Overall, I believe the novelty of the paper and the potential impact of the framework might overcome its weaknesses.
Decision: Accept (poster)